Erlangen program at large

Key words and phrases. Erlangen program, SL(2,R), special linear group, Heisenberg group, symplectic group, Hardy space, Segal-Bargmann space, Clifford algebra, dual numbers, double numbers, Cauchy-Riemann-Dirac operator, Möbius transformations, covariant functional calculus, Weyl calculus (quantization), quantum mechanics, Schrödinger representation, metaplectic representation

2000 Mathematics Subject Classification. Primary 43A85; Secondary 30G30, 42C40, 46H30, 47A13, 81R30, 81R60.

Preface

The idea of Sophus Lie and Felix Klein was that geometry is the theory of invariants of a transitive transformation group. It was used as main topic of F. Klein inauguration lecture for professorship at Erlangen in 1872 and thus become known as the Erlangen programme (EP). As any great idea it was born ahead of its time: it was much later when theory of groups, especially theory of group representations, was able to make a serious impact. Therefore EP had been marked as “producing only abstract returns” (©Wikipedia) and stored on the shelf.

Meanwhile XX century brought a significant progress in the representation theory, especially linear representations, which was closely connected to achievements in functional analysis. Therefore a “study of invariants” become possible in the linear spaces of functions and associated algebras of operators, e.g. the main objects of modern analysis. This echoed in saying, which Yu.I. Manin attributed to I.M. Gelfand:

This attitude can be encoded as Erlangen programme at large (EPAL). In this book we will systematically apply it to construct geometry of two-dimensional spaces. The further development shall extended it to analytic function theories on such spaces and associated co- and contravariant functional calculi with relevant spectra []. Functional spaces are naturally associated with algebras of coordinates on a geometrical (or point, or commutative) space. An operator (noncommutative) algebra is fashionably treated as a non-commutative space . Therefore EPAL plays the same role for non-commutative geometry as EP for the commutative one [, ].

EPAL provides a systematic tool for discovering hidden gardens, which escaped attention before for various psychological reasons. In a sense [] EPAL works like the periodic table of chemical elements discovered by D.I. Mendeleev: it allows us to see which cells are still empty and suggest where to look for the corresponding objects [].

Mathematical theorem once proved remain true forever. However this does not mean we shall not revise the corresponding theories. Very good examples are Geometry Revisited [] and Elementary Mathematics from an Advanced Standpoint [, ]. Cognition comes through comparison and there are many excellent books about Lobachevsky half-plane which made their exposition through a contrast to the Euclidean geometry . Our book offers a different perspective: it considers the Lobachevsky half-plane as one of thee sisters—elliptic, parabolic and hyperbolic conformal geometries on the upper half-plane.

Exercisers are an integral part of these notes. If a mathematical statement is presented as an exercise, it is not meant to be peripheral, unimportant or without further use. Instead, the label “Exercise” indicates that demonstration of the result is not very difficult and may be useful for understanding. Presentation of mathematical theory through a suitable collection of exercises has a long history starting from the famous Polya and Szegő book [] with many other successful examples to follow, e.g. [, ]. Mathematics is among those enjoyable things which are better to practise yourself rather than watch others doing it.

For some exercises I know only a brute force solution, which is certainly undesirable. A good news is that all of them, marked by the symbol on the margins, can be done through a Computer Algebra System (CAS). The provided DVD contains the full package and Appendix C describes initial instructions. Computer-assisted exercises form also a test-suit for our CAS, which validates the both: mathematical correctness of the library and its practical usefulness.

All figures in the book are printed in black&white to reduce costs. The coloured versions of all pictures are enclosed on the DVD as well, see Appendix C.1 to find them. Reader shall be able to produce even more illustrations him/herself with the enclosed software.

There are many classical objects, e.g. pencils of cycles , or power of a point , which oftenly reoccur in this book under different contexts. The detailed Index shall help to trace most of such places.

Chapter 1 serves as an overview and a gentle introduction, thus we do not give a description of the book content here. Reader is invited to start his/her journey into Möbius invariant geometries now.

Lecture 1 Erlangen Programme: Preview

The simplest objects with non-commutative (but still associative) multiplication may be 2× 2 matrices with real entries. The subset of matrices of determinant one has the following properties:

In other words those matrices form a group, the SL₂(ℝ) group []—one of the two most important Lie groups in analysis. The other group is the Heisenberg group []. By contrast the ax+b group , which is often used to build wavelets, is only a subgroup of SL₂(ℝ), see the numerator in (1).

The simplest non-linear transforms of the real line—linear-fractional or M öbius map]Moebius_transformationMöbius maps—may also be associated with 2× 2 matrices, cf. []*Ch. 13:

An enjoyable calculation shows that the composition of two transforms (1) with different matrices g₁ and g₂ is again a Möbius transform with matrix the product g₁ g₂. In other words (1) it is a (left) action of SL₂(ℝ).

According to F. Klein’s Erlangen programme (which was influenced by S. Lie) any geometry is dealing with invariant properties under a certain transitive group action. For example, we may ask: What kinds of geometry are related to the SL₂(ℝ) action (1)?

The Erlangen programme has probably the highest rate of praised/actually used among mathematical theories not only due to the big numerator but also due to undeserving small denominator. As we shall see below Klein’s approach provides some surprising conclusions even for such over-studied objects as circles.

1.1 Make a Guess in Three Attempts

It is easy to see that the SL₂(ℝ) action (1) makes sense also as a map of complex numbers z=x+i y, i²=−1 assuming the denominator is non-zero. Moreover, if y>0 then g· z has a positive imaginary part as well, i.e. (1) defines a map from the upper half-plane to itself.

However there is no need to be restricted to the traditional route of complex numbers only. Less-known double and dual numbers, see []*Suppl. C and Appendix B.1, have also the form z=x+ι y but different assumptions on the imaginary unit ι : ι²=0 or ι²=1 correspondingly. We will write ε and є instead of ι within dual and double numbers respectively. Although the arithmetic of dual and double numbers is different from the complex ones, e.g. they have divisors of zero , we are still able to define their transforms by (1) in most cases.

Three possible values −1, 0 and 1 of σ:=ι² will be refereed to here as elliptic, parabolic and hyperbolic cases respectively. We repeatedly meet such a division of various mathematical objects into three classes. They are named by the historically first example—the classification of conic sections —however the pattern persistently reproduces itself in many different areas: equations, quadratic forms, metrics, manifolds, operators, etc. We will abbreviate this separation as EPH classification. The common origin of this fundamental division of any family with one-parameter can be seen from the simple picture of a coordinate line split by zero into negative and positive half-axes:

Connections between different objects admitting EPH-classification are not limited to this common source. There are many deep results linking, for example, the ellipticity of quadratic forms, metrics and operators, e.g. the Atiyah-Singer index theorem . On the other hand there are still a lot of white spots, empty cells, obscure gaps and missing connections between some subjects as well.

To understand the action (1) in all EPH cases we use the Iwasawa decomposition []*§ III.1 of SL₂(ℝ)=ANK into three one-dimensional subgroups A, N and K:

Subgroups A and N act in (1) irrespectively to value of σ: A makes a dilation by α², i.e. z↦ α²z, and N shifts points to left by ν, i.e. z↦ z+ν.

By contrast, the action of the third matrix from the subgroup K sharply depends on σ, see Fig. 1.2. In elliptic, parabolic and hyperbolic cases K-orbits are circles, parabolas and (equilateral) hyperbolas correspondingly. Thin traversal lines in Fig. 1.2 join points of orbits for the same values of φ and grey arrows represent “local velocities”—vector fields of derived representations.

Definition 1 The common name cycle [] is used to denote circles, parabolas and hyperbolas (as well as straight lines as their limits) in the respective EPH case.

It is well known that any cycle is a conic sections and an interesting observation is that corresponding K-orbits are in fact sections of the same two-sided right-angle cone, see Fig. 1.3. Moreover, each straight line generating the cone, see Fig. 1.3(b), is crossing corresponding EPH K-orbits at points with the same value of parameter φ from (3). In other words, all three types of orbits are generated by the rotations of this generator along the cone.

K-orbits are K-invariant in a trivial way. Moreover since actions of both A and N for any σ are extremely “shape-preserving” we find natural invariant objects of the Möbius map:

Theorem 2 The family of all cycles from Definition 1 is invariant under the action (1).

According to Erlangen ideology we should now study invariant properties of cycles.

1.2 Covariance of FSCc

Fig. 1.3 suggests that we may get a unified treatment of cycles in all EPH cases by consideration of a higher dimension spaces. The standard mathematical method is to declare objects under investigations (cycles in our case, functions in functional analysis, etc.) to be simply points of some bigger space. This space should be equipped with an appropriate structure to hold externally information which were previously inner properties of our objects.

A generic cycle is the set of points (u,v)∈ℝ² defined for all values of σ by the equation

This equation (and the corresponding cycle) is defined by a point (k, l, n, m) from a projective space ℙ³, since for a scaling factor λ ≠ 0 the point (λ k, λ l, λ n, λ m) defines an equation equivalent to (4). We call ℙ³ the cycle space and refer to the initial ℝ² as the point space.

In order to get a connection with Möbius action (1) we arrange numbers (k, l, n, m) into the matrix

with a new hypercomplex unit ι_c and an additional parameter s usually equal to ± 1. The values of σ_c:=ι_c² is −1, 0 or 1 independently from the value of σ. The matrix (5) is the cornerstone of an extended Fillmore–Springer–Cnops construction (FSCc) [].

The significance of FSCc in Erlangen framework is provided by the following result:

Theorem 1 The image S _σc^s of a cycle C_σc^s under transformation (1) with g∈SL₂(ℝ) is given by similarity of the matrix (5):

S _σc^s= gC_σc^sg⁻¹. (6)

In other words FSCc (5) intertwines Möbius action (1) on cycles with linear map (6).

There are several ways to prove (6): either by a brute force calculation (fortunately performed by a CAS) [] or through the related orthogonality of cycles [], see the end of the next section 1.3.

The important observation here is that our extended version of FSCc (5) uses an imaginary unit ι_c, which is not related to ι defining the appearance of cycles on plane. In other words any EPH type of geometry in the cycle space ℙ³ admits drawing of cycles in the point space ℝ² as circles, parabolas or hyperbolas. We may think on points of ℙ³ as ideal cycles while their depictions on ℝ² are only their shadows on the wall of Plato’s cave .

Fig. 1.5(a) shows the same cycles drawn in different EPH styles. Points c_e,p,h=(l/k, −σ_c n/k) are their respective e/p/h-centres . They are related to each other through several identities:

Fig. 1.5(b) presents two cycles drawn as parabolas, they have the same focal length n/2k and thus their e-centres are on the same level. In other words concentric parabolas are obtained by a vertical shift, not scaling as an analogy with circles or hyperbolas may suggest.

which are independent of the sign of s. If a cycle is depicted as a parabola then h-focus, p-focus, e-focus are correspondingly geometrical focus of the parabola, its vertex , and the point on the directrix nearest to the vertex.

As we will see, cf. Theorems 2 and 2, all three centres and three foci are useful attributes of a cycle even if it is drawn as a circle.

1.3 Invariants: Algebraic and Geometric

We use known algebraic invariants of matrices to build appropriate geometric invariants of cycles. It is yet another demonstration that any division of mathematics into subjects is only illusive.

For 2× 2 matrices (and thus cycles) there are only two essentially different invariants under similarity (6) (and thus under Möbius action (1)): the trace and the determinant. The latter was already used in (8) to define cycle’s foci. However due to projective nature of the cycle space ℙ³ the absolute values of trace or determinant are irrelevant, unless they are zero.

Alternatively we may have a special arrangement for normalisation of quadruples (k,l,n,m). For example, if k≠0 we may normalise the quadruple to (1,l/k,n/k,m/k) with highlighted cycle’s centre. Moreover in this case −detC_σc^s is equal to the square of cycle’s radius, cf. Section 1.6. Another normalisation detC_σc^s=±1 is used in [] to get a nice condition for touching circles.

We still get important characterisation even with non-normalised cycles, e.g., invariant classes (for different σ_c) of cycles are defined by the condition detC_σc^s=0. Such a class is parametrises only by two real numbers and as such is easily attached to certain point of ℝ². For example, the cycle C_σc^s with detC_σc^s=0, σ_c=−1 drawn elliptically represent just a point (l/k,n/k), i.e. (elliptic) zero-radius circle . The same condition with σ_c=1 in hyperbolic drawing produces a null-cone originated at point (l/k,n/k):

In general for every notion there are (at least) nine possibilities: three EPH cases in the cycle space times three EPH realisations in the point space. Such nine cases for “zero radius” cycles is shown on Fig. 1.6. For example, p-zero-radius cycles in any implementation touch the real axis.

This “touching” property is a manifestation of the boundary effect in the upper-half plane geometry. The famous question on hearing drum’s shape has a sister: Can we see/feel the boundary from inside a domain?

Both orthogonality relations described below are “boundary aware” as well. It is not surprising after all since SL₂(ℝ) action on the upper-half plane was obtained as an extension of its action (1) on the boundary.

According to the categorical viewpoint internal properties of objects are of minor importance in comparison to their relations with other objects from the same class. As an illustration we may put the proof of Theorem 1 sketched at the end of of the next section. Thus from now on we will look for invariant relations between two or more cycles.

1.4 Joint Invariants: Orthogonality

The most expected relation between cycles is based on the following Möbius invariant “inner product” build from a trace of product of two cycles as matrices:

Here S _σc^s means complex conjugation of elements of the matrix S _σc^s. By the way, an inner product of this type is used, for example, in GNS construction to make a Hilbert space out of C^*-algebra. The next standard move is given by the following definition.

Definition 1 Two cycles are called σ_c-orthogonal if ⟨ C_σc^s,S _σc^s ⟩=0.

For the case of σ_c σ=1, i.e. when geometries of the cycle and point spaces are both either elliptic or hyperbolic, such an orthogonality is the standard one, defined in terms of angles between tangent lines in the intersection points of two cycles. However in the remaining seven (=9−2) cases the innocent-looking Definition 1 brings unexpected relations.

Elliptic (in the point space) realisations of Definition 1, i.e. σ=−1 is shown in Fig. 1.7. The left picture corresponds to the elliptic cycle space, e.g. σ_c=−1. The orthogonality between the red circle and any circle from the blue or green families is given in the usual Euclidean sense. The central (parabolic in the cycle space) and the right (hyperbolic) pictures show non-local nature of the orthogonality. There are analogues pictures in parabolic and hyperbolic point spaces as well, see Section 6.1.

This orthogonality may still be expressed in the traditional sense if we will associate to the red circle the corresponding “ghost” circle, which shown by the dashed line in Fig. 1.7. To describe ghost cycle we need the Heaviside function χ(σ):

Theorem 2 A cycle is σ_c-orthogonal to cycle C_σc^s if it is orthogonal in the usual sense to the σ-realisation of “ghost” cycle G _σc^s, which is defined by the following two conditions:

χ(σ)-centre of G _σc^s coincides with σ_c-centre of C_σc^s.
Cycles G _σc^s and C_σc^s have the same roots, moreover detG _σ¹= detC_σ^χ(σ_c).

The above connection between various centres of cycles illustrates their meaningfulness within our approach.

One can easy check the following orthogonality properties of the zero-radius cycles defined in the previous section:

Proof.[Sketch of proof of Theorem 1] The validity of Theorem 1 for a zero-radius cycle

with the centre z=x+ι y is straightforward. This implies the result for a generic cycle with the help of Möbius invariance of the product (9) (and thus the orthogonality) and the above relation (2) between the orthogonality and the incidence. See Exercise 2 for details.

1.5 Higher Order Joint Invariants: Focal Orthogonality

With appetite already wet one may wish to build more joint invariants. Indeed for any polynomial p(x₁,x₂,…,x_n) of several non-commuting variables one may define an invariant joint disposition of n cycles ^jC_σc^s by the condition:

However it is preferable to keep some geometrical meaning of constructed notions.

An interesting observation is that in the matrix similarity of cycles (6) one may replace element g∈SL₂(ℝ) by an arbitrary matrix corresponding to another cycle. More precisely the product C_σc^sS _σc^sC_σc^s is again the matrix of the form (5) and thus may be associated to a cycle. This cycle may be considered as the reflection of S _σc^s in C_σc^s.

Definition 1 A cycle C_σc^s is f-orthogonal to a cycle S _σc^s if the reflection of S _σc^s in C_σc^s is orthogonal (in the sense of Definition 1) to the real line. Analytically this is defined by:

tr(C_σc^s S _σc^sC_σc^sR_σc^s)=0. (11)

Due to invariance of all components in the above definition f-orthogonality is a Möbius invariant condition. Clearly this is not a symmetric relation: if C_σc^s is f-orthogonal to S _σc^s then S _σc^s is not necessarily f-orthogonal to C_σc^s.

Fig. 1.8 illustrates f-orthogonality in the elliptic point space. By contrast with Fig. 1.7 it is not a local notion at the intersection points of cycles for all σ_c. However it may be again clarified in terms of the appropriate f-ghost cycle, cf. Theorem 2.

Theorem 2 A cycle is f-orthogonal to a cycle C_σc^s if its orthogonal in the traditional sense to its f-ghost cycle S _σc^σ_c = C_σc^χ(σ) ℝ_σc^σ_c C_σc^χ(σ), which is the reflection of the real line in C_σc^χ(σ) and χ is the Heaviside function (10). Moreover

χ(σ)-Centre of S _σc^σ_c coincides with the σ_c-focus of C_σc^s, consequently all lines f-orthogonal to C_σc^s are passing the respective focus.
Cycles C_σc^s and S _σc^σ_c have the same roots.

Note the above intriguing interplay between cycle’s centres and foci. Although f-orthogonality may look exotic it will naturally appear in the end of next Section again.

Of course, it is possible to define another interesting higher order joint invariants of two or even more cycles.

1.6 Distance, Length and Perpendicularity

Geometry in the plain meaning of this word deals with distances and lengths. Can we obtain them from cycles?

We mentioned already that for circles normalised by the condition k=1 the value −det C_σc^s=−1/2⟨ C_σc^s,C_σc^s ⟩ produces the square of the traditional circle radius. Thus we may keep it as the definition of the σ_c-radius for any cycle. But then we need to accept that in the parabolic case the radius is the (Euclidean) distance between (real) roots of the parabola, see Fig. 1.9(a).

Having radii of circles already defined we may use them for other measurements in several different ways. For example, the following variational definition may be used:

Definition 1 The distance between two points is the extremum of diameters of all cycles passing through both points, see Fig. 1.9(b).

If σ_c=σ this definition gives in all EPH cases the following expression for a distance d_e,p,h(u,v) between endpoints of any vector w=u+i v:

The parabolic distance d_p²=u², see Fig. 1.9(b), algebraically sits between d_e and d_h according to the general principle (2) and is widely accepted []. However one may be unsatisfied by its degeneracy.

An alternative measurement is motivated by the fact that a circle is the set of equidistant points from its centre. However the choice of “centre” is now rich: it may be either point from three centres (7) or three foci (8).

Definition 2 The length of a directed interval AB is the radius of the cycle with its centre (denoted by l_c(AB)) or focus (denoted by l_f(AB)) at the point A which passes through B.

This definition is less common and have some unusual properties like non-symmetry: l_f(AB)≠ l_f(BA). However it comfortably fits the Erlangen programme due to its SL₂(ℝ)-conformal invariance:

Theorem 3 ([]) Let l denote either the EPH distances (12) or any length from Definition 2. Then for fixed y, y′∈ℝ^σ the limit:

lim

t→ 0

l(g· y, g·(y+ty′))

l(y, y+ty′)

, where g∈SL₂(ℝ),

exists and its value depends only from y and g and is independent from y′.

We may return from lengths to angles noting that in the Euclidean space a perpendicular is the shortest route from a point to a line, cf. Fig. 1.10.

Definition 4 Let l be a length or distance. We say that a vector AB is l-perpendicular to a vector CD if function l(AB+ε CD) of a variable ε has a local extremum at ε=0.

A pleasant surprise is that l_f-perpendicularity obtained thought the length from focus (Definition 2) coincides with already defined in Section 1.5 f-orthogonality as follows from Theorem 1.

All these study are waiting to be generalised to high dimensions, quaternions and Clifford algebras provide a suitable language for this [, ].

1.7 Erlangen Programme at Large

As we already mentioned the division of mathematics into areas is only apparent. Therefore it is unnatural to limit Erlangen programme only to “geometry”. We may continue to look for SL₂(ℝ) invariant objects in other related fields. For example, transform (1) generates unitary representations on certain L₂ spaces, cf. (1):

For m=1, 2, …the invariant subspaces of L₂ are Hardy and (weighted) Bergman spaces of complex analytic functions. All main objects of complex analysis (Cauchy and Bergman integrals, Cauchy-Riemann and Laplace equations, Taylor series etc.) may be obtaining in terms of invariants of the discrete series representations of SL₂(ℝ), cf. []*§ 3. Moreover two other series (principal and complimentary []) play the similar rôles for hyperbolic and parabolic cases [, ].

Moving further we may observe that transform (1) is defined also for an element x in any algebra A with a unit 1 as soon as (cx+d1)∈A has an inverse. If A is equipped with a topology, e.g. is a Banach algebra, then we may study a functional calculus for element x [] in this way. It is defined as an intertwining operator between the representation (13) in a space of analytic functions and a similar representation in a left A-module.

In the spirit of Erlangen programme such functional calculus is still a geometry, since it is dealing with invariant properties under a group action. However even for a simplest non-normal operator, e.g. a Jordan block of the length k, the obtained space is not like a space of point but is rather a space of k-th jets []. Such non-point behaviour is oftenly attributed to non-commutative geometry and Erlangen programme provides an important input on this fashionable topic [].

Of course, there is no reasons to limit Erlangen programme to SL₂(ℝ) group only, other groups may be more suitable in different situations. However SL₂(ℝ) still possesses an potential and is a good object to start with.

Lecture 2 Groups and Homogeneous Spaces

The group theory and the representation theory are two enormous and interesting subjects themselves. However they are auxiliary in our consideration and we are forced to restrict our consideration to a brief overview.

Besides introduction to that areas presented in [, ] we recommend additionally the books [, ]. The representation theory intensively uses tools of functional analysis and on the other hand inspires its future development. We use the book [] for references on functional analysis here and recommend it as a nice reading too.

2.1 Groups and Transformations

We start from the definition of the central object, which formalizes the universal notion of symmetries []*§ 2.1.

Definition 1 A transformation group G is a nonvoid set of mappings of a certain set X into itself with the following properties:

the identical map is included in G;
if g₁ ∈ G and g₂ ∈ G then g₁g₂∈ G;
if g∈ G then g⁻¹ exists and belongs to G.

Exercise 2 List all transformation groups on a set of three elements.

Exercise 3 Verify that the following sets are transformation groups:

The group of permutations of n elements;
The group of rotations of the unit circle T;
The groups of shifts of the real line ℝ and the plane ℝ²;
The group of one-to-one linear maps of an n-dimensional vector space over a field F onto itself;
The group of linear-fractional (Möbius) transformations

⎛
⎜
⎝
a b

c d

⎞
⎟
⎠

: z ↦
az+b

cz+d

, (1)

of the extended complex plane such that ad−bc≠ 0.

It is worth (and oftenly done) to push abstraction one level up and to keep the group alone without the underlying space:

Definition 4 An abstract group (or simply group) is a nonvoid set G on which there is a law of group multiplication (i.e. mapping G × G→ G) with the properties

associativity: g₁(g₂g₃)=(g₁g₂)g₃;
the existence of the identity: e∈ G such that eg=ge=g for all g∈ G;
the existence of the inverse: for every g∈ G there exists g⁻¹∈ G such that g g⁻¹=g⁻¹g=e.

Exercise 5 Check that any transformation group is an abstract group.

If we forget the nature of elements of a transformation group G as transformations of a set X then we need to supply a separate “multiplication table” for elements of G. An advantage of transition to abstract groups is that the same abstract group can act by transformations of apparently different sets.

Exercise 6 Check that the following transformation groups (cf. Example 3) have the same law of multiplication, i.e. are equivalent as abstract groups:

The group of isometric mapping of an equilateral triangle onto itself;
The group of all permutations of a set of free elements;
The group of invertible matrix of order 2 with coefficients in the field of integers modulo 2;
The group of linear fractional transformations of the extended complex plane generated by the mappings z↦ z⁻¹ and z↦ 1−z.

Exercise^* 7 Expand the list in the above exercise.

Definition 8 A group G is commutative (or abelian ) if for all g₁, g₂∈ G, we have g₁g₂=g₂g₁.

Exercise 9 Which groups among listed in Exercises 2 and 3 are commutative?

Groups may have some additional analytical structures, e.g. they can be a topological space with a corresponding notion of limit and respective continuity. We also assume that our topological groups are always locally compact []*§ 2.4, that is there exists a compact neighbourhood of every point. It is common to assume that the topological and group structures are in agreement:

Definition 10 If for a group G the group multiplication and the taking of inverse are continuous mappings then G is continuous group .

Exercise 11

Describe topologies which make groups from Exercises 2 and 3 continuous.
Show that a continuous group is locally compact if there exists a compact neighbourhood of its identity.

Even a better structure could be found among Lie groups []*§ 6, e.g. groups with a differentiable law of multiplication. Investigating such groups we could employ the whole arsenal of analytical tools, thereafter most of groups studied here will be Lie groups.

Exercise 12 Check that the following are noncommutative Lie (and thus continuous) groups:

The ax+b group (or the affine group ) []*Ch. 7 of the real line: the set of elements (a,b), a∈ ℝ₊, b∈ ℝ with the group law:
(a, b) * (a′, b′) = (aa′, ab′+b).

The identity is (1,0) and (a,b)⁻¹=(a⁻¹,−b/a).
The Heisenberg group ℍ¹ [] []*Ch. 1: a set of triples of real numbers (s,x,y) with the group multiplication:
(s,x,y)*(s′,x′,y′)=(s+s′+
1

2

(x′y−xy′),x+x′,y+y′). (2)

The identity is (0,0,0) and (s,x,y)⁻¹=(−s,−x,−y).

The SL₂(ℝ) group [, ]: a set of 2× 2 matrices (

a	b
c	d

) with real entries a, b, c, d∈ℝ and the determinant det=ad−bc equal to 1 and the group law coinciding with matrix multiplication:

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

⎛
⎜
⎝

a′	b′
c′	d′

⎞
⎟
⎠

⎛
⎜
⎝

aa′+bc′	ab′+bd′
ca′+dc′	cb′+dd′

⎞
⎟
⎠

The identity is the unit matrix and

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

−1

⎛
⎜
⎝

d	−b
−c	a

⎞
⎟
⎠

The above three groups are behind many important results of real and complex analysis [, , , ] and we meet them many times later.

2.2 Subgroups and Homogeneous Spaces

A study of any mathematical object is facilitated by a decomposition into smaller or simpler blocks. In the case of groups we need the following:

Definition 1 A subgroup of a group G is subset H⊂ G such that the restriction of multiplication from G to H makes H a group itself.

Exercise 2 Show that the ax+b group is a subgroup of SL₂(ℝ).
Hint: Consider matrices 1/√a(

a	b
0	1

).⋄

While abstract group are a suitable language for investigation of their general properties we meet groups in applications as transformation groups acting on a set X. We will describe the connections between those two viewpoints. It can be approached it either way: having a homogeneous space build the class of isotropy subgroups or having a subgroup define respective homogeneous space. The next two subsections explore both directions in details.

2.2.1 From a Homogeneous Space to the Isotropy Subgroup

Let X be a set and let for a group G we define an operation G: X→ X of G on X. We say that a subset S⊂ X is G-invariant if g· s∈ S for all g∈ G and s∈ S.

Exercise 3 Show that if S⊂ X is G-invariant then its complement X∖ S is G-invariant as well.

Thus if X has non-trivial invariant subset we can split X into disjoint parts. The finest such a decomposition is obtained from the following equivalence relation on X, say, x₁∼ x₂ if and only if there exists g∈ G such that gx₁=x₂, with respect to which X is a disjoint union of distinct orbits []*§ I.5, that is subsets of all gx₀ with a fixed x₀∈ X and arbitrary g∈ G.

Exercise 4 Let action of SL₂(ℝ) group on ℂ by means of linear-fractional transformations (1). Show that there three orbits: the real axis ℝ, the upper (lower) half-plane ℝ_±ⁿ:

ℝ_±²={ x± iy ∣ x,y∈ ℝ, y>0}.

Thus from now on, without lost of a generality, we assume that the action of G on X is transitive, i.e. for every x∈ X we have

Exercise 5 Show that either of the following conditions define a transitive action of G on X:

For arbitrary two points x₁, x₂∈ X there exists g∈ G such that g x₁ =x₂.
Let a fixed point x₀∈ X is given, then for arbitrary point x∈ X there exists g∈ G such that g x₀ = x.

Exercise 6 Show that for any group G we could define its action on X=G as follows:

The conjugation g: x ↦ g x g⁻¹.
The left shift Λ(g): x ↦ g x and the right shift R(g): x ↦ x g⁻¹.

Above actions define group homomorphisms from G to the transformation group of G, however the conjugation is trivial for all commutative groups.

Exercise 7 Show that:

The set of elements G_x={g∈ G ∣ gx=x} for fixed a point x∈ X forms a subgroup of G, which is called the isotropy (sub)group of x in G []*§ I.5.
For any x₁, x₂∈ X isotropy subgroups G_x1 and G_x2 are conjugated , that is there is g∈ G such that G_x1=g⁻¹G_x2g.

This provides a transition from a G-action on a homogeneous space X to a subgroup of G, or even to an equivalence class of such subgroups under conjugation.

Exercise 8 Find a subgroup which correspond to the given action of G on X:

Action of ax+b group on ℝ by the formula: (a,b): x ↦ ax+b for the point x=0.
Action of SL₂(ℝ) group on one of three orbit from Exercise 4 with respective points x=0, i and −i.

2.2.2 From a Subgroup to the Homogeneous Space

We can go in the opposite direction as well: having a subgroup of G find the corresponding homogeneous space. Let G be a group and H be its subgroup. Let us define the space of cosets X=G/H by the equivalence relation: g₁∼ g₂ if there exists h ∈ H such that g₁=g₂h.

The space X=G / H is a homogeneous space under the left G-action g: g₁↦ gg₁. For practical purposes it is more convenient to have a parametrisation of X and express the above G-action through those parameters. We will do it now.

We define a continuous function (section) []*§ 13.2 s: X→ G such that it is a left inverse to the natural projection p: G→ G/H , i.e. p(s(x))=x for all x∈ X.

Exercise 9 Check that for any g∈ G we have s(p(g))=gh for some h∈ H depending from g.

Then any g∈ G has a unique decomposition of the form g=s(x)h, where x=p(g)∈ X and h∈ H. We define a map r associated to s through the identities:

Exercise 10 Show that:

X is a left G-homogeneous space with the G-action defined in terms of maps s and p as follows:
g: x ↦ g· x=p(g* s(x)), (3)

where * is the multiplication on G. This is illustrated by the diagram:
G <.5ex>[d]^p [r]^g*G <.5ex>[d]^p X <.5ex>[u]^s [r]^g·X <.5ex>[u]^s (4)
The above action of G: X→ X is transitive on X.
The choice of a section s is inessential in the following sense. Let s₁ and s₂ be two smooth maps, such that p(s_i(x))=x for all x∈ X, i=1, 2. Then p(g*s₁(x))=p(g*s₂(x)) for all g∈ G.

Thus starting from a subgroup H of a group G we can define a G-homogeneous space X=G/H.

2.3 Differentiation on Lie Groups and Lie Algebras

The differentiation is naturally defined for Lie groups . If G is a Lie group and G_x be its closed subgroup, then the considered above homogeneous space G/G_x is a smooth manifold (and a loop as an algebraic object) for every x∈ X []*Theorem 2 in § 6.1. Therefore the one-to-one mapping G/G_x → X: g↦ gx induces a structure of C^∞-manifold on X. Thus the class C₀^∞(X) of smooth functions with compact supports on x has the evident definition.

For every Lie group G there is an associated Lie algebra g. This algebra can be realised in many different ways, we will use the following two out of four listed in []*§ 6.3.

2.3.1 One-parameter Subgroups and Lie Algebras

For the first realisation we consider a one-dimensional continuous subgroup x(t) of G as a group homomorphism of x: (ℝ,+)→ G. For such a homomorphism x we have x(s+t)=x(s)x(t) and x(0)=e.

Exercise 1 Check that the following subsets of elements parametrised by t∈ℝ are one-parameter subgroups.

For the affine group: a(t)=(e^t,0) and n(t)=(0,t).
For the Heisenberg group ℍ¹:
s(t)=(t,0,0), x(t)=(0,t,0) and y(t)=(0,0,t).

For the group SL₂(ℝ):

a(t)

⎛
⎜
⎝

e^−t/2	0
0	e^t/2

⎞
⎟
⎠

n(t)

⎛
⎜
⎝

1	t
0	1

⎞
⎟
⎠

(5)

b(t)

⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

cosh

sinh

cosh

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

z(t)

⎛
⎜
⎝

cost	sint
−sint	cost

⎞
⎟
⎠

(6)

One-parameter subgroup x(t) defines a tangent vector X=x′(0) belonging to the tangent space T_e of G at e=x(0). The Lie algebra g can be identified with this tangent space. The important exponential map exp: g → G works in the opposite direction and is defined by expX=x(1) in the previous notations. For the case of a matrix group the exponent map can be explicitly realised through the exponentiation of the matrix representing a tangent vector:

Exercise 2 Check that subgroups a(t), n(t), b(t) and z(t) from Exercise 3 are generated by the exponent map of the following zero-trace matrices:

a(t)

=exp

⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

−

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

n(t)

=exp

⎛
⎜
⎝

0	t
0	0

⎞
⎟
⎠

(7)

b(t)

=exp

⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

z(t)

=exp

⎛
⎜
⎝

0	t
−t	0

⎞
⎟
⎠

(8)

2.3.2 Invariant Vector Fields and Lie Algebras

In the second realisation of the Lie algebra g is identified with the left (right) invariant vector fields on the group G, that is first order differential operators X defined at every point of G and invariant under the left (right) shits : XΛ = Λ X (XR=RX). This realisation is particularly usable for a Lie group with an appropriate parametrisation. The following examples describes different techniques in finding such invariant fields.

Example 3 Let us build left (right) invariant vector fields on G—the ax+b group using the plain definition. Take the basis {∂_a, ∂_b} ({−∂_a, −∂_b}) of the tangent space T_e to G at its identity. We will propagate these vectors to an arbitrary point through the invariance under shits. That is, to find the value of the invariant field at the point g=(a,b) we

make the left (right) shift by g;
apply a differential operator from the basis of T_e;
make the inverse left (right) shift by g⁻¹=(1/a,−b/a).

Thus we will obtain the following invariant vector fields:

A^l=a∂_a, N^l=a∂_b; and A^r=−a∂_a−b∂_b, N^r=−∂_b. (9)

Example 4 An alternative calculation for the same Lie algebra can be done as follows. The Jacobians at g=(a,b) of the left and the right shifts

Λ(u,v):f(a,b)↦ f

⎛
⎜
⎜
⎝

b−v

⎞
⎟
⎟
⎠

, and R(u,v): f(a,b)↦ f(ua, va+b)

by h=(u,v) are:

J_Λ(h)=

⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

, and J_R(h)=

⎛
⎜
⎝

u	0
v	1

⎞
⎟
⎠

Then the invariant vector fields are obtained by the transpose of Jacobians:

⎛
⎜
⎝

A^l

N^l

⎞
⎟
⎠

J_Λ^t(g⁻¹)

⎛
⎜
⎝

∂_a

∂_b

⎞
⎟
⎠

⎛
⎜
⎝

a	0
0	a

⎞
⎟
⎠

⎛
⎜
⎝

∂_a

∂_b

⎞
⎟
⎠

⎛
⎜
⎝

a∂_a

a∂_b

⎞
⎟
⎠

⎛
⎜
⎝

A^r

N^r

⎞
⎟
⎠

J_R^t(g)

⎛
⎜
⎝

−∂_a

−∂_b

⎞
⎟
⎠

⎛
⎜
⎝

a	b
0	1

⎞
⎟
⎠

⎛
⎜
⎝

−∂_a

−∂_b

⎞
⎟
⎠

⎛
⎜
⎝

−a∂_a−b∂_b

−∂_b

⎞
⎟
⎠

This rule is a very special case of the general theorem on the change of variables in the calculus of pseudo-differential operators (PDO), cf. []*§ 4.2 []*Theorem 18.1.17.

Example 5 Finally we calculate the invariant vector fields on the ax+b group through a connection to the above one-parameter subgroups. The left-invariant vector field corresponding to the subgroup a(t) from Exercise 1 is obtained through the differentiation of the right action of this subgroup:

[A^l f](a,b)

f((a,b)*(e^t,0))

⎪
⎪
⎪
⎪

t=0

f(a e^t,b)

⎪
⎪
⎪
⎪

t=0

= af′_a(a,b),

[N^l f](a,b)

f((a,b)*(1,t))

⎪
⎪
⎪
⎪

t=0

f(a,at+b)

⎪
⎪
⎪
⎪

t=0

=af′_b(a,b).

Similarly the right-invariant vector fields are obtained by the derivation of the left action:

[A^r f](a,b)

f((e^−t,0)*(a,b))

⎪
⎪
⎪
⎪

t=0

f(e^−t a ,e^−tb)

⎪
⎪
⎪
⎪

t=0

= −af′_a(a,b)− bf′_b(a,b) ,

[N^r f](a,b)

f((1,−t)*(a,b))

⎪
⎪
⎪
⎪

t=0

f(a,b−t)

⎪
⎪
⎪
⎪

t=0

=−f′_b(a,b).

Exercise 6 Use the above techniques to calculate the following left (right) invariant vector fields on the Heisenberg group :

S^l(r)=±∂_s, X^l(r)=±∂_x−

y∂_s, Y^l(r)=±∂_y+

x∂_s. (10)

2.3.3 Commutator in Lie Algebras

The important operation on a Lie algebra is a commutator. If the Lie algebra of a matrix group is realised by matrices, e.g. Exercise 2, then the commutator is defined by the expression [A,B]=AB−BA in term of the respective matrix operations. If the Lie algebra is realised through left (right) invariant first order differential operators then the commutator [A,B]=AB−BA again define a left (right) invariant first order operator—an element of the same Lie algebra.

Among important properties of the commutator are its anti-commutativity ([A,B]=−[B,A]) and the Jacobi identity:

Exercise 7 Check the following commutation relations.

For the Lie algebra (9) of the ax+b group:
[A^l(r),N^l(r)]=N^l(r).
For the Lie algebra (10) of Heisenberg group:
[X^l(r),Y^l(r)]=S^l(r), [X^l(r),S^l(r)]= [Y^l(r),S^l(r)]=0. (12)

These are the celebrated Heisenberg commutation relations , which are very important in quantum mechanics.
Denote by A, B and Z the generators of the one-parameter subgroups a(t), b(t) and z(t) in (7) and (8), the commutation relations in the Lie algebra sl₂ are:
[Z,A]=2B, [Z,B]=−2A, [A,B]=−
1

2

Z. (13)

The procedure from Example 5 can be used to calculate derived action of a G-action on a homogeneous space as well.

Example 8 Consider the action of the ax+b group on the real line associated with group’s name:

(a,b): x↦ ax+b, x∈ℝ.

Then the derived action on the real line is:

[A^d f](x)

f(e^−t x)

⎪
⎪
⎪
⎪

t=0

=−xf′(x),

[N^d f](a,b)

f(x−t)

⎪
⎪
⎪
⎪

t=0

=−f′(x).

2.4 Integration on Groups

In order to perform an integration we need a suitable measure. A smooth measure dµ on X is called (left) invariant measure with respect to an operation of G on X if

Exercise 1 Show that measure y⁻²dy dx on the upper half-plane ℝ₊² is invariant under action from Exercise 4.

Left invariant measures on X=G is called the (left) Haar measure. It always exists and is uniquely defined up to a scalar multiplier []*§ 0.2. An equivalent formulation of (14) is: G operates on L₂(X,dµ) by unitary operators. We will transfer the Haar measure dµ from G to g via the exponential map exp: g→ G and will call it as the invariant measure on a Lie algebra g.

Exercise 2 Check that the following are Haar measures for corresponding groups:

The Lebesgue measure dx on the real line ℝ.
The Lebesgue measure dφ on the unit circle T.
dx/x is a Haar measure on the multiplicative group ℝ₊;
dx dy/(x²+y²) is a Haar measure on the multiplicative group ℂ∖ {0}, with coordinates z=x+iy.
a⁻² da db and a⁻¹ da db are the left and right invariant measure on ax+b group .
The Lebesgue measure ds dx dy of ℝ³ for the Heisenberg group ℍ¹.

In this notes we assume all integrations on groups performed over the Haar measures.

Exercise 3 Show that invariant measure on a compact group G is finite and thus can be normalised to total measure 1.

The above simple result has surprisingly important consequences for representation theory of compact groups.

Definition 4 The left convolution f₁*f₂ of two functions f₁(g) and f₂(g) defined on a group G is

f₁*f₂(g)=

∫

f₁(h) f₂(h⁻¹g) dh

Exercise 5 Let k(g)∈ L₁(G,dµ) and operator K on L₁(G,dµ) is the left convolution operator with k, .i.e. K: f ↦ k*f. Show that K commutes with all right shifts on G.

The following Lemma characterizes linear subspaces of L₁(G,dµ) invariant under shifts in the term of ideals of convolution algebra L₁(G,dµ) and is of the separate interest.

Lemma 6 A closed linear subspace H of L₁(G,dµ) is invariant under left (right) shifts if and only if H is a left (right) ideal of the right group convolution algebra L₁(G,dµ).

Proof. Of course we consider only the “right-invariance and right-convolution” case. Then the other three cases are analogous. Let H be a closed linear subspace of L₁(G,dµ) invariant under right shifts and k(g)∈ H. We will show the inclusion

But the last sum is simply a linear combination of vectors k(hg_j)∈ H (by the invariance of H) with coefficients f(g_j). Therefore sum (16) belongs to H and this is true for integral (15) by the closeness of H.

Otherwise, let H be a right ideal in the group convolution algebra L₁(G,dµ) and let φ_j(g)∈L₁(G,dµ) be an approximate unit of the algebra []*§ 13.2, i.e. for any f∈L₁(G,dµ) we have

from the first expression is tensing to k(hh′) and from the second one belongs to H (as a right ideal). Again the closeness of H implies k(hh′)∈ H that proves the assertion.

Lecture 3 Homogeneous Spaces from the Group SL(2,R)

Now we specialise the previous theoretical constructions for the particular case of the group SL₂(ℝ). We are going to describe all homogeneous spaces SL₂(ℝ)/H, where H is a subgroup of SL₂(ℝ), see Section 2.2.2. To warm-up we start from the two-dimensional subgroup.

3.1 The Affine Group and the Real Line

The affine group of the real line, also known as the ax+b group , can be identified with either subgroup of lower- or upper-triangular matrices:

These subgroups are obviously conjugated each other and we can consider only the subgroup F here.

The corresponding homogeneous space X=SL₂(ℝ)/F is one-dimensional and can be parametrised by a real number. Following the construction from Section 2.2.2 and using its notations we defined the natural projection p as follows:

Exercise 1

Check that the derived action , see Exercise 8, associated to the one-parameter subgroups a(t), b(t) and z(t) from Exercise 2 respectively are:
      A_F=x
d

dx

,     B_F=
x²−1

2

d

dx

,      Z_F=−(x²+1)
d

dx

.
Verify that the above operators satisfy to the commutator relations for the Lie algebra sl₂, cf. (13):
[Z_F,A_F]=2B_F, [Z_F,B_F]=−2A_F, [A_F,B_F]= −
1

2

Z_F. (5)

3.2 One-dimensional Subgroups of SL₂(ℝ)

Any element of the Lie algebra sl₂ defines a one-parameter subgroup of SL₂(ℝ). We listed four such subgroups in Exercise 3 already and can provide further examples, e.g. the subgroup of lower-triangular matrices. However there are only three different types of subgroups under the matrix similarity A↦ MAM⁻¹.

Proposition 1 Any continuous one-parameter subgroup of SL₂(ℝ) is conjugate to one of the following subgroups:

⎧
⎨
⎩

⎛
⎜
⎝

e^−t/2	0
0	e^t/2

⎞
⎟
⎠

=exp

⎛
⎜
⎝

−t/2	0
0	t/2

⎞
⎟
⎠

, t∈ℝ

⎫
⎬
⎭

(6)

⎧
⎨
⎩

⎛
⎜
⎝

1	t
0	1

⎞
⎟
⎠

=exp

⎛
⎜
⎝

0	t
0	0

⎞
⎟
⎠

t∈ℝ

⎫
⎬
⎭

(7)

⎧
⎨
⎩

⎛
⎜
⎝

cost	sint
−sint	cost

⎞
⎟
⎠

= exp

⎛
⎜
⎝

0	t
−t	0

⎞
⎟
⎠

, t∈(−π,π]

⎫
⎬
⎭

(8)

Proof. Any one-parameter subgroup is obtained through the exponential map , see Section 2.3:

of an element X of the Lie algebra sl₂ of SL₂(ℝ). Such X is a 2× 2 matrix with the zero trace. The behaviour of the Taylor expansion (9) depends from properties of powers Xⁿ. This can be classified by a straightforward calculation:

Lemma 2 The square X² of a traceless matrix X= (

a	b
c	−a

) is the identity matrix times a²+bc=−detX. The factor can be negative, zero or positive, which corresponds to the three different types of the Taylor expansion (9) of e^tX.

It is a simple exercise on characteristic polynomials to see that through the matrix similarity we can obtain from X a generator

The determinant is invariant under the similarity, thus these cases are distinct.

Exercise 3 Find matrix conjugations of the following two subgroups to A and N respectively:

⎧
⎨
⎩

⎛
⎜
⎝

cosht	sinht
sinht	cosht

⎞
⎟
⎠

=exp

⎛
⎜
⎝

0	t
t	0

⎞
⎟
⎠

, t∈ℝ

⎫
⎬
⎭

(10)

N′

⎧
⎨
⎩

⎛
⎜
⎝

1	0
t	1

⎞
⎟
⎠

=exp

⎛
⎜
⎝

0	0
t	0

⎞
⎟
⎠

t∈ℝ

⎫
⎬
⎭

(11)

We will oftenly use subgroups and N′ as representatives of the corresponding equivalence classes under matrix conjugation.

An interesting property of the subgroups A, N and K is their appearance in the Iwasawa decomposition []*§ III.1 of SL₂(ℝ)=ANK in the following sense. Any element of SL₂(ℝ) can be represented as the product:

Exercise 4 Check that the values of parameters in the above decomposition are as follows:

α=

√

c²+d²

, ν=ac+bd, φ = −arctan

Consequently cosφ=d/√c²+d² and sinφ=−c/√c²+d².

The Iwasawa decomposition shows once more that SL₂(ℝ) is a three-dimensional manifold. A similar decomposition G=ANK is possible for any semisimple Lie group G, where K is the maximal compact group, N is nilpotent and A normalises N. Although the Iwasawa decomposition will be used here on several occasion it does not play a crucial role in present consideration. Rather Proposition 1 will be the cornerstone of our construction.

3.3 Two-dimensional Homogeneous Spaces

Here we calculate action of SL₂(ℝ) (3) (see § 2.2.2) on X=SL₂(ℝ)/H for all three possible one-dimensional subgroups H=, N′ or K. Counting dimensions 3−1=2 suggests that the corresponding homogeneous spaces are two-dimensional manifolds. In fact we identify X in each case with a subset of ℝ² as follows. First, for every equivalence class SL₂(ℝ)/H we chose a representative, which is an upper-triangular matrix

Exercise 1 Show that there is at most one upper triangular matrix in every equivalence class SL₂(ℝ)/H, where H=, N′ or K, where in the last case uniqueness is up to the constant factor ± 1.
Hint: The identity matrix is the only upper-triangular matrix in those three subgroups, where again the uniqueness for the subgroup K is understood up to the scalar factor ±1.⋄

The existence of such triangular matrix will be demonstrated in each case separately. Now we define the projection p:SL₂(ℝ)→ X assigning p(g)=(ab,a²), where (

) is the only upper-triangular matrix representing the equivalence class of g. We also choose []*p. 108 the map s: X→ G in the form:

3.3.1 From the Subgroup K

The homogeneous space SL₂(ℝ)/K is is the most traditional case in the representation theory. The above defined maps p and s produce the following decomposition g=s(p(g))r(g):

Then the SL₂(ℝ)-action defined by the formula g· x=p(g*s(x)) (3) takes the form:

Exercise 2 Use CAS to check the above formula, as well as analogous formulae (18) and (21) below. See Appendix C.3 for CAS usage.

Obviously, it preserves the upper half-plane v>0 . The expression (15) is very cumbersome and a relief provided by the complex imaginary unit i²=−1, which reduces (15) to the a Möbius transformation:

We need to assign a meaning to the case cw+d=0 and this can be done by the addition of an infinite point ∞ to the set of complex numbers, see, for example, []*Definition 13.1.3 for details.

3.3.2 From the Subgroup N′

We consider the subgroup of lower-triangular matrices N′ (11). For this subgroup the representative of cosets among the upper triangular matrices will be different, therefore we receive a apparently different decomposition g=s(p(g))r(g), cf. (14):

We postpone the treatment of the exceptional case d=0 till Section 8.1. The SL₂(ℝ)-action (3) takes now the form:

This map preserves the upper half-plane v>0 just like the case of the subgroup K. The expression (18) is simpler than (15), yet we can again rewrite it as a linear-fractional transformation with the help of the dual numbers unit ε ²=0:

We briefly review the algebra of dual numbers in Appendix B.1. Since they have zero divisors the fraction is not properly defined out of the box for all cu+d=0. The proper treatment will be considered in Section 8.1 since it is not as simple as in the case of complex numbers.

3.3.3 From the Subgroup

We will again treat the exceptional situation d=± c in the Section 8.1. The SL₂(ℝ)-action (3) takes the form:

Notably, this time the map does not preserve the upper half-plane v>0: the sign of ( c u+d)² −(cv)² is not determined. To express this map as a Möbius transformation we require the double numbers (also known as split-complex numbers) unit є ²=1:

3.3.4 Unifying All Three Cases

There is an obvious similarity in the formulae obtained in each of the above cases. To present them in a unified way we introduce the parameter σ which is equal −1, 0 or 1 for the subgroups K, N′ or respectively. Then decompositions (14), (17) and (20) are:

The respective SL₂(ℝ)-actions on the homogeneous space SL₂(ℝ)/H, where H=A, N′ or K are given by:

Finally this action becomes the linear-fractional (Möbius) transformation for hypercomplex numbers in two-dimensional commutative associative algebra (see Appendix B.1) spanned by 1 and ι:

Thus a comprehensive study of SL₂(ℝ)-homogeneous spaces naturally introduces three number systems. Obviously only one case (complex numbers) belongs to the mainstream mathematics. We start to discover empty cells in our periodic table.

Remark 3 As we can see now the dual and double numbers naturally appear in relation with the group SL₂(ℝ) and thus their introduction in [, ] was not “a purely generalistic attempt”, cf. the remark on quaternions in []*p. 4.

Remark 4 A different choice of the map s: G/H→ G will produce different (but isomorphic) geometric models. In this way we will obtain three types of “unit disks” in Chapter 9.

3.4 Elliptic, Parabolic and Hyperbolic

As we have seen in the previous Section there is no need to be restricted to the traditional route of complex numbers only. The arithmetic of dual and double numbers is different from the complex ones mainly in the following aspects:

We have agreed in Section 1.1 that, three possible values −1, 0 and 1 of σ:=ι² will be refereed to here as elliptic, parabolic and hyperbolic cases respectively. This separation into three cases will be refereed as EPH classification. Unfortunately, there is here a clash with already established label for the Lobachevsky geometry. It is oftenly called hyperbolic geometry because it can be realised as a Riemann geometry on a two-sheet hyperboloid. However within our framework the Lobachevsky geometry shall be called elliptic and it will have a true hyperbolic sister.

An initial correspondence of subgroups, number systems and other objects is given by the following table:

Notation 1 We denote the space ℝ² of vectors u+v ι by ℝ^e, ℝ^p or ℝ^h to highlight which of number system is used in the present context. The notation ℝ^σ used for a generic case.

Remark 2 Introducing the parabolic objects on a common ground with elliptic and hyperbolic ones we should warn against some common prejudices suggested by picture (2):

The parabolic case is unimportant (has “zero measure”) in comparison to the elliptic and hyperbolic ones. As we shall see (e.g. Remark 5 and 2) some geometrical features are richer in parabolic case.
The parabolic case is a limiting situation or an intermediate position between the elliptic and hyperbolic ones: all properties of the former can be guessed or obtained as a limit or an average from the latter two. Particularly this point of view is implicitly supposed in [].
Although there are some confirmations of this (e.g. Fig. 9.3(E)–(H)), we shall see (e.g. Remark 6) that some properties of the parabolic case cannot be straightforwardly guessed from a combination of elliptic and hyperbolic cases.
All three EPH cases are even less disjoint than it is usually thought. For example, there are meaningful notions of centre of a parabola (3) or focus of a circle (2).
A (co-)invariant geometry is believed to be “coordinate free” which sometimes is pushed to an absolute mantra. However our study within the Erlangen programme framework reveals two useful notions (Definition 3 and (2)) mentioned above which are defined by coordinate expressions and look very “non-invariant” on the first glance.

3.5 Orbits of the Subgroup Actions

on the hypercomplex numbers w=u+ι v from a description of orbits produced by the subgroups A, N and K. Due to the Iwasawa decomposition (12) any Möbius transformation can be represented as a superposition of those three actions.

The actions of subgroups A and N for any kind of hypercomplex numbers on the plane are the same as on the real line: A dilates and N shifts, see Fig. 1.1 for illustrations. Thin traversal lines in Fig. 1.1 join points of orbits obtained from the vertical axis by the same values of t and grey arrows represent “local velocities”—vector fields of derived representations.

Exercise 1 Check that:

The matrix (
e^−t 0

0 e^t

)=exp(
−t 0

0 t

) from A makes a dilation by e^−2t, i.e. z↦ e^−2t z. The respective derived action , see Example 8, is twice the Euler operator u∂_u+v∂_v.
The matrix (
1 t

0 1

)=exp(
0 t

0 0

) from N shifts points horizontally by t, i.e. z↦ z+t=(u+t)+ι v. The respective derived action is ∂_u.
The subgroup of SL₂(ℝ) generated by A and N is isomorphic to the ax+b group , which acts transitively on the upper half-plane.
Hint: Note that the matrix (
         √

a

b/ √

a

0
1/ √

a

)= (
        1 b

0 1

) (
         √

a

0

0
1/ √

a

) maps ι to aι +b and use Exercise 2.⋄

By contrast, the action of the third matrix from the subgroup K sharply depends on σ=ι² as illustrated by the Fig. 1.2. In elliptic, parabolic and hyperbolic cases K-orbits are circles, parabolas and (equilateral) hyperbolas respectively. The meaning of traversal lines and vector fields is the same as on the previous figure.

Exercise 2 The following properties characterise K-orbits:

The derived action of the subgroup K is given by:
K_σ^d(u,v)=(1+u²+σ v²)∂_u+2uv∂_v, σ=ι². (25)

Hint: Use the explicit formula for Möbius transformation of the components (23). An alternative with CAS is provided as well, see Appendix C.3 for usage.⋄
A K-orbit in ℝ^σ passing the point (0,s) has the following equation:
(u²−σ v²)−2v
s⁻¹−σ s

2

+1=0. (26)

Hint: Note, that the equation (26) defines contour lines of the function F(u,v)=(u²−σ v²+1)/v, that is solve the equations F(u,v)=const. Then apply the operator (25) to obtain K_σ^dF=0.⋄
The curvature of a K-orbit at point (0,s) is equal to
κ=
2s

1+σ s²

. (27)
The transverse line obtained from the vertical axis has the equations:
(u²−σ v²)+2cot(2φ) u−1=0,    for g=
⎛
⎜
⎝
        cosφ sinφ

−sinφ cosφ

⎞
⎟
⎠

∈ K.     (28)

Hint: A direct calculation for a point (0,s) in the formula (23) is doable but demanding. A computer symbolic calculation is provided as well.⋄

Much more efficient proofs will be given later (see Exercise 2), when suitable tools will be in our disposal. It will explain also why K-orbits, which are circles, parabolas and hyperbolas, are defined by the same equation (26). Meanwhile these formulae allow to produce geometric characterisation of K-orbits in term of classical notions of conic sections, cf. Appendix B.2.

Exercise 3 Check the following properties of K-orbits (26):

For the elliptic case the orbits of K are circles. A circle with centre at (0, (s+s⁻¹)/2) passing through two points (0,s) and (0,s⁻¹).
For the parabolic case the orbits of K are parabolas with the vertical axis V. A parabola passing through (0,s) has horizontal directrix passing through (0, s−1/(4s)) and focus at (0,s+1/(4s)).
For the hyperbolic case the orbits of K are hyperbolas with asymptotes parallel to lines u=± v. A hyperbola passing the point (0,s) has the second branch passing (0,−s⁻¹) and asymptotes crossing at the point (0,(s−s⁻¹)/2). Foci of this hyperbola are:
f_1,2= ⎛
⎜
⎜
⎝ 0,
1

2

⎛
⎜
⎝ (1± √

2

)s−(1∓ √

2

)s⁻¹ ⎞
⎟
⎠ ⎞
⎟
⎟
⎠ .

The amount of similarities between orbits in three EPH cases suggests that they shall be unified one way or another. We start such attempts in the next section.

3.6 Unifying EPH Cases: The First Attempt

It is well known that any above K-orbit is a conic sections and an interesting observation is that corresponding K-orbits are in fact sections of the same two-sided right-angle cone. More precisely we define the family of double-sided right-angle cones be parametrised by s>0:

Therefore vertices of cones belong to the hyperbola {x=0, y²−z²=1}, see Fig. 1.3.

Exercise 1 Derive equations for the K-orbits described in Lemma 3 by calculation of intersection of a cone (29) with the following planes:

elliptic K-orbits are sections of cones (29) by the plane z=0 (EE′ on Fig. 1.3);
parabolic K-orbits are sections of (29) by the plane y=± z (PP′ on Fig. 1.3);
hyperbolic K-orbits are sections of (29) by the plane y=0 (HH′ on Fig. 1.3);

Moreover, each straight line generating the cone, see Fig. 1.3(b), is crossing corresponding EPH K-orbits at points with the same value of parameter φ from (12). In other words, all three types of orbits are generated by the rotations of this generator along the cone.

Exercise 2 Verify that the rotation of a cone’s generator corresponds to the Möbius transformations in three planes.

Hint: I do not know a smart way to check this, thus a CAS solution is provided.⋄

From the above algebraic and geometric descriptions of the orbits we can make several observations.

Remark 3

The values of all three vector fields dK_e, dK_p and dK_h coincide on the “real” U-axis v=0, i.e. they are three different extensions into the domain of the same boundary condition. Another source of this: the axis U is the intersection of planes EE′, PP′ and HH′ on Fig. 1.3.
The hyperbola passing through the point (0,1) has the shortest focal length √2 among all other hyperbolic orbits since it is the section of the cone x²+(y−1)²+z²=0 closest from the family to the plane HH′.
Two hyperbolas passing through (0,v) and (0,v⁻¹) have the same focal length since they are sections of two cones with the same distance from HH′. Moreover, two such hyperbolas in the lower- and upper half-planes passing the points (0,v) and (0,−v⁻¹) are sections of the same double-sided cone. They are related to each other as explained in Remark 2.

We make a generalisation to all EPH cases of the following notion, which is well-known for circles []*§ 2.3 and parabolas []*§ 10:

Definition 4 A power p of a point (u,v) with respect to a conic section given by the equation x²−σ y²−2lx−2ny+c=0 is defined by the identity:

p=u²−σ v²−2lu−2nv+c. (30)

Exercise 5 Check the following properties:

A conic section is the collection of points having zero power with respect to this section.
The collection of points having the same power with respect to two given conic sections of the above type is either empty or the straight line. This line is called radical axis of the two sections.
All K-orbits are coaxal []*§ 2.3 with the real line being their joint radical axis, that is for a given point on the real line its power with respect to any K-orbit is the same.
All transverse lines (28) are coaxal with the vertical line u=0 being the respective radial axis.

3.7 Isotropy Subgroups

In Section 2.2 we described two-sided connection between homogeneous spaces and subgroups. The Section 3.3 uses it in one direction: from subgroups to homogeneous spaces. The following Exercise does it in the opposite way.

Exercise 1 Let SL₂(ℝ) acts by Möbius transformations (24) on the three number systems. Show that the isotropy subgroupsof the point ι are:

The subgroup K in the elliptic case. Thus the elliptic upper half-plane is a model for the homogeneous space SL₂(ℝ)/K.

The subgroup N′ (11) of matrices

⎛
⎜
⎝

1	0
ν	1

⎞
⎟
⎠

⎛
⎜
⎝

0	−1
1	0

⎞
⎟
⎠

⎛
⎜
⎝

1	ν
0	1

⎞
⎟
⎠

⎛
⎜
⎝

0	1
−1	0

⎞
⎟
⎠

(31)

in the parabolic case. It also fixes any point ε v on the vertical axis, which is the set of zero divisors in dual numbers. The subgroup N′ is conjugate to subgroup N, thus the parabolic upper half-plane is a model for the homogeneous space SL₂(ℝ)/N.

The subgroup (10) of matrices

⎛
⎜
⎝

coshτ	sinhτ
sinhτ	coshτ

⎞
⎟
⎠

⎛
⎜
⎝

1	−1
1	1

⎞
⎟
⎠

⎛
⎜
⎝

e^τ	0
0	e^−τ

⎞
⎟
⎠

⎛
⎜
⎝

1	1
−1	1

⎞
⎟
⎠

, (32)

in the hyperbolic case. Those transformations also fix the light cone centred at є, that is consisting of є+zero divisors . The subgroup is conjugate to the subgroup A, thus two copies of the upper half-plane (see Section 8.2) is a model for SL₂(ℝ)/A.

Fig. 3.1 shows actions of the above isotropic subgroups on the respective numbers. Note that in parabolic and hyperbolic cases they fix larger sets.

Exercise 2 Check further properties of rotations around ι, that is the actions of isotropic subgroups:

Vectors fields of the isotropy subgroup actions are
(u²+σ(v²−1))∂_u+ 2uv∂_v, where σ=ι².
Orbits of the isotropy subgroups , N′ and K satisfy to the equation:
(u²−σ v²)−2lv−σ=0, where l∈ℝ. (33)

Hint: See method used in Exercise 2. An alternative derivation will be available at Exercise 8.⋄
The isotropy subgroups of ι in all EPH cases are uniformly express by matrices of the form:

⎛
⎜
⎝
        a σ b

        b a

⎞
⎟
⎠

,      where a²−σ b²=1.
The isotropy subgroup of a point u+ι v consists of matrices

⎛
⎜
⎜
⎜
⎜
⎜
⎝
       √

1+σ v²c²

+uc

c(σ v²−u²)

      c
√

1+σ v²c²

−uc

⎞
⎟
⎟
⎟
⎟
⎟
⎠

∈SL₂(ℝ)

and describe admissible values of the parameter c.
Hint: Use the previous item and the transitive action of the ax+b from Exercise 3.⋄

Definition 3 In the hyperbolic case we extend the subgroup to a subgroup A″ by the element (

0	1
−1	0

)

This additional elements flips upper and lower half-planes of double numbers, see Section 8.2. Therefor the subgroup A″_h fixes the set {e₁,−e₁}.

Lemma 4 Möbius action of SL₂(ℝ) in each EPH case is generated by action the corresponding isotropy subgroup (A″_h in the hyperbolic case) and actions of the ax+b group, e.g. subgroups A and N.

Proof. The ax+b group transitively acts on the upper or lower half-planes . Thus for any g∈SL₂(ℝ) there is h in ax+b group such that h⁻¹g either fixes e₁ or sends it to −e₁. Thus h⁻¹g is in the corresponding isotropy subgroup.

Lecture 4 Extended Fillmore–Springer–Cnops Construction

Cycles—circles, parabolas and hyperbolas—are invariant families under respective Möbius transformations. We will proceed now with a study of invariant properties of cycles according to Erlangen programme. A very powerful tool used in this notes is the representation of cycles by appropriate 2× 2 matrices.

4.1 Invariance of Cycles

K-orbits, shown in Fig. 1.2, are K-invariant in a trivial way. Moreover since actions of both A and N for any σ are extremely ‘shape-preserving’, see Exercise 1, we meet natural invariant objects of the Möbius map:

Definition 1 The common name cycle [] is used to denote circles, parabolas with horizontal directrices and equivilaterial hyperbolas with vertical axes of symmetry (as well as straight lines as the limiting cases of any from above) in the respective EPH case.

where k, l, n and m are real parameters, such that not all of them equal to zero. Using hypercomplex numbers we can write the same equation as, cf. []*Supl. C(42a):

where w= u+ι v, K=ι k, L=n+ι l, M=ι m and conjugation is defined by w= u−ι v.

Exercise 2 Check that such cycles mean for certain k, l, n, m straight lines and (depending from the case) one of the following:

in the elliptic case: circles with centre (l/k,n/k) and squared radius m−l²+n²/k;
in the parabolic case: parabolas with horizontal directrices and focus at (l/k, m/2n−l²/2nk+n/2k);
in the hyperbolic case: rectangular hyperbolas with centre (l/k,−n/k) and vertical axes of symmetry.

Thereafter words parabola and hyperbola always assume only the above described types. Straight lines are also called flat cycles .

All three EPH types of cycles are enjoying many common properties, sometimes even beyond that we normally expect. For example, the following definition is quite intelligible even when extended from the above elliptic and hyperbolic cases to the parabolic one.

Definition 3 σ_r-Centre of the σ-cycle (1) for any EPH case is the point (l/k, −σ_rn/k)∈ℝ^σ. Notions of e-centre, p-centre, h-centre are used along the adopted EPH notations.

Centres of straight lines are at infinity, see Subsection 8.1.

Remark 4 Here we use a signature σ_r=−1, 0 or 1 of a number system which does not coincides with the signature σ of the space ℝ^σ. We will need also a third signature σ_c to describe the geometry of cycles in Definition 4.

Exercise 5 Note that some quadruples (k,l,n,m) may correspond through the equation (1) to the empty set on certain point space ℝ^σ. Give the full description of parameters (k,l,n,m) and σ which lead to the empty set. Hint: Using coordinates of cycle’s centre and completion to full squares to show that the empty set may appear only in the elliptic point space. Such circles are usually called imaginary.⋄

The meaningfulness of this definition even in the parabolic case will be justified, for example, by:

Exercise 6 Show that, in all EPH cases, the locus of points having a fixed power with respect to a given cycle C is a cycle concentric with C.

This property is classical for circles []*§ 2.3 and also known for parabolas []*§ 10. However for parabolas Yaglom used the word ‘concentric’ in quotes, since he did not define centres of a parabola explicitly.

The family of all cycles from Definition 1 is invariant under Möbius transformations (24) in all EPH cases, that was already stated in Theorem 2. The only gap in its proof was a demonstration that we can always transform a cycle to a K-orbit.

Exercise 7 Let C be a cycle in ℝ^σ with its centre on the vertical axis. Show that there is the unique scaling w ↦ a²w which maps C to a K-orbit. Hint: Check, that a cycle in ℝ^σ with its centre belonging to the vertical axis is completely defined by the point of its intersection with the vertical axis and its curvature at this point. Then find the value of a for a scaling such that the image of C will satisfies to the relation (27).⋄

We fully describe how cycles are transformed by Möbius transformations in Theorem 1.

4.2 Projective Spaces of Cycles

Fig. 1.3 suggests that we may get a unified treatment of cycles in all EPH by consideration of a higher dimension spaces. The standard mathematical method is to declare objects under investigations to be simply points of some bigger space.

Example 1 In functional analysis sequences or functions are considered as points (vectors) of certain linear spaces . Linear operations (addition and multiplication by a scalar) on vectors (that is functions) are defined point-wise.

If an object is considered as a point (in a new space) all information about its inner structure is lost, of course. Thus the space should be equipped with an appropriate enhancement to hold externally information which were previously inner properties of our objects. That is the inner structure of an object is now revealed through its relations to its peers¹.

Example 2 Take the linear space of continuous real-valued functions on the interval [0,1] and introduce the inner product of two functions by the formula:

⟨ f,g ⟩=

∫

f(t) g(t) dt.

It allows us to define the norm of a function and orthogonality of two functions. These are building blocks of Hilbert space theory, which recovers a lot of Euclidean geometry in spaces of functions.

We will utilise the above fundamental approach for cycles. A generic cycle from Definition 1 is the set of points (u,v)∈ℝ^σ defined for the respective values of σ by the equation

This equation (and the corresponding cycle) is completely determined by a point (k, l, n, m)∈ ℝ⁴. However this is not a one-to-one correspondence: for a scaling factor λ ≠ 0 the point (λ k, λ l, λ n, λ m) defines the an equivalent equation to (3). Thus we prefer to consider the projective space ℙ³, that is ℝ⁴ factorised by the equivalence relation (k, l, n, m)∼ (λ k, λ l, λ n, λ m) for any real λ ≠ 0. A good introductory reading on projective spaces is []*Ch. 10.

Definition 3 We call ℙ³ the cycle space and refer to the initial ℝ^σ as the point space . The correspondence which associate a point of the cycle space to a cycle equation (3) is called map Q .

We also note that the equation (2) of a cycle can be written as a quadratic form

in the homogeneous coordinates (w₁,w₂) such that w=w₁/w₂. Since quadratic forms are related to square matrices, see Section 4.4, we define another map on the cycle space as follows.

Definition 4 We arrange numbers (k, l, n, m) into the cycle matrix

C_σc^s=

⎛
⎜
⎝

l+ι_c s n	−m
k	−l+ι_c s n

⎞
⎟
⎠

, (5)

with a new hypercomplex unit ι_c and an additional real parameter s usually equal to ± 1. If we omit it in the cycle notation C_σc, then the value s=1 is assumed.

The values of σ_c:=ι_c² is −1, 0 or 1 independently from the value of σ. The parameter s=±1 oftenly (but not always) is equal to σ. Matrices different by a real non-zero factor are considered as equivalent.

We denote by M such a map from ℙ³ to the projective space of 2× 2 matrices.

The matrix (5) is the cornerstone of (extended) Fillmore–Springer–Cnops construction (FSCc []) and closely related to technique recently used by A.A. Kirillov to study the Apollonian gasket []. A hint for the composition of this matrix is provided by the following exercise.

Exercise 5 Let the space A² is equipped with a product of symplectic type

[w,w′]=w₁w′₂−w₂ w′₁,

where w^(′)=(w₁^(′),w₂^(′))∈A². Let C=(

L	−M
K	L

), check that the equation of the cycle (4) is given by the expression: [w, Cw]=0 .

The both identifications Q and M are straightforward. Indeed, a point (k, l, n, m)∈ℙ³ equally well represents (as soon as σ , σ_c and s are already fixed) both the equation (3) and the line of matrix (5). Thus for fixed σ , σ_c and s one can introduce the correspondence between quadrics and matrices shown by the horizontal arrow on the following diagram:

4.3 Covariance of FSCc

On the first glance the horizontal arrow in (6) seems to be of a little practical interest since it depends from too many different parameters (σ, σ_c and s). However the following result demonstrates that it is compatible with easy calculations of cycles’ images under the Möbius transformations.

Theorem 1 The image S _σc^s of a cycle C_σc^s under transformation (24) in ℝ^σ with g∈SL₂(ℝ) is given by similarity of the matrix (5):

S _σc^s= gC_σc^sg⁻¹. (7)

In other words FSCc (5) intertwines Möbius action (24) on cycles with linear map (7). Explicitly it means:

⎛
⎜
⎝

l′+ι_c s n′	−m′
k′	−l′+ι_c s n′

⎞
⎟
⎠

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

⎛
⎜
⎝

l+ι_c s n	−m
k	−l+ι_c s n

⎞
⎟
⎠

⎛
⎜
⎝

d	−b
−c	a

⎞
⎟
⎠

. (8)

Proof. There are several ways to prove (7), for now we present a brute force calculation, fortunately performed by a CAS) []. See Appendix C for information for:

Firstly, we build a cycle passing a given point P=[u, v]. For this a generic cycle C with parameters (k,l,n,m) is bounded by the corresponding condition:

We also find the image of P under the Möbius transformation the same element of g∈SL₂(ℝ) but a different hypercomplex unit e:

Finally we check that the conjugated cycle C3 passes the Möbius transform P1. A simplification based on the determinant value 1 and s=±1 will be helpful:

Thus we got the confirmation that the theorem is true in the stated generality. One may wish that every mathematical calculation can be done as simply as that.

Remark 2 There is a bit of cheating in the above proof. In fact, the library does not the hypercomplex form (5) of FSCc matrices. Instead it operates with non-commuting Clifford algebras , which make it usable at any dimension, see [, ]. An equivalent form of FSCc for two dimensions based

The above proof cannot satisfy everyone’s aesthetic feeling. For this reason an alternative route based on orthogonality of cycles [] will be given later, see Exercise 2.

It is worth to notice that the image S _σc^s under similarity (7) is independent of values s and σ_c. This in particularly follows from the following exercise.

Exercise 3 Check that the image (k′,l′,n′,m′) of the cycle (k,l,n,m) under similarity with g= (

a	b
c	d

)∈SL₂(ℝ) is:

(k′,l′,n′,m′)= (k d²+2 l c d+m c², k b d+l( bc+a d)+m a c, n, k b²+2 l a b+m a²).

This can be also presented through matrix multiplication:

⎛
⎜
⎜
⎜
⎝

k′

l′

n′

m′

⎞
⎟
⎟
⎟
⎠

⎛
⎜
⎜
⎜
⎝

d²	2 c d	0	c²
b d	bc+a d	0	a c
0	0	1	0
b²	2 a b	0	a²

⎞
⎟
⎟
⎟
⎠

⎛
⎜
⎜
⎜
⎝

⎞
⎟
⎟
⎟
⎠

Now we have an efficient tool to investigate properties of some notable cycles, which have appeared before.

Exercise 4 Use the similarity formula (7) for the following:

Show that the real axis v=0 is represented by the line coming through (0,0,1,0) and a matrix (

sι_c	0
0	sι_c

). For any (

a	b
c	d

)∈ SL₂(ℝ) we have:

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

⎛
⎜
⎝

sι_c	0
0	sι_c

⎞
⎟
⎠

⎛
⎜
⎝

d	−b
−c	a

⎞
⎟
⎠

⎛
⎜
⎝

sι_c	0
0	sι_c

⎞
⎟
⎠

i.e. the real line is SL₂(ℝ)-invariant.

Write matrices which describe cycles represented by A, N and K-orbits shown in Fig.1.1 and 1.2. Verify that matrices representing those cycles are invariant under the similarity with elements of the respective subgroups A, N and K.
Show that cycles (1,0,n,σ), which are orbits of isotropy groups as described in Exercise 2, are invariant under the respective matrix similarity for the respective values of σ=ι² and any real n.
Find the cycles, which are transverse to orbits of the isotropy subgroups, that is are obtained from the vertical axis by the corresponding actions.

4.4 Origins of FSCc

Fillmore–Springer–Cnops construction (in the generalised form) will play the central rôle in our subsequent investigation. Thus it is worth to look on its roots and the origins before we will cultivate it. So far it appeared in from the thin air, but can we intentionally invent it? Are there further useful generalisations of FSCc? All these are important questions and we will make an attempt to approach them here.

As follows from its name FSCc was developed in stages. Moreover, it appeared independently in a different form in recent work of Kirillov []. This indicates the naturalness and objectivity of the construction. We are interested for now in the flow of ideas rather than exact history or proper credits. For later reader may consult the original works [, , , ] and []*Ch. 18 as well as references therein. Here we treat the simplest two dimensional case, in higher dimensions non-commutative Clifford algebras are helpful with some specific adjustments.

4.4.1 Projective Coordiantes and Homegeneous Polynomials

An old important observation is that Möbius maps appear from linear transformations of homogeneous (projective) coordinates, see []*Ch. 1 for this in a context of invariant theory. This leads to FSCc in several steps:

Another route was used in a later book of Cnops []: a predefined geometry of spheres, specifically their orthogonality, was encoded in the respective matrices of the type (5). Similar connections between geometry of cycles and matrices lead Kirillov, see []*§ 6.3 and the end of this section. He arrived to an identification of disks with Hermitian matrices (which is similar to FSCc) through the geometry of the Minkowski space-time and intertwining property of actions of SL₂(ℂ).

There is one more derivation of FSCc based on projective coordinates. We can observe that the homogeneous form (4) of cycle’s equation (2) can be written using matrices as follows:

In other words, we obtained the usual FSCc with the intertwining property of the type (7). However the generalised form FSCc does not fall out of this consideration yet.

Then the intertwining relation similar to (12) holds if matrix similarity is replaced by the matrix congruence:

This identity provides background to the Kirillov correspondence between circles and matrices, see []*§ 6.3. Clearly it is essentially equivalent to FSCc and either of them may be used with a convenience. It does not hint about the generalised form of FSCc either.

We will mainly work with FSCc stating equivalent forms our results for the Kirillov correspondence occasionally.

4.4.2 Coadjoint Representation

In the above construction the identity (11) requires the same imaginary unit to be used in the quadratic form (the left hand side) and FSCc matrix (the right hand side). How can we arrive to the generalised FSCc directly without an intermediate step of standard FSCc with the same type EPH geometry used in the point cycle spaces? We will consider a route appearing from the representation theory.

Any group G acts on itself by the conjugation g: x ↦ g x g⁻¹, see 1. This map obviously fixes the group identity e. For a Lie group G the tangent space at e can be identified with its Lie algebra g, see Subsection 2.3.1. Then the derived map for the conjugation at e will be a linear map g → g. This is an adjoint representation of a Lie group G on its Lie algebra. This is the departure point for Kirillov’s orbit method closely connected to induced representations , see [, ].

Example 1 In the case of G=SL₂(ℝ) the group operation is the multiplication of 2× 2 matrices. The Lie algebra g can be identified with of traceless matrices, which we can write in the form (

l	−m
k	−l

) for k, l, m∈ℝ. Then the coadjoint action of SL₂(ℝ) on sl₂ is:

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

⎛
⎜
⎝

l	−m
k	−l

⎞
⎟
⎠

↦

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

−1

⎛
⎜
⎝

l	−m
k	−l

⎞
⎟
⎠

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

. (13)

) which are scalar multiples of the identity matrix. Thus we can consider the conjugated action (13) of SL₂(ℝ) on the pairs of matrices, or intervals in the matrix space of the following type:

In other words we obtained the generalised FSCc, cf. (5), and the respective action of SL₂(ℝ). However a connection of the above pairs of matrices with the cycles on a plane did follows from the above consideration and requires some further insights.

Remark 2 Note that scalar multiplies of the identity matrix, which are invariant under similarity correspond in FSCc to the real line, which is Möbius invariant as well.

Lecture 5 Indefinite Product Space of Cycles

In the previous Chapter we represented cycles by points of the projective space ℙ³ or lines of 2× 2 matrices. The later was justified so far only by the similarity formula (7). Now we shall investigate connections between cycles and vector space structure. Thereafter we will use the special form of FSCc matrices to introduce very important additional structure in the cycle space.

5.1 Cycles: an Appearance and the Essence

Our extension (5) of FSCc adds two new elements in comparison with the standard one []:

Such a possibility of an extension exists because elements of SL₂(ℝ) are matrices with real entries, for generic Möbius transformations with hypercomplex valued matrices considered in [] it is impossible.

Indeed, the similarity formula (8) does not contain any squares of hypecomplex units, so their type is irrelevant for this purpose. At the moment the hypercomplex unit ι_c serves only as a placeholder which keeps components l and n separated. However the rôle of ι_c will be greatly extended thereafter. On the other hand the hypercomplex unit ι defines the appearance of cycles on the plane, that is any element (k,l,n,m) in the cycle space ℙ³ admits its drawing as circles, parabolas or hyperbolas in the point space ℝ^σ. We may think on points of ℙ³ as ideal cycles while their depictions on ℝ^σ are only their shadows on the wall of Plato’s cave . More prosaically we can consider cones and their (conic) sections as in Fig. 1.3.

Of course, some elliptic shadow may be imaginary , see Exercise 5. But in most cases we are able correctly guess a cycle from its σ-drawing.

Exercise 1 Describe all pairs of different cycles C_σc^s and S _σc^s such that their σ-drawing for some σ are exactly the same sets. Hint: Note, that the vertical axis is p-drawing of two different cycles (1,0,0,0) and (0,1,0,0) and make a proper generalisation of this observation.⋄

Fig. 1.5(a) shows the same cycles drawn in different EPH styles. Points c_σ=(l/k, −σ n/k) are their respective e/p/h-centres from Definition 3. They are related to each other through several identities:

From analytic geometry we know, that a parabola with the equation ku²−2lu−2nv+m=0 has a focal length, that is the distance from its focus to the vertex, equal to n/2k. As we can see, it is a half of the second coordinate in the e-centre. Fig. 1.5(b) presents two cycles drawn as parabolas, they have the same focal length n/2k and thus their e-centres are on the same level. In other words concentric parabolas are obtained by a vertical shift, not scaling as an analogy with circles or hyperbolas may suggest.

We already extended the definition of centres from circles and hyperbolas to parabolas. It is time for a courtesy payback: parabolas share with other types of cycles their focal attributes.

Definition 2 A σ_r-focus , where σ_r is a variable of EPH type, of a cycle C=(k,j,n,m) is the point in ℝ^σ

f_σr=

⎛
⎜
⎜
⎝

detC_σr^s

2nk

⎞
⎟
⎟
⎠

or explicitly f_σr =

⎛
⎜
⎜
⎝

mk−l²+σ_r n²

2nk

⎞
⎟
⎟
⎠

. (2)

We also use names e-focus , p-focus , h-focus and σ_r-focus, in line with the previous conventions.

Focal length of the cycle C is n/2k.

Note that values of all centres, foci and the focal length are independent from a choice of a quadruple of reals (k,l,n,m) which represents a point in the projective space ℙ³.

Fig. 1.5(b) presents e/p/h-foci of two parabolas with the same focal length. If a cycle is depicted as a parabola then h-focus, p-focus, e-focus are correspondingly geometrical focus of the parabola, vertex of parabola , and the point on the directrix of the parabola nearest to the vertex.

As we will see, cf. Propositions 4 and 7, all three centres and three foci are useful attributes of a cycle even if it is drawn as a circle.

Remark 3 Such a variety of choices is a consequence of the usage of SL₂(ℝ)—a smaller group of symmetries in comparison to the all Möbius maps of ℝ^σ. The SL₂(ℝ) group fixes the real line and consequently a decomposition of vectors into “real” (1) and “imaginary” (ι) parts is obvious. This permits to assign an arbitrary value to the square of the hypercomplex unit ι.

The exceptional rôle of the real line can be viewed in many places. For example, geometric invariants defined below, e.g. orthogonalities in sections 6.1 and 6.6, demonstrate “awareness” of the real line invariance in one way or another. We will call this the boundary effect in the upper half-plane geometry. The famous question on hearing drum’s shape has a sister:

Can we see/feel the boundary from inside a domain?

Remarks 3, 4, 5 and 2 provide hints for positive answers.

Exercise 4 Check that different realisations of the same cycle have this properties in common:

All implementations have the same vertical axis of symmetries;
Intersections with the real axis (if exist) coincide, see r₀ and r₁ for the left triple of cycles in Fig. 1.5(a).
Centres of circle c_e and corresponding hyperbolas c_h are mirror reflections of each other in the real axis with the parabolic centre be the middle point.
σ_c-focus of the elliptic K-orbits , see 1, is the reflection in the real axis of the midpoint between σ_c-centre of this orbit and inverse of its h-centre.

Exercise 3 gives another example of similarities between different implementations of the same cycles defined by the equation (26).

5.2 Cycles as Vectors

Elements of the projective space ℙ³ are lines in the linear space ℝ⁴. Would it be possible to pick up a single point on each line as its “label”? We may wish to do this for the following reasons:

However the general scheme of projective spaces does not permit such universal unique representation, otherwise the cumbersome construction with lines in vector spaces would not be needed. Nevertheless there are several partial possibilities which have certain advantages and disadvantages. We will consider two such opportunities, calling them normalisation of a cycle.

The first is very obvious: we try to have the coefficient k in front of squares in the cycle equation (1) equal to 1. The second normalises the value of determinant of cycle’s matrix. More formally:

Definition 1 A FSCc matrix representing a cycle is said to be k-normalised if its (2,1)-element is 1 and it is det_σc-normalised if its σ_c-determinant is equal ±1.

Exercise 2 Check that:

Element (1,1) of k-normalised C_σc¹ matrix is the σ_c-centre of the cycle.
Any cycle (k,l,n,m) with k≠ 0 has an equivalent k-normalised cycle. Cycles which do not admit k-normalisation correspond to single straight lines in any point space.
det-normalisation is preserved by matrix similarity with SL₂(ℝ) element, as in Theorem 1, while the k-normalisation is not.
Any cycle with a non-zero σ_c-determinant admits det_σc-normalisation. Cycles which cannot be det_σc-normalised will be studied in Section 5.4. What are cycles which cannot be det_σc-normalised for any value of σ_c?

Thus each normalisation may be preferred in particular circumstances. The det-normalisation was used, for example, in [] to get a nice condition for tangent circles, cf. Exercise 2. On the other hand, we will see in Section 7.1 that det C_σc^s of a k-normalised cycle is equal to the square of cycle radius.

Remark 3 It is straightforward to check that there is one more cycle normalisation which is invariant under similarity (7). It is given by the condition n=1. However we will not use it at all in this work.

Cycle’s normalisation is connected with scaling of vectors. Now we turn to the second linear operation: addition. Any two different lines define a unique two-dimensional plane passing through them. Vectors from the plane are linear combinations of two vectors spanning each line. If we consider circles corresponding to elements of the linear span we will obtain a pencil of circles, see []*§ 2.3 and []*§ 10.10. As usual, there is no need to be restricted to circles only:

Definition 4 A pencil of cycles is the linear span (in the sense of ℝ⁴) of two cycles.

Fig. 5.1 shows appearance of such pencils as circles, parabolas and hyperbolas. The elliptic case is very well-known in the classical literature, see []*Figs. 2.2(A,B,C) and 2.3A, for example. The appearance of pencils is visually different for two cases: either spanning cycles are intersecting or disjoint. These two possibilities are presented by the left and the right columns in Fig. 5.1. We shall return to these situations in Corollary 3.

Exercise 5 Investigate the following:

What happens with an elliptic pencil if one of spanning circles is imaginary ? Both spanning circles are imaginary?
How does a pencil look if one spanning cycle is a straight line? If both cycles are straight lines?
A pair of circles is symmetric in the line joining their centres. Thus circular pencils looks similar regardless from direction of this line of centres. Our hyperbolas and parabolas have a special orientation: their axes of symmetry are vertical. Thus horizontal or vertical line joining the centres of two hyperbolas/parabolas make a special arrangement, and it was used for Fig. 5.1. How do hyperbolic pencils look if the line of centres is not horizontal?

Exercise 6 Show the following:

The image of a pencil of cycles under a Möbius transformation is again a pencil of cycles.
A pencil spanned by two concentric cycles consists of concentric cycles.
All cycles in a pencil are coaxal , see Ex. 3 for the definition. The joint radical axis , see Exercise 2, is the only straight line in the pencil. This is also visible from Fig. 5.1.
Let two pencils of cycles do not have any common cycle. Then any cycle from ℙ³ belong to a new pencil spanned by two cycles from two given pencils.

This Section demonstrated that there are numerous connections between the linear structure of the cycle space ℙ³ and the geometrical property of cycles in the point space ℝ^σ.

5.3 Invariant Cycle Product

We are looking for a possibility to enrich the geometry of the cycle space through the FSCc matrices. Many important relations between cycles are based on the following Möbius invariant cycle product.

Definition 1 The cycle σ_c-product of two cycles is given by:

⟨ C_σc^s,S _σc^s ⟩ = −tr(C_σc^s

S _σc^s

), (3)

that is negative trace of the matrix product of C_σc^s and hypercomplex conjugation of S _σc^s.

As we already mentioned an inner product of type (3) is used, for example, in Gelfand–Naimark–Segal (GNS) construction to make a Hilbert space out of C^*-algebra. This may be more than a simple coincidence since FSCc matrices can be considered as linear operators on a two-dimensional vector space. However a significant difference with the Hilbert space inner product is that the cycle product is indefinite, see Exercise 4 for details. Thus cycles form a Pontrjagin or Krein space [] rather than Euclidean or Hilbert. Geometrical interpretation of the cycle product will be given in Exercise 3.

Exercise 2 Check that

The value of cycle product (3) will remain the same if both matrices will be replaces by their image under similarity (7) with the same element g∈SL₂(ℝ).

The σ_c-cycle product (3) of cycles defined by quadruples (k,l,n,m) and (k′,l′,n′,m′) is given by

−2l′l+2σ_c s²n′n+k′m+m′k.

(4)

More specifically and also taking into account the value s=± 1 it is

−2n′n−2l′l+k′m+m′k,			(5)
−2l′l+k′m+m′k,			(6)
2n′n−2l′l+k′m+m′k			(7)

in the elliptic, parabolic and hyperbolic cases respectively.

Let C_σc and S _σc be two cycles defined by e-centres (u,v) and (u′, v′) with σ-determinants −r² and −r′² respectively. Then their σ_c-cycle product explicitly is:
⟨ C_σc,S _σc ⟩= (u−u′)²−σ(v−v′)²−2(σ−σ_c)vv′−r²−r′². (8)
The cycle product is symmetric bilinear functions of two cycles. It is indefinite in the sense that there are both cycles C_σc and S _σc such that
⟨ C_σc,C_σc ⟩>0 and ⟨ S _σc,S _σc ⟩<0.

Hint: It is easy to show symmetry of the cycle product from the explicit value (4). This is not so obvious from the initial definition (3) due to the presence of hypercomplex conjugation.⋄

A simple but interesting observation is that for FSCc matrices representing cycles we obtain the second classical invariant (determinant) under similarities (7) from the first (trace) as follows:

Therefore it is not surprising that the determinant of a cycle enters the Definition 2 of the foci and the following definition.

Definition 3 We say that a C_σc is σ_c-zero-radius cycle , σ_c-positive cycle or σ_c-negative cycle if the value of the cycle product of C_σc with itself (9) (and thus its determinant) is zero, positive or negative respectively.

Exercise 4 The zero-radius, positive and negative cycles form three non-empty disjoint Möbius invariant families. Hint: For non-emptiness see Exercise 4 or use the explicit formula (9).⋄

The following relations are useful for description of those three classes of cycles.

Exercise 5 Check that:

For any cycle C_σc^s there is the following relation between its determinants evaluated with elliptic, parabolic and hyperbolic unit ι_c:
detC_e^s ≤ detC_p^s ≤ detC_h^s.
The negative of parabolic determinant −detC_p^s of a cycle (k,l,n,m) coincides with the discriminant of the quadratic equation:
ku²−2lu+m=0, (10)

which defines intersection of the cycle with the real line (in any EPH presentation).

We already illustrated zero-radius cycles in Fig. 1.6. This shall be compared with positive and negative cycles for various values of σ_c in Fig. 5.2. Positiveness-negativeness of cycles has clear geometric manifestation for proper combinations of σ and σ_c at least.

Exercise 6 Verify the following statements:

Circles corresponding to e-positive cycles are imaginary . All regular circles are e-negative.
Parabolas drawn from p- and e-negative cycles have two points of intersection with the real axis, and drawn from p- and h-positive cycles does not intersect the real axis.
Hyperbolas representing h-negative cycles have “vertical” branches and representing h-positive cycles—“horizontal” ones. Hint: Write the condition for a hyperbola defined by the equation (1) to intersect the vertical line passing its centres.⋄

Of course, manifestations of the indefinite nature of the cycle product (3) are not limited to the above examples and we will meet more of them on several other occasions.

5.4 Zero-radius Cycles

Due to the projective nature of the cycle space ℙ³ the absolute values of the cycle product (3) on non-normalised matrices are irrelevant, unless it is zero. There are many reasons to take a closer look on cycles with the zero value of the product—zero-radius cycles, for example:

To highlight that certain cycle is σ_c-zero-radius we will denote it by Z_σc^s. A justification of the chosen name for such cycles is in Exercise 1, further connections will be provided in Section 7.1.

As it often happens in our study, we again have nine different possibilities: for cycles drawn in three EPH types in the point space (parametrised by σ) there are three independent conditions detC_σc^s=0 in the cycle space (parametrised by σ_c). This is illustrated in Fig. 1.6.

Exercise 1 Show that

Any σ_c-zero-radius cycle admitting k-normalisation can be represented by the matrix

Z_σc^s=

⎛
⎜
⎝

z	−zz
1	−z

⎞
⎟
⎠

⎛
⎜
⎝

z	z
1	1

⎞
⎟
⎠

⎛
⎜
⎝

1	−z
1	−z

⎞
⎟
⎠

⎛
⎜
⎝

u₀+ ι_c s v₀	u₀²−σ_c v₀²
1	−u₀+ι_c s v₀

⎞
⎟
⎠

, (11)

for some z=u₀+ ι_c s v₀ being its centre and s=± 1. Compare with (10).

Describe all σ_c-zero-radius cycles which cannot be k-normalised.
σ_c-focus of the σ_c-zero-radius cycle belongs to the real axis, see Fig. 1.6.
Let Z_σc^s and T_σc^s are two k-normalised σ_c-zero radius cycles of the form (11) with e-centres (u₀,v₀) and (u₁,v₁) then:
⟨ Z_σc^s,T_σc^s ⟩= (u₀−u₁)²−σ_c(v₀−v₁)². (12)

As follows from (11) the class of σ_c-zero-radius cycles is parametrises by two real numbers (u,v) only and as such is easily attached to the respective point of z=u+ι v∈ ℝ^σ, at least in the elliptic and hyperbolic point spaces. In ℝ^p a connection between a σ_c-zero-radius cycle and its centre is obscure.

Exercise 2 Prove the following observations from Fig. 1.6:

The cycle Z_σc^s (11) with det Z_σc^s=0, σ_c=−1 drawn elliptically is just a single point (u₀,v₀), i.e. (elliptic) zero-radius circle.
The same condition detZ_σc^s=0 with σ_c=1 in the hyperbolic drawing produces a light cone originated at point (u₀,v₀):
(u−u₀)²−(v−v₀)²=0,

i.e. a zero-radius cycle in hyperbolic metric, see Section 7.1.
p-zero-radius cycles in any implementation touch the real axis.
The e-centre of the transformation gZ_σc^sg⁻¹, g∈SL₂(ℝ) of the σ_c-zero-radius cycle (11) coincides with the Möbius action g· z in ℝ^σ_c, where z is e-centre of Z_σc^s. Hint: The result can be easily obtained along the lines from Subsection 4.4.1. A CAS solution is provided as well.⋄

Remark 3 The above “touching” property 3 is a manifestation of the already mentioned boundary effect in the upper half-plane geometry, see Remark 3.

The previous Exercise shows that σ_c-zero-radius cycles “encode” points into the “cycle language”. The following reformulation of Exercises 4 stresses that this encoding is Möbius invariant as well.

Lemma 4 The conjugate g⁻¹Z_σc^s(y)g of a σ_c-zero-radius cycle Z_σc^s(y) with g∈SL₂(ℝ) is a σ_c-zero-radius cycle Z_σc^s(g· y) with centre at g· y—the Möbius transform of the centre of Z_σc^s(y).

Furthermore we can extend the relation between a zero-radius cycle Z_σc and points through the following connection of Z_σc with the power of its centre.

Exercise 5 Let Z_σ be the zero-radius cycle (11) defined by z=u−ισ v, that is with the σ-centre at the point (u, v). Show that in the elliptic and hyperbolic (but not parabolic) cases a power of a point, see Definition 4, (u,v)∈ℝ^σ with respect to a cycle C_σ is equal to the cycle product ⟨ Z_σ,C_σ ⟩, where both cycles are k-normalised.

It is noteworthy that the notion of point’s power is not Möbius invariant even despite its definition through the invariant cycle product. It is due to the presence of non-invariant k-normalisation. This suggests to extend the notion of the power from a point (that is a zero-radius cycle) to an arbitrary cycles:

Definition 6 A (σ,σ_c)-power of a cycle C with respect to another cycle S is equal to their cycle product ⟨ C_σc,S _σc ⟩, where both matrices are det_σ-normalised.

Since both elements of this definition—the cycle product and det-normalisation—are Möbius invariant, the resulting value is preserved by Möbius transformations as well. Of course, the (e,e)-variant of this notion is well known in the classical theory.

Exercise 7 Show that the (e,e)-power of a circle C with respect to another circle S has an absolute value equal to the inversive distance between C and S , see []*Defn. 3.2.2, []*§ 4, []*§ 5.8 and Thm. 5.91:

d(C, S )=

⎪
⎪
⎪
⎪

⎪
⎪

c₁−c₂

⎪
⎪

²−r₁²−r₂²

2r₁r₂

⎪
⎪
⎪
⎪

. (13)

Here c_1,2 and r_1,2 are e-centres and radii of the circles. We will develop this theme in Exercise 8.

As another illustration of the technique based on zero-radius cycles we return to orbits of isotropy subgroups, cf. Exercise 2.

Exercise 8 Fulfil the following steps.

Write coefficients of σ-zero-radius cycle Z_σ(ι) in ℝ^σ with e-centre at the hypercomplex unit ι=(0,1).
According to Exercise 4 Z_σ(ι) is invariant under the similarity Z_σ(ι) ↦ hZ_σ(ι)h⁻¹ with h in the respective isotropy subgroups K, N′ and of ι. Check this directly.
Write coefficients of a generic cycle in the pencil spanned by Z_σ(ι) and the real line. Note that the real line is also invariant under the action of the isotropy subgroups (as any other Möbius transformations) and conclude that any cycle from the pencil shall be invariant under the action of the isotropy subgroups as well. In other words, those cycles are orbits of the isotropy subgroups , check that you obtained their equation (33).

We freeze our study of zero-radius cycles for their own but they will repeatedly appear in the following text in relation to other objects.

5.5 Cauchy–Schwartz Inequality and Tangent Cycles

We already noted that the invariant cycle product is a special (and remarkable!) example of an indefinite product in a vector space. Continuing this comparison it will be interesting to look for a role of a Cauchy–Schwartz–type inequality:

First of all, the classical form (14) of this inequality failed in any indefinite product space. This can be seen from examples or an observation that all classical proofs start from the assumption that ⟨ x+ty,x+ty ⟩≥ 0 in an inner product space. In an indefinite product space there are always pairs of vectors, which realise any of three possible relation:

A bit of regularity appears from the fact, that the type of inequality is inherited by linear spans.

Exercise 1 Let two vectors x and y in an indefinite product space satisfy to the inequality:

⟨ x,y ⟩⟨ y,x ⟩ < ⟨ x,x ⟩ ⟨ y,y ⟩ (⟨ x,y ⟩⟨ y,x ⟩ > ⟨ x,x ⟩ ⟨ y,y ⟩),

then any two non-collinear vectors z and w from the real linear span of x and y satisfy to the same type of inequality:

⟨ z,w ⟩⟨ w,z ⟩ < ⟨ z,z ⟩ ⟨ w,w ⟩ (⟨ z,w ⟩⟨ w,z ⟩ > ⟨ z,z ⟩ ⟨ w,w ⟩).

The equality ⟨ x,y ⟩⟨ y,x ⟩ = ⟨ x,x ⟩ ⟨ y,y ⟩ always implies ⟨ z,w ⟩⟨ w,z ⟩ = ⟨ z,z ⟩ ⟨ w,w ⟩.

Exercise 2 Check the following.

Let C_σ and S _σ be two cycles defined by e-centres (u,v) and (u′, v′) with σ-determinants −r² and −r′². The relations
⟨ C_σ^s,S _σ^s ⟩² ⪋ ⟨ C_σ^s,C_σ^s ⟩ ⟨ S _σ^s,S _σ^s ⟩

guaranties that σ-implementations of C_σ and S _σ for σ=±1 are intersecting, tangent or disjoint respectively . What happens for the parabolic value σ=0? Hint: Determine the sign of expression ⟨ C_σ^s,S _σ^s ⟩ −√⟨ C_σ^s,C_σ^s ⟩ ⟨ S _σ^s,S _σ^s ⟩ using the formula (8). For the parabolic case Exercise 2(p) shall be useful.⋄
Let two cycles C_σ and S _σ are in det_σ-normalised form. Deduce from the previous item the following Descartes–Kirillov condition []*Lem. 6.3 for C_σ and S _σ to be externally tangent:
det(C_σ+S _σ)=0 and ⟨ C_σ,S _σ ⟩>0. (15)

Moreover, C_σ+S _σ is the σ-zero-radius cycle at the tangent point of C_σ and S _σ. Hint: The identity in (15) follows from the inequality there and the relation | ⟨ C_σ,S _σ ⟩ | =√⟨ C_σ^s,C_σ^s ⟩ ⟨ S _σ^s,S _σ^s ⟩=1 for tangent cycles from the first item. For the last statement use Exercise 5.⋄

Corollary 3 For a given pencil of cycles, see Definition 4, either all cycles are pairwise disjoint, or every two cycles are tangents, or all of them have at least two common points.

Zero-radius cycles form a two-parameter family (in fact a manifold) in the three dimensional projective cycle space ℙ³. It is not flat, as can be seen from its intersection with projective lines—cycle pencils . Cauchy–Schwartz inequality turns to be relevant here as well.

Exercise 4 A pencil of cycles either contains at most two σ_c-zero-radius cycles or consists of σ_c-zero-radius cycles entirely. Moreover:

A pencil spanned by two different cycles cannot consists of e-zero-radius cycles only. Describe all pencils consisting of p- and h-zero-radius cycles only. Hint: Formula (12) shall be useful to describe pencils consisting of (and thus spanned by) σ_c-zero-radius cycles. Orbits of subgroup N′ shown on the central drawing of Fig. 3.1 are an example of pencil of p-zero-radius cycles drawn as parabolas. You can experiment with σ-drawing of certain σ_c-zero-radius pencils.⋄
A pencil spanned by two different cycles C_σc^s and S _σc^s, which does not consist of σ_c-zero-radius cycles only, has exactly two, one or none σ_c-zero-radius cycles depending which of three possible Cauchy–Schwartz–type relations holds:
⟨ C_σc^s,S _σc^s ⟩² ⪋ ⟨ C_σc^s,C_σc^s ⟩ ⟨ S _σc^s,S _σc^s ⟩.

Hint: Write the expression for the cycle product of the span tC_σc^s+S _σc^s, t∈ℝ with itself in terms of products ⟨ C_σc^s,S _σc^s ⟩, ⟨ C_σc^s,C_σc^s ⟩ and ⟨ S _σc^s,S _σc^s ⟩.⋄

Lecture 6 Joint Invariants of Cycles: Orthogonality

The invariant cycle product, defined in the previous Chapter, allows us to define joint invariants of two (or even more) cycles. Being initially defined in an algebraic fashion they reveal their rich geometrical content as well. We will also see that 2× 2 matrices representing cycles

6.1 Orthogonality of Cycles

According to the categorical viewpoint internal properties of objects are of minor importance in comparison to their relations with other objects from the same class. Such a projection of internal properties into external relations was discussed at the beginning of Section 4.2 also. As a further illustration we may put the proof of Theorem 1 sketched below. Thus we will now look for invariant relations between two or more cycles.

After we defined the invariant cycle product (3) the next standard move is to use the analogy with Euclidean and Hilbert spaces and give the following definition.

Definition 1 Two cycles C_σc^s and S _σc^s are called σ_c-orthogonal if their σ_c-cycle product vanishes:

⟨ C_σc^s,S _σc^s ⟩=0. (1)

Exercise 2 Use Exercise 2 to check the following:

The σ_c-orthogonality condition (1) is invariant under Möbius transformations.
The explicit expression for σ_c-orthogonality of cycles in terms of their coefficients is:
2σ_cn′n−2l′l+k′m+m′k=0. (2)
σ_c-orthogonality of cycles defined by their e-centres (u,v) and (u′, v′) with σ-determinants −r² and −r′² respectively is:
(u−u′)²−σ(v−v′)²−2(σ−σ_c)vv′−r²−r′²=0. (3)
Two circles are e-orthogonal if their tangents at an intersection point form the right angle. Hint: Use the previous formula (3), the inverse of Pythagoras’ theorem and Fig. 6.1(a) for this.⋄

The last item can be reformulated as follows: for circles their e-orthogonality as vectors in the cycle spaces ℙ³ with the cycle product (3) coincides with their orthogonality as geometrical sets in the point space ℝ^e. This is a very strong support for FSCc and the cycle product (3) defined from it. Thereafter it is tempting to find similar interpretations for other types of orthogonality. The next Exercise does the first step in the case of σ-orthogonality in the matching point space ℝ^σ.

Exercise 3 Check the following geometrical meaning for σ-orthogonality of σ-cycles.

Let σ=±, then to cycle in ℝ^σ (that is circles or hyperbolas) are σ-orthogonal if slopes S₁ and S₂ of their tangents at the intersection point satisfy to the condition:
S₁ S₂=σ. (4)

Geometrical meaning of this condition can be given either in terms of angles (A) or centres (C):
1. For the case σ=−1 (circles) equation (4) means orthogonality of the tangents, cf. Exercise 4. For σ=1, two hyperbolas are h-orthogonal if lines with the slopes ± 1 bisect the angle of intersection of the hyperbolas, see Fig. 6.1(b). Hint: Define a cycle C_σ by the condition, that it passes a point (u,v)∈ℝ^σ. Define a second cycle S _σ by both conditions: it passes the same point (u,v) and is orthogonal to C_σ. Then use the implicit derivative formula to find slopes of tangents to C_σ and S _σ at (u,v). A script calculating this in CAS is provided as well.⋄
2. In both cases σ=± 1 the tangent to one cycle at the intersection point passes the centre of another cycle. Hint: This fact is clear for circles from the inspection, say, Fig. 6.1(a). For hyperbolas it is enough to observe that the slop of tangent to a hyperbola y=1/x at a point (x, 1/x) is −1/x² and the slope of the line from the centre (0,0) to the point (x,1/x) is 1/x², so the angle between two lines is bisected by vertical/horizontal line. All our hyperbolas are obtained from y=1/x by rotation ± 45 ^∘ and scaling.⋄
Let σ=0 and a parabola C_p have two real roots u₁ and u₂. If a parabola S _p is p-orthogonal to C_p then the tangent to S _p at a point above one of the roots u_1,2 passes the p-centre (u₁+u₂/2,0) of C_p.

Remark 4 Note that geometric p-orthogonality condition for parabolas is non-local in the sense that it does not direct behaviour of tangents at the intersection points. Moreover orthogonal parabolas need not intersect at all. We shall see more examples of such non-locality later on. The relation 3(p) is also another example of boundary awareness, cf. Remark 3 : we are taking a tangent of one parabola above the point of intersection of other parabola with the boundary of the upper half-plane.

The stated geometrical conditions for orthogonality of cycles are not only necessary but are sufficient as well.

Exercise 5

Prove the converses to two statements of Exercise 3. Hint: To avoid irrationalities in the parabolic case and make the calculations accessible for CAS you may proceed as follows. Define a generic parabola passing (u,v) and use implicit derivation to find its tangent at this point. Define the second parabola passing (u,0) and its centre at the intersection of the tangent of the first parabola at (u,v) and the horizontal axis. Then check p-orthogonality of two parabolas.⋄
Let a parabola have two tangents touching it at (u₁,v₁) and (u₂,v₂) and these tangents intersect at a point (u,v). Then u=u₁+u₂/2. Hint: Use the geometric description of p-orthogonality and note that two roots of a parabola are interchangeable in the necessary condition for p-orthogonality.⋄

We found geometrical necessary and sufficient conditions for σ-orthogonality in the matching point space ℝ^σ. The remaining six non-matching cases will be reduced to this in Section 6.3 using an axillary ghost cycle. It will be useful to collect some more properties of orthogonality relations before that.

6.2 Orthogonality Miscellanea

The explicit formulae (2) and (3) allow us to obtain several simple and yet useful conclusions.

Exercise 1 Show that

A cycle is σ_c-self-orthogonal (isotropic) if and only if σ_c-zero-radius cycle (11).
For σ_c=± 1 there is no non-trivial cycle orthogonal to all other non-trivial cycles.
For σ_c=0 only the horizontal axis v=0 is orthogonal to all other non-trivial cycles.
A cycle C_σ^s is σ-orthogonal to a zero-radius cycle Z_σ^s (11) if and only if σ-implementation of C_σ^s passes through the σ-centre of Z_σ^s, or analytically:
k(u² − σ v²) − 2⟨ (l,n),(u, σ_c v) ⟩+m =0, (5)
For σ_c=± 1 any cycle is uniquely defined by the family of cycles orthogonal to it, i.e. (C_σc^s⊥)^⊥={C_σc^s}.
For σ_c= 0 the set (C_σc^s⊥)^⊥ is the pencil spanned by C_σc^s and the real line. In particular if C_σc^s has real roots then all cycles in (C_σc^s⊥)^⊥ have those roots.

The connection between orthogonality and incidence from Exercise 3 allow us to combine techniques of zero-radius cycles and orthogonality in an efficient tool.

Exercise 2 Fill all gaps in the following proof:

Proof.[Sketch of an alternative proof of Theorem 1] We already mentioned in Subsection 4.4.1 that the validity of Theorem 1 for a zero-radius cycle (11)

Z_σc^s=

⎛
⎜
⎝

z	−zz
1	−z

⎞
⎟
⎠

⎛
⎜
⎝

z	z
1	1

⎞
⎟
⎠

⎛
⎜
⎝

1	−z
1	−z

⎞
⎟
⎠

with the centre z=x+i y is a straightforward calculation, see also Exercise 4. This implies the result for a generic cycle with the help of

Möbius invariance of the product (3) (and thus orthogonality);
the above relation 3 between the orthogonality and the incidence.

The idea of such a proof is borrowed from [] and details can be found therein.

The above demonstration suggests a generic technique for extrapolation of results from zero-radius cycles to the entire cycle space. We will formulate it with the help of map Q from the cycle space to conics in the point space from Definition 3.

Proposition 3 Let T: ℙ³ → ℙ³ is an orthogonality preserving map of the cycles space, i.e. ⟨ C_σc^s,S _σc^s ⟩=0 ⇔ ⟨ TC_σc^s,TS _σc^s ⟩=0. Then for σ≠ 0 there is a map T_σ: ℝ^σ →ℝ^σ, such that Q intertwines T and T_σ:

Q T_σ= T Q. (6)

Proof. If T preserves orthogonality (i.e. the cycle product (3) and consequently the determinant, see (9)) then by the image TZ_σc^s(u,v) of a zero-radius cycle Z_σc^s(u,v) is again a zero-radius cycle Z_σc^s(u₁,v₁) and we can define T_σ by the identity T_σ: (u,v)↦ (u₁,v₁).

To prove the intertwining property (6) we need to show that if a cycle C_σc^s passes through (u,v) then the image TC_σc^s passes through T_σ(u,v). However for σ≠ 0 this is a consequence of the T-invariance of orthogonality and the expression of the point-to-cycle incidence through orthogonality from Exercise 3.

Exercise 4 Let T_i: ℙ³ → ℙ³, i=1,2 are two orthogonality preserving maps of the cycles space. Show that if they coincide on the subspace of σ_c-zero-radius cycles, σ_c≠ 0, then they are identical in the whole ℙ³.

We defined orthogonality from an inner product, which is linear in each component. Thus orthogonality respects linearity in its turn as well.

Exercise 5 Check the following relations between orthogonality and pencils:

Let a cycle C_σc be σ_c-orthogonal to two different cycles S _σc and G _σc, then C_σc is σ_c-orthogonal to every cycle in the pencil spanned by S _σc and G _σc.
Check that all cycles σ_c-orthogonal with σ_c=± 1 to two different cycles S _σc and G _σc belong to a single pencil. Describe such a family for σ_c=0. Hint: For the case σ_c=0 the family is spanned by an additional cycle, which was mentioned in Exercise 2.⋄
If two circles are non-intersecting, then the orthogonal pencil is passing through two points, which are the only to e-zero-radius cycles in the pencil. And vice verse: a pencil orthogonal to two intersecting circles consists of disjoint circles . Tangent circles have the orthogonal pencils of circles all of them being tangent at the same point, cf. Corollary 3.

Exercise 2 describes two orthogonal pencils such that each cycle in one pencil is orthogonal to every cycle in the second. In terms of indefinite linear algebra, see []*§ 2.2, we are speaking about the orthogonal complement of a two dimensional subspace in a four dimensional space and it turns up to be two-dimensional as well. For circles this construction is well-known, see []*§ 5.7 and []*§ 10.10. An illustration in three cases is provided by Fig. 6.2. Reader may wish to play more with orthogonal complements to various parabolic and hyperbolic pencils see Fig. 5.1 and Exercise 5.

Such orthogonal pencils naturally appears in many circumstances and we already met them on several occasions. We know from Exercise 3 and 4 that K-orbits and transverse lines make coaxal pencils which turn to be in a relation:

Exercise 6 Check that any K-orbit (26) in ℝ^σ is σ-orthogonal to any transverse line (28). Fig. 1.2 provides an illustration. Hint: There are several possibility to check this. A direct calculation based on the explicit expressions for cycles is not difficult. Alternatively we can observe that the pencil of transverse lines is generated by K-action from the vertical axis and orthogonality is Möbius invariant.⋄

We may describe a finer structure of the cycle space through Möbius invariant subclasses of cycles. Three such families—zero-radius, positive and negative cycles—were already considered in Section 5.3 and 5.4. They were defined through properties of cycle product with itself. Another important class of cycles is given by the value of its cycle product with the real line.

Definition 7 A cycle C_σc^s is called self-adjoint if it is σ_c-orthogonal with σ_c≠ 0 to the real line, i.e. it is defined by the condition ⟨ C_σc^s,R_σc^s ⟩=0, where R_σc^s=(0,0,1,0) corresponds to the horizontal axis v=0.

The following algebraic properties of self-adjoint cycles easily follow from the definition.

Exercise 8 Show that:

Self-adjoint cycles make a Möbius invariant family.
Explicitly a self-adjoint cycle C_σc^s is defined by n=0 in (1) for both values σ_c=± 1.
Any of the following conditions are necessary and sufficient for a cycle to be self-adjoint:
- All three centres of the cycle coincide.
- At least two centres of the cycle belongs to the real line.

From these analytic conditions we can derive geometric characterisation of self-adjoint cycles:

Exercise 9 Show that self-adjoint cycles have the following implementations in the point space ℝ^σ:

(e,h) circles or hyperbolas with their geometric centres on the real line;
(p) vertical lines, consisting of points (u,v) such that | u−u₀ |=r for some u₀∈ℝ, r∈ℝ₊. This cycles are also given by the ||x−y||=r² in the parabolic metric defined below in (3).

Notably, self-adjoint cycles in the parabolic point space were labelled as “parabolic circles” by Yaglom, see []*§ 7. On the other hand Yaglom used the term “parabolic cycle” for our p-cycle with non-zero k and n.

Exercise 10 Show that

Any cycle C_σc^s=(k,l,n,m) belongs to a pencil spanned by a self-adjoint cycle HC_σc^s and the real line:
C_σc^s=HC_σc^s+nR_σc^s, where HC_σc^s=(k,l,0,m). (7)

This identity is a definition of linear orthogonal projection H from the cycle space to its subspace of self-adjoint cycles.
The decomposition of a cycle into the linear combination of a self-adjoint cycle and the real line is Möbius invariant:
g·C_σc^s=g· HC_σc^s+nR_σc^s.

Hint: Two first items are small bits of linear algebra in an indefinite product space, see []*§ 2.2.⋄
Cycles C_σc^s and HC_σc^s have the same real roots.

We are now equipped to consider geometrical meaning of all nine sorts of cycle orthogonality.

6.3 Ghost Cycles and Orthogonality

For the case of σ_cσ=1, i.e. when geometries of the cycle and point spaces are both either elliptic or hyperbolic, σ_c-orthogonality can be expressed locally through tangents to cycles at the intersection points, see Exercise 3(A). Semi-local condition exists as well: the tangent to one cycle at the intersection point passes the second cycle centre, see Exercise 3(C). We may note that the pure parabolic case σ=σ_c=0 the geometric orthogonality condition from Exercise 3(p) can be restate with a help from Exercise 1 as follows:

Corollary 1 Two p-cycles C_p and S _p are p-orthogonal if the tangent to C_p at its intersection point with the projection HS _p (7) of S _p to self-adjoint cycles passes the p-centre of S _p.

Hint: To reformulate Exercise 3(p) to the present form it is enough to use Exercises 9(p) and 3.⋄

The three cases with matching geometries in point and cycle spaces are now pretty unified. Would it be possible to extend such a geometric interpretation of orthogonality to the remaining six (=9−3) cases?

Elliptic (in the point space) realisations of Definition 1, i.e. σ=−1 was shown in Fig. 1.7 and form the first row in Fig. 6.3. The left picture in this row corresponds to the elliptic cycle space, e.g. σ_c=−1. The orthogonality between the red circle and any circle from the blue or green families is given in the usual Euclidean sense described in Exercise 3(e,h). In other words we can decide on orthogonality of circles observing angles between their tangents at the arbitrary small neighbourhood of the intersection point. Therefore, all circles from the either green or blue families, which are orthogonal to the red circle, have the common tangents at points a and b respectively.

The central (parabolic in the cycle space) and the right (hyperbolic) pictures show non-local nature of orthogonality if σ≠σ_c. The blue family has the intersection point b with the red circle, and tangents to blue circles at b are different. However we may observe that all of them are passing the second point d, this property will be used in Section 6.5 to define the inversion in a cycle. A further investigation of Fig. 6.3 reveals that circles from the green family have the common tangent at point a, however this point does not belong to the red circle. Moreover, in line with the geometric interpretation from 3(C) the common tangent to green family at a passes the p-centre (on the central–parabolic drawing) or h-centre (on the right–hyperbolic drawing).

There are analogues pictures in parabolic and hyperbolic point spaces as well, they are presented in the second and third rows of Fig. 6.3. The behaviour of green and blue families of cycles at point a, b and d is similar up to the obvious modification: the matching EPH cases of the point and cycle spaces are central and right drawing for the second and third rows respectively.

Therefore we will clarify the nature of orthogonality if the locus of such points a with tangents passing other cycle’s σ_c-centre will be described. We are going to demonstrate that this locus is a cycle, which we shall call “ghost”. The ghost cycle is shown by the dashed lines in Fig. 6.3. To give an analytic description we need the Heaviside function χ(σ):

More specifically we note the relations: χ(σ)=σ if σ=±1 and χ(σ)=1 if σ=0. Thus Heaviside function will be used to avoid the degeneracy of the parabolic case.

Definition 2 For a cycle C_σc in σ-implementations we define the associated (σ_c-)ghost cycle G _σc by the following two conditions:

χ(σ)-centre of G _σc coincides with σ_c-centre of C_σc.
Determinant of G _σ¹ is equal to determinant of C_σ^σ_c.

Exercise 3 Verify the following properties of a ghost cycle:

G _σ coincides with C_σ if σ σ_c=1;
G _σ has common roots (real or imaginary) with C_σ;
For a cycle C_σc its p-ghost cycle G _σc and the non-selfadjoint part HC_σc (7) coincide.
All straight lines σ_c-orthogonal to a cycle pass its σ_c-centre.

The significance of the ghost cycle is: σ_c-orthogonality between two cycles in ℝ^σ is reduced to σ-orthogonality to the ghost cycle.

Proposition 4 Let cycles C_σc and S _σc be σ_c-orthogonal in ℝ^σ and let G _σc be the ghost cycle of C_σc. Then

S _σc and G _σc are σ-orthogonal in ℝ^σ for seven cases except two cases σ=0 and σ_c=± 1.
In the σ-implementation the tangent line to S _σc at a point of its intersection with
1. the ghost cycle G _σ, if σ=± 1,
2. non-selfadjoint part HC_σc (7) of the cycle C_σc, if σ=0
passes the σ_c-centre C_σc, which coincides with σ-centre of G _σ.

Proof. The statement 1 can be shown by algebraic manipulation, possibly in CAS. Then the non-parabolic case 2(a) follows from the first part 1, which reduces non-matching orthogonality to matching one with the ghost cycle, and the geometric description of matching orthogonality from Exercise 3. Therefore we only need to provide a new calculation for the parabolic case 2(b). Note that in the case σ=σ_c=0 there is no a disagreement between the first and second parts of the proposition since HC_σc=S _σc due to 3.

Consideration of ghost cycles does present orthogonality in the geometric terms however it hides the symmetry of this relation. Indeed it is not obvious that S _σc^s relates to the ghost of C_σc^s in the same way as C_σc^s relates to the ghost of S _σc^s.

Remark 5 Elliptic and hyperbolic ghost cycles are symmetric in the real line, the parabolic ghost cycle has its centre on it, see Fig. 6.3. This is an illustration to the boundary effect from Remarks 3.

Finally we note that Proposition 4 expresses σ_c-orthogonality through σ_c centre of cycles. It illustrates their meaningfulness of various centres within our approach, that may be not so obvious at the beginning.

6.4 Actions of FSCc Matrices

Definition 4 associates a 2× 2-matrix to any cycle. Those matrices can be treated analogously to elements of SL₂(ℝ) in many respects. Similarly to the SL₂(ℝ) action (24) we can consider a fraction-linear transformation on the point space ℝ^σ defined by a cycle and its FSCc matrix:

Exercise 1 Check that w=u+ι v∈ℝ^σ is a fixed point of the map C_σ^−σ (9) if and only if σ-implementation of C_σ^−σ passes w. If detS _σ^s≠ 0 then the second iteration of the map is the identity.

We can also extend from SL₂(ℝ) to cycles the conjugated action (7) on cycle space. Indeed a cycle S _σc^s in the matrix form acts on another cycle C_σc^s by the σ_c-similarity:

The similarity can be considered as a transformation of the cycle space ℙ³ to itself due to the following result.

Exercise 2 Check that:

The cycle σ_c-similarity (10) with a cycle S _σc^s, where det S _σc^s≠ 0, preserves the structure of FSCc matrices and S _σc^s₁ is its fixed point. In a non-singular case detS _σc^s≠ 0 the second iteration of similarity is the identity map.
The σ_c-similarity with a σ_c-zero-radius cycle Z_σc^s always produces this cycle.

σ_c-Similarity with a cycle (k,l,n,m) is a linear transformation of the cycle space ℝ⁴ with the matrix

⎛
⎜
⎜
⎜
⎝

km−detC_σc	−2 k l	2 σ_c k n	k²
l m	−2l²−detC_σc	2 σ_c l n	k l
n m	−2 n l	2 σ_c n²−detC_σc	k n
m²	−2 ml	2σ_c m n	km−detC_σc

⎞
⎟
⎟
⎟
⎠

⎛
⎜
⎜
⎜
⎝

⎞
⎟
⎟
⎟
⎠

⎛
⎝

−2l

2σ_c n

⎞
⎠

−det(C_σc)· I_4× 4 .

Note the apparent regularity of its entries.

Remark 3 Here is another example where usage of complex (dual or double) numbers is different from Clifford algebras . To use commutative hypercomplex numbers we require the complex conjugation for the cycle product (3), linear-fractional transformation (9) and cycle similarity (10). Non-commutativity of Clifford algebras allows us to avoid complex conjugation in all those formulae, see Appendix B.5. For example, the reflection in the real line (complex conjugation) is given by matrix similarity with the corresponding matrix (

e₁	0
0	−e₁

A comparison of Exercises 1 and 2 suggests that there is a connection between two actions (9) and (10) of cycles, which is similar to the relation SL₂(ℝ) actions on points and cycles from Lemma 4.

Exercise 4 Let detS _σc^s ≠ 0, show that:

The σ_c-similarity (10) σ_c-preserves the orthogonality relation (1). More specifically, if G _σc^s and G _σc^s are matrix similarity (10) of cycles C_σc^s and G _σc^s respectively with the cycle S _σc^s₁, then:
⟨ G _σc^s,G _σc^s ⟩= ⟨ C_σc^s,G _σc^s ⟩ (detS _σc^s)².

Hint: Note that S _σc^sS _σc^s= −det(S _σc^s) I, where I is the identity matrix. This is a particular case of Vahlen condition,see []*Prop. 2. Thus we have
G _σc^s

G _σc^s

= −S _σc^s₁C_σc^s

G _σc^s

S _σc^s₁

· det S _σc^s.

The final step uses the invariance of trace under the matrix similarity. A CAS calculation is provided as well.⋄
The image T_σ^s=C_σ^s₂ Z_σ^s₁C_σ^s₂ of a σ-zero-radius cycle T_σ^s₁ under the similarity (10) is a σ-zero-radius cycle T_σ^s₁. The (s₁s₂)-centre of T_σc^s is the linear-fractional transformation (9) of (s₂/s₁)-centre of Z_σc^s.
Both formulae (9) and (10) define the same transformation of the point space ℝ^σ, with σ=σ_c≠ 0. Consequently, the linear-fractional transformation (9) maps cycles to cycles in those case. Hint: This part follows from the first two items and Proposition 3.⋄
There is a cycle C_σ^s such that neither map of the parabolic point space ℝ^p represents similarity with C_σ^s. Hint: Consider S _σ^s=(1,0,1/2,−1) and a cycle C_σ^s passing point (u,v). Then the similarity of C_σ^s with S _σ^s passes the point T(u,v)=(1+v/u,v+v²/u²) if and only if either
- C_σ^s is a straight line; or
- (u,v) belongs to S _σ^s and is fixed by the above map T.
That is the map T of the point space ℝ^p serves flat cycles and S _σ^s but no others. Thus there is no a map of the point space, which is compatible with the cycle similarity for an arbitrary cycle.⋄

To get a feeling of inversion we provide Fig. 6.4. The initial setup is shown on Fig. 6.4(a): the red unit circle and the grid of horizontal (green) and vertical (blue) straight lines. It is very convenient in this case that the grid is formed by two orthogonal pencils of cycles , which can be consider of any EPH type. Fig. 6.4(b) shows e-inversion of the grid in the unit circle, which is the locus of fixed points, of course. Straight lines of the greed are transformed to circles, but orthogonality between them is preserved, see Exercise 1.

Similarly Fig. 6.4(c) presents the result of p-inversion in the degenerated parabolic cycle u²−1=0. This time the grid is mapped to two orthogonal pencils of parabolas and vertical lines. By the way, due to the known optical illusion we perceive those vertical straight lines as being bended.

Finally, Fig. 6.4(d) demonstrates h-inversion in the unit hyperbola u²−v²−1=0. We again obtained two pencils of orthogonal hyperbolas. The bold blue cycles—the dot at the origin in (b), parabola in (c) and two lines (the light cone ) on (d)—will be explained in Section 8.1. Further details are provided by the following Exercise.

Exercise 5 Check that the above rectangular grid is produced by horizontal and vertical lines given by quadruples (0,0,1,m) and (0,1,0,m) respectively.

The similarity with the cycle (1,0,0,−1) sends a cycle (k,l,n,m) to (m,l,n,k). In particular the image of the grid are cycles (m,0,1,0) and (m,1,0,0).

We conclude this section by an observation, that cycle similarity is similar to a mirror reflection, which preserves directions of vectors parallel to the mirror and revert vectors which are orthogonal.

Exercise 6 Let detS _σc^s≠ 0, then for similarity (10) with S _σc^s:

Verify the identities:
−S _σc^s₁

S _σc^s

S _σc^s₁

=

det

σ_c

(S _σc^s₁)· S _σc^s₁

−S _σc

C_σc

S _σc

=
−

det

σ_c

(S _σc)·C_σc,

where C_σc^s is a cycle σ_c-orthogonal to S _σc. Note the difference in the signs in the right-hand sides of both identities.
Describe all cycles which are fixed (as points in the projective space ℙ³) by the similarity with the given cycle S _σc^s. Hint: Use a decomposition of a generic cycle into a sum S _σc^s and a cycle orthogonal to S _σc^s similar to (7).⋄

As we will see in the next Section those orthogonal reflections in the cycle space correspond to a “bended” reflections in the point space.

6.5 Inversions and Reflections in Cycles

The maps in point and cycle spaces considered in the previous section was introduced from the action of FSCc matrices of cycles. They can be also approached from the more geometrical viewpoint. There are at least two natural ways to define an inversion in a cycle :

Definition 1 For a given cycle C_σ^s we define two maps of the point space ℝ^σ associated to it:

An σ_c-inversion in a σ-cycle C_σ^s sends a point b∈ℝ^σ to the second point d of intersection of all σ-cycles σ_c-orthogonal to C_σ^s and passing through b, see Fig. 6.3.
A σ_c-reflection in a σ-cycle C_σ^s is given by M⁻¹RM, where M is a σ_c-similarity (10) sends the σ-cycle C_σ^s into the horizontal axis and R is the mirror reflection of ℝ^σ in that axis.

We are going to see that inversions are given by (9) and reflections are expressed through (10), thus they are essentially the same for EH cases in light of Exercise 1. However some facts are easier to establish using the inversion and others are more natural in terms of reflection. Thus it is advantageous to keep both notions. Since we have three different EPH orthogonality between cycles at every type of point spaces, there are also three different inversions in each of them.

Exercise 2 Prove the following properties of inversion:

If a cycle S _σc^s is σ_c-orthogonal to a cycle C_σc^s=(k,l,n,m) then for any point u₁+ι v₁∈ ℝ^σ (ι²=σ) belonging to σ-implementation of S _σc^s also passes through the image
u₂+ι v₂ =
⎛
⎜
⎝
        l+ισ_c n −m

        k −l+ισ_c n

⎞
⎟
⎠

(u₁−ι v₁)     (11)

under the Möbius transform (9) defined by the matrix C_σ^σ_c. Thus the point u₂+ι v₂ =C_σ^σ_c(u₁−ι v₁ ) is the inversion of u₁+ι v₁ in C_σc^s.
Conversely, if a cycle S _σc^s passes two different points u₁+ι v₁ and u₂+ι v₂ related through (11), then S _σc^s is σ_c-orthogonal to C_σc^s.
If a cycle S _σc^s is σ_c-orthogonal to a cycle C_σc^s, then the σ_c-inversion in C_σc^s sends S _σc^s to itself.
σ_c-inversion in the σ-implementation of a cycle C_σc^s coincides with σ-inversion in its σ_c-ghost cycle G _σc^s.

Note the interplay between parameters σ and σ_c in the above statement 1. Although we are speaking about σ_c-orthogonality, we take the Möbius transformation (11) with the imaginary unit ι such that ι²=σ (as the signature of the point space). On the other hand, the value σ_c is used there as the s-parameter for the cycle C_σ^σ_c.

Proposition 3 The reflection 2 of a zero-radius cycle Z_σc^s in a cycle C_σc^s is given by the similarity: C_σc^sZ_σc^sC_σc^s.

Proof. Let a cycle S _σc^s has the property S _σc^s C_σc^s S _σc^s = R_σc^s, where R_σc^s is the cycle representing the real line. Then S _σc^s R_σc^s S _σc^s = C_σc^s since S _σc^sS _σc^s= S _σc^sS _σc^s = −det S _σc^sI. The mirror reflection in the real line is given by the similarity with R_σc^s, therefore the transformation described in 2 is a similarity with the cycle S _σc^s R_σc^s S _σc^s = C_σc^s and thus coincides with (11).

Corollary 4 The σ_c-inversion with a cycle C_σc^s in the point space ℝ^σ coincides with σ_c-reflection in C_σc^s.

The auxullary cycle S _σc^s from the above proof of Prop. 3 is of a separate interest and can be characterised in the elliptic and hyperbolic cases as follows.

Exercise 5 Let C_σc^s=(k, l, n,m) be a cycle such that σ_c detC_σc^s>0 for σ_c≠ 0. Let us define the cycle S _σc^s by the quadruple (k, l, n±√σ_c detC_σc^s,m). Then

S _σc^s C_σc^s S _σc^s = ℝ and S _σc^s ℝ S _σc^s = C_σc^s
S _σc^s and C_σc^s have common roots.
In the σ_c-implementation the cycle C_σc^s passes the centre of S _σc^s.

Hint: One can directly observe 2 for real roots, since they are fixed points of the inversion. Also the transformation of C_σc^s to a flat cycle implies that C_σc^s is passing the centre of inversion, hence 3. There is a CAS calculation for this as well.⋄

Inversions are helpful to transform pencils of cycles to the simplest possible form.

Exercise 6 Check the following:

Let the σ-implementation of a cycle C_σ^s passes σ-centre of a cycle S _σ^s, then σ-reflection of C_σ^s in S _σ^s is a straight line.
Let to cycles C_σ^s and S _σ^s intersects in two points P, P′∈ℝ^σ such that P−P′ is not a divisor of zero in the respective number system. Then there is an inversion which maps the pencil of cycles orthogonal to C_σ^s and S _σ^s (see Exercise 2) into a pencil of concentric cycles. Hint: Make an inversion into a cycle with σ-centre P, then C_σ^s and S _σ^s will be transformed into straight lines due to the previous item. Those straight lines will intersect in a finite point P″ which is the image of P′ under the inversion. The pencil orthogonal to C_σ^s and S _σ^s will be transformed to a pencil orthogonal to those two straight lines. A CAS calculations shows, that all cycles from the pencil have σ-centre at P″.⋄

A classical source of the above result in the inversive geometry []*Thm 5.71 tells that an inversion can put any pair of non-intersecting circles to concentric ones. This is due to the fact that a orthogonal pencil to the pencil generated by two non-intersecting circles always pass two special points, see Exercise 3 for further development.

Finally we compare our consideration for the parabolic point space with the Yaglom’s book. The Möbius transformation (9) and respective inversion illustrated by Fig. 6.4(c) essentially coincide with the inversion of first kind from []*§ 10. Yaglom also introduces the inversion of second kind, see []*§ 10: for a parabola v=k(u−l)²+m he defined the map of the parabolic point space:

i.e. the parabola bisects the vertical line joining a point and its image. There are also other geometric characterisations of this map in [], which make it very similar to the Euclidean inversion in a circle. Here is the result expression this transformation through the usual inversion in parabolas:

Exercise 7 The inversion of second kind (12) is a composition of three Möbius transformations (9) defined by cycles (1,l,2m,l²+m/k), (1,l,0,l²+m/k) and the real line in the parabolic point space ℝ^p.

Möbius transformations (9) and similarity (12) with FSCc matrices map cycles to cycles just like matrices from SL₂(ℝ) do. It is natural to ask for a general types of matrices sharing this property. See works [, , ] dealing with more general elliptic and hyperbolic (but not parabolic) cases. It is beyond the scope of our consideration since it derails from the geometry of upper half-plane. We only mention the rôle of the Vahlen condition, C_σc^sC_σc^s= −det(C_σc^s) I, used in Exercise 1.

6.6 Higher Order Joint Invariants: Focal Orthogonality

Considering Möbius action (1) there is no need to be restricted to joints invariants of two cycles only. Indeed for any polynomial p(x₁,x₂,…,x_n) of several non-commuting variables one may define an invariant joint disposition of n cycles ^jC_σc^s by the condition:

where the polynomial of FSCc matrices is defined through the standard matrix algebra. To create a Möbius invariant which is not affected by the projectivity in the cycle space we can either:

Let us construct some lower order realisations of (13). To be essentially different from the previously considered orthogonality such invariants may either contain non-linear powers of the same cycle, or accommodate more than two cycles. In this respect consideration of higher order invariants is similar to a transition from Riemannian geometry to Finsler one [, ]. The later is based on the replacement of quadratic line element g^ij dx_i dx_j in the tangent space by a more complicated function.

A further observation is that we can simultaneously study several invariants of various orders and link one to another by some operations. There are some standard procedures changing orders of invariants working in both directions:

Consider both operations in an example. We already know that a similarity of a cycle with another cycle produces a new cycle. The cycle product of the later with a third cycle creates a joint invariant of those three cycles:

which is build from the second-order invariant ⟨ ·,· ⟩. Now we can reduce the order of this invariant by fixing ³C_σc^s to be the real line, since it is SL₂(ℝ) invariant. This invariant deserves a special consideration. Its geometrical meaning connected to the matrix similarity of cycles (10) (inversion in cycles) and orthogonality.

Definition 1 A cycle S _σc^s is σ_c-focal orthogonal (or f_σc-orthogonal) to a cycle C_σc^s if the σ_c-reflection of C_σc^s in S _σc^s is σ_c-orthogonal (in the sense of Definition 1) to the real line. We denote it by S _σc^s ⊣ C_σc^s.

Remark 2 This definition is explicitly based on the invariance of the real line and is an illustration to the boundary value effect from Remark 3.

Exercise 3 f-orthogonality is equivalent to any of the following:

The cycle S _σc^s C_σc^sS _σc^s is a self-adjoint cycle , see Definition 7.
Analytical condition:
⟨ S _σc^s

C_σc^s

S _σc^s,R_σc^s ⟩= tr(S _σc^s

C_σc^s

S _σc^sR_σc^s)=0. (15)

Remark 4 It is easy to observe the following

f-orthogonality is not a symmetric: C_σc^s ⊣ S _σc^s does not implies S _σc^s ⊣ C_σc^s;
Since the horizontal axis R_σc^s and orthogonality (1) are SL₂(ℝ)-invariant objects f-orthogonality is also SL₂(ℝ)-invariant.

However an invariance of f-orthogonality under inversion of cycles required some study since the real line is not invariant under such transformations in general.

Exercise 5 The image G _σc^s₁ R_σc^s G _σc^s₁ of the real line under inversion in G _σc^s₁=(k,l,n,m) with s≠ 0 is the cycle:

(2 s s₁ σ_c k n, 2 s s₁ σ_c l n, −det(C_σc^s₁), 2 s s₁ σ_c m n).

It is the real line again if det(C_σc^s₁)≠0 and either

s₁ n=0, in this case it is a composition of SL₂(ℝ)-action by (
l −m

k −l

) and the reflection in the real line; or
σ_c=0, i.e. the parabolic case of the cycle space.

If either of two conditions is satisfied then f-orthogonality S _σc^s⊣ C_σc^s is preserved by the σ_c-similarity with G _σc^s₁.

The following explicit expressions of f-orthogonality reveal further connections with cycles’ invariants.

Exercise 6 f-orthogonality of S _σc^s to C_σc^s is given by either of the following equivalent identities

s n (l′²+σ_c s₁²n′²− m′ k′) + s₁n′(m k′ −2l l′+ k m′ )	=	0, or
n det(S _σc¹) + n′⟨ C_σc¹,S _σc¹ ⟩	=	0, if s=s₁=1.

The f-orthogonality may be again related to the usual orthogonality through an appropriately chosen f-ghost cycle, cf. Proposition 4:

Proposition 7 Let C_σc^s be a cycle, then its f-ghost cycle G _σc^σ_c = C_σc^χ(σ) ℝ_σc^σ_c C_σc^χ(σ) is the reflection of the real line in C_σc^χ(σ), where χ(σ) is the Heaviside function (8). Then

Cycles C_σc¹ and G _σc^σ_c have the same roots.
χ(σ)-Centre of G _σc^σ_c coincides with the (−σ_c)-focus of C_σc^s, consequently all straight lines σ_c-f-orthogonal to C_σc^s pass its (−σ_c)-focus.
f-inversion in C_σc^s defined from f-orthogonality (see Definition 1) coincides with usual inversion in G _σc^σ_c.

Note the above intriguing interplay between cycle’s centres and foci. It also explains our choice of name for focal orthogonality, cf. Definition 1. f-Orthogonality and respective f-ghost cycles are presented on Fig. 6.5, which uses the same outline and legend as Fig. 6.3.

The definition of f-orthogonality may look rather extravagant at a first glance. However it will get a new support when will appear again from consideration of lengths and distances in the next Chapter. In will be also useful for infinitesimal cycles, cf. Section 7.5.

Of course, it is possible and meaningful to define other interesting higher order joint invariants of two or even more cycles.

Lecture 7 Metric Invariants in Upper Half-Planes

So far we discussed only invariants like orthogonality, which are related to angles. However geometry in the plain meaning of this word deals with measurements of distances and lengths. We will derive metrical quantities from cycles in a way which shall be Möbius invariant.

7.1 Distances

Cycles are covariant objects performing as “circles” in our three EPH geometries. Now we play the traditional mathematical game: turn some properties of classical objects into definitions of new ones.

Definition 1 The σ_c-radius r_σc of a cycle C_σc^s if squared is equal to the minus σ_c-determinant of cycle’s k-normalised (see Definition 1) matrix, i.e.

r_σc²= −

⟨ C_σc^s,C_σc^s ⟩

2k²

= −

detC_σc^s

k²

l²− σ_c s²n²−km

k²

. (1)

As usual, the σ_c-diameter of a cycles is two times its radius.

The expression (1) for radius though the invariant cycle product resembles the definition of a norm of vector in an inner product space []*§ 5.1.

Exercise 2 Check the following geometrical content of the formula (1).

The value of (1) is the usual radius of a circle or hyperbola given by the equation k(u²−σ v²)−2lu−2nv+m=0;
The diameter of a parabola is the (Euclidean) distance between its (real) roots, i.e. solutions of ku²−2lu+m=0, or roots of its “adjoint” parabola −ku²+2lu+m−2l²/k=0 (see Fig. 1.9(a)).

Exercise 3 Check the following relations:

The σ_c-radius of a cycle C_σc^s is equal to 1/k, where k is (2,1)-entry of det-normalised FSCc matrix (see Definition 1) of the cycle.
Let u_σc be the second coordinate of a cycle’s σ_c-focus and f be its focal length , then square of cycle’s σ_c-radius is:
r_σc²=−4fu_σc.
Cycles (11) have zero σ_c-radius, thus Definitions 3 and 1 agree.

An intuitive notion of a distance in both mathematics and the everyday life is usually of a variational nature. We natural perceive the shortest distance between two points delivered by the straight lines and only then can define it for curves through an approximation. This variational nature echoes also in the following definition.

Definition 4 The (σ,σ_c)-distance d_σ,σc(P,P′) between two points P and P′ is the extremum of σ_c-diameters for all σ-implementations of cycles passing P and P′.

Lemma 5 The distance between two points P=u+ι v and P′=u′+ι v′ in the elliptic or hyperbolic spaces is

d_σ,σc²(P, P′) =

σ_c ((u−u′)²−σ(v− v′)²) +4(1−σσ_c) v v′

(u− u′)² σ_c−(v−v′)²

((u−u′)² −σ(v− v′)²), (2)

and in parabolic case it is (see Fig. 1.9(b) and also []*(5), p. 38)

d_p,σc²(y, y′) = (u−u′)². (3)

Proof. Let C_s^σ(l) be the family of cycles passing through both points (u, v) and (u′, v′) (under the assumption v≠ v′) and parametrised by its coefficient l, which is the first coordinate of the cycle centre. By a symbolic calculation in CAS we found that the only critical point of det(C_s^σ(l)) is:

Note, that in the case σσ_c=1, i.e. both point and cycle spaces are simultaneously either elliptic or hyperbolic, this expression reduces to the expected midpoint l₀=1/2(u+u′). Since in the elliptic or hyperbolic case the parameter l can take any real value, the extremum of det(C_s^σ(l)) is reached in l₀ and is equal to (2), again calculated by CAS. A separate calculation for the case v= v′ gives the same answer.

In the parabolic case possible values of l are either in (−∞, 1/2(u+u′)), or (1/2(u+u′),∞), or the only value is l=1/2(u+u′) since for that value a parabola should flip between upward and downward directions of its branches. In any of those cases the extremum value corresponds to the boundary point l=1/2(u+u′) and is equal to (3).

i.e. these are familiar expressions for distances in the elliptic and hyperbolic spaces. However four other cases (σσ_c=−1 or 0) give quite different results. For example, d_σ,σc²(P, P′) does not tense to 0 if P→ P′ in the usual sense.

Exercise 6 Show that:

In the three cases σ=σ_c=−1, 0 or 1, which were typically studied in the literature, the above distances are conveniently defined through the arithmetic of corresponding numbers:
d_σ,σ²(u+ι v)=(u+vι)(u−vι)=ww. (5)
Unless σ=σ_c the parabolic distance d_p,σc (3) is not received from (2) by the substitution σ =0.
If cycles C_σc^s and S _σc^s are k-normalised then
      ⟨ C_σc^s−S _σc^s,C_σc^s−S _σc^s ⟩ =2det(C_σc^s−S _σc^s) =−2d_σc,σc²(P,P′).

where P and P′ are e- or h-centres of C_σc^s and S _σc^s. Therefore we can rewrite the relation (8) for the cycle product
      ⟨ C_σc^s,S _σc^s ⟩ = d_σc,σc²(P,P′)−r_σc²−r′_σc²,

using r_σc and r′_σc—the σ_c-radii of the respective cycles. In particular, cf. (12):
      ⟨ Z_σc^s,T_σc^s ⟩ = d_σc,σc²(P,P′),

for k-normalised σ_c-zero-radius cycles Z_σc^s and T_σc^s with centres P and P′. From Exercise 5 we can also derive that d_σc,σc²(P,P′) is equal to the power of the point P (P′) with respect to the cycle T_σc^s (Z_σc^s).

The distance allows to expand for all EPH cases the result, which is well-known in the cases of circles []*§ 2.1 and parabolas []*§ 10.

Exercise 7 Show that the power of a point W with respect to a cycle (see Definition 4) is the product d_σ(W,P)· d_σ(W,P′) of (σ,σ)-distances (5), where P and P′ any two points of the cycle collinear with W. Hint: Take P=(u,v) and P′=(u′,v′), then any collinear W is (tu+(1−t)u′,tv+(1−t)v′) for some t∈ℝ. Furthermore a simple calculation shows that d_σ(y,z)· d_σ(z,y′)= t(t−1)d_σ²(y,y′). Te last expression is equal to the power of W with respect to the cycle, this step can be done by CAS.⋄

Exercise 8 Let two cycles have e-centres P and P′ with σ_c-radii r_σc and r_σc′, then the (σ_c,σ_c)-power of one cycle with respect to another from Definition 6 is, cf. the elliptic case in (13):

d_σc,σc²(P,P′)−r_σc²−r_σc^′ 2

2r_σc r_σc′

7.2 Lengths

During geometry classes we often make measurements with a compass, which is based on the idea that a circle is locus of points equidistant from its centre. We can expand it for all cycles in the following definition:

Definition 1 The (σ,σ_c,σ_r)-length from the σ_r-centre (from the σ_r-focus ) of a directed interval AB is the σ_c-radius of the σ-cycle with its σ_r-centre (σ_r-focus) at the point A which passes through B. These lengths are denoted by l_c(AB) and l_f(AB) correspondingly.

It is easy to be confused by the triple of parameters σ, σ_c and σ_r in this definition. However we will rarely operate in such a generality, some special relations between sigmas will be assumed oftenly. We also do not attach the triple (σ,σ_c,σ_r) to l_c(AB) and l_f(AB) in formulae, their values shall be clear from surrounding text.

Exercise 2 Check the following properties of the lengths.

The length is not a symmetric functions of two points (unlike the distance).
A cycle is uniquely defined by its elliptic or hyperbolic centre and a point which it passes. However the parabolic centre is not so useful. Consequently lengths from parabolic centre are not properly defined, thus we always assume σ_r=± 1 for lengths from a centre.
A cycle is uniquely defined by any focus and a point which it passes.

Lemma 3 For two points P=u+ι v, P′=u′+ι v′∈ℝ^σ:

the σ_c-length from the σ_r-centre for σ_r=±1 between P and P′ is
l_cσc²(
▸

PP′

) = (u−u′)²−σ v′²+2σ_r v v′ −σ_c v²; (6)

the σ_c-length from the σ_r-focus between P and P′ is

l_fσc²(

	▸
PP′

) = (σ_r−σ_c) p²−2vp, (7)

where:

σ_r

⎛
⎜
⎝

−(v′−v)±

√

σ_r(u′−u)²+(v′−v)²−σσ_r v′²

⎞
⎟
⎠

, if σ_r≠0;

(8)

(u′−u)²−σ v′²

2(v′−v)

, if σ_r=0.

(9)

Proof. Identity (6) is verified in CAS. For the second part we observe that a cycle with the σ_r-focus (u,v) passing through (u′,v′)∈ℝ^σ has the following parameters:

Exercise 4 Check that:

The value of p in (8) is the focal length of either of the two cycles, which are in the parabolic case upward or downward parabolas (corresponding to the plus or minus signs) with focus at (u, v) and passing (u′, v′).
In the case σσ_c=1 the length from centre (6) became the standard elliptic or hyperbolic distance (u−u′)²−σ (v−v′)² obtained in (2). Since these expressions appeared both as distances and lengths they are widely used.
On the other hand, in the parabolic point space we get three additional lengths besides of distance (3):
l_cσc²(y, y′) = (u−u′)²+2 v v′−σ_c v²

parametrised by σ_c (cf. Remark 1).
The parabolic distance (3) can be expressed as
d²(y, y′) = p²+2(v−v′)p

in terms of the focal length p (8), it is an expression similar to (7).

7.3 Conformal Properties of Moebius Maps

All lengths l(AB) in ℝ^σ from Definition 1 are such that for a fixed point A every contour line l(AB)=c is a corresponding σ-cycle, which is covariant object in the appropriate geometry. This is also true for distances if σ=σ_c. Thus we can expect some covariant properties of distances and lengths.

Definition 1 We say that a distance or a length d is SL₂(ℝ)-conformal if for fixed y, y′∈ℝ^σ the limit:

lim

t→ 0

d(g· y, g·(y+ty′))

d(y, y+ty′)

, where g∈SL₂(ℝ), (10)

exists and its value depends only from y and g and is independent from y′.

Informally rephrasing this definition we can say that a distance/length is SL₂(ℝ)-conformal if Möbius map scales all small intervals originated at a point by the same factor. And since a scaling preserves the shape of cycles we can restate the SL₂(ℝ)-conformality once more in the familiar terms: small cycles are mapped to small cycles. To complete analogy with conformality in complex plane we note, that preservation of angles (well, at least orthogonality) by Möbius transformations is automatic.

Exercise 2 Show SL₂(ℝ)-conformality in the following cases.

The distance (2) is conformal if and only if the type of point and cycle spaces are the same, i.e. σσ_c=1. The parabolic distance (3) is conformal only in the parabolic point space.
The lengths from centres (6) are conformal for any combination of values of σ, σ_c and σ_r.
The lengths from foci (7) are conformal for σ_r≠ 0 and any combination of values of σ and σ_c.

The conformal property of the distance (2)–(3) from Proposition 1 is well-known, of course, see [, ]. However the same property of non-symmetric lengths from Proposition 2 and 3 could be hardly expected. As a reason for it one remarks that the smaller group SL₂(ℝ) (in comparison to all linear-fractional transforms of the whole ℝ²) admits a larger number of conformal metrics, cf. Remark 3.

The exception of the case σ_r=0 from the conformality in 3 looks disappointing on the first glance, especially in the light of the parabolic Cayley transform considered later in § 9.3. However a detailed study of algebraic structure invariant under parabolic rotations, see Chapter 11, will remove obscurity from this case. Indeed our Definition 1 of conformality heavily depends on the underlying linear structure in ℝ^σ: we measure a distance between points y and y+ty′ and intuitively expect that it is always small for small t. As explained in § 11.4 the standard linear structure is incompatible with the parabolic rotations and thus should be replaced by a more relevant one. More precisely, instead of limits y′→ y along the straight lines towards y we need to consider limits along vertical lines, as illustrated on Fig. 9.1, Fig. 9.1 and Definition 3.

We will return to the parabolic case of conformality in Proposition 4. An approach to the parabolic point space and related conformality based on infinitesimal cycles will be considered in § 7.6.

Remark 3 The expressions of lengths (6)–(7) are generally non-symmetric and this is a price one should pay for its non-triviality. All symmetric distances lead only to nine two-dimensional Cayley-Klein geometries, see []*App. B and [, , , ]. In the parabolic case a symmetric distance of a vector (u,v) is always a function of u alone, cf. Remark 6. For such a distance a parabolic unit circle consists from two vertical lines (see dotted vertical lines in the second rows on Figs. 6.3 and 6.5), which is not aesthetically attractive. On the other hand the parabolic “unit cycles” defined by lengths (6) and (7) are parabolas, which makes the parabolic Cayley transform (see Section 9.3) very natural.

We can also consider a distance between points in the upper half-plane which has a stronger property than SL₂(ℝ)-conformality. Namely, the metric shall be preserved by SL₂(ℝ) action or, in other words, Möbius transformations are isometries for it. We study such a metric in Chapter 10.

7.4 Perpendicularity and Orthogonality

In a Euclidean space the shortest distance from a point to a straight line is provided by the corresponding perpendicular. Since we have already defined various distances and lengths we may use them for a definition of respective notions of perpendicularity.

Definition 1 Let l be a length or a distance. We say that a vector AB is l-perpendicular to a vector CD if function l(AB+ε CD) of a variable ε has a local extremum at ε=0, cf. Fig. 1.10. This is denoted by AB⋋_l CD or simply AB⋋ CD if the meaning of l is clear from the context.

Exercise 2 Check that:

The l-perpendicularity is not a symmetric notion (i.e. AB⋋ CD does not imply CD⋋ AB) similarly to f-orthogonality, see section 6.6.
l-perpendicularity is linear in CD, i.e. AB⋋ CD implies AB⋋ rCD for any real non-zero r. However l-perpendicularity is not generally linear in AB, i.e. AB⋋ CD does not necessarily imply rAB⋋ CD.

Lemma 3 Let AB be l_c-perpendicular (l_f-perpendicular) to a vector CD for the length from centre (from focus) defined by the triple (σ,σ_c,σ_r). Then the flat cycle (straight line) AB, is σ_r-(f-)orthogonal to the σ-cycle C_σ^s with σ_r-centre ((−σ_r)-focus) at A passing through B. The vector CD is tangent at B to σ -implementation of C_σ^s.

Proof. Consider the cycle C_σ^s with its σ_r-centre at A and passing B in its σ-implementation. This cycle C_σ^s is a contour line for a function l(X)=l_c(AX) the triple (σ,σ_c,σ_r). Therefore the cycle separates regions where l(X)< l_c(AB) and l(X)> l_c(AB). The tangents line to C_σ^s at B (or, at least, its portion in the vicinity of B) belongs to one of these two regions, thus l(X) has a local extremum at B. Thus AB is l_c-perpendicular to the tangent line. The line AB is also σ_r-orthogonal to the cycle C_σc^s since it passes its σ_r-centre A, cf. Exercise 4.

The second case of l_f(AB) and f-orthogonality can be considered similarly with the obvious swing of centre to focus, cf. Proposition 2.

Obviously, perpendicularity turn to be familiar in the elliptic case, cf. Lemma 1 below. For two other cases the description is given as follows:

Exercise 4 Let A=(u,v) and B=(u′,v′). Then

d-perpendicular (in the sense of (2)) to AB in the elliptic or hyperbolic cases is a multiple of the vector

⎛
⎜
⎝
        σ (v−v′)³−(u−u′)² (v+v′(1−2 σ σ_c))

        σ_c(u−u′)³−(u−u′)(v− v′)(−2 v′ +(v+v′) σ_c σ)

⎞
⎟
⎠

,

which for σσ_c=1 reduces to the expected value (v−v′, σ(u−u′)).
d-perpendicular (in the sense of (3)) to AB in the parabolic case is (0, t), t∈ℝ which coincides with the Galilean orthogonality defined in []*§ 3.
l_cσc-perpendicular (in the sense of (6)) to AB is a multiple of (σ v′−σ_r v, u−u′).
l_fσc-perpendicular (in the sense of (7)) to AB is a multiple of (σ v′+p, u−u′), where p is defined either by (8) or by (9) for corresponding values of σ_r.

Hint: Contour lines for all distances and lengths are cycles. Use implicit derivation of (1) to determine the tangent vector to a cycle at a point. Then apply the formula to a cycle which passes (u′,v′) and is a contour line for a distance/length from (u,v).⋄

It is worth to have an idea about different types of perpendicularity in the terms of the standard Euclidean geometry . Here are some examples.

Exercise 5 Let AB=u +ι v and CD=u′+ι v′, then:

In the elliptic case the d-perpendicularity for σ_c=−1 means that AB and CD form a right angle, or analytically u u′+v v′=0.
In the parabolic case the l_fσc-perpendicularity for σ_c=1 means that AB bisect the angle between CD and the vertical direction or analytically:
u′u−v′p=u′u−v′( √

u²+v²

−v)=0, (11)

where p is the focal length (8)
In the hyperbolic case the d-perpendicularity for σ_c=−1 means that the angles between AB and CD are bisected by lines parallel to u=± v, or analytically u′ u−v′ v=0. Compare with Exercise 3(A).

Remark 6 If one attempts to devise a parabolic length as a limit or an intermediate case for the elliptic l_e=u²+v² and hyperbolic l_p=u²−v² lengths then the only possible guess is l′_p=u² (3), which is too trivial for an interesting geometry.

Similarly the only orthogonality conditions linking the elliptic u₁ u₂+v₁ v₂=0 and the hyperbolic u₁ u₂−v₁ v₂=0 cases from the above Exercise seems to be u₁ u₂=0 (see []*§ 3 and 2), which is again too trivial. This support our Remark 2.

7.5 Infinitesimal Radius Cycles

Although parabolic zero-radius cycles defined in 3 do not satisfy our expectations for “smallness” but they are often technically suitable for the same purposes as elliptic and hyperbolic ones. Yet we may want to find something which fits better for our intuition on “zero sized” object. Here we present an approach based on non-Archimedean (non-standard) analysis , see, for example, [, ] for a detailed exposition.

Apart from this inequalities infinitesimals obey all other properties of real numbers. In particular in our CAS computations an infinitesimal will represented by a positive real symbol and we replace some its powers by zero if their order of infinitesimality will admit this. The existence of infinitesimals in the standard real analysis is explicitly excluded by the Archimedean axiom, thus the theory operating with infinitesimals is known as non-standard or non-Archimedean analysis. We assume from now on that there exists an infinitesimal number є.

Definition 1 A cycle C_σc^s such that detC_σc^s is an infinitesimal number is called an infinitesimal radius cycle .

Exercise 2 Let σ_c and σ_r be two metric signs and let a point (u₀,v₀)∈ℝ^p with v₀>0. Consider a cycle C_σc^s defined by

C_σc^s=(1, u₀, n, u₀² −σ_r n²+є²), (13)

where

⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩

v₀−

√

v₀²−(σ_r−σ_c)є²

σ_r−σ_c

if σ_r≠ σ_c;

є²

2v₀

if σ_r=σ_c.

(14)

Then

The point (u₀, v₀) is σ_r-focus of the cycle.
The square of σ_c-radius is exactly −є², i.e. (13) defines an infinitesimal radius cycle.
The focal length of the cycle is an infinitesimal number of order є².

Hint: Combining two quadratic, defining the squared σ_c-radius and focus v coordinate, we found that n shall satisfy to the equation:

Moreover only the root from (14) of the quadratic case σ_r−σ_c≠ 0 gives an infinitesimal focal length. Then we can find the m component of the cycle. The answer is also supported by CAS calculations.⋄

The graph of cycle (13) in the parabolic space drawn at the scale of real numbers looks like a vertical ray started at its focus, see Fig. 7.1(a), due to the following result.

Exercise 3 Infinitesimal cycle (13) consists of points, which are infinitesimally close (in the sense of length from focus (7)) to its focus F=(u₀, v₀):

(u₀+є u, v₀+v₀u²+((σ_c−σ_r) u²−σ_r)

є²

4v₀

+O(є³)). (15)

Note that points below of F (in the ordinary scale) are not infinitesimally close to F in the sense of length (7), but are in the sense of distance (3). Thus having the set of points on the infinitesimal distance from an unknown point F we are not able to identify F, however this is possible from the set of points on the infinitesimal length from F. Figure 7.1(a) shows elliptic, hyperbolic concentric and parabolic confocal cycles of decreasing radii which shrink to the corresponding infinitesimal radius cycles.

It is easy to see that infinitesimal radius cycles has properties similar to zero-radius ones, cf. Lemma 4.

Exercise 4 The image of SL₂(ℝ)-action on an infinitesimal radius cycle (13) by conjugation (7) is an infinitesimal radius cycle of the same order. Image of an infinitesimal cycle under cycle conjugation is an infinitesimal cycle of the same or lesser order.

The consideration of infinitesimal numbers in the elliptic and hyperbolic case should not bring any advantages since the (leading) quadratic terms in these cases are non-zero. However non-Archimedean numbers in the parabolic case provide a more intuitive and efficient presentation. For example, zero-radius cycles are not helpful for the parabolic Cayley transform (see section 9.3) but infinitesimal cycles are their successful replacements. Another illustration is the second part of the following result as a useful substitution for Exercise 3.

Exercise 5 Let C_σc^s be the infinitesimal cycle (13) and G _σc^s=(k,l,n,m) be a generic cycle. Then

The both orthogonality condition C_σc^s⊥ G _σc^s (1) and the f-orthogonality G _σc^s ⊣ C_σc^s (15) are given by:
ku₀²−2lu₀+m=O(є).

In other words, the cycle G _σc^s has the real root u₀.
The f-orthogonality (15) C_σc^s ⊣ G _σc^s is given by:
ku₀²−2lu₀−2nv₀+m=O(є). (16)

In other words, the cycle G _σc^s passes focus (u₀,v₀) of the infinitesimal cycle in the p-implementation.

It is interesting to note that the exotic f-orthogonality became a matching replacement of the usual one for the infinitesimal cycles. Unfortunately, f-orthogonality is more fragile: for example, it is not invariant under a generic cycle conjugation (Exercise 5), consequently we cannot use infinitesimal radius cycle to define a new parabolic inversion besides shown on Fig. 6.4(c).

7.6 Infinitesimal Conformality

An intuitive idea of conformal maps, which is oftenly provided in the complex analysis textbooks for illustration purposes, is “they send small circles into small circles with respective centres”. Using infinitesimal cycles one can turn it into a precise definition.

Definition 1 A map of a region of ℝ^σ to another region is l-infinitesimally conformal for a length l (in the sense of Definition 1) if for any l-infinitesimal cycle:

Its image is an l-infinitesimal cycle of the same order;
The image of its centre/focus is displaced from the centre/focus of its image by an infinitesimal number of a greater order than its radius.

Remark 2 Note that in comparison with Definition 1 we now work “in the opposite direction”: former we had the fixed group of motions and looked for corresponding conformal lengths/distances, now we take a distance/length (encoded in the infinitesimally equidistant cycle) and check which motions respect it.

Natural conformalities for lengths from centre in the elliptic and parabolic cases are already well studied. Thus we are mostly interested here in conformality in the parabolic case, where lengths from focus are more relevant. The image of an infinitesimal cycle (13) under SL₂(ℝ)-action is a cycle, moreover its is again an infinitesimal cycle of the same order by Exercise 4. This provides the first condition of Definition 1. The second part fulfils as well.

Exercise 3 Let G _σc^s be the image under g∈SL₂(ℝ) of an infinitesimal cycle C_σc^s from (13). Then σ_r-focus of G _σc^s is displaced from g(u₀,v₀) by infinitesimals of order є² (while both cycles have σ_c-radius of order є).

Consequently SL₂(ℝ)-action is infinitesimally conformal in the sense of Definition 1 with respect to the length from focus (Definition 1) for all combinations of σ, σ_c and σ_r.

Infinitesimal conformality seems to be intuitively close to Definition 1. Thus it is desirable to understand a reason for the absence of exclusion clauses in Exercise 3 in comparison to Exercise 3.

Exercise 4 Show that for lengths from foci (7) and σ_r= 0 the limit (10) at point y₀=u₀+ι v₀ does exist but depends from the direction y=u+ι v:

lim

t→ 0

d(g· y₀, g·(y₀+ty))

d(y₀, y₀+ty)

(d+cu₀)²+σ c² v₀² −2 K c v₀(d+c u₀)

, (17)

where K=u/v and g= (

a	b
c	d

). Thus the length is not conformal.

However if we consider points (15) of the infinitesimal cycle then K=є u/v₀ u²= є/v₀ u. Thus the value of the limit (17) at the infinitesimal scale is independent from y=u+ι v. It also coincides (up to an infinitesimal number) with the value in (26), which is defined through a different conformal condition.

Infinitesimal cycles are also a convenient tool for calculations of invariant measures, Jacobians, etc.

Remark 5 There are further connections between the infinitesimal number є and the dual unit ε. Indeed, in non-standard analysis є² is a higher order infinitesimal than є and effectively can be treated as 0 at infinitesimal scale of є. Thus it is simply a bit more relaxed version of the defining property of the dual unit ε²=0. This explains why many results of differential calculus can be naturally deduced within dual numbers framework [], which naturally absorbs many proofs from non-standard analysis.

Using this analogy between є and ε we can think about the parabolic point space ℝ^p as a model for a subset of hyperreal numbers ℝ^* having the representation x+є y with x and y being real. Then a vertical line in ℝ^p (a special line, in Yaglom’s terms []) represents a monad , that is the equivalence class of hyperreals, which are different by an infinitesimal number. Then a Möbius transformation of ℝ^p are analytic extensions of the Möbius map from the real line to the subset of hyperreals.

The graph of a parabola correspondence to a “smooth” choice of a hypereal representative from each monad. Geometric properties of parabolas studied in this work correspond to results about such choices of representatives and their invariants under Möbius transformations. It will be interesting to push this analogy further and look for a flow of ideas in the opposite direction: from non-standard analysis to the parabolic geometry.

Lecture 8 Global Geometry of Upper Half-Planes

So far we were interested in individual properties of cycles and (relatively) localised properties of the point space. Now we describe some global properties which are related to the set of cycles as the whole.

8.1 Compactification of the Point Space

Giving Definitions 3 and 4 of the maps Q and M on the cycle space we did not properly consider their domains and ranges. For example, the point (0,0,0,1)∈ℙ³ is transformed by Q to the equation 1=0, which is not a valid equation of a conic section in any point space ℝ^σ. We also did not accurately investigate yet singular points of the Möbius map (24). It turns out that the both questions are connected.

One of the standard approaches []*§ 1 to deal with singularities of Möbius maps is to consider projective coordinates on the real line. More specifically we assign, cf. Section 4.4.1, a point x∈ℝ to a vector (x,1), then linear-fractional transformations of the real line correspond to linear transformations of two-dimensional vectors, cf. (1) and (9). All vectors with a non-zero second component can be mapped back to the real line. However vectors (x,0) do not correspond to real numbers and represent the ideal element, see []*Ch. 10 for a pedagogical introduction. The union of the real line with the ideal element produce the compactified real line. The similar construction is known for Möbius transformations of the complex plain and its compactification.

Since we have already a projective space of cycles, we may use it as a model for compactification of point spaces as well, it turns out to be even more appropriate and uniform in all EPH cases. The identification of points with zero-radius cycles, cf. Exercise 2, plays an important rôle here.

Definition 1 The only irregular point (0,0,0,1)∈ℙ³ of the map Q is called the zero-radius cycle at infinity and denoted by Z_∞.

Exercise 2 Check the following:

Z_∞ is the image of the zero-radius cycle Z_(0,0)=(1,0,0,0) at the origin under reflection (inversion) into the unit cycle (1, 0,0,−1), see blue cycles in Fig. 6.4(b)-(d).
The following statements are equivalent
1. A point (u,v)∈ℝ^σ belongs to the zero-radius cycle Z_(0,0) centred at the origin;
2. The zero-radius cycle Z_(u,v) is σ-orthogonal to zero-radius cycle Z_(0,0);
3. The inversion z↦ 1/z in the unit cycle is singular in the point (u,v);
4. The image of Z_(u,v) under inversion in the unit cycle is orthogonal to Z_∞.
If any from the above is true we also say that image of (u,v) under inversion in the unit cycle belongs to zero-radius cycle at infinity.

Hint: These can be easily obtained by direct calculations (even without CAS).⋄

In the elliptic case the compactification is done by addition to ℝ^e a single point ∞ (infinity), which is the elliptic zero-radius cycle, of course. However in the parabolic and hyperbolic cases singularities of the inversion z↦ 1/z are not localised in a single point, indeed the denominator is a zero divisor for the whole zero-radius cycle. Thus in each EPH case the correct compactification is made by the union ℝ^σ∪Z_∞.

It is common to identify the compactification ℝ^e of the space ℝ^e with a Riemann sphere. This model can be visualised by the stereographic projection (or polar projection ) as follows, see Fig. 8.1(a) and []*§ 18.1.4 for further details. Consider a unit sphere with a centre at the origin of ℝ³ and the horizontal plane passing the centre. Any non-tangential line passing the north pol S shall intersect the sphere in another point P and meet the plane at a point Q. This defines a one-to-one correspondence of the plane and the sphere without point S. If point Q moves far from the origin the point P shall approach S. Thus it is natural to associate S to the infinity.

A similar model can be provided for the parabolic and hyperbolic spaces as well, see Fig. 8.1(b),(c) and further discussion in []*§ 10, []. Indeed the space ℝ^σ can be identified with a corresponding surface of the constant curvature: the sphere (σ=−1), the cylinder (Fig. 8.1(b), σ=0), or the one-sheet hyperboloid (Fig. 8.1(c), σ=1). The map of a surface to ℝ^σ is given by the polar projection, see Fig. 8.1(a-c) as well as []*Figs. 129, 135, 179 and []*Fig. 1. The ideal elements which do not correspond to any point of the plane are shown in red on Fig. 8.1. As we may note this is exactly zero-radus cycles at each case: the point (E), the line (P) and two lines—the light cone (H) at infinity. These surfaces provide “compact” model of the corresponding ℝ^σ in the sense that Möbius transformations which are lifted from ℝ^σ the constant curvature surface by the polar projection are not singular on these surfaces.

However the hyperbolic case has its own caveats which may be easily oversight as in the paper [], for example. A compactification of the hyperbolic space ℝ^h by a light cone (which the hyperbolic zero-radius cycle) at infinity will indeed produce a closed Möbius invariant object or a model of 2D conformal space-time. However it will not be satisfactory for some other reasons explained in the next section.

8.2 (Non)-Invariance of The Upper Half-Plane

Exercise 1 In the elliptic and parabolic cases the upper halfplane in ℝ^σ is preserved by Möbius transformations from SL₂(ℝ). However in the hyperbolic case any point (u,v) with v>0 can be mapped to an arbitrary point (u′,v′) with v′≠ 0.

This is illustrated by Fig. 1.3: any cone from the family (29) is intersecting the both planes EE′ and PP′ over a connected curve (K-orbit —a circle and parabola respectively) belonging to a half-plane. However the intersection of a two-sided cone with the plane HH′ is two branches of a hyperbola in different half-planes (only one of them is shown on Fig. 1.3). Thus a rotation of the cone produces a transition of the intersection point from one half-plane to another and back again.

The lack of invariance of the half-planes in the hyperbolic case has many important consequences in seemingly different areas, for example:

All the above problems can be resolved in the following way, see []*§ III.4 and []*§ A.3. We take two copies ℝ₊^h and ℝ₋^h of the hyperbolic point space ℝ^h, depicted by the squares ACA′C″ and A′C′A″C″ in Fig. 8.3 correspondingly. The boundaries of these squares are light cones at infinity and we glue ℝ₊^h and ℝ₋^h in such a way that the construction is invariant under the natural action of the Möbius transformation. That is achieved if the letters A, B, C, D, E in Fig. 8.3 are identified regardless of the number of primes attached to them.

This aggregate, denoted by ℝ′^h , is a two-fold cover of ℝ^h. The hyperbolic “upper” half-plane ℍ′^h in ℝ′^h consists of the upper halfplane in ℝ₊^h and the lower one in ℝ₋^h, it is shown as a shaded region in Fig. 8.3(a). It is Möbius invariant and has a matching complement in ℝ′^h . More formally:

The hyperbolic “upper” half-plane is bounded by two disjoint “real” axes denoted by AA′ and C′C″ in Fig. 8.3(a).

Remark 2 The hyperbolic orbit of the subgroup K in the ℝ′^h consists of two branches of the hyperbola passing through (0,v) in ℝ₊^h and (0,−v⁻¹) in ℝ₋^h, see the sin-like curve in Fig. 8.3(a). If we watch the continuous rotation of a straight line generating a cone (29) then its intersection with the plane HH′ on Fig. 1.3(b) will draw the both branches. As mentioned in Remark 2 they have the same focal length and form a single K-orbit.

The corresponding model through a stereographic projection is presented in Fig. 8.4. In comparison with the single hyperboloid in Fig. 8.1(c) we add the second hyperboloid intersecting the first one over the light cone at the infinity. A crossing of the light cone in any direction shall imply a swap of hyperboloids, cf. flat map on Fig. 8.3. A similar conformally invariant two-fold cover of the Minkowski space-time was constructed in []*§ III.4 in connection with the red shift problem in extragalactic astronomy, see Section 8.4 for future information.

8.3 Optics and Mechanics

We already used a lot of physical terms (light cone, space-time, etc.) to describe the hyperbolic point space. It will be useful to outline more physical connections for all EPH case. Our list may not be exhaustive, but it illustrates that SL₂(ℝ) not only presents in some distinct areas but also link them in a fruitful way.

8.3.1 Optics

Consider an optical system of centred lens, the propagation of rays, which are close to the symmetry axis, through such a device is a subject of the paraxial optics. See []*Ch. 2 for a pedagogical presentation of matrix methods in this area, we give only a briefly outline here. A a ray at a certain point of the symmetry axis A can be described by a pair of numbers P=(y, V), see Fig. 8.5. Here y is the height (positive or negative) of the ray above the axis A and V=ncosv, where v is the angle of the ray with the axis and n is the refractive index of the media.

The paraxial approximation to geometric optic provides the straightforward recipe to evaluate output components out of the given data:

If two paraxial systems are aggregated one after another then the composed devise is described by the product of the respective transfer matrices of the subsystems. In other words we obtained an action of the group SL₂(ℝ) on the space of rays. More complicated optical system can be locally approximated by paraxial models.

There is a covariance of the theory generated by the conjugation automorphism g: g′↦ gg′g⁻¹ of SL₂(ℝ). Indeed we can simultaneously replace rays by g P and system’s matrices by gAg⁻¹ for any fixed g∈SL₂(ℝ). Another important invariant can be constructed as follows. For matrices from SL₂(ℝ) we note the remarkable relation:

where P₁=AP and P′₁=AP′. In other words the symplectic form is an invariant of the covariant action of SL₂(ℝ) on the optical system.

Example 1 The matrix J (3) belongs the subgroup K. It is a transfer matrix between two focal planes of a system, cf. []*§ II.8.2. It swaps components of the vector (y, V), thus the ray height y₂ at the second focal plane depends only on the ray angle V₁ in the first, and vise verse.

8.3.2 Classical Mechanics

A Hamiltonian formalism in classical mechanics was motivated by an analogy between optics and mechanics, see []*§ 46. For one-dimensional system it replaces the description of rays through (y,V) by a point (q,p) in the phase space ℝ². The component q gives coordinate of a particle and p is its momentum.

Paraxial optics corresponds to transformations of the phase space over a fixed period of time t under quadratic Hamiltonians . They are also represented by linear transformations of ℝ² with matrices from SL₂(ℝ), preserve the symplectic form (3) and covariant under the linear changes of coordinates in the phase space with matrices from SL₂(ℝ).

For a generic Hamiltonian we can approximate it by a quadratic one at the infinitesimal scale of phase space and time interval t. Thus the symplectic form becomes an invariant object at the tangent space of the phase space. There is a wide and important class of non-linear transformations of the phase space whose derived form preserves the symplectic form at the on the tangent space to every point. They are called canonical transformations. In particular the Hamiltonian dynamics is a one-parameter group of canonical transformations.

Example 2 The transformation of the phase space defined by the matrix J (3) is provided by the quadratic Hamiltonian q²+p² of the harmonic oscillator . Similarly to the optical Example 1 it swaps the coordinate and momenta of the system, rotating the phase space by 90 degrees..

8.3.3 Quantum Mechanics

Having a transformation φ of a set X we can always extend it to a linear transformation φ^* in a space of function defined on X through the “change of variables”: [φ^* f](x)=f(φ(x)). Using this for transformations of the phase space we obtain a language to work with statistical ensembles: functions on X can describe probability distribution on the set.

However there is an important development of this scheme for the case of a homogeneous space X=G / H. We use maps p:G→ G/H , s: X → G and r: G → H defined in Subsection 2.2.2. Let χ: H → B(V) be a linear representation of H in a vector space V. Then χ induces a linear representation of G in a space of V-valued functions on X given by the formula (cf. []*§ 13.2.(7)–(9)):

where g∈ G, x∈ X and h∈ H; * denotes multiplication on G and · denotes the action (3) of G on X from the left.

One can build induced representations for the action SL₂(ℝ) on the classical phase space, as a result the quantum mechanics emerges out of the classic one [, ]. The main distinction between two mechanics is encoded in the factor χ(r(g⁻¹ * s(x))) in (5). This term switch on self-interaction of such functions in linear combinations, the effect is natural for wave packets rather than the classical statistical distributions.

Example 3 Let us come back to the matrix J (3) and its action on the phase space from the Example 2. In the standard quantum mechanics the representation (5) is induced by a complex valued character of the subgroup K. Consequently the action of J is [ρχ(J) f](p,q)= i f(−p,q). There are eigenfunctions w_n(q,p)=(q+i p)ⁿ of this actions, the respective eigenvalues compose the energy spectrum of a quantum harmonic oscillator . A similar discreteness is responsible for appearance of spectral lines in the light emission by atoms of chemical elements.

8.4 Relativity of the Space-Time

Relativity describes an invariance of a kinematics with respect to a group of transformations, generated by transitions from one admissible reference system to another. Obviously it is a counterpart of the Erlangen programme in physics and can be equivalently stated: a physical theory studies invariants under a group of transformations, acting transitively on the set of admissible observers. One shall admit that group invariance is much more respected in physics than in mathematics.

An example of SL₂(ℝ) (symplectic) invariance we saw in the previous Section. The main distinction is that the transformations in kinematics relativity involve time component of the space-time, while mechanical covariance was formulated for the phase space.

There are many good sources with a comprehensive discussion of relativity, see for example [, ]. We will briefly outline main principle only restricting ourselves to the toy two-dimensional space-time with one-dimensional spatial component. We also highlight the role of subgroups N′ and in the relativity formulation to make a closer connection to the origin of our development, cf. Section 3.3.

Example 1 (Galilean relativity of classical mechanics) Denote by (t,x) coordinates in ℝ², which is identified with the space-time. Specifically t denotes time and x—spatial component. Then Galilean relativity principle tells us that laws of mechanics shall be invariant under the shifts of the reference point and the following linear transformations :

⎛
⎜
⎝

t′

x′

⎞
⎟
⎠

⎛
⎜
⎝

x+vt

⎞
⎟
⎠

=G_v

⎛
⎜
⎝

⎞
⎟
⎠

, where G=

⎛
⎜
⎝

1	0
v	1

⎞
⎟
⎠

∈ SL₂(ℝ). (6)

This map translates events to another reference system, which is moving with a constant speed v with respect to the first one. The matrix G_v in (6) belong to the subgroup N′ (11).

It is easy to see directly that parabolic cycles make an invariant family under the transformations (6). Those parabolas are graphs of particles moving with a constant acceleration, which is reciprocal of the focal length up to a factor. Thus movements with a constant acceleration a form an invariant class in Galilean mechanics. A particular case is a=0, that is a uniform motion, which is represented by a non-vertical straight lines. Each such line can be mapped to another by a Galilean transformation.

The class of vertical lines, representing sets of simultaneous events, is invariant under Galilean transformations as well. In other words, Galilean mechanics posses the absolute time which is independent from spatial coordinates and their transformations. See []*§ 2 for a detailed exposition of Galilean relativity.

A different type class of transformations, discovered by Lorentz and Poincare , is admitted in Minkowski space-time .

Example 2 (Lorentz–Poincare) We again take the space-time ℝ² with coordinates (t,x) but consider linear transformations associated to elements of subgroup :

⎛
⎜
⎝

t′

x′

⎞
⎟
⎠

⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝

t−vx

√

1−v²

−vt+x

√

1−v²

⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠

=L_v

⎛
⎜
⎝

⎞
⎟
⎠

where L_x=

√

1−v²

⎛
⎜
⎝

1	−v
−v	1

⎞
⎟
⎠

. (7)

The physical meaning of the transformation is the same: they provide a transition from a given reference system to a new one moving with a velocity v with respect to the initial one. The relativity principle again requests laws of mechanics to be invariant under Lorentz–Poincare transformations generated by (7) and shifts of the reference point.

The admissible values v∈(−1,1) for transformations (7) are bounded by 1, which serves as speed of light. Velocities greater than speed of light are not considered in this theory. The fundamental object preserved by (7) is the light cone:

The light cone C₀ is obviously the collection of points d_h(t,x)=0, for other values we obtain hyperbolas with asymmptotes formed by C₀. The light cone separates areas of the space-time such where d_h(t,x)>0 or d_h(t,x)<0. The first consists of time-like intervals and second—space-like ones . They can be transformed by (7) to pure time (t,0) or pure space (0,x) intervals respectively. However no mixing between intervals of different kinds is admitted by Lorentz–Poincare transformations.

Furthermore for a time-like intervals there is a preferred direction, it assigns the meaning of the future (also known as arrow of time ) to one half of the cone consisting of time-like intervals, e.g. if t-component is positive. This causal orientation []*§ II.1 is required by the real-world observation that we cannot remember a future states of a physical system but may affect them. Conversely, we may have a record of the system’s past but cannot change it from now.

Exercise 3 Check that such a separation of the time-like cone d_h(t,x)>0 into the future (t>0) and the past (t<0) halves is compatible with the group of Lorentz–Poincare transformations generated by the hyperbolic rotations (7).

However the causal orientation is not preserved if the group of admissible transformations is extended to conformal maps by an addition of inversions, see Fig. 8.2. Such a extension is motivated by a study of the red shift in astronomy . Namely spectral lines (see Example 3) of chemical elements observed from remote stars are shifted toward the red part of the spectrum in comparison to values known from our laboratory measurements. Conformal (rather than Lorentz–Poincare) invariance produces much better correlation to experimental data [] than the school textbooks’ explanation of the red shift based on the Doppler principle and expanding Universe.

Lecture 9 Conformal Unit Disk

The upper half-plane is a universal object for all three EPH case, which was obtained in a uniform fashion considering two-dimensional homogeneous spaces of SL₂(ℝ). However universal models are rarely best suited to all particular circumstances. For example, it is more convenient for many reasons to consider the compact unit disk in ℂ rather than the unbounded upper half-plane:

Of course, both models are conformally isomorphic through the Cayley transform.

We produce similar constructions for parabolic and hyperbolic cases in this Chapter. However we shall see that there is no a “universal unit disk”, instead we obtain something specific in each EPH case from the same upper half-plane. As it already happened on several occasions the elliptic and hyperbolic cases are rather similar and this is the parabolic case who requires a special treatment.

9.1 Elliptic Cayley Transforms

In the elliptic and hyperbolic cases the Cayley transform is given by the matrix Y_σ=(

), where σ=ι² and detY_σ =2. The matrix Y_σ produces the respective Möbius transform ℝ^σ→ ℝ^σ:

The same matrix Y_σ acts by conjugation g_Y=1/2Y_σgY_σ⁻¹ on an element g∈SL₂(ℝ):

The connection between two forms (1) and (2) of the Cayley transform is given by g_Y Y_σw= Y_σ(gw), i.e. Y_σ intertwines the actions of g and g_Y.

The Cayley transform in the elliptic case is very important []*§ IX.3, []*Ch. 8, (1.12) both for complex analysis and representation theory of SL₂(ℝ). The transformation g↦ g_Y (2) is an isomorphism of the groups SL₂(ℝ) and SU(1,1) namely in complex numbers we have

The group SU(1,1) acts transitively on the elliptic unit disk. The images of elliptic actions of subgroups A, N, K are given in Fig. 9.3(E). Any other subgroup is conjugated to one of them and its class can be easily distinguished in this model by the number of fixed points on the boundary: two, one and zero correspondingly. A closer inspection demonstrates that there are always two fixed points on the whole plane. They are either:

Consideration of Fig. 7.1(b) shows that the parabolic subgroup N is like a phase transition between the elliptic subgroup K and hyperbolic A, cf. (2). Indeed, if a fixed point of a subgroup conjugated to K approaches to a place on the boundary, then the other fixed point shall move to the same place on the unit disk from the opposite side. After they collide to a parabolic double point on the boundary, they may decouple into two distinct fixed points on the unit disk representing a subgroup conjugated to A.

In some sense the elliptic Cayley transform swaps complexities: by contrast to the upper half-plane the K-action is now simple but A and N are not. The simplicity of K orbits is explained by diagonalisation of the corresponding matrices:

These diagonal matrices generate Möbius transformations which are multiplication by unimodular scalar e^{2i φ}. Geometrically they are isometric rotations , namely they preserve distances d_e,e (2) and length l_ce.

Exercise 1 Check in the elliptic case that the real axis is transformed to the unit circle and the upper half-plane is mapped to the elliptic unit disk:

ℝ={(u,v) ∣ v = 0}	→	T^e={ (u,v) ∣ l_ce²(u,v)= u²+v²=1},	(5)
ℝ₊^e={(u,v) ∣ v > 0}	→	ⅅ^e={(u,v) ∣ l_ce²(u,v)= u²+v²<1},	(6)

where the length from centre l_ce² is given by (6) for σ=σ_c=−1 and coincides with the distance d_e,e (2).

On both sets SL₂(ℝ) acts transitively and the unit circle is generated, for example, by the point (0, 1) and the unit disk is generated by (0,0).

9.2 Hyperbolic Cayley transform

A hyperbolic version of the Cayley transform was used in []. The above formula (2) in ℝ^h becomes as follows:

with some subtle differences in comparison with (3). The corresponding A, N and K orbits are given on Fig. 9.3(H). However there is an important distinction between the elliptic and hyperbolic cases similar to one discussed in Section 8.2.

Exercise 1 Check in the hyperbolic case that the real axis is transformed to the cycle:

ℝ={(u,v) ∣ v = 0} → T^h={ (u,v) ∣ l_ch²(u,v)= u²−v²=−1}, (8)

where the length from centre l_ch² is given by (6) for σ=σ_c=1 and coincides with the distance d_h,h (2). On the hyperbolic unit circle SL₂(ℝ) acts transitively and it is generated, for example, by point (0,1).

SL₂(ℝ) acts also transitively on the whole complement

{(u,v) ∣ l_ch²(u,v)≠ −1}

to the unit circle, i.e. on its “inner” and “outer” parts together.

Recall from Section 8.2 that we defined ℝ′^h to be the two-fold cover of the hyperbolic point space ℝ^h consisting of two isomorphic copies ℝ₊^h and ℝ₋^h glued together, cf. Fig. 8.3. The conformal version of the hyperbolic unit disk in ℝ′^h is, cf. the upper half-plane from (1):

Exercise 2 Verify that that

ⅅ′^h is conformally invariant and has a boundary T′^h —two copies of the unit hyperbolas in ℝ₊^h and ℝ₋^h.
The hyperbolic Cayley transform is a one-to-one map between the hyperbolic upper half-plane ℍ′^h and hyperbolic unit disk ⅅ′^h .

We call T′^h the hyperbolic unit cycle in ℝ^h. Fig. 8.3(b) illustrates the geometry of the hyperbolic unit disk in ℝ′^h in comparison with the upper half-plane. We also can say (a bit informal) that hyperbolic Cayley transform maps the “upper” half plane onto the “inner” part of the unit disk.

One may wish that the hyperbolic Cayley transform diagonalises the action of subgroup A, or its some conjugate, in a fashion similar to the elliptic case (4) for K. Geometrically it will correspond to hyperbolic rotations of hyperbolic unit disk around the origin. Since the origin is the image of the point ι in the upper half-plane under the Cayley transform, we will use the isotropy subgroup . Under the Cayley map (7) an element of the subgroup becomes:

where e^{є t} = cosht +є sinht. The corresponding Möbius transformations is multiplication by e^{2є t}, obviously corresponds to isometric hyperbolic rotations of ℝ^h for distance d_h,h and length l_ch. This is illustrated on Fig.9.1(H:A’).

9.3 Parabolic Cayley Transforms

The parabolic case benefits from a bigger variety of choices. The first natural attempt is to define a Cayley transform from the same formula (1) with the parabolic value σ=0. The corresponding transformation is defined by the matrix (

However within the extended FSCc a more general version of parabolic Cayley transform is possible as well. It is given by the matrix

Here σ_c=−1 corresponds to the parabolic Cayley transform P_e with the elliptic flavour, σ_c=1 — to the parabolic Cayley transform P_h with the hyperbolic flavour. Finally the parabolic-parabolic transform P_p is given by an upper-triangular matrix mentioned at the beginning of this Section.

Fig. 9.3 presents these transforms in rows P_e, P_p and P_h correspondingly. The row P_p almost coincides with Figs. 1.1 and parabolic case in Fig. 1.2. Consideration of Fig. 9.3 by columns from top to bottom gives an impressive mixture of many common properties (e.g. the number of fixed point on the boundary for each subgroup) with several gradual mutations.

The description of the parabolic unit disk admits several different interpretations in terms lengths from Definition 1.

Exercise 1 Parabolic Cayley transform P_σc as defined by the matrix Y_σc (11) acts on the V-axis always as a shift one unit down.

If σ_c≠0 then P_σc transforms the real axis to the parabolic unit cycle such that

T^p_σc= { (u,v)∈ℝ^p ∣ l²(B, (u,v))·(−σ_c)=1}, (12)

and the image of upper halfplane is:

ⅅ^p_σc= {(u,v)∈ℝ^p ∣ l²(B, (u,v))·(−σ_c)< 1}, (13)

where the length l and the point B can be either of the following:

l²=l_ce²(B, (u,v))= u²+σ_c v is the (p,p,e)-length (6) from the e-centre B_e=(0,−σ_c/2).
l²=l_fh²(B, (u,v)) is the (p,p,h)-length (8) from the h-focus B=(0,−1−σ_c/4).
l²=l_fp²(B, (u,v))=u²/v+1 is the (p,p,p)-length (9) from the p-focus B=(0,−1).

Hint: The statements are a bit tautological, since the p-cycles are loci of points with mentioned lengths from their respective centre/focus by definition.⋄

Remark 2 Note that the both elliptic (5) and hyperbolic (8) unit cycles can be also presented in the form similar to (12)

ⅅ^σ={ (u′,v′) ∣ l_cσ²(B,(u,v))·(−σ )=1}

in term of the (σ,σ)-length from σ-centre B=(0,0) just like Exercise 1.

The above descriptions 1 and 3 are attractive for reasons given in the following two Exercises. Firstly, we note that K-orbits in the elliptic case (Fig. 9.1(E:K)) and the -orbits in the hyperbolic case (Fig. 9.1(H:A’)) of Cayley transform are concentric.

Exercise 3 N-orbits in the parabolic cases (Fig. 9.3(P_e:N, P_p:N, P_h:N)) are concentric parabolas (or straight lines) in the sense of Definition 3 with e-centres at (0,1/2), (0,∞), (0,−1/2) correspondingly. Consequently, the N-orbits are loci of equidistant points in the sense of the (p,p,e)-length from those centres.

Secondly, we observe that Cayley images of the isotropy subgroups’ orbits in elliptic and hyperbolic spaces in Fig. 9.1(E:K) and (H:A) are equidistant from the origin in the corresponding metrics.

Exercise 4 The Cayley transform of orbits of the parabolic isotropy subgroup in Fig. 9.1(P_e:N′) are confocal parabolas consisting of points on the same l_fp-length (7) from the point (0,−1), cf. 3.

We will introduce linear structures preserved by actions of the subgroups N and N′ on the parabolic unit disk in Chapter 11.

Remark 5 We see that the varieties of possible Cayley transforms in the parabolic case is bigger than in the two other cases. It is interesting that this parabolic richness is a consequence of the parabolic degeneracy of the generator ε²=0. Indeed for both the elliptic and the hyperbolic signs in ι²=± 1 only one matrix (1) out of two possible (

1	ι
± σ ι	1

) has a non-zero determinant. And only for the degenerate parabolic value ι²=0 all these matrices are non-singular.

Orbits of the isotropy subgroups , N′ and K from Exercise 1 under the Cayley transform are shown on Fig. 9.1, which should be compared with the action of the same subgroup on the upper half-plane in Fig. 3.1.

9.4 Cayley Transforms of Cycles

The next natural step within the FSCc is to expand the Cayley transform to the space of cycles .

Exercise 1 Let C_a^s be a cycle in ℝ^σ. Check that:

In the elliptic or hyperbolic cases the Cayley transform of the cycle C_σ is R_σG _σC_σG _σR_σ—the composition of the similarity (10) by the cycle G _σ^s=(σ ,0,1,1) and the similarity by the real line (see the first and last drawings on Fig. 9.2).
In the parabolic case the Cayley transform maps a cycle (k,l,n,m) to the cycle (k−2σ_c n, l, n ,m−2 n).

Hint: We can use the similar path to the proof of Theorem 1. Alternatively, for the first part we can notice that the matrix Y_σ of the Cayley transform and FSCc matrix of the cycle G _σ^s are different by a factor. The reflection in the real line compensates the effect of complex conjugation in the similarity (10).⋄

Exercise 2 Investigate, what are images under the Cayley transform of zero-radius cycles, self-adjoint cycles, orthogonal cycles and f-orthogonal cycles.

The above extensions of the Cayley transform to the cycles space is linear, however in the parabolic case it is not expressed as a similarity of matrices (reflections in a cycle). This can be seen, for example, from the fact that the parabolic Cayley transform does not preserve the zero-radius cycles represented by matrices with zero p-determinant.

Since orbits of all subgroups in SL₂(ℝ) as well as their Cayley images are cycles in the corresponding metrics we may use Exercises 2(p) and 2 to prove the following statements (in addition to Exercise 3):

Exercise 3 Verify that:

A-orbits in transforms P_e and P_h are segments of parabolas with the focal length 1/4 passing through (0,−1). Their p-foci (i.e. vertices) belong to two parabolas v=(−u²−1) and v=(u²−1) correspondingly, which are boundaries of parabolic circles in P_h and P_e (note the swap!).
K-orbits in transform P_e are parabolas with focal length less than 1/4 and in transform P_h—with reciprocal of the focal length bigger than −4.

Since the action of parabolic Cayley transform on cycles does not preserve zero-radius cycles one shall better use infinitesimal-radius cycles from § 7.5 instead. Indeed, the Cayley transform preserves infinitesimality:

Exercise 4 Show that images of infinitesimal cycles under parabolic Cayley transform are infinitesimal cycles again.

We recall a useful expression of concurrence with infinitesimal cycle focus through f-orthogonality from Exercise 2. A caution is required since f-orthogonality of generic cycles is not preserved by the parabolic Cayley transform, just like it is not preserved by cycle similarity in Exercise 2(p). A remarkably exclusion happens for the infinitesimal cycles:

Exercise 5 An infinitesimal cycle C_σc^a (13) is f-orthogonal (in the sense of Exercise 2) to a cycle S _σc^a if and only if the Cayley transform 2(p) of C_σc^a is f-orthogonal to the Cayley transform of S _σc^a.

Lecture 10 Geodesics

The Euclidean metric is not preserved by automorphisms of the Lobachevsky half-plane. Instead one has only a weaker property of conformality. However it is possible to find such a metric on the Lobachevsky half-plane that Möbius transformations will be isometries. Similarly, in Chapter ?? we describe a variety of distances and lengths and many of them have conformal property with respect to SL₂(ℝ) action. However it is worth to find a metric which is preserved by Möbius transformations. We will do it now following the paper [] very close.

Our consideration will be based on equidistant orbits, which physically correspond to wavefronts with a constant velocity. For example, if you drop a stone in the pond the ripples you see will be waves, which travelled the same distance from a dropping point assuming the constant velocity of waves. A dual description to wavefronts uses rays—the paths along which waves travels, i.e. the geodesics in the case of a constant velocity. The duality between wavefronts and rays is provided by Huygens’ principle, see []*§ 46.

Geodesics also play a central role in differential geometry generalising the notion of a straight lines. They are closely related to a metric : geodesics are often defined as curves which extremize length of curves. As a consequence, along geodesics the metric is additive.

10.1 Metric, Curves’ Length and Extrema

Definition 1 A metric on a set X is a function d: X × X → ℝ^+ such that:

d(x, y) ≥ 0 (positivity),
d(x, y)=0 iff x=y (non-degeneracy),
d(x, y)=d(y, x) (symmetry),
d(x,y)≤ d(x,z)+d(z,y) (the triangle inequality ),

for all x, y, z ∈ X.

Although adequate in many cases, the definition does not cover all interesting metrics. Examples include the non-symmetric lengths from Section 7.2 or distances (5) in the Minkowski space with the reverse triangle inequality d(x,y)≥ d(x,z)+d(z,y).

Recall the established procedure of constructing geodesics in Riemannian geometry (two-dimensional case) from []*§ 7:

Exercise 2 Let the quadratic form (1) be SL₂(ℝ)-invariant. Show that the above procedure leads to an SL₂(ℝ)-invariant metric.

We recall from Section 3.7 that an isotropy subgroup H fixing the hypercomplex unit ι under the action of (24) is K (8), N′ (11) and A′ (10) in the corresponding EPH cases. We will refer H-action as EPH rotation around ι. For an SL₂(ℝ) invariant metric the orbits of H will be equidistant points from ι, giving some indication on what the metric should be. But this does not determine the metric entirely since there is freedom in assigning values to the orbits.

Lemma 3 The invariant infinitesimal metric in EPH cases is

d s²=

du²−σ dv²

v²

, (3)

where σ=−1,0,1 respectively.

Proof. In order to calculate the infinitesimal metric consider the subgroups H of Möbius transformations that fix ι. Denote an element of those rotations by E_σ. We require an isometry so:

A metric is invariant under the above rotations if it is preserved under the linear transformation:

To calculate the metric at an arbitrary point w=u+iv we map w to ι by an affine Möbius transformation, which acts transitively on the upper half-plane

hence there is a factor of 1/v², the resulting metric is ds²=du²−σ dv²/v².

Corollary 4 With the notation from above, for an arbitrary curve Γ:

length(Γ)=

∫

(du²−σ dv²)

. (5)

It is invariant under Möbius transformations of the upper half-plane.

The standard tool to find geodesics for a given Riemannian metric is the Euler–Lagrange equations, see []*§ 7.1. For the metric (3) they take the form:

where γ is a smooth curve γ(t)=(γ₁(t),γ₂(t)) and t ∈ (a,b). This general approach can be used In the two non-degenerate cases (elliptic and hyperbolic) and produce curves with the minimum or the maximum lengths respectively. However the SL₂(ℝ)-invariance of metric allows to use more elegant methods in this case. For example, in Lobachevsky half-plane the solutions are well-known: semicircles orthogonal to the real axes or vertical lines, cf. []*§ 15.

Exercise 5 For the elliptic space (σ=−1):

Write a parametric equation of a circle orthogonal to the real line and check that the curve satisfies to Euler–Lagrange equations (6).
Geodesics passing the imaginary unit i are transverse circles to K-orbits from Fig. 1.2 and Exercise 4 with the equation, cf. Fig. 10.3(E):
(u² +v²) −2t u −1 =0, where t ∈ ℝ. (7)
The Möbius invariant metric is:
m(z,w)=sinh⁻¹
⎪
⎪ z−w ⎪
⎪ _e

2 √

ℑ[z]ℑ[w]

(8)

where ℑ[z] is the imaginary part of a complex number z and | z |_e²=u²+v². Hint: One can directly or using CAS verify that this is a Möbius invariant expression. Thus we can transform z and w to i and a point on the imaginary axis by a suitable Möbius transformation without changing the metric. The shortest curve in the Riemannian metric (3) is the vertical line, that is du=0. For a segment of vertical line the expression (8) can be evaluated straightforwardly. See also []*Thm. 7.2.1 for detailed proof and a number of alternative expressions.⋄

Exercise 6 Show similarly that in the hyperbolic case (σ=1):

There are two families of solutions passing the double unit є, one space-like (Fig. 10.3(H_S)), one time-like (Fig. 10.3(H_T)), cf. Section 8.4:
(u²−v²) −2tu +1=0, where ⎪
⎪ t ⎪
⎪ >1 or ⎪
⎪ t ⎪
⎪ <1. (9)

The space-like solutions are obtained by rotation of the vertical axis and the time-like solution is the -image of the cycle (1,0,0,1), cf. Fig. 3.1. They also consist of positive and negative cycles respectively, which are orthogonal to the real axis.

The respective metric in those two cases is:

d(z,w)=

⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩

2 sin⁻¹

⎪
⎪

z−w

⎪
⎪

√

ℑ[z]ℑ[w]

when z−w is time-like;

2 sinh⁻¹

⎪
⎪

z−w

⎪
⎪

√

ℑ[z]ℑ[w]

when z−w is space-like,

(10)

where ℑ[z] is the imaginary part of a double number z and | z |_h²=u²−v². Hint: The hint from the Previous exercise can be used here again with some modification to space-/time-like curves and replacing the minimum of possible curves’ lengths by maximum.⋄

The same geodesic equations can be obtained by Beltrami’s method, see []*§ 8.1. However the parabolic case presents just another disappointment.

Exercise 7 Show that for the parabolic case (σ=0):

The only solution of the Euler–Lagrange equations (6) are vertical lines, as in []*§ 3, which are again orthogonal to the real axes.
Vertical lines minimise the curves’ length between two points w₁, w₂, see Fig. 10.1 for an example of a blue curve approximating the infimum 0. The respective “geodesics” passing the dual unit ε are:
u²−2tu=0. (11)

Similarly a length of the red curve can be arbitrary large if the horizontal path is close enough to the real axis.
The only SL₂(ℝ)-invariant metric obtained from the above extremal consideration is ether identically equal to 0 or infinity.

There is a similarity between all three cases, for example we can uniformly write equations (7), (9) and (11) of geodesics through ι as

However the triviality of the parabolic invariant metric is annoying and we go on to study further the algebraic and geometric invariants to find a more adequate answer.

10.2 Invariant Metrics

We seek all real valued functions f on the half-plane invariant under the Möbius action:

which can be shown by a simple direct calculation, for CAS see Exercise 3. Recall that | z |_σ²=u²−σ v² in analogy with the distance d_σ,σ (5) in EPH geometries and []*App. C. To describe other invariant functions we will need the following definition:

Definition 1 A function f : X × X → ℝ^+ is called a monotonous metric if f(Γ(0), Γ(t)) is a continuous monotonically increasing of t, where Γ : [0,1) → X is a smooth curve with Γ(0)=z₀ that intersects all equidistant orbits of z₀ exactly once.

Exercise 2 Check the following

The function F(z,w) (12) is monotonous.
If h a monotonically increasing continuous real function then f(z, w)=h ∘ F(z,w) is a monotonous SL₂(ℝ)-invariant function.

Theorem 3 A monotonous function f(z,w) is invariant under g ∈ SL₂(ℝ) if and only if there exists a monotonically increasing continuous real function h such that f(z, w)=h ∘ F(z,w).

Proof. Due to the Exercise 2 we show the necessity. Suppose there exists another function with such a property say, H(z,w). Due to invariance under SL₂(ℝ) this can be viewed as a function in one variable if we apply r⁻¹ (cf. (4)) which sends z to ι and w to r⁻¹(w). Now by considering a fixed smooth curve Γ from Definition 1 we can completely define H(z,w) as a function of a single real variable h(t)=H(i, Γ(t)) and similarly for F(z,w):

where h and f are both continuous and monotonically increasing since they represent metric. Hence the inverse f⁻¹ exists everywhere by the inverse function theorem. So:

Note that hf⁻¹ is monotonic as its the composition of two monotonically increasing and this ends the proof.

Remark 4 The above proof carries over to a more general theorem stating: If there exist two monotonous F(u,v) and H(u,v) invariant under a transitive action of a group G then there exists a monotonically increasing real function h such that H(z,w)=h ∘ F(z,w).

As discussed in the previous Section, in elliptic and hyperbolic geometries the function h from above is either sinh⁻¹t or sin⁻¹t (8) (10). Hence it is reasonable to try inverse trigonometric and hyperbolic in the intermediate parabolic case too.

Remark 5 The above result sheds light on the freedom we have; we can either

“label” the equidistant orbits with numbers, i.e choose a function h which will then determine the geodesic; or
choose a geodesic which will then determine h.

Those two approaches are reflected in the next two Sections.

10.3 Geodesics: Additivity of Metric

As pointed out earlier, there might not be a metric function which satisfies all the traditional properties. But we still need the key ones, in light of this we make the following definition:

Definition 1 Geodesics for a metric d are smooth curves along which d is additive, that is d(x,y)+d(y,z)=d(x,z) for any three point of the curve, such that y is between x and z.

Remark 2 It is important that this definition is relevant in all EPH cases, i.e. in the elliptic and hyperbolic cases it would produce the well-known geodesics defined by the extremality condition.

Exercise 3 Let γ be a geodesics for a metric d, write the differential equation for γ. Hint: Consider d(w, w′) as a real function in 4 variables say f(u, v, u′, v′). Then write the infinitesimal version of the additivity condition and get the equation:

δ v

δ u

−f′₃(u, v, u′,v′) +f′₃(u′,v′,u′,v′)

f′₄(u, v, u′,v′) −f′₄(u′,v′,u′,v′)

, (15)

where f′_n stands for partial derivative of f with respect to the n-th variable.⋄

Note that σ_r is independent on σ although it takes the same three values, similarly to the different signature of point and cycle spaces introduced in Chapter ??. It is used to denote the possible sub-cases within the parabolic geometry alone.

Exercise 4 Check that

For u=0, v=1 and the metric d_p,σr (16) the equation (15) is:

δ v

δ u

=
2v

u

−
√

⎪
⎪ σ_r u²+4v ⎪
⎪

u

.
The geodesics through ι for the metric d_p,σr (16) are parabolas:
(σ_r+4t²)u²−8tu−4v+4=0. (18)

Let us verify which properties from Definition 1 are satisfied by the invariant metric derived from (16). Two of the four properties hold: it is clearly symmetric and positive for every two points. But the metric of any point to a point on the same vertical line is zero so d(z, w)=0 does not imply z=w. This can be overcome by introducing a different metric function just for the points on the vertical lines, see []*§ 3. Note that we still have d(z,z)=0 for all z.

The triangle inequality holds only in the elliptic point space, whereas in the hyperbolic point space we have the reverse one: d(w₁,w₂)≥ d(w₁,z)+d(z,w₂). There is an intermediate situation in the parabolic point space:

Lemma 5 Take any SL₂(ℝ) invariant metric function and take two points w₁, w₂ and the geodesic (in the sense of Definition 1) through the points. Consider the strip ℜ[w₁]<u<ℜ[w₂] and take a point z in it. Then the geodesic divides the strip into two regions where d(w₁,w₂)≤ d(w₁,z)+d(z,w₂) and where d(w₁,w₂)≥ d(w₁,z)+d(z,w₂).

Proof. The only possible invariant metric function in parabolic geometry is of the form d(z, w)=h ∘ |ℜ[z−w] |/2√ℑ[z]ℑ[w] where h is a monotonically increasing continuous real function by Theorem 3. Fix two points w₁, w₂ and the geodesic though them. Now consider some point z=a+ib in the strip. The metric function is additive along a geodesic so d(w₁, w₂)= d(w₁, w(a))+d(w(a),w₂) where w(a) is a point on the geodesic with real part equal to a. But if ℑ[w(a)]<b then d(w₁, w(a))>d(w₁, z) and d(w(a), w₂)>d(z, w₂) which implies d(w₁,w₂)> d(w₁,z)+d(z,w₂). Similarly if ℑ[w(a)]>b then d(w₁,w₂)< d(w₁,z)+d(z,w₂).

Remark 6 The reason for the ease with which the result falls out is the fact that the metric function is additive along the geodesics. This justifies the Definition 1 of geodesic in terms of additivity.

To illustrate those ideas look at the region where the converse of the triangular inequality holds for d(z,w)_σr=sin_σr⁻¹ |ℜ[z−w] |/2√ℑ[z]ℑ[w] marked red on Figure 10.2. It is enclosed by two parabolas both of the form (σ_r +4t²)u²−8tu−4v+4=0 (which is the general equation of geodesics) and both go though the two fixed point. They arise from taking ± when solving the quadratic equation to find t. Both of them separate the region where the triangle inequality fails but one of them in between two points and the second outside.

10.4 Geometric Invariants

In the previous section we defined an invariant metric and derive respective geodesics, now we will proceed in the opposite direction. As we discussed in Exercise 7 the parabolic invariant metric obtained from extremality condition is trivial. We work out an invariant metric from the Riemannian metric and predefined geodesics. It is schematically depicted, cf. (14):

A minimal requirements for the family of geodesics is: they should form an invariant subset of an invariant class of curves with no more than one curve joining every two points. Thus if we are looking for SL₂(ℝ) invariant metric it is natural to ask, that geodesic are cycles. An invariant subset of cycles may be characterised by an invariant algebraic condition, e.g. orthogonality. However the ordinary orthogonality is already fulfilled for the trivial geodesics from Exercise 7, thus instead we shall try f-orthogonality to the real axes, Definition 1. Recall that a cycle is f-orthogonality to the real axes if the real axes inverted in a cycle is orthogonal (in the usual sense) to the real axes.

Exercise 1 Check that a parabola ku²−2lu−2nv+m=0 is σ_r-f-orthogonal to the real line if l²+σ_r n²−mk=0, i.e. it is a (−σ_r)-zero radius cycle.

As a starting point consider the cycles that pass though ι. It is enough to specify only one such f-orthogonal cycle; the rest will be obtained by Möbius transformations fixing ι, i.e parabolic rotations. Within those constraints there are three different families of parabolas determined by the value σ_r.

Exercise 2 Check that the main parabolas passing ε:

σ_r u²−4v+4=0, (20)

where σ_r=−1,0,1 are f-orthogonal to the real line. Their rotations by an element (

1	0
t	1

)∈ N′ are: (σ_r +4t²)u²−8tu−4v+4=0.

Note that those are exactly the same geodesics obtained in (18). Hence we already know what the metric function has to be. But it is instructive to make the calculation from the scratch since it does not involve anything from the previous section and is in a way more elementary and intuitive.

Exercise 3 Follow these steps to calculate invariant metric:

Calculate the metric from ε to a point on the main parabola (20). Hint: Depending on whether the discriminant of the denominator in the integral (2) is positive, zero or negative the results are trigonometric, rationals or hyperbolic respectively:

∫

σt²+1

⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩

4log

2+u

2−u

if σ_r=−1;

if σ_r=0;

tan⁻¹

if σ_r=1.

This is another example of EPH classification.⋄

For a generic point (u,v) find the N′ rotation which put the point to the main parabola (20).
For two given points w and w′ combine the Möbius transformation g such that g:w↦ ε with the N′-rotation which put g(w′) to the main parabola.
Deduce from the previous items the invariant metric from ε to (u,v) and check that its a multiple of (16)
The invariant parabolic metric for σ_r=1 is equal to the angle between tangents to the geodesic at its endpoints.

We meet another example of diffusion of the parabolic geometry into three different sub-cases, cf. three types of Cayley transform in Section 9.3 and Fig. 9.3. The respective geodesics and equidistant orbits have been drawn in Fig. 10.3. There is one more gradual transformation between the different geometries. We can see the transitions from the elliptic case to P_e then to P_p to P_h to the hyperbolic light-like and finally to space-like. To link it back we observe a similarity between the final space-like case and the initial elliptic one.

Exercise 4 Show that all parabolic geodesics from ε for given value of σ_r are touching a certain parabola. This parabola can be called horizon because geodesic rays will never reach points outside of it. Note the similar effect for space-like and time-like geodesics in the hyperbolic case.

There is one more pleasant parallel between all the geometries. In the Lobachevsky and Minkowski geometries the centre of geodesics lies on the real axes. In the parabolic geometry the respective σ_r-foci (see Definition 2) of σ_r-geodesic parabolas lie on the real axes. This fact is due to the relations between f-orthogonality and foci, cf. Proposition 7.

Exercise 5 Check that:

(e,h) Elliptic and hyperbolic Cayley transforms (1) send the respective geodesics (7) and (9) passing ι to the straight line passing the origin. The respective invariant metrics on the unit disks are, cf. []*Tab. VI:
sin_σr⁻¹
⎪
⎪ w−w′ ⎪
⎪ _σ

2 √

(1+σ ww)(1+σ w′w′)

, (21)

where σ_r=1 in the elliptic case and has a value depending of space-/light-likeness in the hyperbolic case.
(p) The σ_c-parabolic Cayley transform (11) maps the σ_r-geodesics (18) to the parabolas passing the origin:
      (σ_r−4σ_c+4t²)u²−8tu−4v=0.

The invariant σ_r-metric on σ_c-parabolic unit circle is
sin_σr⁻¹
⎪
⎪ u−u′ ⎪
⎪

√

(1+v+σ_c u²)(1+v′+σ_c u′²)

.     (22)

Lecture 11 Unitary Rotations

One of the important advantages of the elliptic and hyperbolic unit disks introduced in Sections 9.1–9.2 is a simplification of isotropy subgroup actions. Indeed, images of the subgroups K and , which fix the origin in the elliptic and hyperbolic disks respectively, consist of diagonal matrices, see (4) and (10). Those diagonal matrices produce Möbius transformations, which are multiplications by hypercomplex unimodular number and thus are linear. In this Chapter we discuss possibility of similar results in the parabolic unit disks from Section 9.3.

11.1 Unitary Rotations—an Algebraic Approach

Consider the elliptic unit disk zz<1 (6) with the Möbius transformations transferred by the Cayley transform (1) from the upper half-plane. The isotropy subgroup of the origin is conjugated to K and consists of the diagonal matrices (

) (4). Corresponding Möbius transformations are linear and geometrically represented by rotation of ℝ² by the angle 2φ. After identification ℝ²=ℂ this action is given by the multiplication e^{2i phi}. The rotation preserve the (elliptic) distance (5) given by:

Therefore the orbits of rotations are circles, any line passing the origin (a “spoke”) is rotated by the angle 2φ, see Fig. 11.1(E). We can also see that those rotations are isometries for the conformally invariant metric (21) on the elliptic unit disk. Moreover, the rotated “spokes”—the straight lines through the origin—are geodesics for this invariant metric.

A natural attempt is to employ the algebraic side of this construction and translate to two other cases (parabolic and hyperbolic) through the respective hypercomplex numbers.

Exercise 1 Use the algebraic similarity between the three number systems from Fig. B.2 and its geometric depiction from Fig. 11.1 to check the following for each EPH case:

The algebraic EPH disks are defined by the condition d_σ,σ(0,z)<1, where d_σ,σ²(0,z)=zz.
There is the one-parameter group of automorphisms provided by multiplication by e^{ι t}, t∈ℝ. Orbits of these transformations are “rims” d_σ,σ(0,z)=r, where r<1.
The “spokes”, that is the straight lines through the origin, are rotated, in other words the image of one spoke is another spoke.

The value of e^{ι t} can be defined, say from the Taylor expansion of the exponent. In particular for the parabolic case ε^k=0 for all k≥ 2, thus e^{ε t}=1+ε t. Then the parabolic rotations explicitly act on dual numbers as follows:

In other words the value of the imaginary part does not affect transformation of the real one, but not vise verse.This links the parabolic rotations with the Galilean group [] of symmetries of the classic mechanics, with the absolute time disconnected from space, cf. Section 8.4.

The obvious algebraic similarity from Exercise 1 and the connection to classical kinematic is a wide spread justification for the following viewpoint on the parabolic case, cf. [, ]:

Those algebraic analogies are quite explicit and widely accepted as an ultimate source for parabolic trigonometry [, , ]. However we will see shortly that there exist geometric motivation and connection with parabolic equation of mathematical physics.

11.2 Unitary Rotations—a Geometrical Viewpoint

We make another attempt to describe parabolic rotations. The algebraic attempt exploit the representation of rotation by hypercomplex multiplication. However in the case of dual numbers this leads to a degenerate picture. If multiplication (a linear transformation) is not sophisticated enough for this we can advance to the next level of complexity: linear-fractional.

In brief, we change our viewpoint from algebraic to geometric. Elliptic and hypercomplex rotations of the respective unit disks are also Möbius transformations from one-parameter subgroup K and in the respective Cayley transform. Therefore the parabolic counterpart can be Möbius transformations from the subgroup N′.

For a sake of brevity we will treat only the elliptic version P_e of the parabolic Cayley transform from Section 9.3, we use the Cayley transform defined by the matrix:

This is not far from the diagonal forms in the elliptic (4) and hyperbolic (10), however, the off-diagonal (1,2)-term destroys harmony. Nevertheless we will continue a unitary parabolic rotation to be the Möbius transformation with the matrix (6), which is not be a multiplication by a scalar anymore. For the subgroup N′ the matrix is obtained by transposition of (6).

Exercise 1 Check that the parabolic rotations with the upper-triangular matrices from the subgroup N becomes:

⎛
⎜
⎝

e^ε t	t
0	e^−ε t

⎞
⎟
⎠

: −ε ↦ t −ε(1−t²). (9)

This coincides with the cyclic rotations defined in []*§ 8. A comparison with the Euler formula seemingly confirms that sinp t=t, but suggests a new expression for cosp t:

The identity (10) is also less trivial than the version cosp² t =1 from (3)–(4), see also []. Ranges of the cosine and sine functions in all cases:

There is the second option to define parabolic rotations for the lower-triangular matrices from the subgroup N′. The important difference now is: the reference point cannot be −ε since it is a fixed point (as well as any point on the vertical axis). Instead we take ε⁻¹, which is an ideal element (a point at infinity) since ε is a divisor of zero . The proper treatment is base on the projective coordinates, where point ε⁻¹ is represented by a vector (1, ε), see Section 8.1.

Exercise 2 Check the map of reference point ε⁻¹ for the subgroup N′:

⎛
⎜
⎝

e^−ε t	0
t	e^ε t

⎞
⎟
⎠

↦

+ ε

⎛
⎜
⎜
⎝

1−

t²

⎞
⎟
⎟
⎠

. (11)

A comparison with (9) shows that this form is obtained by the substitution t↦ t⁻¹. The same transformation gives new expressions for parabolic trigonometric functions. The parabolic “unit cycle” is defined by the equation u²−v=1 for both subgroups, see Fig. 9.1(P) and (P′) and Exercise 1. However other orbits are different and we will give their description in the next Section. Fig. 9.1 illustrates Möbius actions of matrices (7), (8) and (6) on the respective “unit disks”, which are images of the upper half-planes under respective Cayley transforms 9.1–9.3.

At this point a reader may suspect that structural analogy mentioned at the Section beginning is insufficient motivation to call transformations (9) and (11) “parabolic rotation” and the rest of the Chapter is a kind of post-modern deconstruction. To dispel the doubts we present the following example.

Example 3 (The heat equation and kernel) The differential equation, which describes the dynamics f(x,t) of heat distribution over a one-dimensional infinite string is modelled by the partial differential equation:

(∂_t −k∂_x²) f(x,t)=0, where x∈ℝ, t∈ℝ₊. (12)

For the initial value problem with the data f(x,0)=g(x) the solution is given by the convolution:

u(x,t)=

√

4π kt

∞

∫

−∞

exp

⎛
⎜
⎜
⎝

−

(x−y)²

4kt

⎞
⎟
⎟
⎠

g(y) dy ,

with the function exp(−x²/4kt), which is called heat kernel .

Exercise 4 Check that the Möbius transformations:

⎛
⎜
⎝

1	0
c	1

⎞
⎟
⎠

: x+ε t ↦

x+ε t

c(x+ε t)+1

from the subgroup N′ do not change the heat kernel. Hint: N′-orbits from Fig. 3.1 are contour line of the function exp(−u²/t).⋄

The last Example hints on the further works linking the parabolic geometry with parabolic partial differential equations.

11.3 Rebuilding Algebraic Structures from Geometry

Rotations in elliptic and hyperbolic cases are given by products of complex or double numbers respectively and thus are linear. However non-trivial parabolic rotations (9) and (11) (Fig. 9.1(P) and (P′)) are not linear.

Can we find algebraic operations for dual numbers, which linearise those Möbius transformations? To this end we will “revert a theorem into a definition” and use this systematically to recover a compatible algebraic structure.

11.3.1 Modulus and Argument

In the elliptic and hyperbolic cases orbits of rotations are points with the constant norm (modulus): either x²+y² or x²−y². In the parabolic case we employed this point of view already treated orbits of the subgroup N′ as equidistant points for certain metric in Chapter 10, we shall do this again.

Definition 1 Orbits of actions (9) and (11) are contour lines for the following functions which we call respective moduli (norms ):

for N:

⎪
⎪

u+ε v

⎪
⎪

=u²−v, for N′:

⎪
⎪

u+ε v

⎪
⎪

′=

u²

v+1

. (13)

Remark 2 The definitions are supported by the following observations:

The expression | (u,v) |=u²−v represents a parabolic distance from (0,1/2) to (u,v), see Exercise 3, which is in line with the parabolic Pythagoras’ identity (10).
Modulus for N′ expresses the parabolic focal length from (0,−1) to (u,v) as described in Exercise 4.

The only straight lines preserved by both the parabolic rotations N and N′ are vertical lines, thus we will treat them as “spokes” for parabolic “wheels”. Elliptic spokes in mathematical terms are “points on the complex plane with the same argument”, thus we again use this for the parabolic definition:

Definition 3 Parabolic arguments are defined as follows:

for N: arg(u+ε v)=u, for N′: arg′(u+ε v)=

. (14)

Both Definitions 1 and 3 possess natural properties with respect to parabolic rotations:

Exercise 4 Let w_t be a parabolic rotation of w by an angle t in (9) or in (11). Then:

⎪
⎪

w_t

⎪
⎪

^(′)=

⎪
⎪

^(′), arg^(′) w_t=arg^(′) w+t,

where primed versions are used for subgroup N′.

Remark 5 Note that in the commonly accepted approach, cf. []*App. C(30’), parabolic modulus and argument are given by expressions (5), which are, in a sense, opposite to our present agreements.

11.3.2 Rotation as Multiplication

We revert again theorems into definitions to assign multiplication. In fact, we consider parabolic rotations as multiplications by unimodular numbers thus we define multiplication through an extension of properties from Exercise 4:

Definition 6 The product of vectors w₁ and w₂ is defined by the following two conditions:

arg^(′)(w₁ w₂)=arg^(′) w₁ + arg^(′) w₂;
| w₁ w₂ |^(′) =| w₁ |^(′)· | w₂ |^(′).

Here and thereafter primed versions of formulae correspond to the case of subgroup N′ and unprimed—to the subgroup N.

We also need a special form of parabolic conjugation, which coincides with sign reversion of the argument.

Definition 7 Parabolic conjugation is given by

u+ε v

=−u+ε v. (15)

Obviously we have the properties: | w |^(′)=| w |^(′) and arg^(′)w=−arg^(′) w. A combination of Definitions 1, 3 and 6 uniquely determine expressions for products.

Exercise 8 Check the explicit expressions for the parabolic products:

for

(u,v)*(u′,v′)

= (u+u′,(u+u′)²−(u²−v)(u′²−v′));

(16)

for

N′:

(u,v)*(u′,v′)

⎛
⎜
⎜
⎝

uu′

u+u′

(v+1)(v′+1)

(u+u′)²

−1

⎞
⎟
⎟
⎠

(17)

Although the both above expressions look unusual they have many familiar properties, which are easier to demonstrate out of the implicit definition rather than the explicit formulae.:

Exercise 9 Check that the both products (16) and (17) satisfy the following conditions:

They are commutative and associative;
The respective rotations (9) and (11) are given by multiplications with a dual number with the unit norm.
The product w₁w₂ is invariant under respective rotations (9) and (11).
For any dual number w the following identity holds:
⎪
⎪ ww ⎪
⎪ = ⎪
⎪ w ⎪
⎪ ².

We defined multiplication though the modulus and argument spelt out in the previous Subsection. Our notion of the norm is rotational invariant and unique up to composition with a monotonic function of a real argument, see discussion in Section 10.2. However argument can be defined with a bigger freedom. For example, we may note that level curves for argument in elliptic and hyperbolic cases are geodesics in respective metric, see Exercise 5(e,h).

Exercise 10 Define the parabolic argument to be constant along geodesics from Exercise 5(p) drawn by blue lines on Fig. 10.3. What is the respective multiplication formula?

11.4 Invariant Linear Algebra

Now we wish to define a linear structure on ℝ² which would be invariant under point multiplication from the previous Subsection (and thus under the parabolic rotations, cf. Exercise 2). Multiplication by a real scalar is straightforward (at least for a positive scalar): it should preserve the argument and scale the norm of a vector. Thus we have formulae for a>0:

On the other hand the addition of vectors can be done in several different ways. We present two possibilities: one is tropical and another—exotic.

11.4.1 Tropical form

Exercise 1 (Tropical mathematics) Consider the so-called max-plus algebra ℝ_max, namely the field of real numbers together with the minus infinity: ℝ_max= ℝ∪ {−∞}. Define operations x⊕ y = max(x, y) and and x ⊙ y = x + y. Check that:

the addition ⊕ and the multiplication ⊙ are associative;
the addition ⊕ is commutative;
the multiplication ⊙ is distributive with respect to the addition ⊕;
−∞ is the neutral element for ⊕.

Define similarly ℝ_min = ℝ ∪ {+∞} with the operations ⊕ = min, ⊙ = +.

The above example is fundamental in the broad area of tropical mathematics or idempotent mathematics , also known as Maslov dequantisation algebras, see [] for a comprehensive survey.

Exercise 2 Check that above relation is transitive.

One can define functions min and max of a pair of points on ℝ² respectively. Then an addition of two vectors can be defined either as their minimum or maximum.

Exercise 3 Check that such an addition is commutative, associative and distributive with respect to scalar multiplications (18)—(19) and consequently is invariant under parabolic rotations.

Although it looks promising to investigate this framework we do not study it further for now.

11.4.2 Exotic form

Addition of vectors for both subgroups N and N′ can be defined by the following common rules, where subtle differences are hidden within corresponding Definitions 1 (norms) and 3 (arguments).

Definition 4 Parabolic addition of vectors is defined by the following formulae:

arg^(′)(w₁+w₂)

arg^(′) w₁·

⎪
⎪

w₁

⎪
⎪

^(′) +arg^(′) w₂·

⎪
⎪

w₂

⎪
⎪

^(′)

⎪
⎪

w₁+w₂

⎪
⎪

^(′)

(20)

⎪
⎪

w₁+w₂

⎪
⎪

^(′)

⎪
⎪

w₁

⎪
⎪

^(′)±

⎪
⎪

w₂

⎪
⎪

^(′),

(21)

where primed versions are used for the subgroup N′.

The rule for the norm of sum (21) may look too trivial at the first glance. We should say in its defence that it nicely sits in between the elliptic | w+w′ |≤ | w |+| w′ | and hyperbolic | w+w′ |≥ | w |+| w′ | triangle inequalities for norms , see Section 10.3 for their discussion.

The rule (20) for argument of the sum is not arbitrary as well. From the Sine Theorem in the Euclidean geometry we can deduce that:

where ψ^(′)=argw^(′) and φ=arg (w+w^(′)). Using parabolic expression (3) for the sine sinp θ=θ we obtain the arguments addition formula (20).

A proper treatment of zeros in denominator of (20) can be achieved through a representation of a dual number w=u+ε v as a pair of homogeneous polar coordinates [a,r]=[ | w |^(′) · arg^(′) w, | w |^(′)] (dashed version for the subgroup N′). Then the above addition is defined component-wise in the homogeneous coordinates:

The multiplication from Definition 6 is given in the homogeneous polar coordinates by:

Thus homogeneous coordinates linearise the addition (20)–(21) and multiplication by a scalar (18).

Both formulae (20)–(21) together uniquely define explicit expressions for addition of vectors. However those expressions are rather cumbersome and not really much needed. Instead we list properties of these operations:

Exercise 5 Verify that the vector additions for subgroups N and N′ defined by (20)–(21) satisfy the following conditions:

They are commutative and associative.
They are distributive for multiplications (16) and (17); consequently:
They are parabolic rotationally invariant;
They are distributive in both ways for the scalar multiplications (18) and (19) respectively:
a·(w₁+w₂)=a· w₁+a· w₂, (a+b)· w=a· w+b· w.

To complete the construction we need to define the zero vector and the inverse. The inverse of w has the same argument as w and the opposite norm.

Exercise 6 Check that for corresponding subgroups we have:

(N) The zero vector is (0,0) and consequently the inverse of (u,v) is (u,2u²−v).
(N′) The zero vector is (∞,−1) and consequently the inverse of (u,v) is (u,−v−2).

Thereafter we can check that scalar multiplications by negative reals are given by the same identities (18) and (19) as for positive ones.

11.5 Linearisation of the exotic form

Some useful information can be obtained from the transformation between the parabolic unit disk and its linearised model. In such linearised coordinates (a,b) the addition (20)–(21) is done in the usual coordinate-wise manner: (a,b)+(a′,b′)=(a+a′,b+b′).

Exercise 1 Calculate values of a and b in the linear combination (u,v)=a·(1,0)+b·(−1,0) and check the following:

For the subgroup N the relations are:

        u
=
a−b

a+b

,

v
=
(a−b)²

(a+b)²

−(a+b),

    (22)

a
=
u²−v

2

(1+u),

b
=
u²−v

2

(1−u).

    (23)
For the subgroup N′ such a transformations are:

      u
=
a+b

a−b

,

v
=
(a+b)

(a−b)²

−1,

a
=
u(u+1)

2(v+1)

,

b
=
u(u−1)

2(v+1)

.

We also note that both norms (13) have exactly the same value a+b in the respective (a,b)-coordinates. It is not difficult to transfer parabolic rotations from (u,v)-plane to (a,b)-coordinates.

Exercise 2 Show that:

The expression for N action (9) in (a,b) coordinates is:

⎛
⎜
⎝
e^ε t t

0 e^−ε t

⎞
⎟
⎠

: (a,b) ↦ ⎛
⎜
⎜
⎝ a+
t

2

(a+b),b−
t

2

(a+b) ⎞
⎟
⎟
⎠ . (24)

Hint: Use identities (22).⋄
After (euclidean) rotation by 45^∘ given by
(a,b)↦(a+b,a−b) (25)

formula (24) coincides with the initial parabolic rotation (2) shown on Fig. 11.1(P₀).
The composition of transformations (22) and (25) maps algebraic operations from Definitions 6 and 4 to corresponding operations on dual numbers.

This should not be surprising since any associative and commutative two dimensional algebra is formed either by complex, dual or double numbers []. However it does not trivialise our construction, since the above transition is essentially singular and shall be treated withing birational geometry framework [].

11.5.1 Retrospective: Parabolic Conformality

The irrelevance of the standard linear structure for parabolic rotations manifests itself in many different ways, e.g. in an apparent “non-conformality” of lengths from parabolic foci, that is with the parameter σ_r=0 in Proposition 3. An adjustment of notions to the proper framework restores the clear picture.

The initial Definition 1 of conformality considers the usual limit y′→ y along a straight line, i.e. “spoke” in terms of Fig. 11.1. This is justified in the elliptic and hyperbolic cases. However in the parabolic setting the proper “spokes” are vertical lines, see Definition 3 of argument and illustration on Fig. 9.1(P) and (P′). Therefore the parabolic limit should be taken along the vertical lines.

Definition 3 We say that a length l is p-conformal if for any given P=u+ι v and another point P′=u′+ι v′ the following limit exist and independent from u′:

lim

v′→ ∞

	▸
QQ′

)

	▸
PP′

)

, where g∈SL₂(ℝ), Q=g· P, Q′=g · P′.

Exercise 4 Let the focal length is given by the identity (7) with σ=σ_r=0, e.g.:

l_fσc²(

	▸
PP′

) = −σ_c p²−2vp, where p =

(u′−u)²

2(v′−v)

Check that l_fσc is p-conformal and moreover:

lim

v′→ ∞

l_fσc(

	▸
QQ′

)

l_fσc(

	▸
PP′

)

(cu+d)²

, where g=

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

, (26)

and Q=g· P, Q′=g· P′

Another application of the exotic linear algebra is the construction of linear representations of SL₂(ℝ) induced from characters of subgroups N realised as parabolic rotations [].

Lecture 12 Representation Theory

12.1 Representations

Objects unveil their nature in actions. Groups act on other sets by means of representations. A representation of a group G is a group homomorphism of G in a transformation group of a set. It is a fundamental observation that linear objects are easer to study. Therefore we begin from linear representations of groups.

Definition 1 A linear continuous representation of a group G is a continuous function T(g) on G with values in the group of non-degenerate linear continuous transformation in a linear space H (either finite or infinite dimensional) such that T(g) satisfies to the functional identity:

T(g₁ g₂) =T(g₁) T(g₂). (1)

Remark 2 If we have a representation of a group G by its action on a set X we can use the following linearization procedure . Let us consider a linear space L(X) of functions X→ ℂ which may be restricted by some additional requirements (e.g. integrability, boundedness, continuity, etc.). There is a natural representation of G on L(X) which produced by its action on X:

g: f(x) ↦ ρ_g f(x)= f(g⁻¹· x), where g∈ G, x∈ X. (2)

Clearly this representation is already linear. However in many practical cases the formula for linearization (2) has some additional terms which are required to make it, for example, unitary.

Exercise 3 Show that T(g⁻¹)=T⁻¹(g) and T(e)=I, where I is the identity operator on H.

Exercise 4 Show that these are linear continuous representations of corresponding groups:

Operators T(x) such that [T(x) f](t)=f(t+x) form a representation of ℝ in L₂(ℝ).
Operators T(n) such that T(n) a_k=a_k+n form a representation of ℤ in l₂.
Operators T(a,b) defined by
[T(a,b) f](x)= √

a

f(ax+b), a ∈ ℝ₊, b∈ℝ (3)

form a representation of ax+b group in L₂(ℝ).
Operators T(s,x,y) defined by
[T(s,x,y) f] (t)=e
i(2s− √

2

yt+xy)

f(t− √

2

x) (4)

form Schrödinger representation of the Heisenberg group ℍ¹ in L₂(ℝ).
Operators T(g) defined by
[T(g) f](t) =
1

ct+d

f ⎛
⎜
⎜
⎝
at+b

ct+d

⎞
⎟
⎟
⎠ ,     where g⁻¹=
⎛
⎜
⎝
      a b

      c d

⎞
⎟
⎠

,     (5)

form a representation of SL₂(ℝ) in L₂(ℝ).

In the sequel a representation always means linear continuous representation. T(g) is an exact representation (or faithful representation if T(g)=I only for g=e. The opposite case when T(g)=I for all g∈ G is a trivial representation. The space H is representation space and in most cases will be a Hilber space [, § III.5]. If dimensionality of H is finite then T is a finite dimensional representation, in the opposite case it is infinite dimensional representation.

We denote the scalar product on H by ⟨ ·,· ⟩. Let {e_j} be an (finite or infinite) orthonormal basis in H, i.e.

where δ_jk is the Kroneker delta, and linear span of {e_j} is dense in H.

Definition 5 The matrix elements t_jk(g) of a representation T of a group G (with respect to a basis {e_j} in H) are complex valued functions on G defined by

t_jk(g) = ⟨ T(g)e_j,e_k ⟩. (6)

Exercise 6 Show that [, § 1.1.3]

T(g) e_k=∑_j t_jk(g) e_j.
t_jk(g₁g₂)=∑_n t_jn(g₁) t_nk(g₂).

It is typical mathematical questions to determine identical objects which may have a different appearance. For representations it is solved in the following definition.

Definition 7 Two representations T₁ and T₂ of the same group G in spaces H₁ and H₂ correspondingly are equivalent representations if there exist a linear operator A: H₁ → H₂ with the continuous inverse operator A⁻¹ such that:

T₂(g)= A T₁(g) A⁻¹, ∀ g∈ G.

Exercise 8 Show that representation T(a,b) of ax+b group in L₂(ℝ) from Exercise 3 is equivalent to the representation

[T₁(a,b) f] (x)=

√

⎛
⎜
⎜
⎝

⎞
⎟
⎟
⎠

. (7)

The relation of equivalence is reflexive, symmetric, and transitive. Thus it splits the set of all representations of a group G into classes of equivalent representations. In the sequel we study group representations up to their equivalence classes only.

Exercise 9 Show that equivalent representations have the same matrix elements in appropriate basis.

Definition 10 Let T be a representation of a group G in a Hilbert space H The adjoint representation T′(g) of G in H is defined by

T′(g)=

⎛
⎝

T(g⁻¹)

⎞
⎠

^*,

where ^* denotes the adjoint operator in H.

Exercise 11 Show that

T′ is indeed a representation.
t′_jk(g)=t_kj(g⁻¹).

Exercise 12 Show that UU^*=I.

Definition 13 T is a unitary representation of a group G in a space H if T(g) is a unitary operator for all g∈ G. T₁ and T₂ are unitary equivalent representations if T₁=UT₂U⁻¹ for a unitary operator U.

Exercise 14

Show that all representations from Exercises 4 are unitary.
Show that representations from Exercises 3 and 8 are unitary equivalent.

Exercise 15 Show that if a Lie group G is represented by unitary operators in H then its Lie algebra g is represented by self-adjoint (possibly unbounded) operators in H.

Definition 16 A character of representation T is equal χ(g)= tr(T(g)), where tr is the trace [, § III.5.2 (Probl.)] of operator.

Exercise 17 Show that

Characters of a representation T are constant on the adjoint elements g⁻¹hg, for all g∈ G.
Character is an algebra homomorphism from an algebra of representations with Kronecker’s (tensor) multiplication [, § 1.9] to complex numbers.

Proof.[Hint] Use that tr(AB)=tr(BA), tr(A+B)=trA + trB, and tr( A ⊗ B)= trA trB.

For infinite dimensional representation characters could be defined either as distributions [, § 11.2] or in infinitesimal terms of Lie algebras [, § 11.3].

The characters of a representation should not be confused with the following notion.

Definition 18 A character of a group G is a one-dimensional representation of G.

Exercise 19

Let χ be a character of a group G. Show that a character of representation χ coincides with it and thus is a character of G.
A matrix element of a group character χ coincides with χ.
Let χ₁ and χ₁ be characters of a group G. Show that χ₁ ⊗ χ₂ =χ₁χ₂ and χ′(g)=χ₁(g⁻¹) are again characters of G. In other words characters of a group form a group themselves.

12.2 Decomposition of Representations

The important part of any mathematical theory is classification theorems on structural properties of objects. Very well known examples are:

The similar structural results in the representation theory are very difficult. The easiest (but still rather difficult) questions are on classification of unitary representations up to unitary equivalence.

Definition 1 Let T be a representation of G in H. A linear subspace L⊂ H is invariant subspace for T if for any x∈ L and any g∈ G the vector T(g)x again belong to L.

There are always two trivial invariant subspaces: the null space and entire H. All other are nontrivial invariant subspaces.

Definition 2 If there are only two trivial invariant subspaces then T is irreducible representation . Otherwise we have reducible representation .

For any nontrivial invariant subspace we could define the restriction of representation of T on it. In this way we obtain a subrepresentation of T.

Example 3 Let T(a), a∈ℝ₊ be defined as follows: [T(a)]f(x)=f(ax). Then spaces of even and odd functions are invariant.

Definition 4 If the closure of liner span of all vectors T(g) v is dense in H then v is called cyclic vector for T.

Exercise 5 Show that for an irreducible representation any non zero vector is cyclic.

The following important result of representation theory of compact groups is a consequence of the Exercise 3 and we state here it without a proof.

Theorem 6 [, § 9.2]

Every topologically irreducible representation of a compact group G is finite-dimensional and unitarizable.
If T₁ and T₂ are two inequivalent irreducible representations, then every matrix element of T₁ is orthogonal in L₂(G) to every matrix element of T₂.
For a compact group G its dual space Ĝ is discrete.

Exercise 7 Let a unitary representation T has an invariant subspace L⊂ H, then its orthogonal completion L^⊥ is also invariant.

Definition 8 A representation on H is called decomposable if there are two non-trivial invariant subspaces H₁ and H₂ of H such that H=H₁⊕ H₂.

If a representation is not decomposable then its primary.

Theorem 9 [, § 8.4] Any unitary representation T of a locally compact group G could be decomposed in a (continuous) direct sum irreducible representations: T=∫_X T_x dµ(x).

Exercise 10 Let T be a representation of ℝ in L₂(ℝ) as follows: [T(a)f](x)=e^iaxf(x). Show that

Any measurable set E⊂ ℝ define an invariant subspace of functions vanishing outside E.
T does not have invariant irreducible subrepresentations.

Definition 11 The set of equivalence classes of unitary irreducible representations of a group G is denoted by Ĝ and called dual object (or dual space) of the group G.

Definition 12 A left regular representation Λ(g) of a group G is the representation by left shifts in the space L₂(G) of square-integrable function on G with the left Haar measure

Λg: f(h) ↦ f(g⁻¹h). (8)

The main problem of representation theory is to decompose a left regular representation Λ(g) into irreducible components.

12.3 Schur’s Lemma

It is a pleasant feature of an abstract theory that we obtain important general statements from simple observations. Finiteness of invariant measure on a compact group is one such example. Another example is Schur’s Lemma presented here.

To find different classes of representations we need to compare them each other. This is done by intertwining operators.

Definition 1 Let T₁ and T₂ are representations of a group G in a spaces H₁ and H₂ correspondingly. An operator A: H₁ → H₂ is called an intertwining operator if

A T₁(g) = T₂(g) A, ∀ g∈ G.

If T₁=T₂=T then A is interntwinig operator or commuting operator for T.

Exercise 2 Let G, H, T(g), and A be as above. Show the following: [, § 1.3.1]

Let x∈ H be an eigenvector for A with eigenvalue λ. Then T(g)x for all g∈ G are eigenvectors of A with the same eigenvalue λ.
All eigenvectors of A with a fixed eigenvalue λ for a linear subspace invariant under all T(g), g∈ G.
If an operator A is commuting with irreducible representation T then A=λ I.

Proof.[Hint] Use the spectral decomposition of selfadjoint operators [, § V.2.2].

Lemma 3 (Schur) [, § 8.2] If two representations T₁ and T₂ of a group G are irreducible, then every intertwining operator between them is either zero or invertible.

Exercise 4 Show that

Two irreducible representations are either equivalent or disjunctive.
All operators commuting with an irreducible representation form a field.
Irreducible representation of commutative group are one-dimensional.
If T is unitary irreducible representation in H and B(·,·) is a bounded semi linear form in H invariant under T: B(T(g)x,T(g)y)=B(x,y) then B(·,·)=λ⟨ ·,· ⟩.

12.4 Induced Representations

The general scheme of induced representations is as follows, see [, § 13.2], [, Ch. 5], [, Ch. 6], [, § 3.1] and subsection 2.2.2. Let G be a group and let H be its subgroup. Let X=G / H be the corresponding left homogeneous space and s: X → G be a continuous function (section) [, § 13.2] which is a left inverse to the natural projection p:G→ G/H.

Then any g∈ G has a unique decomposition of the form g=s(x)h where x=p(g)∈ X and h∈ H. We define the map r: G→ H:

Note that X is a left homogeneous space with the G-action defined in terms of p and s as follows, see Ex. 10:

Let χ: H → B(V) be a linear representation of H in a vector space V, e.g. by unitary rotations in the algebra of either complex, dual or double numbers. Then χ induces a linear representation of G, which is known as induced representation in the sense of Mackey [, § 13.2]. This representation has the canonical realisation ρ in a space of V-valued functions on X. It is given by the formula (cf. [, § 13.2.(7)–(9)]):

where g∈ G, x∈ X, h∈ H and r: G → H, s: X → G are maps defined above; * denotes multiplication on G and · denotes the action (9) of G on X from the left.

In the case of complex numbers this representation automatically becomes unitary in the space L₂(X) of the functions square integrable with respect to a measure dµ if instead of the representation χ one uses the following substitute:

However in our study the unitarity of representations or its proper replacements is a more subtle issue and we will consider it separately.

An alternative construction of induced representations is realised on the space of functions on G which have the following property:

This space is invariant under the left shifts. The restriction of the left regular representation to this subspace is equivalent to the induced representation described above.

Exercise 1

Write the intertwining operator for this equivalence.
Define the corresponding inner product on the space of functions 12 in such a way that the above intertwining operator becomes unitary.
Proof.[Hint] Use the map s: X → G.

Lecture 13 Wavelets on Groups

A matured mathematical theory looks like a tree. There is a solid trunk which supports all branches and leaves but could not be alive without them. In the case of group approach to wavelets the trunk of the theory is a construction of wavelets from a square integrable representation [], [, Chap. 8]. We begin from this trunk which is a model for many different generalisations and will continue with some smaller “generalising” branches later.

13.1 Wavelet Transform on Groups

Let G be a group with a left Haar measure dµ and let ρ be a unitary irreducible representation of a group G by operators ρ_g, g ∈ G in a Hilbert space H.

Definition 1 Let us fix a vector w₀∈ H. We call w₀ ∈ H a vacuum vector or a mother wavelet (other less-used names are ground state , fiducial vector , etc.). We will say that set of vectors w_g=ρ(g) w₀, g∈ G form a family of coherent states (wavelets).

Exercise 2 If ρ is irreducible then w_g, g∈ G is a total set in H, i.e. the linear span of these vectors is dense in H.

The wavelet transform could be defined as a mapping from H to a space of functions over G via its representational coefficients (also known as matrix coefficients):

Exercise 3 Show that the wavelet transform W is a continuous linear mapping and the image of a vector is a bounded continuous function on G. The liner space of all such images is denoted by W(G).

Exercise 4 Let a Hilbert space H has a basis e_j, j∈ℤ and a unitary representation ρ of G=ℤ defined by ρ(k)e_j=e_j+k. Write a formula for wavelet transform with w₀=e₀ and characterise W(ℤ).

Exercise 5 Let G be ax+b group and ρ is given by (cf. (3)):

[T(a,b) f](x)=

√

⎛
⎜
⎜
⎝

x−b

⎞
⎟
⎟
⎠

, (2)

in L₂(ℝ). Show that

The representation is reducible and describe its irreducible components.
for w₀(x)=1/2π i (x+i) coherent states are v_(a,b)(x)=√a/2π i (x−(b−ia)).
Wavelet transform is given by
v(a,b)=
√

a

2π i

∫

ℝ

v(x)

x−(b+ia)

dx,

which resembles the Cauchy integral formula.
Give a characteristic of W(G).
Write the wavelet transform for the same representation of the group ax+b and the GaussianGauss function e^−x2/2 (see Fig. 13.1) as a mother wavelets.

Proposition 6 The wavelet transform W intertwines ρ and the left regular representation Λ (8) of G:

W ρ(g) = Λ(g) W.

Corollary 7 The function space W(G) is invariant under the representation Λ of G.

Wavelet transform maps vectors of H to functions on G. We can consider a map in the opposite direction sends a function on G to a vector in H.

Definition 8 The inverse wavelet transform M associated with a vector w′₀∈ H maps L₁(G) to H and is given by the formula:

M: L₁(G) → H: v(g) ↦ M [v(g)]

∫

v(g) w′_g dµ(g)

∫

v(g) ρ(g) dµ(g) w′₀,

(3)

where in the last formula the integral express an operator acting on vector w′₀.

Exercise 9 Write inverse wavelet transforms for Exercises 4 and 5.

Lemma 10 If the wavelet transform W and inverse wavelet transform M are defined by the same vector w₀ then they are adjoint operators: W^*=M.

where the scalar product in the first line is on H and in the last line is on L₂(G). Now the result follows from the totality of coherent states w_g in H.

Proposition 11 The inverse wavelet transform M intertwines the representation Λ (8) on L₂(G) and ρ on H:

M Λ(g) = ρ(g) M.

Corollary 12 The image M(L₁(G))⊂ H of subspace under the inverse wavelet transform M is invariant under the representation ρ.

The following proposition explain the usage of the name “inverse” (not “adjoint” as it could be expected from Lemma 10) for M.

Theorem 13 The operator

P= M W: H → H (4)

maps H into its linear subspace for which w′₀ is cyclic. Particularly if ρ is an irreducible representation then P is cI for some constant c depending from w₀ and w′₀.

Proof. It follows from Propositions 6 and 11 that operator MW: H → H intertwines ρ with itself. Then Corollaries 7 and 12 imply that the image MW is a ρ-invariant subspace of H containing w₀. From irreducibility of ρ by Schur’s Lemma [, § 8.2] one concludes that MW=cI on C for a constant c∈ℂ.

Remark 14 From Exercises 4 and 9 it follows that irreducibility of ρ is not necessary for MW=cI, it is sufficient that w₀ and w′₀ are cyclic only.

Theorem 15 Operator WM is up to a complex multiplier a projection of L₁(G) to W(G).

13.2 Square Integrable Representations

So far our consideration of wavelets was mainly algebraic. Usually in analysis we wish that the wavelet transform could preserve an analytic structure, e.g. values of scalar product in Hilbert spaces. This accomplished if a representation ρ possesses the following property.

Definition 1 [, § 9.3] Let a group G with a left Haar measure dµ have a unitary representation ρ: G → L(H). A vector w∈ H is called admissible vector if the function ŵ(g)=⟨ ρ(g)w,w ⟩ is non void and square integrable on G with respect to dµ:

0<c²=

∫

⟨ ρ(g)w,w ⟩ ⟨ w,ρ(g)w ⟩ dµ(g) < ∞. (5)

If an admissible vector exists then ρ is a square integrable representation .

Square integrable representations of groups have many interesting properties (see [, § 14] for unimodular groups and [], [, Chap. 8] for not unimodular generalisation) which are crucial in the construction of wavelets. For example, for a square integrable representation all functions ⟨ ρ(g)v₁,v₂ ⟩ with an admissible vector v₁ and any v₂∈ H are square integrable on G; such representation belong to dicrete series; etc.

Exercise 2 Show that

Admissible vectors form a linear space.
For an irreducible ρ the set of admissible vectors is dense in H or empty.

Proof.[Hint] The set of all admissible vectors is an ρ-invariant subspace of H.

Exercise 3

Find a condition for a vector to be admissible for the representation (2) (and therefore the representation is square integrable).
Show that w₀(x)=1/2π i (x+i) is admissible for ax+b group.
Show that the Gaussian e^−x2 is not admissible for ax+b group.

For an admissible vector w we take its normalisation w₀=||w||/c w to obtain:

Proposition 4 If both wavelet transform W and inverse wavelet transform M for an irreducible square integrable representation ρ are defined by the same admissible vector w₀ then the following three statements are equivalent:

w₀ satisfy (6);
MW=I;
for any vectors v₁, v₂∈ H:
⟨ v₁,v₂ ⟩= ∫

G

v₁(g)

v₂(g)

dµ(g). (7)

Proof. We already knew that MW=cI for a constant c∈ ℂ. Then (6) exactly says that c=1. Because W and M are adjoint operators it follows from MW=I on H that:

which is exactly the isometry of W (7). Finally condition (6) is a partticular case of general isometry of W for vector w₀.

Exercise 5 Write the isometry conditions (7) for wavelet transforms for ℤ and ax+b groups (Exercises 4 and 5).

Wavelets from square integrable representation closely related to the following notion:

Definition 6 A reproducing kernel on a set X with a measure is a function K(x,y) such that:

K(x,x)

0, ∀ x∈ X,

(8)

K(x,y)

K(y,x)

(9)

K(x,z)

∫

K(x,y)K(y,z) dy.

(10)

Proposition 7 The image W(G) of the wavelet transform W has a reproducing kernel K(g,g′)=⟨ w_g,w_g′ ⟩. The reproducing formula is in fact a convolution:

v(g′)

∫

K(g′,g) v(g) dµ(g)

∫

ŵ₀(g⁻¹g′) v(g) dµ(g)

(11)

with a wavelet transform of the vacuum vector ŵ₀(g)= ⟨ w₀,ρ(g)w₀ ⟩.

Exercise 8 Write reproducing kernels for wavelet transforms for ℤ and ax+b groups (Exercises 4 and 5.

Exercise^* 9 Operator (11) of convolution with ŵ₀ is an orthogonal projection of L₂(G) onto W(G).

Proof.[Hint] Use that an left invariant subspace of L₂(G) is in fact an right ideal in convolution algebra, see Lemma 6.

Remark 10 To possess a reproducing kernel—is a well-known property of spaces of analytic functions. The space W(G) shares also another important property of analytic functions: it belongs to a kernel of a certain first order differential operator with Clifford coefficients (the Dirac operator) and a second order operator with scalar coefficients (the Laplace operator) [, , , ], which we will consider that later too.

We consider only fundamentals of the wavelet construction here. There are much results which could be stated in an abstract level. To avoid repetition we will formulate it later on together with an interesting examples of applications.

The construction of wavelets from square integrable representations is general and straightforward. However we could not use it everywhere we may wish:

To be vivid the trunk of the wavelets theory should split into several branches adopted to particular cases and we describe some of them in the next lectures.

13.3 Fundamentals of Wavelets on Homogeneous Spaces

Let G be a group and H be its closed normal subgroup. Let X=G/H be the corresponding homogeneous space with a left invariant measure dµ. Let s: X → G be a Borel section in the principal bundle of the natural projection p: G → G/H. Let ρ be a continuous representation of a group G by invertible unitary operators ρ(g), g ∈ G in a Hilbert space H.

For any g∈ G there is a unique decomposition of the form g=s(x)h, h∈ H, x=p(g)∈ X. We will define r: G → H: r(g)=h=(s(p(g)))⁻¹g from the previous equality. Then there is a geometric action of G on X → X defined as follows

Example 1 As a subgroup H we select now the center of ℍⁿ consisting of elements (t,0). Of course X=G/H isomorphic to ℂⁿ and mapping s: ℂⁿ → G simply is defined as s(z)=(0,z). The Haar measure on ℍⁿ coincides with the standard Lebesgue measure on ℝ²ⁿ⁺¹ [, § 1.1] thus the invariant measure on X also coincides with the Lebesgue measure on ℂⁿ. Note also that composition law p(g· s(z)) reduces to Euclidean shifts on ℂⁿ. We also find p((s(z₁))⁻¹· s(z₂))=z₂−z₁ and r((s(z₁))⁻¹· s(z₂))= 1/2 ℑ z₁z₂.

Let ρ: G → L(V) be a unitary representation of the group G by operators in a Hilbert space V.

Definition 2 Let G, H, X=G/H, s: X → G, ρ: G → L(V) be as above. We say that w₀ ∈ H is a vacuum vector if it satisfies to the following two conditions:

ρ(h) w₀ = χ(h) w₀, χ(h) ∈ ℂ, for all h∈ H;

(13)

∫

⎪
⎪

⟨ w₀,ρ(s(x))w₀ ⟩

⎪
⎪

² dx =

⎪⎪
⎪⎪

w₀

⎪⎪
⎪⎪

².

(14)

We will say that set of vectors w_x=ρ(x) w₀, x∈ X form a family of coherent states.

Note that mapping h → χ(h) from (13) defines a character of the subgroup H. The condition (14) could be easily achieved by a renormalisation w₀ as soon as we sure that the integral in the left hand side is finite and non-zero.

Convention 3 In that follow we will usually write x∈ X and x⁻¹∈ X instead of s(x)∈ G and s(x)⁻¹∈ G correspondingly. The right meaning of “x” could be easily found from the context (whether an element of X or G is expected there).

Example 4 As a “vacuum vector” we will select the original vacuum vector of quantum mechanics—the Gauss function w₀(q)=e^−q2/2 (see Figure 13.1), which belongs to all L₂(ℝⁿ). Its transformations are defined as follow:

w_g(q)=[ρ_(s,z) w₀](q)

i(2s−

√

xq+xy)

−

(q−

√

²/2

e^{2is−(x2+y2)/2} e

((x+iy)²−q²)/2−

√

i(x+iy)q

e^2is−zz/2e

(z²−q²)/2−

√

i z q

Particularly [ρ_(t,0) w₀](q)=e^−2itw₀(q), i.e., it really is a vacuum vector in the sense of our definition with respect to H.

Exercise 5 Check the square integrability condition (14) for w₀(q)=e^−q2/2.

The wavelet transform (similarly to [eq:wavelet-transform]the group case) could be defined as a mapping from V to a space of bounded continuous functions over G via representational coefficients

Due to (13) such functions have simple transformation properties along H-orbits:

Thus the wavelet transform is completely defined by its values indexed by points of X=G/H. Therefore we prefer to consider so called induced wavelet transform.

Remark 6 In the earlier papers [], [] we use name reduced wavelet transform since it produces functions on a homogeneous space rather than the entire group. From now on we prefer the name induced wavelet transform due to its explicit connection with induced representations.

Definition 7 The induced wavelet transform W from a Hilbert space H to a space of function W(X) on a homogeneous space X=G/H defined by a representation ρ of G on H, a vacuum vector w₀ is given by the formula

W: H → W(X): v ↦ v(x)= [Wv] (x)=⟨ ρ(x⁻¹) v,w₀ ⟩= ⟨ v,ρ(x)w₀ ⟩. (15)

Example 8 The transformation (15) with the kernel [ρ_(0,z) w₀](q) is an embedding L₂(ℝⁿ) → L₂(ℂⁿ) and is given by the formula

f(z)

⟨ f,ρ_s(z)f₀ ⟩

π^−n/4

∫

ℝⁿ

f(q) e^−zz/2 e

− (z²+q²)/2+

√

e^−zz/2π^−n/4

∫

ℝⁿ

f(q) e

− (z²+q²)/2+

√

dq .

(16)

Then f(g) belongs to L₂( ℂⁿ , dg) or its preferably to say that function f(z)=e^zz/2f(t₀,z) belongs to space L₂( ℂⁿ , e^{− | z |2}dg) because f(z) is analytic in z. Such functions form the Segal-Bargmann space F₂( ℂⁿ, e^{−
| z |2}dg) of functions [, ], which are analytic by z and square-integrable with respect to the Gaussian measure Gauss measure e^{− | z |2}dz. We use notation W for the mapping v ↦ v(z)=e^zz/2Wv. Analyticity of f(z) is equivalent to the condition ( ∂ / ∂z_j + 1/2 z_j I ) f(z)=0 . The integral in (16) is the well-known Segal-Bargmann transform [, ].

Exercise 9 Check that w₀(z)=1 for the vacuum vector w₀(q)=e^−q2/2.

There is a natural representation of G in W(X). It could be obtained if we first lift functions from X to G, apply the left regular representation Λ and then pul them back to X. The result defines a representation λ(g): W(X) → W(X) as follow

We recall that χ(h) is a character of H defined in (13) by the vacuum vector w₀. Of course, for the case of trivial H={e} (17) becomes the left regular representation Λ(g) of G.

Proposition 10 The induced wavelet transform W intertwines ρ and the representation λ (17) on W(X):

W ρ(g) = λ(g) W.

Corollary 11 The function space W(X) is invariant under the representation λ of G.

Example 12 Integral transformation (16) intertwines the Schrödinger representation (4) with the following realisation of representation (17):

λ(s,z) f(u)	=	f₀(z⁻¹· u) χ(s+r(z⁻¹· u))
	=	f₀(u−z)e^is+iℑ(zu)	(18)

Exercise 13

Using relation W=e^{−| z |2/2}W derive from above that W intertwines the Schrödinger representation with the following:
λ(s,z) f(u) = f₀(u−z) e
2is−zu− ⎪
⎪ z ⎪
⎪ ²/2

.
Show that infinitesimal generators of representation λ are:
∂λ(s,0,0)=iI, ∂λ(0,x,0)=−∂_u−uI, ∂λ(0,0,y)=i(−∂_z+zI)

Definition 14 The inverse wavelet transform M from W(X) to H is given by the formula:

M: W(X) → H: v(x) ↦ M [v(x)]

∫

v(x) w_x dµ(x)

∫

v(x) ρ(x) dµ(x) w₀.

(19)

Proposition 15 The inverse wavelet transform M intertwines the representation λ on W(X) and ρ on H:

M λ(g) = ρ(g) M.

Corollary 16 The image M(W(X))⊂ H of subspace W(X) under the inverse wavelet transform M is invariant under the representation ρ.

Example 17 Inverse transformation to (16) is given by a realisation of (19):

f(q)

∫

ℂⁿ

f(z) f_s(z)(q) dz

∫

ℂⁿ

f(x,y) e

iy(x−

√

−

(q−

√

²/2

dx dy

(20)

∫

ℂⁿ

f(z) e

− (z²+q²)/2+

√

−

⎪
⎪

dz.

The transformation (20) intertwines the representations (18) and the Schrödinger representation (4) of the Heisenberg group.

Theorem 18 The operator

P= M W: H → H (21)

is a projection of H to its linear subspace for which w₀ is cyclic. Particularly if ρ is an irreducible representation then the inverse wavelet transform M is a left inverse operator on H for the wavelet transform W:

MW=I.

Proof. It follows from Propositions 10 and 15 that operator MW: H → H intertwines ρ with itself. Then Corollaries 11 and 16 imply that the image MW is a ρ-invariant subspace of H containing w₀. Because of MWw₀=w₀ we conclude that MW is a projection.

From irreducibility of ρ by Schur’s Lemma [, § 8.2] one concludes that MW=cI on H for a constant c∈ℂ. Particularly

From the condition (14) it follows that ⟨ cw₀,w₀ ⟩=⟨ MW w₀,w₀ ⟩=⟨ w₀,w₀ ⟩ and therefore c=1.

Theorem 19 Operator WM is a projection of L₁(X) to W(X).

Corollary 20 In the space W(X) the strong convergence implies point-wise convergence.

Since the wavelet transform is an isometry we conclude that | f(x) |≤ c||f|| for c=||w₀||, which implies the assertion about two types of convergence.

Example 21 The corresponding operator for the Segal-Bargmann space P (21) is an identity operator L₂(ℝⁿ) → L₂(ℝⁿ) and (21) gives an integral presentation of the Dirac delta.

While the orthoprojection L₂( ℂⁿ, e^{−
| z |2}dg) → F₂( ℂⁿ, e^{−
| z |2}dg) is of a separate interest and is a principal ingredient in Berezin quantization [, ]. We could easy find its kernel from (24). Indeed, f₀(z)=e^{− | z |2}, then the kernel is

K(z,w)

f₀(z⁻¹· w) χ(r(z⁻¹· w))

f₀(w−z)e^iℑ(zw)

exp

⎛
⎜
⎜
⎝

(−

⎪
⎪

w−z

⎪
⎪

² +wz−zw)

⎞
⎟
⎟
⎠

exp

⎛
⎜
⎜
⎝

(−

⎪
⎪

²−

⎪
⎪

²) +wz

⎞
⎟
⎟
⎠

To receive the reproducing kernel for functions f(z)=e^{| z |2} f(z) in the Segal-Bargmann space we should multiply K(z,w) by e^{(−| z |2+
| w |2)/2} which gives the standard reproducing kernel = exp(− | z |² +wz) [, (1.10)].

We denote by W^*: W^*(X) → H and M^*: H → W^*(X) the adjoint (in the standard sense) operators to W and M respectively.

Corollary 22 We have the following identity:

⟨ W v , M^* l ⟩_W(X) = ⟨ v,l ⟩_H, ∀ v, l∈ H, (22)

or equivalently

∫

⟨ ρ(x⁻¹) v,w₀ ⟩ ⟨ ρ(x) w₀,l ⟩ dµ(x) = ⟨ v,l ⟩. (23)

Proof. We show the equality in the first form (23) (but we will apply it often in the second one):

Corollary 23 The space W(X) has the reproducing formula

v(y)=

∫

v(x) b₀(x⁻¹· y) dµ(x), (24)

where b₀(y)=[Ww₀] (y) is the wavelet transform of the vacuum vector w₀.

Lecture 14 Linear Representations

A consideration of the symmetries in analysis is natural to start from linear representations. The previous geometrical actions (??) can be naturally extended to such representations by induction []*§ 13.2 []*§ 3.1 from a representation of a subgroup H. If H is one-dimensional then its irreducible representation is a character, which is always supposed to be a complex valued. However hypercomplex number naturally appeared in the SL₂(ℝ) action (??), see Subsection ?? and [], why shall we admit only i²=−1 to deliver a character then?

14.1 Hypercomplex Characters

As we already mentioned the typical discussion of induced representations of SL₂(ℝ) is centred around the case H=K and a complex valued character of K. A linear transformation defined by a matrix (8) in K is a rotation of ℝ² by the angle t. After identification ℝ²=ℂ this action is given by the multiplication e^{i t}, with i²=−1. The rotation preserve the (elliptic) metric given by:

Therefore the orbits of rotations are circles, any line passing the origin (a “spoke”) is rotated by the angle t, see Fig. 11.1.

Dual and double numbers produces the most straightforward adaptation of this result.

Proposition 1 The following table show correspondences between three types of algebraic characters:

Elliptic	Parabolic	Hyperbolic
i²=−1	ε²=0	є²=1
w=x+i y	w=x+ε y	w=x+є y
w=x−i y	w=x−ε y	w=x−є y
e^{i t} = cost +i sint	e^{ε t} = 1 +ε t	e^{є t} = cosht +є sinht
\| w \|_e²=ww=x²+y²	\| w \|_p²=ww=x²	\| w \|_h²=ww=x²−y²
argw = tan⁻¹ y/x /	argw = y/x	argw = tanh⁻¹ y/x
unit circle \| w \|_e²=1	“unit” strip x=± 1	unit hyperbola \| w \|_h²=1

Geometrical action of multiplication by e^{ι t} is drawn in Fig. 11.1 for all three cases.

Explicitly parabolic rotations associated with e^{ε t} acts on dual numbers as follows:

This links the parabolic case with the Galilean group [] of symmetries of the classic mechanics, with the absolute time disconnected from space.

The obvious algebraic similarity and the connection to classical kinematic is a wide spread justification for the following viewpoint on the parabolic case, cf. [, ]:

Those algebraic analogies are quite explicit and widely accepted as an ultimate source for parabolic trigonometry [, , ]. Moreover, those three rotations are all non-isomorphic symplectic linear transformations of the phase space, which makes them useful in the context of classical and quantum mechanics [, ], see Section 18.1. There exist also alternative characters [] based on Möbius transformations with geometric motivation and connections to equations of mathematical physics.

14.2 Induced Representations

Let G be a group, H be its closed subgroup with the corresponding homogeneous space X=G/H with an invariant measure. We are using notations and definitions of maps p: G→ X, s:X→ G and r: G→ H from Subsection ??. Let χ be an irreducible representation of H in a vector space V, then it induces a representation of G in the sense of Mackey []*§ 13.2. This representation has the realisation ρχ in the space L₂(X) of V-valued functions by the formula []*§ 13.2.(7)–(9):

where g∈ G, x∈ X, h∈ H and r: G → H, s: X → G are maps defined above; * denotes multiplication on G and · denotes the action (9) of G on X.

Consider this scheme for representations of SL₂(ℝ) induced from characters of its one-dimensional subgroups. We can notice that only the subgroup K requires a complex valued character due to the fact of its compactness. For subgroups N′ and we can consider characters of all three types—elliptic, parabolic and hyperbolic. Therefore we have seven essentially different induced representations. We will write explicitly only three of them here.

Example 1 Consider the subgroup H=K, due to its compactness we are limited to complex valued characters of K only. All of them are of the form χ_k:

χ_k

⎛
⎜
⎝

cost	sint
−sint	cost

⎞
⎟
⎠

=e⁻ⁱ^k t, where k∈ℤ. (7)

Using the explicit form (??) of the map s we find the map r given in (??) as follows:

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

√

c²+d²

⎛
⎜
⎝

d	−c
c	d

⎞
⎟
⎠

∈ K.

Therefore:

r(g⁻¹ * s(u,v)) =

√

(c u+d)² +(cv)²

⎛
⎜
⎝

cu+d	−cv
cv	cu+d

⎞
⎟
⎠

, where g⁻¹=

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

Substituting this into (7) and combining with the Möbius transformation of the domain (??) we get the explicit realisation ρk of the induced representation (5):

ρk(g) f(w)=

⎪
⎪

cw+d

⎪
⎪

(cw+d)^k

⎛
⎜
⎜
⎝

aw+b

cw+d

⎞
⎟
⎟
⎠

, where g⁻¹=

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

, w=u+i v. (8)

This representation acts on complex valued functions in the upper half-plane ℝ₊²=SL₂(ℝ)/K and belongs to the discrete series []*§ IX.2. It is common to get rid of the factor | cw+d |^k from that expression in order to keep analyticity and we will follow this practise for a convenience as well.

Example 2 In the case of the subgroup N there is a wider choice of possible characters.

Traditionally only complex valued characters of the subgroup N are considered, they are:
χ_τ^ℂ
⎛
⎜
⎝
        1 0

        t 1

⎞
⎟
⎠

=eⁱ^τ t,       where τ∈ℝ.     (9)

A direct calculation shows that:
      r
⎛
⎜
⎝
        a b

c d

⎞
⎟
⎠

=
⎛
⎜
⎜
⎜
⎜
⎝
        1 0

c

d

1

⎞
⎟
⎟
⎟
⎟
⎠

∈ N′.

Thus:
r(g⁻¹*s(u,v))=
⎛
⎜
⎜
⎜
⎜
⎝
        1 0

cv

d+cu

1

⎞
⎟
⎟
⎟
⎟
⎠

,    where g⁻¹=
⎛
⎜
⎝
        a b

c d

⎞
⎟
⎠

.     (10)

A substitution of this value into the character (9) together with the Möbius transformation (??) we obtain the next realisation of (5):
      ρ[ℂ]τ(g) f(w)= exp ⎛
⎜
⎜
⎝ i
τ c v

cu+d

⎞
⎟
⎟
⎠ f ⎛
⎜
⎜
⎝
aw+b

cw+d

⎞
⎟
⎟
⎠ ,    where w=u+ε v,   g⁻¹=
⎛
⎜
⎝
a b

c d

⎞
⎟
⎠

.

The representation acts on the space of complex valued functions on the upper half-plane ℝ₊², which is a subset of dual numbers as a homogeneous space SL₂(ℝ)/N′. The mixture of complex and dual numbers in the same expression is confusing.
The parabolic character χ_τ with the algebraic flavour is provided by multiplication (2) with the dual number:
      χ_τ
⎛
⎜
⎝
        1 0

        t 1

⎞
⎟
⎠

=e^ε τ t=1+ε τ t,       where τ∈ℝ.

If we substitute the value (10) into this character, then we receive the representation:
      ρτ(g) f(w)= ⎛
⎜
⎜
⎝ 1+ε
τ c v

cu+d

⎞
⎟
⎟
⎠ f ⎛
⎜
⎜
⎝
aw+b

cw+d

⎞
⎟
⎟
⎠ ,

where w, τ and g are as above. The representation is defined on the space of dual numbers valued functions on the upper half-plane of dual numbers. Thus expression contains only dual numbers with their usual algebraic operations. Thus it is linear with respect to them.

All characters in the previous Example are unitary. Then the general scheme of induced representations []*§ 13.2 implies their unitarity in proper senses.

Theorem 3 ([]) Both representations of SL₂(ℝ) from Example 2 are unitary on the space of function on the upper half-plane ℝ₊² of dual numbers with the inner product:

⟨ f₁,f₂ ⟩=

∫

ℝ₊²

f₁(w) f₂(w)

du dv

v²

, where w=u+ε v, (11)

and we use the conjugation and multiplication of functions’ values in algebras of complex and dual numbers for representations ρ[ℂ]τ and ρτ respectively.

The inner product (11) is positive defined for the representation ρ[ℂ]τ but is not for the other. The respective spaces are parabolic cousins of the Krein spaces [], which are hyperbolic in our sense.

14.3 Similarity and Correspondence: Ladder Operators

From the above observation we can deduce the following empirical principle, which has a heuristic value.

Principle 1 (Similarity and correspondence)

Subgroups K, N′ and play a similar rôle in the structure of the group SL₂(ℝ) and its representations.
The subgroups shall be swapped simultaneously with the respective replacement of hypercomplex unit ι.

The first part of the Principle (similarity) does not look sound alone. It is enough to mention that the subgroup K is compact (and thus its spectrum is discrete) while two other subgroups are not. However in a conjunction with the second part (correspondence) the Principle have received the following confirmations so far, see [] for details:

Remark 2 The principle of similarity and correspondence resembles supersymmetry between bosons and fermions in particle physics, but we have similarity between three different types of entities in our case.

Let us give another illustration to the Principle. Consider the Lie algebra sl₂ of the group SL₂(ℝ). Pick up the following basis in sl₂ []*§ 8.1:

Let ρ be a representation of the group SL₂(ℝ) in a space V. Consider the derived representation dρ of the Lie algebra sl₂ []*§ VI.1 and denote X′=dρ(X) for X∈sl₂. To see the structure of the representation ρ we can decompose the space V into eigenspaces of the operator X′ for some X∈ sl₂, cf. the Taylor series in Section 16.4.

Example 3 It would not be surprising that we are going to consider three cases:

Let X=Z be a generator of the subgroup K (8). Since this is a compact subgroup the corresponding eigenspaces Z′ v_k=i k v_k are parametrised by an integer k∈ℤ. The raising/lowering or ladder operators L^± []*§ VI.2 []*§ 8.2 are defined by the following commutation relations:

[Z′,L^±]=λ_±L^±. (14)

In other words L^± are eigenvectors for operators adZ of adjoint representation of sl₂ []*§ VI.2.

Remark 4 The existence of such ladder operators follows from the general properties of Lie algebras if the element X∈sl₂ belongs to a Cartan subalgebra. This is the case for vectors Z and B, which are the only two non-isomorphic types of Cartan subalgebras in sl₂. However the third case considered in this paper, the parabolic vector B+Z/2, does not belong to a Cartan subalgebra, yet a sort of ladder operators is still possible with dual number coefficients. Moreover, for the hyperbolic vector B, besides the standard ladder operators an additional pair with double number coefficients will also be described.

From the commutators (14) we deduce that L⁺ v_k are eigenvectors of Z′ as well:

Z′(L⁺ v_k)	=	(L⁺Z′+λ₊L⁺)v_k=L⁺(Z′v_k)+λ₊L⁺v_k* =i k L⁺v_k+λ₊L⁺v_k*
	=	(i k+λ₊)L⁺v_k.

Thus action of ladder operators on respective eigenspaces can be visualised by the diagram:

1 … <.4ex>[r]^L+ V_i k−λ <.4ex>[l]^L⁻<.4ex>[r]^L⁺ V_i k <.4ex>[l]^L⁻ <.4ex>[r]^L⁺ V_i_k+ λ <.4ex>[l]^L− <.4ex>[r]^L+ …<.4ex>[l]^L− (15)

Assuming L⁺=aA′+bB′+cZ′ from the relations (13) and defining condition (14) we obtain linear equations with unknown a, b and c:

c=0, 2a=λ₊ b, −2b=λ₊ a.

The equations have a solution if and only if λ₊²+4=0, and the raising/lowering operators are L^±=±i A′+B′.

Consider the case X=2B of a generator of the subgroup (10). The subgroup is not compact and eigenvalues of the operator B′ can be arbitrary, however raising/lowering operators are still important []*§ II.1 []*§ 1.1. We again seek a solution in the form L_h⁺=aA′+bB′+cZ′ for the commutator [2B′,L_h⁺]=λ L_h⁺. We will get the system:

4c=λ a, b=0, a=λ c.

A solution exists if and only if λ²=4. There are obvious values λ=± 2 with the ladder operators L_h^±=±2A′+Z′, see []*§ II.1 []*§ 1.1. Each indecomposable sl₂-module is formed by a one-dimensional chain of eigenvalues with a transitive action of ladder operators.

Admitting double numbers we have an extra possibility to satisfy λ²=4 with values λ=±2є. Then there is an additional pair of hyperbolic ladder operators L_є^±=±2єA′+Z′, which shift eigenvectors in the “orthogonal” direction to the standard operators L_h^±. Therefore an indecomposable sl₂-module can be parametrised by a two-dimensional lattice of eigenvalues on the double number plane, see Fig. 14.2

=2.5em@C=1.5em@M=.5em … <.4ex>[d]^L_є+ … <.4ex>[d]^L_є+ … <.4ex>[d]^L_є+
… <.4ex>[r]⁻L_h⁺ V_{(n−2)+є (k−2)} <.4ex>[l]⁻L_h⁻<.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+ V_{n+є (k−2)} <.4ex>[l]^L_h− <.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+ V_{(n+2)+є (k−2)} <.4ex>[l]^L_h− <.4ex>[r]⁻L_h⁺ <.4ex>[u]^L_є− <.4ex>[d]^L_є+ …<.4ex>[l]⁻L_h⁻
… <.4ex>[r]⁻L_h⁺ V_{(n−2)+є k} <.4ex>[l]⁻L_h⁻<.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+ V_{n+є k} <.4ex>[l]^L_h− <.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+ V_{(n+2)+є k} <.4ex>[l]^L_h− <.4ex>[r]⁻L_h⁺ <.4ex>[u]^L_є− <.4ex>[d]^L_є+ …<.4ex>[l]⁻L_h⁻
… <.4ex>[r]⁻L_h⁺ V_{(n−2)+є (k+2)} <.4ex>[l]⁻L_h⁻<.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+ V_{n+є (k+2)} <.4ex>[l]^L_h− <.4ex>[r]^L_h+ <.4ex>[u]^L_є− <.4ex>[d]^L_є+ V_{(n+2)+є (k+2)} <.4ex>[l]^L_h− <.4ex>[r]⁻L_h⁺ <.4ex>[u]^L_є− <.4ex>[d]^L_є+ …<.4ex>[l]⁻L_h⁻
… <.4ex>[u]^L_є− … <.4ex>[u]^L_є− … <.4ex>[u]^L_є−
Figure 14.2: The action of hyperbolic ladder operators on a 2D lattice of eigenspaces. Operators L_h^± move the eigenvalues by 2, making shifts in the horizontal direction. Operators L_є^± change the eigenvalues by 2є, shown as vertical shifts.

Finally consider the case of a generator X=−B+Z/2 of the subgroup N′ (11). According to the above procedure we get the equations:
b+2c=λ a, −a=λ b,
a

2

=λ c,

which can be resolved if and only if λ²=0. If we restrict ourselves with the only real (complex) root λ=0, then the corresponding operators L_p^±=−B′+Z′/2 will not affect eigenvalues and thus are useless in the above context. However the dual number roots λ =± ε t, t∈ℝ lead to the operators L_ε^±=± ε tA′−B′+Z′/2. These operators are suitable to build an sl₂-modules with a one-dimensional chain of eigenvalues.

Remark 5 It is noteworthy that:

the introduction of complex numbers is a necessity for the existence of ladder operators in the elliptic case;
in the parabolic case we need dual numbers to make ladder operators useful;
in the hyperbolic case double numbers are not required neither for the existence or for the usability of ladder operators, but they do provide an enhancement.

We summarise the above consideration with a focus on the Principle of similarity and correspondence:

Proposition 6 Let a vector X∈sl₂ generates the subgroup K, N′ or , that is X=Z, B−Z/2, or B respectively. Let ι be the respective hypercomplex unit.

Then raising/lowering operators L^± satisfying to the commutation relation:

[X,L^±]=±ι L^±, [L⁻,L⁺]=2ι X.

are:

L^±=±ι A′ +Y′.

Here Y∈sl₂ is a linear combination of B and Z with the properties:

Y=[A,X].
X=[A,Y].
Killings form K(X,Y) []*§ 6.2 vanishes.

Any of the above properties defines the vector Y∈span{B,Z} up to a real constant factor.

The usability of the Principle of similarity and correspondence will be illustrated by more examples below.

Lecture 15 Covariant Transform

A general group-theoretical construction [, , , , , , ] of wavelets (or coherent state) starts from an irreducible square integrable representation —in the proper sense or modulo a subgroup. Then a mother wavelet is chosen to be admissible. This leads to a wavelet transform which is an isometry to L₂ space with respect to the Haar measure on the group or (quasi)invariant measure on a homogeneous space.

The importance of the above situation shall not be diminished, however an exclusive restriction to such a setup is not necessary, in fact. Here is a classical example from complex analysis: the Hardy space H₂(T) on the unit circle and Bergman spaces B₂ⁿ(ⅅ), n≥ 2 in the unit disk produce wavelets associated with representations ρ₁ and ρ_n of the group SL₂(ℝ) respectively []. While representations ρ_n, n≥ 2 are from square integrable discrete series, the mock discrete series representation ρ₁ is not square integrable []*§ VI.5 []*§ 8.4. However it would be natural to treat the Hardy space in the same framework as Bergman ones. Some more examples will be presented below.

15.1 Extending Wavelet Transform

To make a sharp but still natural generalisation of wavelets we give the following definition.

Definition 1 [] Let ρ be a representation of a group G in a space V and F be an operator from V to a space U. We define a covariant transform W from V to the space L(G,U) of U-valued functions on G by the formula:

W: v↦ v(g) = F(ρ(g⁻¹) v), v∈ V, g∈ G. (1)

Operator F will be called fiducial operator in this context.

We borrow the name for operator F from fiducial vectors of Klauder and Skagerstam [].

Remark 2 We do not require that fiducial operator F shall be linear. Sometimes the positive homogeneity, i.e. F(t v)=tF(v) for t>0, alone can be already sufficient, see Example 8.

Remark 3 Usefulness of the covariant transform is in the reverse proportion to the dimensionality of the space U. The covariant transform encodes properties of v in a function Wv on G. For a low dimensional U this function can be ultimately investigated by means of harmonic analysis. Thus dimU=1 (scalar-valued functions) is the ideal case, however, it is unattainable sometimes, see Example 5 below. We may have to use a higher dimensions of U if the given group G is not rich enough.

As we will see below covariant transform is a close relative of wavelet transform. The name is chosen due to the following common property of both transformations.

Theorem 4 The covariant transform (1) intertwines ρ and the left regular representation Λ on L(G,U):

W ρ(g) = Λ(g) W.

Here Λ is defined as usual by:

Λ(g): f(h) ↦ f(g⁻¹h). (2)

Proof. We have a calculation similar to wavelet transform []*Prop. 2.6. Take u=ρ(g) v and calculate its covariant transform:

Corollary 5 The image space W(V) is invariant under the left shifts on G.

Remark 6 A further generalisation of the covariant transform can be obtained if we relax the group structure. Consider, for example, a cancellative semigroup ℤ₊ of non-negative integers. It has a linear presentation on the space of polynomials in a variable t defined by the action m: tⁿ ↦ t^m+n on the monomials. Application of a linear functional l, e.g. defined by an integration over a measure on the real line, produces umbral calculus l(tⁿ)=c_n, which has a magic efficiency in many areas, notably in combinatorics [, ]. In this direction we also find fruitful to expand the notion of an intertwining operator to a token [].

15.2 Examples of Covariant Transform

In this Subsection we will provide several examples of covariant transforms. Some of them will be expanded in subsequent sections, however a detailed study of all aspects will not fit into the present work. We start from the classical example of the group-theoretical wavelet transform:

Example 1 Let V be a Hilbert space with an inner product ⟨ ·,· ⟩ and ρ be a unitary representation of a group G in the space V. Let F: V → ℂ be a functional v↦ ⟨ v,v₀ ⟩ defined by a vector v₀∈ V. The vector v₀ is oftenly called the mother wavelet in areas related to signal processing or the vacuum state in quantum framework.

Then the transformation (1) is the well-known expression for a wavelet transform []*(7.48) (or representation coefficients):

W: v↦ v(g) = ⟨ ρ(g⁻¹)v,v₀ ⟩ = ⟨ v,ρ(g)v₀ ⟩, v∈ V, g∈ G. (3)

The family of vectors v_g=ρ(g)v₀ is called wavelets or coherent states. In this case we obtain scalar valued functions on G, thus the fundamental rôle of this example is explained in Rem. 3.

This scheme is typically carried out for a square integrable representation ρ and v₀ being an admissible vector [, , , , ]. In this case the wavelet (covariant) transform is a map into the square integrable functions [] with respect to the left Haar measure. The map becomes an isometry if v₀ is properly scaled.

However square integrable representations and admissible vectors does not cover all interesting cases.

Example 2 Let G=Aff be the “ax+b” (or affine) group []*§ 8.2: the set of points (a,b), a∈ ℝ₊, b∈ ℝ in the upper half-plane with the group law:

(a, b) * (a′, b′) = (aa′, ab′+b) (4)

and left invariant measure a⁻² da db . Its isometric representation on V=L_p(ℝ) is given by the formula:

[ρp(g) f](x)= a

⎛
⎝

ax+b

⎞
⎠

, where g⁻¹=(a,b). (5)

We consider the operators F_±:L₂(ℝ) → ℂ defined by:

F_±(f)=

2π i

∫

ℝ

f(t) dt

x∓ i

. (6)

Then the covariant transform (1) is the Cauchy integral from L_p(ℝ) to the space of functions f(a,b) such that a^−1/pf(a,b) is in the Hardy space in the upper/lower half-plane H_p(ℝ_±²). Although the representation (7) is square integrable for p=2, the function 1/x± i used in (6) is not an admissible vacuum vector. Thus the complex analysis become decoupled from the traditional wavelet theory. As a result the application of wavelet theory shall relay on an extraneous mother wavelets [].

Many important objects in complex analysis are generated by inadmissible mother wavelets like (6). For example, if F:L₂(ℝ) → ℂ is defined by F: f ↦ F₊ f + F₋f then the covariant transform (1) reduces to the Poisson integral . If F:L₂(ℝ) → ℂ² is defined by F: f ↦( F₊ f, F₋f) then the covariant transform (1) represents a function f on the real line as a jump:

f(z)=f₊(z)−f₋(z), f_±(z)∈ H_p(ℝ_±²) (7)

between functions analytic in the upper and the lower half-planes. This makes a decomposition of L₂(ℝ) into irreducible components of the representation (7). Another interesting but non-admissible vector is the Gaussian e^−x2.

Example 3 For the group G=SL₂(ℝ) [] let us consider the unitary representation ρ on the space of square integrable function L₂(ℝ₊²) on the upper half-plane through the Möbius transformations (??):

ρ(g): f(z) ↦

(c z + d)²

⎛
⎜
⎜
⎝

a z+ b

c z +d

⎞
⎟
⎟
⎠

, g⁻¹=

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

. (8)

This is a representation from the discrete series and L₂(ⅅ) and irreducible invariant subspaces are parametrised by integers. Let F_k be the functional L₂(ℝ₊²)→ ℂ of pairing with the lowest/highest k-weight vector in the corresponding irreducible component (Bergman space) B_k(ℝ_±²), k≥ 2 of the discrete series []*Ch. VI. Then we can build an operator F from various F_k similarly to the previous Example. In particular, the jump representation (7) on the real line generalises to the representation of a square integrable function f on the upper half-plane as a sum

f(z)=

∑

a_k f_k(z), f_k∈B_n(ℝ_±²)

for prescribed coefficients a_k and analytic functions f_k in question from different irreducible subspaces.

Covariant transform is also meaningful for principal and complementary series of representations of the group SL₂(ℝ), which are not square integrable [].

Example 4 Let G=SU(2)× Aff be the Cartesian product of the groups SU(2) of unitary rotations of ℂ² and the ax+b group Aff. This group has a unitary linear representation on the space L₂(ℝ,ℂ²) of square-integrable (vector) ℂ²-valued functions by the formula:

ρ(g)

⎛
⎜
⎝

f₁(t)

f₂(t)

⎞
⎟
⎠

⎛
⎜
⎝

α f₁(at+b)+ β f₂(at+b)

γ f₁(at+b)+δ f₂(at+b)

⎞
⎟
⎠

where g= (

α	β
γ	δ

)× (a,b)∈SU(2)× Aff. It is obvious that the vector Hardy space, that is functions with both components being analytic, is invariant under such action of G.

As a fiducial operator F: L₂(ℝ,ℂ²) → ℂ we can take, cf. (6):

⎛
⎜
⎝

f₁(t)

f₂(t)

⎞
⎟
⎠

2π i

∫

ℝ

f₁(t) dt

x− i

. (9)

Thus the image of the associated covariant transform is a subspace of scalar valued bounded functions on G. In this way we can transform (without a loss of information) vector-valued problems, e.g. matrix Wiener–Hopf factorisation [], to scalar question of harmonic analysis on the group G.

Example 5 A straightforward generalisation of Ex. 1 is obtained if V is a Banach space and F: V → ℂ is an element of V^*. Then the covariant transform coincides with the construction of wavelets in Banach spaces [].

Example 6 The next stage of generalisation is achieved if V is a Banach space and F: V → ℂⁿ is a linear operator. Then the corresponding covariant transform is a map W: V → L(G,ℂⁿ). This is closely related to M.G. Krein’s works on directing functionals [], see also multiresolution wavelet analysis [], Clifford-valued Fock–Segal–Bargmann spaces [] and []*Thm. 7.3.1.

Example 7 Let F be a projector L_p(ℝ)→ L_p(ℝ) defined by the relation (Ff) (λ )=χ(λ)f(λ), where the hat denotes the Fourier transform and χ(λ) is the characteristic function of the set [−2,−1]∪[1,2]. Then the covariant transform L_p(ℝ)→ C(Aff, L_p(ℝ)) generated by the representation (7) of the affine group from F contains all information provided by the Littlewood–Paley operator []*§ 5.1.1.

Example 8 A step in a different direction is a consideration of non-linear operators. Take again the “ax+b” group and its representation (7). We define F to be a homogeneous but non-linear functional V→ ℝ₊:

F (f) =

∫

−1

⎪
⎪

f(x)

⎪
⎪

dx.

The covariant transform (1) becomes:

[W_p f](a,b) = F(ρp(a,b) f) =

∫

−1

⎪
⎪
⎪

⎛
⎝

ax+b

⎞
⎠

⎪
⎪
⎪

dx = a

b+a

∫

b−a

⎪
⎪

⎛
⎝

⎞
⎠

⎪
⎪

dx. (10)

Obviously M_f(b)=max_a[W_∞f](a,b) coincides with the Hardy maximal function , which contains important information on the original function f. From the Cor. 7 we deduce that the operator M: f↦ M_f intertwines ρp with itself ρpM=M ρp.

Of course, the full covariant transform (10) is even more detailed than M. For example, ||f||=max_b[W_∞f](1/2,b) is the shift invariant norm [].

Example 9 Let V=L_c(ℝ²) be the space of compactly supported bounded functions on the plane. We take F be the linear operator V→ ℂ of integration over the real line:

F: f(x,y)↦ F(f)=

∫

ℝ

f(x,0) dx.

Let G be the group of Euclidean motions of the plane represented by ρ on V by a change of variables. Then the wavelet transform F(ρ(g)f) is the Radon transform [].

15.3 Symbolic Calculi

There is a very important class of the covariant transforms which maps operators to functions. Among numerous sources we wish to single out works of Berezin [, ]. We start from the Berezin covariant symbol.

Example 1 Let a representation ρ of a group G act on a space X. Then there is an associated representation ρB of G on a space V=B(X,Y) of linear operators X→ Y defined by the identity [, ]:

(ρB(g) A)x=A(ρ(g⁻¹)x), x∈ X, g∈ G, A ∈ B(X,Y). (11)

Following the Remark 3 we take F to be a functional V→ℂ, for example F can be defined from a pair x∈ X, l∈ Y^* by the expression F: A↦ ⟨ Ax,l ⟩. Then the covariant transform is:

W: A ↦ Â(g)=F(ρB(g) A).

This is an example of covariant calculus [, ].

There are several variants of the last Example which are of a separate interest.

Example 2 A modification of the previous construction is obtained if we have two groups G₁ and G₂ represented by ρ1 and ρ2 on X and Y^* respectively. Then we have a covariant transform B(X,Y)→ L(G₁× G₂, ℂ) defined by the formula:

W: A ↦ Â(g₁,g₂)=⟨ Aρ1(g₁)x,ρ2(g₂)l ⟩.

This generalises the above Berezin covariant calculi [].

Example 3 Let us restrict the previous example to the case when X=Y is a Hilbert space, ρ1=ρ2=ρ and x=l with ||x||=1. Than the range of the covariant transform:

W: A ↦ Â(g)=⟨ Aρ(g)x,ρ(g)x ⟩

is a subset of the numerical range of the operator A. As a function on a group Â(g) provides a better description of A than the set of its values—numerical range.

Example 4 The group SU(1,1)≃ SL₂(ℝ) consists of 2× 2 matrices of the form (

α	β
β	α

) with the unit determinant []*§ IX.1. Let T be an operator with the spectral radius less than 1. Then the associated Möbius transformation

g: T ↦ g· T =

α T+β I

βT+αI

, where g=

⎛
⎜
⎝

α	β
β	α

⎞
⎟
⎠

∈ SL₂(ℝ), (12)

produces a well-defined operator with the spectral radius less than 1 as well. Thus we have a representation of SU(1,1).

Let us introduce the defect operators D_T=(I−T^*T)^1/2 and D_T^*=(I−TT^*)^1/2. For the fiducial operator F=D_T^* the covariant transform is, cf. []*§ VI.1, (1.2):

[W T](g)=F(g· T)=−eⁱ^φ Θ_T(z) D_T, for g=

⎛
⎜
⎝

e^iφ/2	0
0	e^−iφ/2

⎞
⎟
⎠

⎛
⎜
⎝

1	−z
−z	1

⎞
⎟
⎠

where the characteristic function Θ_T(z) []*§ VI.1, (1.1) is:

Θ_T(z) = −T+D_T^* (I−zT^*)⁻¹ z D_T.

Thus we approached the functional model of operators from the covariant transform. In accordance with Remark 3 the model is most fruitful for the case of operator F=D_T^* being one-dimensional.

The intertwining property in the previous examples was obtained as a consequence of the general Prop. 6 about the covariant transform. However it may be worth to select it as a separate definition:

Definition 5 A covariant calculus , also known as symbolic calculus , is a map from operators to functions, which intertwines two representations of the same group in the respective spaces.

There is a dual class of covariant transforms acting in the opposite direction: from functions to operators. The prominent examples are the Berezin contravariant symbol [, ] and symbols of a pseudodifferential operators (PDO) [, ].

Example 6 The classical Riesz–Dunford functional calculus []*§ VII.3 []*§ IV.2 maps analytical functions on the unit disk to the linear operators, it is defined through the Cauchy-type formula with the resolvent. The calculus is an intertwining operator [] between the Möbius transformations of the unit disk, cf. (22), and the actions (12) on operators from the Example 4. This topic will be developed in Subsection 17.2.

In line with the Defn. 5 we can directly define the corresponding calculus through the intertwining property [, ]:

Definition 7 A contravariant calculus , also know as functional calculus , is a map from functions to operators, which intertwines two representations of the same group in the respective spaces.

The duality between co- and contravariant calculi is the particular case of the duality between covariant transform and the inverse covariant transform defined in the next Subsection. In many cases a proper choice of spaces makes covariant and/or contravariant calculus a bijection between functions and operators. Subsequently only one form of calculus, either co- or contravariant, is defined explicitly, although both of them are there in fact.

15.4 Inverse Covariant Transform

An object invariant under the left action Λ (2) is called left invariant. For example, let L and L′ be two left invariant spaces of functions on G. We say that a pairing ⟨ ·,· ⟩: L× L′ → ℂ is left invariant if

Remark 1

We do not require the pairing to be linear in general.
If the pairing is invariant on space L× L′ it is not necessarily invariant (or even defined) on the whole C(G)× C(G).
In a more general setting we shall study an invariant pairing on a homogeneous spaces instead of the group. However due to length constraints we cannot consider it here beyond the Example 4.
An invariant pairing on G can be obtained from an invariant functional l by the formula ⟨ f₁,f₂ ⟩=l(f₁f₂).

For a representation ρ of G in V and v₀∈ V we fix a function w(g)=ρ(g)v₀. We assume that the pairing can be extended in its second component to this V-valued functions, say, in the weak sense.

Definition 2 Let ⟨ ·,· ⟩ be a left invariant pairing on L× L′ as above, let ρ be a representation of G in a space V, we define the function w(g)=ρ(g)v₀ for v₀∈ V. The inverse covariant transform M is a map L → V defined by the pairing:

M: f ↦ ⟨ f,w ⟩, where f∈ L. (14)

Example 3 Let G be a group with a unitary square integrable representation ρ. An invariant pairing of two square integrable functions is obviously done by the integration over the Haar measure:

⟨ f₁,f₂ ⟩=

∫

f₁(g)f₂(g) dg.

For an admissible vector v₀ [] []*Chap. 8 the inverse covariant transform is known in this setup as a reconstruction formula .

Example 4 Let ρ be a square integrable representation of G modulo a subgroup H⊂ G and let X=G/H be the corresponding homogeneous space with a quasi-invariant measure dx. Then integration over dx with an appropriate weight produces an invariant pairing. The inverse covariant transform is a more general version []*(7.52) of the reconstruction formula mentioned in the previous example.

Let ρ be not a square integrable representation (even modulo a subgroup) or let v₀ be inadmissible vector of a square integrable representation ρ. An invariant pairing in this case is not associated with an integration over any non singular invariant measure on G. In this case we have a Hardy pairing. The following example explains the name.

Example 5 Let G be the “ax+b” group and its representation ρ (7) from Ex. 2. An invariant pairing on G, which is not generated by the Haar measure a⁻²da db, is:

⟨ f₁,f₂ ⟩=

lim

a→ 0

∞

∫

−∞

f₁(a,b) f₂(a,b) db. (15)

For this pairing we can consider functions 1/2π i (x+i) or e^−x2, which are not admissible vectors in the sense of square integrable representations. Then the inverse covariant transform provides an integral resolutions of the identity.

Similar pairings can be defined for other semi-direct products of two groups. We can also extend a Hardy pairing to a group, which has a subgroup with such a pairing.

Example 6 Let G be the group SL₂(ℝ) from the Ex. 3. Then the “ax+b” group is a subgroup of SL₂(ℝ), moreover we can parametrise SL₂(ℝ) by triples (a,b,θ), θ∈(−π,π] with the respective Haar measure []*III.1(3). Then the Hardy pairing

⟨ f₁,f₂ ⟩=

lim

a→ 0

∞

∫

−∞

f₁(a,b,θ) f₂(a,b,θ) db dθ. (16)

is invariant on SL₂(ℝ) as well. The corresponding inverse covariant transform provides even a finer resolution of the identity which is invariant under conformal mappings of the Lobachevsky half-plane.

Lecture 16 Analytic Functions

We saw in the first section that an inspiring geometry of cycles can be recovered from the properties of SL₂(ℝ). In this section we consider a realisation of the function theory within Erlangen approach [, , , ]. The covariant transform will be our principal tool in this construction.

16.1 Induced Covariant Transform

The choice of a mother wavelet or fiducial operator F from Section 15.1 can significantly influence the behaviour of the covariant transform. Let G be a group and H be its closed subgroup with the corresponding homogeneous space X=G/H. Let ρ be a representation of G by operators on a space V, we denote by ρH the restriction of ρ to the subgroup H.

Definition 1 Let χ be a representation of the subgroup H in a space U and F: V→ U be an intertwining operator between χ and the representation ρH:

F(ρ(h) v)=F(v)χ(h), for all h∈ H, v∈ V. (1)

Then the covariant transform (1) generated by F is called the induced covariant transform

Example 2 Consider the traditional wavelet transform as outlined in Ex. 1. Chose a vacuum vector v₀ to be a joint eigenvector for all operators ρ(h), h∈ H, that is ρ(h) v₀=χ(h) v₀, where χ(h) is a complex number depending of h. Then χ is obviously a character of H.

The image of wavelet transform (3) with such a mother wavelet will have a property:

v(gh) = ⟨ v,ρ(gh)v₀ ⟩ = ⟨ v,ρ(g)χ(h)v₀ ⟩ =χ(h)v(g).

Thus the wavelet transform is uniquely defined by cosets on the homogeneous space G/H. In this case we previously spoke about the reduced wavelet transform []. A representation ρ0 is called square integrable mod H if the induced wavelet transform [Wf₀](w) of the vacuum vector f₀(x) is square integrable on X.

Thus it is enough to know the value of the covariant transform only at a single element in every coset G/H in order to reconstruct it for the entire group G by the representation χ. Since coherent states (wavelets) are now parametrised by points homogeneous space G/H they are referred sometimes as coherent states which are not connected to a group [], however this is true only in a very narrow sense as explained above.

Example 3 To make it more specific we can consider the representation of SL₂(ℝ) defined on L₂(ℝ) by the formula, cf. (8):

ρ(g): f(z) ↦

(c x + d)

⎛
⎜
⎜
⎝

a x+ b

c x +d

⎞
⎟
⎟
⎠

, g⁻¹=

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

Let K⊂SL₂(ℝ) be the compact subgroup of matrices h_t= (

cost	sint
−sint	cost

). Then for the fiducial operator F_± (6) we have F_±∘ρ(h_t)=e^∓ⁱ^tF_±. Thus we can consider the covariant transform only for points in the homogeneous space SL₂(ℝ)/K, moreover this set can be naturally identified with the ax+b group. Thus we do not obtain any advantage of extending the group in Ex. 2 from ax+b to SL₂(ℝ) if we will be still using the fiducial operator F_± (6).

Functions on the group G, which have the property v(gh)=v(g)χ(h) (2), provide a space for the representation of G induced by the representation χ of the subgroup H. This explains the choice of the name for induced covariant transform.

Remark 4 Induced covariant transform uses the fiducial operator F which passes through the action of the subgroup H. This reduces information which we obtained from this transform in some cases.

There is also a simple connection between a covariant transform and right shifts:

Proposition 5 Let G be a Lie group and ρ be a representation of G in a space V. Let [Wf](g)=F(ρ(g⁻¹)f) be a covariant transform defined by the fiducial operator F: V → U. Then the right shift [Wf](gg′) by g′ is the covariant transform [W′f](g)=F′(ρ(g⁻¹)f)] defined by the fiducial operator F′=F∘ρ(g⁻¹).

In other words the covariant transform intertwines right shifts on the group G with the associated action ρB (11) on fiducial operators.

Although the above result is obvious, its infinitesimal version has interesting consequences.

Corollary 6 ([]) Let G be a Lie group with a Lie algebra g and ρ be a smooth representation of G. We denote by dρB the derived representation of the associated representation ρB (11) on fiducial operators.

Let a fiducial operator F be a null-solution, i.e. A F=0, for the operator A=∑_J a_j dρ[X_j]B, where X_j∈g and a_j are constants. Then the covariant transform [W f](g)=F(ρ(g⁻¹)f) for any f satisfies:

D F(g)= 0, where D=

∑

ā_jL^X_j.

Here L^X_j are the left invariant fields (Lie derivatives) on G corresponding to X_j.

Example 7 Consider the representation ρ (7) of the ax+b group with the p=1. Let A and N be the basis of the corresponding Lie algebra generating one-parameter subgroups (e^t,0) and (0,t). Then the derived representations are:

[dρ[A] f](x)= f(x)+xf′(x), [dρ[N]f](x)=f′(x).

The corresponding left invariant vector fields on ax+b group are:

L^A =a ∂_a, L^N=a∂_b.

The mother wavelet 1/x+i is a null solution of the operator dρ[A] +i dρ[N]=I+(x+i)d/dx. Therefore the covariant transform with the fiducial operator F₊ (6) will consist with the null solutions to the operator L^A−iL^N=−i a(∂_b+i∂_a), that is in the essence the Cauchy-Riemann operator in the upper half-plane.

There is a statement which extends the previous Corollary from differential operators to integro-differential ones. We will formulate it for the wavelets setting.

Corollary 8 Let G be a group and ρ be a unitary representation of G, which can be extended to a vector space V of functions or distributions on G. Let a mother wavelet w∈ V′ satisfy the equation

∫

a(g) ρ(g) w dg=0,

for a fixed distribution a(g) ∈ V and a (not necessarily invariant) measure dg. Then any wavelet transform F(g)= W f(g)=⟨ f,ρ(g)w₀ ⟩ obeys the condition:

DF=0, where D=

∫

ā(g) R(g) dg,

with R being the right regular representation of G.

Clearly, the Corollary 6 is a particular case of the Corollary 8 with a distribution a, which is a combination of derivatives of Dirac’s delta functions. The last Corollary will be illustrated at the end of Section 17.2.

Remark 9 We note that Corollaries 6 and 8 are true whenever we have an intertwining property between ρ with the right regular representation of G.

16.2 Induced Wavelet Transform and Cauchy Integral

We again use the general scheme from Subsection 14.2. The ax+b group is isomorphic to a subgroups of SL₂(ℝ) consisting of the lower-triangular matrices:

The corresponding homogeneous space X=SL₂(ℝ)/F is one-dimensional and can be parametrised by a real number. The natural projection p:SL₂(ℝ)→ ℝ and its left inverse s: ℝ→ SL₂(ℝ) can be defined as follows:

Therefore the action of SL₂(ℝ) on the real line is exactly the Möbius map (??):

To build an induced representation we need a character of the affine group. A generic character of F is a power of its diagonal element:

The only freedom remaining by the scheme is in a choice of a value of number κ and the corresponding functional space where our representation acts. At this point we have a wider choice of κ than it is usually assumed: it can belong to different hypercomplex systems.

One of the important properties which would be nice to have is the unitarity of the representation (5) with respect to the standard inner product:

A change of variables x=au+b/cu+d in the integral suggests the following property is necessary and sufficient for that:

A mother wavelet for an induced wavelet transform shall be an eigenvector for the action of a subgroup H′ of SL₂(ℝ), see (1). Let us consider the most common case of H′=K and take the infinitesimal condition with the derived representation: dρ[Z]nw₀ =λ w₀, since Z (12) is the generator of the subgroup K. In other word the restriction of w₀ to a K-orbit should be given by e^{λ t} in the exponential coordinate t along the K-orbit. However we usually need its expression in other “more natural” coordinates. For example [], an eigenvector of the derived representation of dρ[Z]n should satisfy the differential equation in the ordinary parameter x∈ℝ:

The equation does not have singular points, the general solution is globally defined (up to a constant factor) by:

To avoid multivalent functions we need 2π-periodicity along the exponential coordinate on K. This implies that the parameter m=−iλ is an integer. Therefore the solution becomes:

The corresponding wavelets resemble the Cauchy kernel normalised to the invariant metric in the Lobachevsky half-plane:

Therefore the wavelet transform (3) from function on the real line to functions on the upper half-plane is:

According to the general theory this wavelet transform intertwines representations ρ[F]κ (5) on the real line (induced by the character a^κ of the subgroup F) and ρ[K]m (8) on the upper half-plane (induced by the character e^{i m t} of the subgroup K).

16.3 The Cauchy-Riemann (Dirac) and Laplace Operators

Ladder operators L^±=±i A +B act by raising/lowering indexes of the K-eigenfunctions w_m,κ (8), see Subsection 14.3. More explicitly []:

There are two possibilities here: m±κ is zero for some m or not. In the first case the chain (11) of eigenfunction w_m,κ terminates on one side under the transitive action (15) of the ladder operators; otherwise the chain is infinite in both directions. That is, the values m=∓κ and only those correspond to the maximal (minimal) weight function w_{∓κ ,κ}(x)=1/(x±i)^κ ∈ L₂(ℝ), which are annihilated by L^±:

By the Cor. 6 for the mother wavelets w_{∓κ ,κ}, which are annihilated by (12), the images of the respective wavelet transforms are null solutions to the left-invariant differential operator D_±=L^L±:

This is a conformal version of the Cauchy–Riemann equation . The second order conformal Laplace-type operators Δ₊=L^L−L^L+ and Δ₋=L^L+L^L− are:

For the mother wavelets w_m,κ in (12) such that m=∓κ the unitarity condition κ+κ=2, see (6), together with m∈ℤ implies κ=∓ m=1. In such a case the wavelet transforms (10) are:

for w_−1,1 and w_1,1 respectively. The first one is the Cauchy integral formula up to the factor 2πi √ℑ z. Clearly, one integral is the complex conjugation of another. Moreover, the minimal/maximal weight cases can be intertwined by the following automorphism of the Lie algebra sl₂:

As explained before f^±(w) are null solutions to the operators D_± (13) and Δ_± (14). These transformations intertwine unitary equivalent representations on the real line and on the upper half-plane, thus they can be made unitary for proper spaces. This is the source of two faces of the Hardy spaces: they can be defined either as square-integrable on the real line with an analytic extension to the half-plane, or analytic on the half-plane with square-integrability on an infinitesimal displacement of the real line.

For the third possibility, m±κ≠ 0, there is no an operator spanned by the derived representation of the Lie algebra sl₂ which kills the mother wavelet w_m,κ. However the remarkable Casimir operator C=Z²−2(L⁻L⁺+L⁺L⁻), which spans the centre of the universal enveloping algebra of sl₂ []*§ 8.1 []*§ X.1, produces a second order operator which does the job. Indeed from the identities (11) we get:

Thus we get dρ[C]κ w_m,2=0 for κ=2 or 0. The mother wavelet w_0,2 turns to be the Poisson kernel []*Ex. 1.2.17. The associated wavelet transform

consists of null solutions of the left-invariant second-order Laplacian , image of the Casimir operator, cf. (14):

Another integral formula producing solutions to this equation delivered by the mother wavelet w_m,0 with the value κ=0 in (16):

Furthermore, we can introduce higher order differential operators. The functions w_{∓ 2m+1,1} are annihilated by n-th power of operator dρ[L^±]κ with 1≤ m≤ n. By the Cor. 6 the the image of wavelet transform (10) from a mother wavelet ∑₁ⁿ a_m w_{∓ 2m,1} will consist of null-solutions of the n-th power D_±ⁿ of the conformal Cauchy–Riemann operator (13). They are a conformal flavour of polyanalytic functions [].

We can similarly look for mother wavelets which are eigenvectors for other types of one dimensional subgroups. Our consideration of subgroup K is simplified by several facts:

For both subgroups and N′ this will not be true. The further consideration will be given in [].

16.4 The Taylor Expansion

Consider an induced wavelet transform generated by a Lie group G, its representation ρ and a mother wavelet w which is an eigenvector of a one-dimensional subgroup H′⊂ G. Then by Prop. 5 the wavelet transform intertwines ρ with a representation ρ[H′] induced by a character of H′.

If the mother wavelet is itself in the domain of the induced wavelet transform then the chain (15) of H′-eigenvectors w_m will be mapped to the similar chain of their images ŵ_m. The corresponding derived induced representation dρ[H′] produces ladder operators with the transitive action of the ladder operators on the chain of ŵ_m. Then the vector space of “formal power series”:

Coming back to the case of the group G=SL₂(ℝ) and subgroup H′=K. Images ŵ_m,1 of the eigenfunctions (9) under the Cauchy integral transform (15) are:

They are eigenfunctions of the derived representation on the upper half-plane and the action of ladder operators is given by the same expressions (11). In particular, the sl₂-module generated by ŵ_1,1 will be one-sided since this vector is annihilated by the lowering operator. Since the Cauchy integral produces an unitary intertwining operator between two representations we get the following variant of Taylor series:

For two other types of subgroups, representations and mother wavelets this scheme shall be suitably adapted and detailed study will be presented elsewhere [].

16.5 Wavelet Transform in the Unit Disk and Other Domains

We can similarly construct an analytic function theories in unit disks, including parabolic and hyperbolic ones []. This can be done simply by an application of the Cayley transform to the function theories in the upper half-plane. Alternatively we can apply the full procedure for properly chosen groups and subgroups. We will briefly outline such a possibility here, see also [].

Elements of SL₂(ℝ) could be also represented by 2× 2-matrices with complex entries such that, cf. Example 6:

This realisations of SL₂(ℝ) (or rather SU(2,ℂ)) is more suitable for function theory in the unit disk. It is obtained from the form, which we used before for the upper half-plane, by means of the Cayley transform []*§ 8.1.

We may identify the unit disk ⅅ with the homogeneous space SL₂(ℝ)/T for the unit circle T through the important decomposition SL₂(ℝ)∼ ⅅ×T with K=T—the compact subgroup of SL₂(ℝ):

Each element g∈SL₂(ℝ) acts by the linear-fractional transformation (the Möbius map) on ⅅ and T H₂(T) as follows:

In the decomposition (20) the first matrix on the right hand side acts by transformation (21) as an orthogonal rotation of T or ⅅ; and the second one—by transitive family of maps of the unit disk onto itself.

The representation induced by a complex-valued character χ_k(z)=z^−k of T according to the Section 14.2 is:

The representation ρ1 is unitary on square-integrable functions and irreducible on the Hardy space on the unit circle.

We choose [, ] K-invariant function v₀(z)≡ 1 to be a vacuum vector. Thus the associated coherent states

are completely determined by the point on the unit disk u=βα⁻¹. The family of coherent states considered as a function of both u and z is obviously the Cauchy kernel []. The wavelet transform [, ] W:L₂(T)→ H₂(ⅅ): f(z)↦ Wf(g)=⟨ f,v_g ⟩ is the Cauchy integral:

This approach can be extended to arbitrary connected simply-connected domain. Indeed, it is known that Möbius maps is the whole group of biholomorphic automorphisms of the unit disk or upper half-plane. Thus we can state the following corollary from the Riemann mapping theorem:

Corollary 1 The group of biholomorphic automorphisms of a connected simply connected domain with at least two points on its boundary is isomorphic to SL₂(ℝ).

If a domain is non-simply connected , then the group of its biholomorphic mapping can be trivial [, ]. However we may look for a rich group acting on function spaces rather than on geometric sets. Let a connected non-simply connected domain D be bounded by a finite collection of non-intersecting contours Γ_i, i=1,…,n. For each Γ_i consider the isomorphic image G_i of the SL₂(ℝ) group which is defined by the Corollary 1. Then define the group G=G₁× G₂× … × G_n and its action on L₂(∂ D)= L₂(Γ₁)⊕ L₂(Γ₂)⊕ … ⊕ L₂(Γ_n) through the Moebius action of G_i on L₂(Γ_i).

Example 2 Consider an annulus defined by r<| z |<R. It is bounded by two circles: Γ₁={z: | z |=r} and Γ₂={z: | z |=R}. For Γ₁ the Möbius action of SL₂(ℝ) is

⎛
⎜
⎝

α	β
β	α

⎞
⎟
⎠

: z↦

α z +β/r

β z/r + α

, where

⎪
⎪

²−

⎪
⎪

²=1,

with the respective action on Γ₂. Those action can be linearised in the spaces L₂(Γ₁) and L₂(Γ₂). If we consider a subrepresentation reduced to analytic function on the annulus, then one copy of SL₂(ℝ) will act on the part of functions analytic outside of Γ₁ and another copy—on the part of functions analytic inside of Γ₂.

Thus all classical objects of complex analysis (the Cauchy-Riemann equation, the Taylor series, the Bergman space , etc.) for a rather generic domain D can be also obtained from suitable representations similarly to the case of the upper half-plane [, ].

Lecture 17 Covariant and Contravariant Calculi

United in the trinity functional calculus , spectrum, and spectral mapping theorem play the exceptional rôle in functional analysis and could not be substituted by anything else.

17.1 Functional Calculus as an Algebraic Homomorphism

Many traditional definitions of functional calculus are covered by the following rigid template based on the algebra homomorphism property:

Definition 1 An functional calculus for an element a∈A is a continuous linear mapping Φ: A→ A such that

Φ is a unital algebra homomorphism
Φ(f · g)=Φ(f) · Φ (g).
There is an initialisation condition: Φ[v₀]=a for for a fixed function v₀, e.g. v₀(z)=z.

The most typical definition of the spectrum is seemingly independent and uses the important notion of resolvent:

Definition 2 A resolvent of element a∈A is the function R(λ)=(a−λ e)⁻¹, which is the image under Φ of the Cauchy kernel (z−λ)⁻¹.

A spectrum of a∈A is the set a of singular points of its resolvent R(λ).

Then the following important theorem links spectrum and functional calculus together.

Theorem 3 (Spectral Mapping) For a function f suitable for the functional calculus:

f( a)= f(a). (1)

However the power of the classic spectral theory rapidly decreases if we move beyond the study of one normal operator (e.g. for quasinilpotent ones) and is virtually nil if we consider several non-commuting ones. Sometimes these severe limitations are seen to be irresistible and alternative constructions, i.e. model theory cf. Example 4 and [], were developed.

Yet the spectral theory can be revived from a fresh start. While three components—functional calculus, spectrum, and spectral mapping theorem—are highly interdependent in various ways we will nevertheless arrange them as follows:

Thus the entire scheme depends from the notion of the functional calculus and our ability to escape limitations of Definition 1. The first known to the present author definition of functional calculus not linked to algebra homomorphism property was the Weyl functional calculus defined by an integral formula []. Then its intertwining property with affine transformations of Euclidean space was proved as a theorem. However it seems to be the only “non-homomorphism” calculus for decades.

The different approach to whole range of calculi was given in [] and developed in [, , , ] in terms of intertwining operators for group representations. It was initially targeted for several non-commuting operators because no non-trivial algebra homomorphism is possible with a commutative algebra of function in this case. However it emerged later that the new definition is a useful replacement for classical one across all range of problems.

In the following Subsections we will support the last claim by consideration of the simple known problem: characterisation a n × n matrix up to similarity. Even that “freshman” question could be only sorted out by the classical spectral theory for a small set of diagonalisable matrices. Our solution in terms of new spectrum will be full and thus unavoidably coincides with one given by the Jordan normal form of matrices. Other more difficult questions are the subject of ongoing research.

17.2 Intertwining Group Actions on Functions and Operators

Any functional calculus uses properties of functions to model properties of operators. Thus changing our viewpoint on functions, as was done in Section 16, we could get another approach to operators. The two main possibilities are encoded in Definitions 5 and 7: we can assign a certain function to the given operator or wise verse. Here we consider the second possibility and treat the first in the Subsection 17.5.

The representation ρ1 (22) is unitary irreducible when acts on the Hardy space H₂. Consequently we have one more reason to abolish the template definition 1: H₂ is not an algebra. Instead we replace the homomorphism property by a symmetric covariance:

Definition 1 ([]) An contravariant analytic calculus for an element a∈A and an A-module M is a continuous linear mapping Φ:A(ⅅ)→ A(ⅅ,M) such that

Φ is an intertwining operator
Φρ₁=ρ_aΦ

between two representations of the SL₂(ℝ) group ρ₁ (22) and ρ_a defined below in (4).
There is an initialisation condition: Φ[v₀]=m for v₀(z)≡ 1 and m∈ M, where M is a left A-module.

Note that our functional calculus released from the homomorphism condition can take value in any left A-module M, which however could be A itself if suitable. This add much flexibility to our construction.

The earliest functional calculus, which is not an algebraic homomorphism, was the Weyl functional calculus and was defined just by an integral formula as an operator valued distribution []. In that paper (joint) spectrum was defined as support of the Weyl calculus, i.e. as the set of point where this operator valued distribution does not vanish. We also define the spectrum as a support of functional calculus, but due to our Definition 1 it will means the set of non-vanishing intertwining operators with primary subrepresentations.

Definition 2 A corresponding spectrum of a∈A is the support of the functional calculus Φ, i.e. the collection of intertwining operators of ρ_a with primary representations []*§ 8.3.

More variations of contravariant functional calculi are obtained from other groups and their representations [, , , , ].

A simple but important observation is that the Möbius transformations (??) can be easily extended to any Banach algebra. Let A be a Banach algebra with the unit e, an element a∈A with ||a||<1 be fixed, then

is a well defined SL₂(ℝ) action on a subset A={g· a ∣ g∈ SL₂(ℝ)}⊂A, i.e. A is a SL₂(ℝ)-homogeneous space. Let us define the resolvent function R(g,a):A→ A:

The last identity is well known in representation theory []*§ 13.2(10) and is a key ingredient of induced representations. Thus we can again linearise (2), cf. (22), in the space of continuous functions C(A,M) with values in a left A-module M, e.g. M=A:

For any m∈ M we can define a K-invariant vacuum vector as v_m(g⁻¹· a)=m⊗ v₀(g⁻¹· a) ∈ C(A,M). It generates the associated with v_m family of coherent states v_m(u,a)=(ue−a)⁻¹m, where u∈ⅅ.

The wavelet transform defined by the same common formula based on coherent states (cf. (23)):

is a version of Cauchy integral , which maps L₂(A) to C(SL₂(ℝ),M). It is closely related (but not identical!) to the Riesz-Dunford functional calculus: the traditional functional calculus is given by the case:

The both conditions—the intertwining property and initial value—required by Definition 1 easily follows from our construction. Finally, we wish to provide an example of application of the Corollary 8.

Example 3 Let a be an operator and φ be a function which annihilates it, i.e. φ(a)=0. For example, if a is a matrix φ can be its minimal polynomial. From the integral representation of the contravariant calculus on G=SL₂(ℝ) we can rewrite the annihilation property like this:

∫

φ(g) R(g,a) dg=0.

Then the vector-valued function [W_m f](g) defined by (5) shall satisfy to the following condition:

∫

φ(g′) [W_m f] (gg′) dg′=0

due to the Corollary 8.

17.3 Jet Bundles and Prolongations

Spectrum was defined in 2 as the support of our functional calculus. To elaborate its meaning we need the notion of a prolongation of representations introduced by S. Lie, see [, ] for a detailed exposition.

Definition 1 []*Chap. 4 Two holomorphic functions have nth order contact in a point if their value and their first n derivatives agree at that point, in other words their Taylor expansions are the same in first n+1 terms.

A point (z,u⁽ⁿ⁾)=(z,u,u₁,…,u_n) of the jet space Jⁿ∼ⅅ×ℂⁿ is the equivalence class of holomorphic functions having nth contact at the point z with the polynomial:

p_n(w)=u_n

(w−z)ⁿ

+⋯+u₁

(w−z)

+u. (6)

For a fixed n each holomorphic function f:ⅅ→ℂ has nth prolongation (or n-jet) j_nf: ⅅ → ℂⁿ⁺¹:

The graph amma_f⁽ⁿ⁾ of j_nf is a submanifold of Jⁿ which is section of the jet bundle over ⅅ with a fibre ℂⁿ⁺¹. We also introduce a notation J_n for the map J_n:f↦amma_f⁽ⁿ⁾ of a holomorphic f to the graph amma_f⁽ⁿ⁾ of its n-jet j_nf(z) (7).

One can prolong any map of functions ψ: f(z)↦ [ψ f](z) to a map ψ⁽ⁿ⁾ of n-jets by the formula

For example such a prolongation ρ₁⁽ⁿ⁾ of the representation ρ₁ of the group SL₂(ℝ) in H₂(ⅅ) (as any other representation of a Lie group []) will be again a representation of SL₂(ℝ). Equivalently we can say that J_n intertwines ρ₁ and ρ₁⁽ⁿ⁾:

Of course, the representation ρ₁⁽ⁿ⁾ is not irreducible: any jet subspace J^k, 0≤ k ≤ n is ρ₁⁽ⁿ⁾-invariant subspace of Jⁿ. However the representations ρ₁⁽ⁿ⁾ are primary []*§ 8.3 in the sense that they are not sums of two subrepresentations.

The following statement explains why jet spaces appeared in our study of functional calculus.

Proposition 2 Let matrix a be a Jordan block of a length k with the eigenvalue λ=0, and m be its root vector of order k, i.e. a^k−1m≠ a^k m =0. Then the restriction of ρ_a on the subspace generated by v_m is equivalent to the representation ρ₁^k.

17.4 Spectrum and Spectral Mapping Theorem

Now we are prepared to describe a spectrum of a matrix. Since the functional calculus is an intertwining operator its support is a decomposition into intertwining operators with primary representations (we could not expect generally that these primary subrepresentations are irreducible).

Recall the transitive on ⅅ group of inner automorphisms of SL₂(ℝ), which can send any λ∈ⅅ to 0 and are actually parametrised by such a λ. This group extends Proposition 2 to the complete characterisation of ρ_a for matrices.

Proposition 1 Representation ρ_a is equivalent to a direct sum of the prolongations ρ₁^(k) of ρ₁ in the kth jet space J^k intertwined with inner automorphisms. Consequently the spectrum of a (defined via the functional calculus Φ=W_m) labelled exactly by n pairs of numbers (λ_i,k_i), λ_i∈ⅅ, k_i∈ℤ₊, 1≤ i ≤ n some of whom could coincide.

Obviously this spectral theory is a fancy restatement of the Jordan normal form of matrices.

Example 2 Let J_k(λ) denote the Jordan block of the length k for the eigenvalue λ. In Fig. 17.1 there are two pictures of the spectrum for the matrix

a=J₃

⎛
⎝

λ₁

⎞
⎠

⊕ J₄

⎛
⎝

λ₂

⎞
⎠

⊕ J₁

⎛
⎝

λ₃

⎞
⎠

⊕ J₂

⎛
⎝

λ₄

⎞
⎠

where

λ₁=

e^iπ/4, λ₂=

e^i5π/6, λ₃=

e^−i3π/4, λ₄=

e^−iπ/3.

Part (a) represents the conventional two-dimensional image of the spectrum, i.e. eigenvalues of a, and (b) describes spectrum a arising from the wavelet construction. The first image did not allow to distinguish a from many other essentially different matrices, e.g. the diagonal matrix

⎛
⎝

λ₁,λ₂,λ₃,λ₄

⎞
⎠

which even have a different dimensionality. At the same time Fig. 17.1(b) completely characterise a up to a similarity. Note that each point of a in Fig. 17.1(b) corresponds to a particular root vector, which spans a primary subrepresentation.

As was mentioned in the beginning of this section a resonable spectrum should be linked to the corresponding functional calculus by an appropriate spectral mapping theorem. The new version of spectrum is based on prolongation of ρ₁ into jet spaces (see Section 17.3). Naturally a correct version of spectral mapping theorem should also operate in jet spaces.

Let φ: ⅅ → ⅅ be a holomorphic map, let us define its action on functions [φ_* f](z)=f(φ(z)). According to the general formula (8) we can define the prolongation φ_*⁽ⁿ⁾ onto the jet space Jⁿ. Its associated action ρ₁^k φ_*⁽ⁿ⁾=φ_*⁽ⁿ⁾ρ₁ⁿ on the pairs (λ,k) is given by the formula:

where deg_λφ denotes the degree of zero of the function φ(z)−φ(λ) at the point z=λ and [x] denotes the integer part of x.

Theorem 3 (Spectral mapping) Let φ be a holomorphic mapping φ: ⅅ → ⅅ and its prolonged action φ_*⁽ⁿ⁾ defined by (9), then

φ(a) = φ_*⁽ⁿ⁾ a.

The explicit expression of (9) for φ_*⁽ⁿ⁾, which involves derivatives of φ upto nth order, is known, see for example []*Thm. 6.2.25, but was not recognised before as form of spectral mapping.

Example 4 Let us continue with Example 2. Let φ map all four eigenvalues λ₁, …, λ₄ of the matrix a into themselves. Then Fig. 17.1(a) will represent the classical spectrum of φ(a) as well as a.

However Fig. 17.1(c) shows mapping of the new spectrum for the case φ has orders of zeros at these points as follows: the order 1 at λ₁, exactly the order 3 at λ₂, an order at least 2 at λ₃, and finally any order at λ₄.

17.5 Functional Model and Spectral Distance

If all eigenvalues λ_i of a (i.e. all roots of µ(z) belong to the unit disk we can consider the respective Blaschke product

such that its numerator coincides with the minimal polynomial µ(z). Moreover, for an unimodular z we have B_a(z)=µ_a(z)µ_a⁻¹(z)z^−m, where m=m₁+… +m_n. We also have the following covariance property:

Proposition 1 The above correspondence a↦ B_a intertwines the SL₂(ℝ) action (2) on the matrices with the action (22) with k=0 on functions.

The result follows from the observation that every elementary product z−λ _i/1−λ_iz is the Moebius transformation of z with the matrix (

). Thus the correspondence a↦ B_a(z) is a covariant (symbolic) calculus in the sense of the Defn. 5. See also the Example 4.

The Jordan normal form of a matrix provides a description, which is equivalent to its contravariant spectrum . From various viewpoints, e.g. numerical approximations, it is worth to consider its stability under a perturbation. It is easy to see, that an arbitrarily small disturbance breaks the Jordan structure of a matrix. However, the result of random small perturbation will not be random, its nature is described by the following remarkable theorem:

Theorem 2 (Lidskii [], see also []) Let J_n be a Jordan block of a length n>1 with zero eigenvalues and K be an arbitrary matrix. Then eigenvalues of the perturbed matrix J_n+εⁿ K admit the expansion

λ_j= ε ξ^1/n +o(ε), j=1,…,n,

where ξ^1/n represents all n-th complex roots of certain ξ∈ℂ.

The left picture in Fig. 17.2 presents a perturbation of a Jordan block J₁₀₀ by a random matrix. Perturbed eigenvalues are close to vertices of a right polygon with 100 vertices. Those regular arrangements occur despite of the fact that eigenvalues of the matrix K are dispersed through the unit disk (the right picture in Fig. 17.2). In a sense it is rather the Jordan block regularises eigenvalues of K than K perturbs the eigenvalue of the Jordan block.

Although the Jordan structure itself is extremely fragile, it still can be easily guessed from a perturbed eigenvalues. Thus there exists a certain characterisation of matrices which is stable under small perturbations. We will describe a sense, in which the covariant spectrum of the matrix J_n+εⁿ K is stable for small ε. For this we introduce the covariant version of spectral distances motivated by the functional model . Our definition is different from other types known in the literature []*Ch. 5.

Definition 3 Let a and b be two matrices with all their eigenvalues sitting inside of the unit disk and B_a(z) and B_b(z) be respective Blaschke products as defined above. The (covariant) spectral distance d(a,b) between a and b is equal to the distance ||B_a−B_b||₂ between B_a(z) and B_b(z) in the Hardy space on the unit circle.

Since the spectral distance is defined through the distance in H₂ all standard axioms of a distance are automatically satisfied. For a Blaschke products we have | B_a(z) |=1 if | z |=1, thus ||B_a||_p=1 in any L_p on the unit circle. Therefore an alternative expression for the spectral distance is:

In particular, we always have 0≤ d(a,b) ≤ 2. We get an obvious consequence of Prop. 1, which justifies the name of the covariant spectral distance:

Corollary 4 For any g∈SL₂(ℝ) we have d(a,b)=d(g· a, g· a), where · denotes the Möbius action (2).

An important property of the covariant spectral distance is its stability under small perturbations.

Theorem 5 For n=2 let λ₁(ε ) and λ₂(ε ) be eigenvalues of the matrix J₂+ε ²· K for some matrix K. Then

⎪
⎪

λ₁(ε )

⎪
⎪

λ₂(ε )

⎪
⎪

=O(ε ), however

⎪
⎪

λ₁(ε )+λ₂(ε )

⎪
⎪

=O(ε ²). (10)

The spectral distance from the 1-jet at 0 to two 0-jets at points λ₁ and λ₂ bounded only by the first condition in (10) is O(ε ²). However the spectral distance between J₂ and J₂+ε ²· K is O(ε ⁴).

In other words, a matrix with eigenvalues satisfying to the Lisdkii condition from the Thm. 2 is much closer to the Jordan block J₂ than a generic one with eigenvalues of the same order. Thus the covariant spectral distance is more stable under perturbation that magnitude of eigenvalues. For n=2 a proof can be forced through a direct calculation. We also conjecture that the similar statement is true for any n≥ 2.

17.6 Covariant Pencils of Operators

Let H be a real Hilbert space, possibly of finite dimensionality. For bounded linear operators A and B consider the generalised eigenvalue problem, that is finding a scalar λ and a vector x∈ H such that:

The standard eigenvalue problem corresponds to the case B=I, moreover for an invertible B the generalised problem can be reduced to the standard one for the operator B⁻¹A. Thus it is sensible to introduce the equivalence relation on the pairs of operators:

If we consider this SL₂(ℝ)-action subject to the equivalence relation (12) then we will arrive to a version of the linear-fractional transformation of the operator defined in (2). There is a connection of the SL₂(ℝ)-action (13) to the problem (11) through the following intertwining relation :

Proposition 1 Let λ and x∈ H solve the generalised eigenvalue problem (11) for the pair (A,B). Then the pair (C,D)=g· (A,B), g∈SL₂(ℝ) has a solution µ and x, where

µ=g· λ=

aλ +b

cλ +d

, for g=

⎛
⎜
⎝

a	b
c	d

⎞
⎟
⎠

∈SL₂(ℝ),

is defined by the Möbius transformation (??).

is another realisation of a covariant calculus in the sense of Defn. 5. The collection of all pairs g· (A,B), g∈SL₂(ℝ) is an example of covariant pencil of operators. This set is a SL₂(ℝ)-homogeneous spaces, thus it shall be within the classification of such homogeneous spaces provided in the Subsection ??.

Example 2 It is easy to demonstrate that all existing homogeneous spaces can be realised by matrix pairs.

Take the pair (O, I) where O and I are the zero and identity n× n matrices respectively. Then any transformation of this pair by a lower-triangular matrix from SL₂(ℝ) is equivalent to (O, I). The respective homogeneous space is isomorphic to the real line with the Möbius transformations (??).
Consider H=ℝ². Using the notations ι from Subsection 3.4 we define three realisations (elliptic, parabolic and hyperbolic) of an operator A_ι:
A_i=
⎛
⎜
⎝
        0 1

−1 0

⎞
⎟
⎠

,      A_ε=
⎛
⎜
⎝
        0 1

0 0

⎞
⎟
⎠

,     A_є=
⎛
⎜
⎝
        0 1

1 0

⎞
⎟
⎠

.     (14)

Then for an arbitrary element h of the subgroup K, N or A the respective (in the sense of the Principle 1) pair h· (A_ι,I) is equivalent to (A_ι,I) itself. Thus those three homogeneous spaces are isomorphic to the elliptic, parabolic and hyperbolic half-planes under respective actions of SL₂(ℝ). Note, that A_ι² =ι² I, that is A_ι is a model for hypercomplex units.
Let A be a direct sum of any two different matrices out of the three A_ι from (14), then the fix group of the equivalence class of the pair (A,I) is the identity of SL₂(ℝ). Thus the corresponding homogeneous space coincides with the group itself.

Hawing homogeneous spaces generated by pairs of operators we can define respective functions on those spaces. The special attention is due the following paraphrase of the resolvent:

Obviously R_(A,B)(g) contains the essential information about the pair (A,B). Probably, the function R_(A,B)(g) contains too much simultaneous information, we may restrict it to get a more detailed view. For vectors u, v∈ H we also consider vector and scalar-valued functions related to the generalised resolvent:

where (cA+dB)⁻¹u is understood as a solution w of the equation u=(cA+dB)w if it exists and is unique, this does not require the full invertibility of cA+dB.

It is easy to see that the map (A,B)↦ R_(A,B)^(u,v)(g) is a covariant calculus as well. It worth to notice that function R_(A,B) can again fall into three EPH cases.

Example 3 For the three matrices A_ι considered in the previous Example we denote by R_ι(g) the resolvetn-type function of the pair (A_ι, I). Then:

R_i(g)=

c²+d²

⎛
⎜
⎝

d	−c
c	d

⎞
⎟
⎠

, R_ε(g)=

d²

⎛
⎜
⎝

d	−c
0	d

⎞
⎟
⎠

, R_є(g)=

d²−c²

⎛
⎜
⎝

d	−c
−c	d

⎞
⎟
⎠

Put u=(1,0)∈ H, then R_ι(g) u is a two-dimensional real vector valued functions with components equal to real and imaginary part of hypercomplex Cauchy kernel considered in [].

Consider the space L(G) of functions spanned by all left translations of R_(A,B)(g). As usual, a closure in a suitable metric, say L_p, can be taken. The left action g: f(h)↦ f(g⁻¹h) of SL₂(ℝ) on this space is a linear representation of this group. Afterwards the representation can be decomposed into a sum of primary subrepresentations .

Example 4 For the matrices A_ι the irreducible components are isomorphic to analytic spaces of hypercomplex functions under the fraction-linear transformations build in Subsection 14.2.

An important observation is that a decomposition into irreducible or primary components can reveal an EPH structure even in the cases hiding it on the homogeneous space level.

Example 5 Take the operator A=A_i⊕ A_є from the Example 2(3). The corresponding homogeneous space coincides with the entire SL₂(ℝ). However if we take two vectors u_i=(1,0)⊕(0,0) and u_є=(0,0)⊕(1,0) then the respective linear spaces generated by functions R_A(g)u_i and R_A(g)u_є will be of elliptic and hyperbolic types respectively.

Let us briefly consider a quadratic eigenvalue problem: for given operators (matrices) A₀, A₁ and A₂ from B(H) find a scalar λ and a vector x∈ H such that

There is a connection with our study of conic sections from Subsection ?? which we will only hint for now. Comparing (15) with the equation of the cycle (3) we can associate the respective Fillmore–Springer–Cnops–type matrix to Q(λ), cf. (??):

Then we can state the following analogue of Thm. ?? for the quadratic eigenvalues:

Proposition 6 Let two quadratic matrix polynomials Q and Q′ are such that their FSC matrices (16) are conjugated C_Q′=gC_Q g⁻¹ by an element g∈SL₂(ℝ). Then λ is a solution of the quadratic eigenvalue problem for Q and x∈ H if and only if µ=g· λ is a solution of the quadratic eigenvalue problem for Q′ and x. Here µ=g· λ is the Möbius transformation (??) associated to g∈SL₂(ℝ).

So quadratic matrix polynomials are non-commuting analogues of the cycles and it would be exciting to extend the geometry from Section ?? to this non-commutative setting as much as possible.

Remark 7 It is beneficial to extend a notion of a scalar in an (generalised) eigenvalue problem to an abstract field or ring. For example, we can consider pencils of operators/matrices with polynomial coefficients. In many circumstances we may factorise the polynomial ring by an ideal generated by a collection of algebraic equations. Our work with hypercomplex units is the most elementary realisation of this setup. Indeed, the algebra of hypercomplex numbers with the hypercomplex unit ι is a realisation of the polynomial ring in a variable t factored by the single quadratic relation t²+σ=0, where σ=ι².

Lecture 18 Harmonic Oscillator and Ladder Operators

18.1 Harmonic Oscillator

Harmonic oscillators are treated in most textbooks on quantum mechanics . This is efficiently done through creation/annihilation (ladder) operators [] []. The underlying structure is the representation theory of the the Heisenberg and symplectic groups []*§ VI.2 []*§ 8.2 [] []. It is also known that quantum mechanics and field theory can benefit from introduction of the Clifford algebra-valued group representations [] [] [] [].

The dynamics of a harmonic oscillator generates the symplectic transformation of the phase space of the elliptic type. The respective parabolic and hyperbolic counterparts are also of interest []*§ 3.8 []. As we will see, they are naturally connected with respective hypercomplex numbers.

We recall that the symplectic group [2] []*§ 1.2 is isomorphic to the group SL₂(ℝ) [] [] [] and provides linear symplectomorphisms of the two-dimensional phase space. It has three types of non-isomorphic one-dimensional continuous subgroups (8-10) with symplectic action on the phase space illustrated by Fig. 11.1. We will refer to them as elliptic, parabolic and hyperbolic subgroups respectively.

On the other hand complex, dual and double numbers present three non-isomorphic types of commutative, associative two-dimensional algebras []*App. C []*§ 5. These units can be also labelled as elliptic, parabolic and hyperbolic.

In the paper [] we considered representations of the Heisenberg group which are induced by hypercomplex characters of its centre. The elliptic case (complex numbers) describes the traditional framework of quantum mechanics, of course.

Double-valued representations, with the hypercomplex unit є²=1, are a natural source of hyperbolic quantum mechanics developed for a while [, , , , ]. The representation acts on a Krein space with an indefinite inner product []. This aroused significant recent interest in connection with PT–symmetric quantum mechanics []. However our approach is different from the classical treatment of Krein spaces: we use the hyperbolic unit є and build the hyperbolic analytic function theory on its own basis [, ]. In the traditional approach the indefinite metric is mapped to a definite inner product through an auxiliary operators.

The representation with values in dual numbers provides a convenient description of the classical mechanics. To this end we do not take any sort of semiclassical limit, rather the nilpotency of the hypercomplex unit (ε²=0) performs the task. This removes the vicious necessity to consider the Planck constant tending to zero. Mixing this with complex numbers we get a convenient tool for modelling the interaction between quantum and classical systems [, ].

Our construction [] provides three different types of dynamics and also generates the respective rules for addition of probabilities. In this paper we analyse the three types of dynamics produced by transformations (8–10) from the symplectic group [2] by means of ladder operators. As a result we obtain further illustrations to the Similarity and Correspondence Principle 1.

In this paper we work with the simplest case of a particle with only one degree of freedom. Higher dimensions and the respective group of symplectomorphisms [2n] may require consideration of Clifford algebras [].

18.2 Heisenberg Group and Its Automorphisms

Let (s,x,y), where s, x, y∈ ℝ, be an element of the one-dimensional Heisenberg group ℍ¹ [, ]. Consideration of the general case of ℍⁿ will be similar, but is beyond the scope of present paper. The group law on ℍ¹ is given as follows:

where the non-commutativity is due to ω—the symplectic form on ℝ²ⁿ, which is the central object of the classical mechanics []*§ 37:

act as a representation of ℍ¹ on a certain linear space of functions. For example, an action on L₂(ℍ,dg) with respect to the Haar measure dg=ds dx dy is the left regular representation , which is unitary.

The Lie algebra hⁿ of ℍ¹ is spanned by left-(right-)invariant vector fields

and all other commutators vanishing. We will sometimes omit the superscript l for left-invariant field.

The group of outer automorphisms of ℍ¹, which trivially acts on the centre of ℍ¹, is the symplectic group [2] defined in the precious section. It is the group of symmetries of the symplectic form x []*Thm. 1.22 []*p. 830. The symplectic group is isomorphic to SL₂(ℝ) [][]*Ch. 8. The explicit action of [2] on the Heisenberg group is:

The Shale–Weil theorem []*§ 4.2 []*p. 830 states that any representation ρℏ of the Heisenberg groups generates a unitary oscillator (or metaplectic) representation ρ[SW]ℏ of the Sp′(2), the two-fold cover of the symplectic group []*Thm. 4.58.

We can consider the semidirect product G=ℍ¹⋊Sp′(2) with the standard group law:

and the stars denote the respective group operations while the action g(h′) is defined as the composition of the projection map Sp′(2)→ Sp(2) and the action (6). This group is sometimes called the Schrödinger group and it is known as the maximal kinematical invariance group of both the free Schrödinger equation and the quantum harmonic oscillator []. This group is of interest not only in quantum mechanics but also in optics [, ].

Consider the Lie algebra sp₂ of the group [2]. We again use the basis A, B, Z (12) with commutators (13). Vectors Z, B−Z/2 and B are generators of the one-parameter subgroups K, N′ and (8–10) respectively.

Furthermore we can consider the basis {S, X, Y, A, B, Z} of the Lie algebra g of the Lie group G=ℍ¹⋊Sp′(2). All non-zero commutators besides those already listed in (12) and (13) are:

The Shale–Weil theorem allows us to expand any representation ρℏ of the Heisenberg group to the representation ρ[2]ℏ=ρℏ⊕ρ[SW]ℏ of the group G.

Example 1 Let ρℏ be the Schrödinger representation []*§ 1.3 of ℍ¹ in L₂(ℝ), that is []*(3.5):

[ρχ(s,x,y) f ](q)=e^{2πiℏ (s−xy/2)
+2πi x q} f(q−ℏ y). (9)

Thus the action of the derived representation on the Lie algebra h₁ is:

ρℏ(X)=2πi q, ρℏ(Y)=−ℏ

, ρℏ(S)=2πiℏ I. (10)

Then the associated Shale–Weil representation of [2] in L₂(ℝ) has the derived action, cf. []*(2.2) []*§ 4.3:

ρ[SW]ℏ(A) =−

−

, ρ[SW]ℏ(B)=−

ℏi

8π

d²

dq²

−

πi q²

2ℏ

, ρ[SW]ℏ(Z)=

ℏi

4π

d²

dq²

−

πi q²

ℏ

. (11)

We can verify commutators (12) and (13,8) for operators (10–11). It is also obvious that in this representation the following algebraic relations hold:

ρ[SW]ℏ(A)

4πℏ

(ρℏ(X)ρℏ(Y)−

ρℏ(S))

(12)

8πℏ

(ρℏ(X)ρℏ(Y)+ρℏ(Y)ρℏ(X) ),

ρ[SW]ℏ(B)

8πℏ

(ρℏ(X)²−ρℏ(Y)²),

(13)

ρ[SW]ℏ(Z)

4πℏ

(ρℏ(X)²+ρℏ(Y)²).

(14)

Thus it is common in quantum optics to name g as a Lie algebra with quadratic generators , see []*§ 2.2.4.

Note that ρ[SW]ℏ(Z) is the Hamiltonian of the harmonic oscillator (up to a factor). Then we can consider ρ[SW]ℏ(B) as the Hamiltonian of a repulsive (hyperbolic) oscillator. The operator ρ[SW]ℏ(B−Z/2)=ℏi/4πd²/dq² is the parabolic analog. A graphical representation of all three transformations is given in Fig. 11.1 and a further discussion of these Hamiltonians can be found in []*§ 3.8.

An important observation, which is often missed, is that the three linear symplectic transformations are unitary rotations in the corresponding hypercomplex algebra []*§ 3. This means, that the symplectomorphisms generated by operators Z, B−Z/2, B within time t coincide with the multiplication of hypercomplex number q+ι p by e^ι
t, see Subsection 14.1 and Fig. 11.1, which is just another illustration of the Similarity and Correspondence Principle 1.

Example 2 There are many advantages of considering representations of the Heisenberg group on the phase space []*§ 1.7 []*§ 1.6 []. A convenient expression for Fock–Segal–Bargmann (FSB) representation on the phase space is []*(3.2):

[ρF(s,x,y) f] (q,p)= e^−2πⁱ^{(ℏ s+qx+py)} f

⎛
⎜
⎜
⎝

q−

ℏ

y, p+

ℏ

⎞
⎟
⎟
⎠

. (15)

Then the derived representation of h₁ is:

ρF(X)=−2πi q+

ℏ

∂_p, ρF(Y)=−2πi p−

ℏ

∂_q, ρF(S)=−2πiℏ I. (16)

This produces the derived form of the Shale–Weil representation:

ρ[SW]F(A) =

⎛
⎝

q∂_q−p∂_p

⎞
⎠

, ρ[SW]F(B)=−

⎛
⎝

p∂_q+q∂_p

⎞
⎠

, ρ[SW]F(Z)=p∂_q−q∂_p. (17)

Note that this representation does not contain the parameter ℏ unlike the equivalent representation (11). Thus the FSB model explicitly shows the equivalence of ρ[SW]ℏ₁ and ρ[SW]ℏ₂ if ℏ₁ ℏ₂>0 []*Thm. 4.57.

As we will also see below the FSB-type representations in hypercomplex numbers produce almost the same Shale–Weil representations.

18.3 Ladder Operators in Quantum Mechanics

Let ρ be a representation of the group G=ℍ¹⋊Sp′(2) in a space V. Consider the derived representation of the Lie algebra g []*§ VI.1 and denote X′=ρ(X) for X∈g. To see the structure of the representation ρ we can decompose the space V into eigenspaces of the operator X′ for some X∈ g. The canonical example is the Taylor series in complex analysis.

We are going to consider three cases corresponding to three non-isomorphic subgroups (8–10) of [2] starting from the compact case. Let H=Z be a generator of the compact subgroup K. Corresponding symplectomorphisms (6) of the phase space are given by orthogonal rotations with matrices (

). The Shale–Weil representation (11) coincides with the Hamiltonian of the harmonic oscillator.

Since this is a double cover of a compact group the corresponding eigenspaces Z′ v_k=i k v_k are parametrised by a half-integer k∈ℤ/2. Explicitly for a half-integer k:

From the point of view of quantum mechanics as well as the representation theory (which may be the same) it is beneficial to introduce the ladder operators L^± (14), known as creation/annihilation in quantum mechanics []*p. 49 or raising/lowering in representation theory []*§ VI.2 []*§ 8.2 []. In other words L^± are eigenvectors for operators adZ of the adjoint representation of g []*§ VI.2. From the commutators (14) we deduce that if v_k is an eigenvector of Z′ then L⁺ v_k is an eigenvector as well:

Thus the action of ladder operators on respective the eigenspaces V_k can be visualised by the diagram:

There are two ways to search for ladder operators: in (complexified) Lie algebras h₁ and sp₂. We will consider them in a sequence.

18.3.1 Ladder Operators from the Heisenberg Group

Assuming L⁺=aX′+bY′ we obtain from the relations (7–8) and (14) the linear equations with unknown a and b:

The equations have a solution if and only if λ₊²+1=0, and the raising/lowering operators are L^±= X′∓iY′.

Remark 1 Here we have an interesting asymmetric response: due to the structure of the semidirect product ℍ¹⋊Sp′(2) it is the symplectic group which acts on ℍ¹, not vise versa. However the Heisenberg group has a weak action in the opposite direction: it shifts eigenfunctions of [2].

The standard treatment of the harmonic oscillator in quantum mechanics, which can be found in many textbooks, e.g. []*§ 1.7 []*§ 2.2.3, is as follows. The vector v_−1/2(q)=e^{−π q2/ℏ} is an eigenvector of Z′ with the eigenvalue −i/2. In addition v_−1/2 is annihilated by L⁺. Thus the chain (20) terminates to the right and the complete set of eigenvectors of the harmonic oscillator Hamiltonian is presented by (L⁻)^k v_−1/2 with k=0, 1, 2, ….

We can make a wavelet transform generated by the Heisenberg group with the mother wavelet v_−1/2, and the image will be the Fock–Segal–Bargmann (FSB) space [] []*§ 1.6. Since v_−1/2 is the null solution of L⁺=X′−i Y′, then by Cor. 6 the image of the wavelet transform will be null-solutions of the corresponding linear combination of the Lie derivatives (10):

which turns out to be the Cauchy–Riemann equation on a weighted FSB-type space.

18.3.2 Symplectic Ladder Operators

We can also look for ladder operators within the Lie algebra sp₂, see []*§ 8. Assuming L₂⁺=aA′+bB′+cZ′ from the relations (13) and defining condition (14) we obtain the linear equations with unknown a, b and c:

The equations have a solution if and only if λ₊²+4=0, and the raising/lowering operators are L₂^±=±i A′+B′. In the Shale–Weil representation (11) they turn out to be:

Since this time λ₊=2i the ladder operators L₂^± produce a shift on the diagram (20) twice bigger than the operators L^± from the Heisenberg group. After all, this is not surprising since from the explicit representations (21) and (23) we get:

18.4 Ladder Operators for the Hyperbolic Subgroup

Consider the case of the Hamiltonian H=2B, which is a repulsive (hyperbolic) harmonic oscillator []*§ 3.8. The corresponding one-dimensional subgroup of symplectomorphisms produces hyperbolic rotations of the phase space. The eigenvectors v_µ of the operator

are Weber–Hermite (or parabolic cylinder) functions v_ν=D_ν−1/2(±2e^{i π/4}√π/ℏ q), see []*§ 8.2 [] for fundamentals of Weber–Hermite functions and [] for further illustrations and applications in optics.

The corresponding one-parameter group is not compact and the eigenvalues of the operator 2B′ are not restricted by any integrality condition, but the raising/lowering operators are still important []*§ II.1 []*§ 1.1. We again seek solutions in two subalgebras h₁ and sp₂ separately. However the additional options will be provided by a choice of the number system: either complex or double.

18.4.1 Complex Ladder Operators

Assuming L_h⁺=aX′+bY′ from the commutators (7–8) we obtain the linear equations:

The equations have a solution if and only if λ₊²−1=0. Taking the real roots λ=±1 we obtain that the raising/lowering operators are L_h^±=X′∓Y′. In the Schrödinger representation (10) the ladder operators are

The null solutions v_±1/2(q)=e^±πi/ℏ
q2 to operators ρℏ(L^±) are also eigenvectors of the Hamiltonian ρ[SW]ℏ(2B) with the eigenvalue ±1/2. However the important distinction from the elliptic case is, that they are not square-integrable on the real line anymore.

We can also look for ladder operators within the sp₂, that is in the form L_2h⁺=aA′+bB′+cZ′ for the commutator [2B′,L_h⁺]=λ L_h⁺. We will get the system:

A solution again exists if and only if λ²=4. Within complex numbers we get only the values λ=± 2 with the ladder operators L_2h^±=±2A′+Z′/2, see []*§ II.1 []*§ 1.1. Each indecomposable h₁- or sp₂-module is formed by a one-dimensional chain of eigenvalues with a transitive action of ladder operators L_h^± or L_2h^± respectively. And we again have a quadratic relation between the ladder operators:

18.4.2 Double Ladder Operators

There are extra possibilities in in the context of hyperbolic quantum mechanics [] [] []. Here we use the representation of ℍ¹ induced by a hyperbolic character e^{є h
t}=cosh(h t)+єsinh(h t), see []*(4.5), and obtain the hyperbolic representation of ℍ¹, cf. (9):

Then the associated Shale–Weil derived representation of sp₂ in the Schwartz space S(ℝ) is, cf. (11):

Note that ρ[SW]h(B) now generates a usual harmonic oscillator, not the repulsive one like ρ[SW]ℏ(B) in (11). However the expressions in the quadratic algebra are still the same (up to a factor), cf. (12–14):

This is due to the Principle 1 of similarity and correspondence: we can swap operators Z and B with simultaneous replacement of hypercomplex units i and є.

The eigenspace of the operator 2ρ[SW]h(B) with an eigenvalue є ν are spanned by the Weber–Hermite functions D_−ν−1/2(±√2/hx), see []*§ 8.2. Functions D_ν are generalisations of the Hermit functions (18).

The compatibility condition for a ladder operator within the Lie algebra h₁ will be (24) as before, since it depends only on the commutators (7–8). Thus we still have the set of ladder operators corresponding to values λ=±1:

Admitting double numbers we have an extra way to satisfy λ²=1 in (24) with values λ=±є. Then there is an additional pair of hyperbolic ladder operators, which are identical (up to factors) to (21):

Pairs L_h^± and L_є^± shift eigenvectors in the “orthogonal” directions changing their eigenvalues by ±1 and ±є. Therefore an indecomposable sp₂-module can be parametrised by a two-dimensional lattice of eigenvalues in double numbers, see Fig. 14.2.

are null solutions to the operators L_h^± and L_є^± respectively. They are also eigenvectors of 2ρ[SW]h(B) with eigenvalues ∓є/2 and ∓1/2 respectively. If these functions are used as mother wavelets for the wavelet transforms generated by the Heisenberg group, then the image space will consist of the null-solutions of the following differential operators, see Cor. 6:

for v_1/2^±h and v_1/2^±є respectively. This is again in line with the classical result (22). However annihilation of the eigenvector by a ladder operator does not mean that the part of the 2D-lattice becomes void since it can be reached via alternative routes. Instead of multiplication by a zero, as it happens in the elliptic case, a half-plane of eigenvalues will be multiplied by the divisors of zero 1±є.

We can also search ladder operators within the algebra sp₂ and admitting double numbers we will again find two sets of them, cf. Section 14.3:

Again the operators L_2h^± and L_2h^± produce double shifts in the orthogonal directions on the same two-dimensional lattice in Fig. 14.2.

18.5 Ladder Operator for the Nilpotent Subgroup

Finally we look for ladder operators for the Hamiltonian B′+Z′/2 or, equivalently, −B′+Z′/2. It can be identified with a free particle []*§ 3.8.

We can look for ladder operators in the representation (10–11) within the Lie algebra h₁ in the form L_ε^±=aX′+bY′. This is possible if and only if

The compatibility condition λ²=0 implies λ=0 within complex numbers. However such a “ladder” operator produces only the zero shift on the eigenvectors, cf. (19).

Another possibility appears if we consider the representation of the Heisenberg group induced by dual-valued characters. On the configurational space such a representation is []*(4.11):

However the Shale–Weil extension generated by this representation is inconvenient. It is better to consider the FSB–type parabolic representation []*(4.9) on the phase space induced by the same dual-valued character, cf. (15):

An advantage of the FSB representation is that the derived form of the parabolic Shale–Weil representation coincides with the elliptic one (17).

Eigenfunctions with the eigenvalue µ of the parabolic Hamiltonian B′+Z′/2=q∂_p have the form

The linear equations defining the corresponding ladder operator L_ε^±=aX′+bY′ in the algebra h₁ are (32). The compatibility condition λ²=0 implies λ=0 within complex numbers again. Admitting dual numbers we have additional values λ=±ελ₁ with λ₁∈ℂ with the corresponding ladder operators

For the eigenvalue µ=µ₀+εµ₁ with µ₀, µ₁∈ℂ the eigenfunction (37) can be rewritten as:

due to the nilpotency of ε. Then the ladder action of L_ε^± is µ₀+εµ₁↦ µ₀+ε(µ₁± λ₁). Therefore these operators are suitable for building sp₂-modules with a one-dimensional chain of eigenvalues.

Finally, consider the ladder operator for the same element B+Z/2 within the Lie algebra sp₂. According to the above procedure we get the equations:

which be can again resolved if and only if λ²=0. There is the only complex root λ=0 with the corresponding operators L_p^±=B′+Z′/2, which does not affect the eigenvalues. However the dual number roots λ =±ελ₂ with λ₂∈ℂ lead to the operators

18.6 Similarity and Correspondence

We wish to summarise our findings. Firstly, the appearance of hypercomplex numbers in ladder operators for h₁ follows exactly the same pattern as was already noted for sp₂, see Rem. 5:

In the spirit of the Similarity and Correspondence Principle 1 we have the following extension of Prop. 6:

Proposition 1 Let a vector H∈sp₂ generates the subgroup K, N′ or , that is H=Z, B+Z/2, or 2B respectively. Let ι be the respective hypercomplex unit. Then the ladder operators L^± satisfying to the commutation relation:

[H,L₂^±]=±ι L^±

are given by:

Within the Lie algebra h₁: L^±=X′∓ι Y′.
Within the Lie algebra sp₂: L₂^±=±ι A′ +E′. Here E∈sp₂ is a linear combination of B and Z with the properties:
- E=[A,H].
- H=[A,E].
- Killings form K(H,E) []*§ 6.2 vanishes.
Any of the above properties defines the vector E∈span{B,Z} up to a real constant factor.

It is worth continuing this investigation and describing in details hyperbolic and parabolic versions of FSB spaces.

Appendix A Open Problems

A reader may already note numerous objects and results, which deserve a further consideration. It may also worth to state some open problems explicitly. In this section we indicate several directions for further work, which go through four main areas described in the paper.

A.1 Geometry

Geometry is most elaborated area so far, yet many directions are waiting for further exploration.

A.2 Analytic Functions

It is known that in several dimensions there are different notions of analyticity, e.g. several complex variables and Clifford analysis. However, analytic functions of a complex variable are usually thought to be the only options in a plane domain. The following seems to be promising:

A.3 Functional Calculus

The functional calculus of a finite dimensional operator considered in Section 17 is elementary but provides a coherent and comprehensive treatment. It shall be extended to further cases where other approaches seems to be rather limited.

A.4 Quantum Mechanics

Due to the space restrictions we only touched quantum mechanics, further details can be found in [] [] [] [] [] []. In general, Erlangen approach is much more popular among physicists rather than mathematicians. Nevertheless its potential is not exhausted even there.

Remark 1 This work is performed within the “Erlangen programme at large” framework [, ], thus it would be suitable to explain the numbering of various papers. Since the logical order may be different from chronological one the following numbering scheme is used:

Prefix	Branch description
“0” or no prefix	Mainly geometrical works, within the classical field of Erlangen programme by F. Klein, see [] []
“1”	Papers on analytical functions theories and wavelets, e.g. []
“2”	Papers on operator theory, functional calculi and spectra, e.g. []
“3”	Papers on mathematical physics, e.g. []

For example, [] is the first paper in the mathematical physics area. The present paper [] outlines the whole framework and thus does not carry a subdivision number. The on-line version of this paper may be updated in due course to reflect the achieved progress.

Appendix B Supplementary Material

B.1 Dual and Double Numbers

Complex numbers form a two-dimensional commutative associative algebra with an identity. Up to a suitable choice of a basis there are exactly three different types of such algebras, see []. They are spanned by a basis consisting of 1 and a hypercomplex unit ι. The square of ι is −1 for complex numbers , 0 for dual numbers and 1 for double numbers . In these cases we write the hypercomplex unit ι as i, ε and є respectively.

Arithmetic of hypercomplex numbers is defined by associative, commutative and distributive laws, e.g., the product of two numbers is:

Despite of significant similarities only complex numbers belong to the mainstream mathematics. Among their obvious advantages is the following:

The first property is not very crucial, zero divisors can be treated through appropriate techniques, e.g. projective coordinates, see Section 8.1. Algebraic closedness never was used in the present work. Thus the absence of these properties is not be insuperable obstacles in the study of hypercomplex numbers. On the other hand, hypercomplex numbers naturally appeared in Section 3.3 from SL₂(ℝ) action on the three different types of homogeneous spaces.

B.2 Classical Properties of Conic Sections

We call cycles three types of curves: circles, parabolas and equilateral hyperbolas. They belong to a large class of conic sections , that is they can be obtained as intersection of a cone with a plane, see Fig. 1.3. Algebraically cycles are defined by a quadratic equation (1) and are a subset of quadrics.

The beauty of conic sections attracts mathematicians for several thousand years already. There is an extensive literature, see []*§ 6 for an entry level and []*Ch. 17 for a comprehensive coverage. We list below the basic definitions only in order to clarify the distinction between the classical foci and centres of conic sections and our usage for cycles .

We use the notation | P₁P₂ | and | P l | for the Euclidean distance between points P₁, P₂ and between a point P and a line l.

The above definition in terms of distances allows to deduce equality of respective angles in each case, see Fig. B.1 and []*§ 6. This implies respective rays reflection : for example, any ray perpendicular to the directrix is reflected by the parabola to pass its focus—“burning point”. There are many applications of this from the legendary burning of the Roman fleet by Archimedes to practical (parabolic) satellite dishes.

B.3 Comparison with Yaglom’s Book

The detailed, profound book of Yaglom [] is already a golden classic appreciated by several generations of mathematician. To avoid confusion, we provide a comparison of our notions and results with Yaglom’s ones.

Firstly, there is the methodological difference: Yaglom started from notions of length and angles and then found out objects (notably parabolas) which carry them out in an invariant way. We work in the opposite direction: take invariant object (FSCc matrices) then found out respective notions and properties which are invariant as well. This leads to significant distinctions in our results which are collected in the following table.

In short: we tried to avoid an overlap with Yaglom’s book []: our results are either new or obtained in a different manner.

B.4 Other Approaches and Results

The development of parabolic and hyperbolic analogs of the complex analysis has a long but sporadic history. In the absence of continuity there are many examples, when a researcher started from a scratch without any knowledge on the previous works. There may be even more forgotten papers in the subject. To improve the situation we list here some papers without a hope to be comprehensive or even representative.

The survey and history of Cayley-Klein geometries is presented in Yaglom’s book [], this shall be completed by the work [] which provides the full axiomatic classification of EPH cases. A search for hyperbolic function theory was attempted several times starting from 1930’s, see for example [, , ]. Despite of some important advances the obtained hyperbolic theory does not as natural and complete yet as complex analysis is. Parabolic geometry was considered in book [] with rather trivial “parabolic calculus” described in []. There is also an interest to this topic in different areas: differential geometry [, , , , , , ], quantum mechanics [, , , ], group representations [, ] space-time geometry [, , , ], hypercomplex analysis [, , , ]. A brief history of the topic is nicely presented in [] and further references can be found in the above papers.

B.5 FSCc with Clifford Algebras

In this section we provide formulae for FSCc which uses Clifford algebras []. Although in this case we need to take care on non-commutativity of numbers, many matrix expressions have a simpler form. Clifford algebras also admit straightforward generalisation to higher dimension and was used to implement our CAS library.

We use four dimensional Clifford algebra C−0.12emℓ(σ) with unit 1, two generators e₀ and e₁, and the fourth element of the basis—their product e₀e₁. The multiplication table is: e₀²=−1, e₁²=σ and e₀e₁=−e₁e₀. Here σ=−1, 0 and 1 in the respective EPH cases. The point space ℝ^σ consists of vectors u e₀+ ve₁. An isomorphic realisation of SL₂(ℝ) is obtained if we replace a matrix (

) for any σ. The Möbius transformation of ℝ^σ→ ℝ^σ for all three algebras C−0.12emℓ(σ) by the same expression, cf. (24):

where EPH type of C−0.12emℓ(σ_c) may be different from the type of C−0.12emℓ(σ). In terms of Clifford values matrice (1) and (2) the similarity with element of SL₂(ℝ) (7) has exactly the same form S _σc^s= gC_σc^sg⁻¹. However the cycle similarity (10) becomes simpler, e.g. there is no need in conjugation:

Appendix C How to Use Software

The enclosed DVD with software is derived from several open source projects, notably the Debian GNU–Linux (http://www.debian.org/), GiNaC library of symbolic calculations [], Asymptote [] and many others. Thus our work is distributed under the GNU General Public License (GPL) 3.0 []. In this Appendix we only briefly outline how to start using the enclosed DVD. As soon as it will be up and running further help may be obtained on the computer screen. We describe also how to get the most of the disk on computers without a DVD drive at the end of Sections C.1, C.2.1 and C.2.2.

C.1 Viewing Colour Graphics

The easiest part is to view colour illustrations on your computer. There are not much hardware and software demands for this task: your computer shall have a DVD drive and be able to render HTML pages. The last task can be done by any Web browser. If these requirements are satisfied do the following steps:

If your computer does not have a DVD drive (e.g. is a netbook) but you can gain a short-time access to a computer with a drive then you can copy the top-level folder doc from the enclosed DVD to a portable media, say, a memory stick. Illustrations (and other documentation) can be accessed by opening index.html file from this folder.

In a similar way reader can access ISO images of bootable disks, software sources and other supplementary information described below.

C.2 Installation of CAS

The method A is straightforward and can bring some performance enhancement. However it requires a hardware compatibility, in particular you shall have a so-called “i386 architecture”. The method B shall run on a much wider set of hardware and you can use CAS from the comfort of your standard desktop. However this may require an additional third-party programme to be installed. We describe details for the above methods now.

C.2.1 Booting from the DVD Disk

It is difficult to give an exact list of hardware requirements for DVD booting, however your computer shall be necessary based on i386 architecture. If you are ready to give a try follow these steps:

If the DVD boots but the graphic X server did not start for any reason and you have the text command prompt only, you still can use the most o the CAS. This is described in the last paragraph of Section C.3.

If your computer does not have a DVD drive you may still boot the CAS on your computer from a spare USB stick of at least 1Gb capacity. For this use UNetbootin (http://unetbootin.sourceforge.net/) or a similar tool to put an ISO image of a booting disk to the memory stick. The ISO image(s) is located at the top-level folder iso-images of the DVD and file README in this folder describes them. You can access this folder as was described in Section C.1.

C.2.2 Running a Linux Emulator

You can also use the enclosed CAS on a wide range of hardware running various operational system, e.g. Linux, Windows, Mac OS, etc. To this end you need to install a so called virtual machine, which can emulate i386 architecture. I would recommend VirtualBox (http://www.virtualbox.org)—a free open source software which works well on many existing platforms. There are many alternatives (including open source), for example:

Here we outline the procedure for VirtualBox, for other emulators you may need to make some adjustments. To use VirtualBox follow these steps:

If you succeeded in this you may proceed to the Section C.3. Some tricks to improve your experience with emulations are described in the detailed electronic manual.

C.2.3 Recompiling the CAS on Your OS

The core of our software is a C++ library which is based on the GiNaC [], see its Web page for the up-to-date information. The later can be compiled and installed on Linux and Windows. Subsequently our library can be compiled on those computers from the provided sources as well. Then the library can be used in your C++ programmes. See the top-level folder src on the DVD and documentation there.

Our interactive tool is based on pyGiNaC []—a Python binding for GiNaC. This may be working on many flavours of Linux as well. Please note, that in order to use pyGiNaC with the recent GiNaC you need to apply my patches to the official version. The DVD contains the whole pyGiNaC source tree which is already patched and is ready to use.

There is also a possibility to use our library interactively with swiGiNaC, which is another Python binding for GiNaC and included in many Linux distribution. The complete sources for binding our library to swiGiNaC are in the corresponding folder of the enclosed DVD. However swiGiNaC does not implement full functionality of our library.

C.3 Using the CAS and Computer Exercises

Once you booted to the graphical user interface with the open CAS window as described in Subsection either C.2.1 or C.2.2 you may need to configure your keyboard (if it is not US one). To install, for example, a Portuguese keyboard you may type at the prompt of the open window the following command:

The keyboard will be switched and the corresponding national flag displayed at the bottom-left corner of the window. For another keyboard you need to use the appropriate two-letter country code instead of pt in the above command. The first exclamation mark tells that the interpreter need to pass this command to the shell.

C.3.1 Warming Up

First few lines on the top of CAS windows suggest several commands to receive a quick introduction or some help on the IPython interpreter. Our CAS was loaded with many objects already predefined, see Section C.5. Let us see what is C, for example:

Thus C is a two-dimensional cycle defined with the quadruple (k,l,n,m). Its determinant is:

Here si stands for σ—the signature of point space metric. Thus the answer reads km−l²+σ n²—the determinant of FSCc matrix of C.

As an exercise reader may follow now the proof of Theorem 1 remembering that the point P and cycle C are already defined. In fact, all statements and exercises marked by the symbol on the margins present on DVD already. For example, to access the proof of Theorem 1 type the following at the prompt:

Here the special %ed calls the external editor jed to visit the file ex.1.py. This file is a Python script containing the same lines as the proof of Theorem 1 in the book. The editor jed may be manipulated from its menu and has command keystrokes compatible with GNU Emacs. For example, to exit the editor press Ctrl-X Ctrl-C. After that the interactive shell executes the visited file and outputs:

For any other CAS-assisted statement or exercises you can also visit the corresponding solution using its number against the symbol on the margin. For example, for Exercise 2 open file ex.2.py. However the next mouse sign marks the item 1, thus you need to visit file ex.1.py in this case. Those files are located on a read-only file system, thus to modify them you need to save them first with a new name (Ctrl-X Ctrl-W), exit the editor, and then use %ed special to edit freshly saved file.

C.3.2 Drawing Cycles

You can visualise cycles instantly. First we open an Asymptote instance and define a picture size:

This cycle depends from a variable sign and it shall be substituted with a numeric value before a visualisation becomes possible:

By now a separate window shall be opened with cycle Cn triply drawn as a circle, parabola and hyperbola. Note that you do not need to retype inputs 12 and 13 from a scratch. Up/down arrows scroll the input history, so you can simply edit the value of sign in the input line 10. And since you are in Linux Tab-key will do a completion for you whenever possible.

The interactive shell evaluates and remember all expressions, thus it may be useful to restart it sometime. It can be closed by Ctrl-D and started from the Main Menu (the left/bottom corner of the screen) Accessories→CAS pycyle. In the same menu folders there are two items, which open documentation about the library in PDF and HTML formats.

C.3.3 Further Usage

There are several batch checks which can be performed with CAS. Open a terminal window from Main Menu → Accessories → LXTerminal. Type at the command prompt:

A comprehensive test of the library will be performed and the end of output shall look like this:

Under normal circumstances the reported total number of errors shall be zero, of course. You can also run all exercises from this book in a batch. From a new terminal window type:

Exercises will be performed one by one with their numbers reported. Numerous graphical windows will be opened to show pencils of cycles . Those windows shall be closed by pressing q-key for each of them. This batch file suppresses all output from Exercises, except containing False string. Under normal circumstances those are only Exercises 1 and 2.

You may access the CAS from a command line as well. This may be required if the graphic X server failed to start for any reason. From the command prompt type the following:

Full capacity of CAS shall be accessible from the command prompt as well except the preview of drawn cycles in a graphical window. However EPS files still can be created with Asymptote, see shipout() method.

C.4 Library for Cycles

Our C++ library defines the class cycle to manipulate cycles of arbitrary dimension in symbolic manner. The derived class cycle2D is tailored to manipulate two-dimensional ones. For the purpose of the book we briefly list here some methods for cycle2D in the pyGiNaC binding form only:

Further information can be obtained from electronic documentation on the enclosed DVD, an inspection of the test file CAS/pycycle/test_pycycle.py and solutions of the Exercises.

C.5 Predefined Object on Initialisation

For a convenience we predefine many GiNaC objects which may be helpful. Here is a brief indication of most used among them:

The solutions of Exercises heavily use those objects. Look for their exact definition in the file CAS/pycycle/init_cycle.py from the home directory.

Index

AliAntGaz00book author=Ali, Syed Twareque, author=Antoine, Jean-Pierre, author=Gazeau, Jean-Pierre, title=Coherent states, wavelets and their generalizations, series=Graduate Texts in Contemporary Physics, publisher=Springer-Verlag, address=New York, date=2000, ISBN=0-387-98908-0, review=MR # 2002m:81092,

Anderson69article author=Anderson, Robert F. V., title=The Weyl functional calculus, date=1969, journal=J. Functional Analysis, volume=4, pages=240267, review=MR # 58 #30405,

Arnold91book author=Arnol’d, V. I., title=Mathematical methods of classical mechanics, series=Graduate Texts in Mathematics, publisher=Springer-Verlag, address=New York, date=1991, volume=60, ISBN=0-387-96890-3, note=Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition, review=MR # 96c:70001,

ArovDym08book author=Arov, Damir Z., author=Dym, Harry, title=J-contractive matrix valued functions and related topics, series=Encyclopedia of Mathematics and its Applications, publisher=Cambridge University Press, address=Cambridge, date=2008, volume=116, ISBN=978-0-521-88300-9, review=MR # MR2474532,

AtiyahSchmid80incollection author=Atiyah, Michael, author=Schmid, Wilfried, title=A geometric construction of the discrete series for semisimple Lie group, date=1980, booktitle=Harmonic analysis and representations of semisimple Lie group, editor=Wolf, J.A., editor=Cahen, M., editor=Wilde, M. De, series=Mathematical Physics and Applied Mathematics, volume=5, publisher=D. Reidel Publishing Company, address=Dordrecht, Holland, pages=317383,

AzizovIokhvidov71aarticle author=Azizov, T. Ja., author=Iohvidov, I. S., title=Linear operators in Hilbert spaces with G-metric, date=1971, ISSN=0042-1316, journal=Uspehi Mat. Nauk, volume=26, number=4 (160), pages=4392, review=MR # 0288613 (44 #5809),

Balk97aincollection author=Balk, M. B., title=Polyanalytic functions and their generalizations [ MR1155418 (93f:30050)], date=1997, booktitle=Complex analysis, I, series=Encyclopaedia Math. Sci., volume=85, publisher=Springer, address=Berlin, pages=195253, review=MR # 1464199,

Bargmann61article author=Bargmann, V., title=On a Hilbert space of analytic functions and an associated integral transform. Part I, date=1961, journal=Comm. Pure Appl. Math., volume=3, pages=215228,

GiNaCmisc author=Bauer, Christian, author=Frink, Alexander, author=Kreckel, Richard, author=Vollinga, Jens, title=GiNaC is Not a CAS, note=http://www.ginac.de/,

Beardon95book author=Beardon, Alan F., title=The geometry of discrete groups, series=Graduate Texts in Mathematics, publisher=Springer-Verlag, address=New York, date=1995, volume=91, ISBN=0-387-90788-2, note=Corrected reprint of the 1983 original, review=MR # MR1393195 (97d:22011),

Beardon05abook author=Beardon, Alan F., title=Algebra and geometry, publisher=Cambridge University Press, address=Cambridge, date=2005, ISBN=0-521-89049-7, review=MR # MR2153234 (2006a:00001),

Beardon07aarticle author=Beardon, Alan F., author=Short, Ian, title=Conformal symmetries of regions, date=2007, ISSN=0791-5578, journal=Irish Math. Soc. Bull., number=59, pages=4960, review=MR # MR2353408 (2008j:30013),

BekkaraFrancesZeghib06article author=Bekkara, Esmaa, author=Frances, Charles, author=Zeghib, Abdelghani, title=On lightlike geometry: isometric actions, and rigidity aspects, date=2006, ISSN=1631-073X, journal=C. R. Math. Acad. Sci. Paris, volume=343, number=5, pages=317321, review=MR # MR2253050 (2007e:53045),

Benz06aincollection author=Benz, Walter, title=Hyperbolic geometry, dimension-free, date=2006, booktitle=Non-Euclidean geometries, series=Math. Appl. (N. Y.), volume=581, publisher=Springer, address=New York, pages=97107, url=http://dx.doi.org/10.1007/0-387-29555-0_4, review=MR # MR2191242 (2007a:51018),

Berezin72article author=Berezin, F. A., title=Covariant and contravariant symbols of operators, date=1972, journal=Izv. Akad. Nauk SSSR Ser. Mat., volume=36, pages=11341167, note=Reprinted in [, pp. 228–261], review=MR # 50 #2996,

Berezin86book author=Berezin, F. A., title=Metod vtorichnogo kvantovaniya, edition=Second, publisher=“Nauka”, address=Moscow, date=1986, note=Edited and with a preface by M. K. Polivanov, review=MR # 89c:81001,

BergerIIbook author=Berger, Marcel, title=Geometry. II, series=Universitext, publisher=Springer-Verlag, address=Berlin, date=1987, ISBN=3-540-17015-4, note=Translated from the French by M. Cole and S. Levy, review=MR # 882916 (88a:51001b),

BergerIbook author=Berger, Marcel, title=Geometry. I, series=Universitext, publisher=Springer-Verlag, address=Berlin, date=1994, ISBN=3-540-11658-3, note=Translated from the 1977 French original by M. Cole and S. Levy, Corrected reprint of the 1987 translation, review=MR # 1295239 (95g:51001),

BernTayl94article author=Bernier, David, author=Taylor, Keith F., title=Wavelets from square-integrable representations, date=1996, ISSN=0036-1410, journal=SIAM J. Math. Anal., volume=27, number=2, pages=594608, note=MR # 97h:22004,

BocCatoniCannataNichZamp07book author=Boccaletti, Dino, author=Catoni, Francesco, author=Cannata, Roberto, author=Catoni, Vincenzo, author=Nichelatti, Enrico, author=Zampetti, Paolo, title=The mathematics of Minkowski space-time and an introduction to commutative hypercomplex numbers, publisher=Springer Verlag, date=2007,

BoetcherKarlovichSpitkovsky02abook author=Böttcher, Albrecht, author=Karlovich, Yuri I., author=Spitkovsky, Ilya M., title=Convolution operators and factorization of almost periodic matrix functions, series=Operator Theory: Advances and Applications, publisher=Birkhäuser Verlag, address=Basel, date=2002, volume=131, ISBN=3-7643-6672-9, review=MR # 1898405 (2003c:47047),

BoyerMiller74aarticle author=Boyer, Charles P., author=Miller, Willard, Jr., title=A classification of second-order raising operators for Hamiltonians in two variables, date=1974, ISSN=0022-2488, journal=J. Mathematical Phys., volume=15, pages=14841489, review=MR # 0345542 (49 #10278),

pyGiNaCmisc author=Brandmeyer, Jonathan, title=PyGiNaC—a Python interface to the C++ symbolic math library GiNaC, date=2004–2007, note=http://sourceforge.net/projects/pyginac/,

BratJorg97aarticle author=Bratteli, Ola, author=Jorgensen, Palle E. T., title=Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N, date=1997, ISSN=0378-620X, journal=Integral Equations Operator Theory, volume=28, number=4, pages=382443, note=E-print: arXiv:funct-an/9612003,

CatoniCannataNichelatti04article author=Catoni, Francesco, author=Cannata, Roberto, author=Nichelatti, Enrico, title=The parabolic analytic functions and the derivative of real functions, date=2004, journal=Advances in Applied Clifford algebras, volume=14, number=2, pages=185190,

CatoniCannataZampetti05article author=Catoni, Francesco, author=Cannata, Vincenzo, Roberto Catoni, author=Zampetti, Paolo, title=N-dimensional geometries generated by hypercomplex numbers, date=2005, journal=Advances in Applied Clifford Algebras, volume=15, number=1,

CerejeirasKahlerSommen05aarticle author=Cerejeiras, P., author=Kähler, U., author=Sommen, F., title=Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains, date=2005, ISSN=0170-4214, journal=Math. Methods Appl. Sci., volume=28, number=14, pages=17151724, review=MR # MR2167561,

Chern96aarticle author=Chern, Shiing-Shen, title=Finsler geometry is just Riemannian geometry without the quadratic restriction, date=1996, ISSN=0002-9920, journal=Notices Amer. Math. Soc., volume=43, number=9, pages=959963, review=MR # MR1400859 (97e:53129),

ChristensenOlafsson09aarticle author=Christensen, Jens Gerlach, author=Ólafsson, Gestur, title=Examples of coorbit spaces for dual pairs, date=2009, ISSN=0167-8019, journal=Acta Appl. Math., volume=107, number=1-3, pages=2548, review=MR # MR2520008,

Cnops94athesis author=Cnops, Jan, title=Hurwitz pairs and applications of Möbius transformations, type=Habilitation Dissertation, address=Universiteit Gent, Faculteit van de Wetenschappen, date=1994, note=See also [],

Cnops02abook author=Cnops, Jan, title=An introduction to Dirac operators on manifolds, series=Progress in Mathematical Physics, publisher=Birkhäuser Boston Inc., address=Boston, MA, date=2002, volume=24, ISBN=0-8176-4298-6, review=MR # 1 917 405,

CnopsKisil97aarticle author=Cnops, Jan, author=Kisil, Vladimir V., title=Monogenic functions and representations of nilpotent Lie groups in quantum mechanics, date=1999, ISSN=0170-4214, journal=Math. Methods Appl. Sci., volume=22, number=4, pages=353373, note=E-print: arXiv:math/9806150. Zbl # 1005.22003, review=MR # MR1671449 (2000b:81044),

Coburn94aincollection author=Coburn, Lewis A., title=Berezin-Toeplitz quantization, date=1994, booktitle=Algebraic mettods in operator theory, publisher=Birkhäuser Verlag, address=New York, pages=101108,

ConstalesFaustinoKrausshar11aarticle author=Constales, Denis, author=Faustino, Nelson, author=Kraußhar, Rolf Sören, title=Fock spaces, Landau operators and the time-harmonic Maxwell equations, date=2011, journal=Journal of Physics A: Mathematical and Theoretical, volume=44, number=13, pages=135303, url=http://stacks.iop.org/1751-8121/44/i=13/a=135303,

CoxeterGreitzerbook author=Coxeter, H.S.M., author=Greitzer, S.L., title=Geometry revisited., language=English, publisher=New York: Random House: The L. W. Singer Company. XIV, 193 p. , date=1967, note=Zbl # 0166.16402,

Devis77book author=Davis, Martin, title=Applied nonstandard analysis, publisher=Wiley-Interscience [John Wiley & Sons], address=New York, date=1977, ISBN=0-471-19897-8, note=Pure and Applied Mathematics, review=MR # MR0505473 (58 #21590),

deGosson08aarticle author=de Gosson, Maurice A., title=Spectral properties of a class of generalized Landau operators, date=2008, ISSN=0360-5302, journal=Comm. Partial Differential Equations, volume=33, number=10-12, pages=20962104, url=http://dx.doi.org/10.1080/03605300802501434, review=MR # MR2475331 (2010b:47128),

Dixmier69book author=Dixmier, Jacques, title=Les C*-algebres et leurs representations, publisher=Gauthier-Villars, address=Paris, date=1964,

Dixmier77book author=Dixmier, Jacques, title=C-algebras, publisher=North-Holland Publishing Co., address=Amsterdam, date=1977, ISBN=0-7204-0762-1, note=Translated from the French by Francis Jellett, North-Holland Mathematical Library, Vol. 15, review=MR # 56 #16388,

DufloMoorearticle author=Duflo, M., author=Moore, Calvin C., title=On the regular representation of a nonunimodular locally compact group, date=1976, journal=J. Functional Analysis, volume=21, number=2, pages=209243, review=MR # 52 #14145,

DunfordSchwartzIbook author=Dunford, Nelson, author=Schwartz, Jacob T., title=Linears operators. part i: General theory, series=Pure and Applied Mathematics, publisher=John Wiley & Sons, Inc., address=New York, date=1957, volume=VII,

EelbodeSommen04aarticle author=Eelbode, D., author=Sommen, F., title=Taylor series on the hyperbolic unit ball, date=2004, ISSN=1370-1444, journal=Bull. Belg. Math. Soc. Simon Stevin, volume=11, number=5, pages=719737, review=MR # MR2130635 (2005k:30092),

EelbodeSommen05aarticle author=Eelbode, D., author=Sommen, F., title=The fundamental solution of the hyperbolic Dirac operator on ℝ^1,m: a new approach, date=2005, ISSN=1370-1444, journal=Bull. Belg. Math. Soc. Simon Stevin, volume=12, number=1, pages=2337, review=MR # MR2134853 (2005k:30093),

ErdelyiMagnusIIbook author=Erdélyi, Arthur, author=Magnus, Wilhelm, author=Oberhettinger, Fritz, author=Tricomi, Francesco G., title=Higher transcendental functions. Vol. II, publisher=Robert E. Krieger Publishing Co. Inc., address=Melbourne, Fla., date=1981, ISBN=0-89874-069-X, note=Based on notes left by Harry Bateman, Reprint of the 1953 original, review=MR # 698780 (84h:33001b),

FeichGroech89aarticle author=Feichtinger, Hans G. and Groechenig, K.H., title=Banach spaces related to integrable group representations and their atomic decompositions, I, date=1989, journal=J. Funct. Anal., volume=86, number=2, pages=307340, note=Zbl # 691.46011,

FillmoreSpringer90aarticle author=Fillmore, Jay P., author=Springer, A., title=Möbius groups over general fields using Clifford algebras associated with spheres, date=1990, ISSN=0020-7748, journal=Internat. J. Theoret. Phys., volume=29, number=3, pages=225246, url=http://dx.doi.org/10.1007/BF00673627, review=MR # 1049005 (92a:22016),

FjelstadGal01article author=Fjelstad, Paul, author=Gal, Sorin G., title=Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers, date=2001, ISSN=0188-7009, journal=Adv. Appl. Clifford Algebras, volume=11, number=1, pages=81107 (2002), review=MR # MR1953513 (2003k:83009),

Folland89book author=Folland, Gerald B., title=Harmonic analysis in phase space, series=Annals of Mathematics Studies, publisher=Princeton University Press, address=Princeton, NJ, date=1989, volume=122, ISBN=0-691-08527-7; 0-691-08528-5, review=MR # 92k:22017,

Folland95book author=Folland, Gerald B., title=A course in abstract harmonic analysis, language=English, publisher=Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. viii, 276 p. $ 61.95 , date=1995,

Fuhr05abook author=Führ, Hartmut, title=Abstract harmonic analysis of continuous wavelet transforms, series=Lecture Notes in Mathematics, publisher=Springer-Verlag, address=Berlin, date=2005, volume=1863, ISBN=3-540-24259-7, review=MR # MR2130226 (2006m:43003),

Garasko09abook author=Garas’ko, G.I., title=Nachala finslerovoi0 geometrii dlya fizikov. (Russian) [Elements of Finsler geometry for physicists], publisher=TETRU, address=Moscow, date=2009, note=268 pp. http://hypercomplex.xpsweb.com/articles/487/ru/pdf/00-gbook.pdf,

Gazeau09abook author=Gazeau, Jean-Pierre, title=Coherent States in Quantum Physics, publisher=Wiley-VCH Verlag, date=2009, ISBN=9783527407095,

Gerrard94abook author=Gerrard, A., author=Burch, J. M., title=Introduction to matrix methods in optics, publisher=Dover Publications Inc., address=New York, date=1994, ISBN=0-486-68044-4, note=Corrected reprint of the 1975 original, review=MR # 1298432 (95h:78001),

GlazmanLjubic06book author=Glazman, I. M., author=Ljubič, Ju. I., title=Finite-dimensional linear analysis, publisher=Dover Publications Inc., address=Mineola, NY, date=2006, ISBN=0-486-45332-4, note=A systematic presentation in problem form, Translated from the Russian and edited by G. P. Barker and G. Kuerti, Reprint of the 1974 edition, review=MR # 2302906 (2007m:46002),

GNUGPLmanual author=GNU, title=General Public License (GPL), edition=version 3, organization=Free Software Foundation, Inc., address=59 Temple Place - Suite 330, Boston, MA 02111-1307, USA, date=29 June 2007, note=http://www.gnu.org/licenses/gpl.html,

GohbergLancasterRodman05abook author=Gohberg, Israel, author=Lancaster, Peter, author=Rodman, Leiba, title=Indefinite linear algebra and applications, publisher=Birkhäuser Verlag, address=Basel, date=2005, ISBN=978-3-7643-7349-8; 3-7643-7349-0, review=MR # 2186302 (2006j:15001),

Grafakos08book author=Grafakos, Loukas, title=Classical Fourier analysis, edition=Second, series=Graduate Texts in Mathematics, publisher=Springer, address=New York, date=2008, volume=249, ISBN=978-0-387-09431-1, review=MR # 2445437,

Gromov90abook author=Gromov, N. A., title=Kontraktsii i analiticheskie prodolzheniya klassicheskikh grupp. Edinyi podkhod. (Russian) [Contractions and analytic extensions of classical groups. Unified approach], publisher=Akad. Nauk SSSR Ural. Otdel. Komi Nauchn. Tsentr, address=Syktyvkar, date=1990, review=MR # MR1092760 (91m:81078),

Gromov90barticle author=Gromov, N. A., title=Transitions: Contractions and analytical continuations of the Cayley-Klein groups, date=1990, ISSN=0020-7748, journal=International Journal of Theoretical Physics, volume=29, pages=607620, url=http://dx.doi.org/10.1007/BF00672035, note=10.1007/BF00672035,

GuentherKuzhel10aarticle author=Günther, Uwe, author=Kuzhel, Sergii, title=PT–symmetry, Cartan decompositions, Lie triple systems and Krein space-related Clifford algebras, date=2010, journal=Journal of Physics A: Mathematical and Theoretical, volume=43, number=39, pages=392002, url=http://stacks.iop.org/1751-8121/43/i=39/a=392002,

GutenmacherVasilyev04abook author=Gutenmacher, Victor, author=Vasilyev, N.B., title=Lines and curves. A practical geometry handbook. Based on an English translation of the original Russian edition by A. Kundu. Foreword by Mark Saul., language=English, publisher=Basel: Birkhäuser. xv, 156 p. EUR 42.00/net; sFr. 70.00 , date=2004,

Asymptotemisc author=Hammerlindl, Andy, author=Bowman, John, author=Prince, Tom, title=Asymptote—powerful descriptive vector graphics language for technical drawing, inspired by MetaPost, date=2004–2011, note=http://asymptote.sourceforge.net/,

Helgason11abook author=Helgason, Sigurdur, title=Integral geometry and Radon transforms, publisher=Springer, address=New York, date=2011, ISBN=978-1-4419-6054-2, review=MR # 2743116,

HerranzOrtegaSantander99aarticle author=Herranz, Francisco J., author=Ortega, Ramón, author=Santander, Mariano, title=Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry, date=2000, ISSN=0305-4470, journal=J. Phys. A, volume=33, number=24, pages=45254551, note=E-print: arXiv:math-ph/9910041, review=MR # MR1768742 (2001k:53099),

HerranzSantander02barticle author=Herranz, Francisco J., author=Santander, Mariano, title=Conformal compactification of spacetimes, date=2002, ISSN=0305-4470, journal=J. Phys. A, volume=35, number=31, pages=66196629, note=E-print: arXiv:math-ph/0110019, review=MR # MR1928852 (2004b:53123),

HerranzSantander02aarticle author=Herranz, Francisco J., author=Santander, Mariano, title=Conformal symmetries of spacetimes, date=2002, ISSN=0305-4470, journal=J. Phys. A, volume=35, number=31, pages=66016618, note=E-print: arXiv:math-ph/0110019, review=MR # MR1928851 (2004a:53091),

Hormander85book author=Hörmander, Lars, title=The analysis of linear partial differential operators III: Pseudodifferential operators, publisher=Springer-Verlag, address=Berlin, date=1985,

HornJohnson94book author=Horn, Roger A., author=Johnson, Charles R., title=Topics in matrix analysis, publisher=Cambridge University Press, address=Cambridge, date=1994, ISBN=0-521-46713-6, note=Corrected reprint of the 1991 original., review=MR # 95c:15001,

Howe80aarticle author=Howe, Roger, title=On the role of the Heisenberg group in harmonic analysis, date=1980, ISSN=0002-9904, journal=Bull. Amer. Math. Soc. (N.S.), volume=3, number=2, pages=821843, review=MR # 81h:22010,

Howe80barticle author=Howe, Roger, title=Quantum mechanics and partial differential equations, date=1980, ISSN=0022-1236, journal=J. Funct. Anal., volume=38, number=2, pages=188254, review=MR # 83b:35166,

HoweTan92book author=Howe, Roger, author=Tan, Eng Chye, title=Non-abelian harmonic analysis: Applications of SL(2,ℝ), series=Universitext, publisher=Springer-Verlag, address=New York, date=1992,

Hudson66athesis author=Hudson, Robin, title=Generalised translation-invariant mechanics, type=D. Phil. thesis, address=Bodleian Library, Oxford, date=1966,

Hudson04aincollection author=Hudson, Robin, title=Translation invariant phase space mechanics, date=2004, booktitle=Quantum theory: reconsideration of foundations—2, series=Math. Model. Phys. Eng. Cogn. Sci., volume=10, publisher=Växjö Univ. Press, Växjö, pages=301314, review=MR # 2111131 (2006e:81134),

Hutnik09aarticle author=Hutník, Ondrej, title=On Toeplitz-type operators related to wavelets, date=2009, ISSN=0378-620X, journal=Integral Equations Operator Theory, volume=63, number=1, pages=2946, url=http://dx.doi.org/10.1007/s00020-008-1647-9, review=MR # MR2480637,

Johansson08aarticle author=Johansson, Andreas, title=Shift-invariant signal norms for fault detection and control, date=2008, ISSN=0167-6911, journal=Systems Control Lett., volume=57, number=2, pages=105111, review=MR # MR2378755 (2009d:93035),

Khrennikov05aarticle author=Khrennikov, A. Yu., title=Hyperbolic quantum mechanics, date=2005, ISSN=0869-5652, journal=Dokl. Akad. Nauk, volume=402, number=2, pages=170172, review=MR # MR2162434 (2006d:81118),

Khrennikov03aarticle author=Khrennikov, Andrei, title=Hyperbolic Quantum Mechanics, language=English, date=2003, journal=Adv. Appl. Clifford Algebr., volume=13, number=1, pages=19, note=E-print: arXiv:quant-ph/0101002,

Khrennikov08aarticle author=Khrennikov, Andrei, title=Hyperbolic quantization, date=2008, ISSN=0188-7009, journal=Adv. Appl. Clifford Algebr., volume=18, number=3-4, pages=843852, review=MR # MR2490591,

KhrennikovSegre07aincollection author=Khrennikov, Andrei, author=Segre, Gavriel, title=Hyperbolic quantization, date=2007, booktitle=Quantum probability and infinite dimensional analysis, series=QP–PQ: Quantum Probab. White Noise Anal., volume=20, publisher=World Sci. Publ., Hackensack, NJ, pages=282287, review=MR # MR2359402,

Kirillov76book author=Kirillov, A. A., title=Elements of the theory of representations, publisher=Springer-Verlag, address=Berlin, date=1976, note=Translated from the Russian by Edwin Hewitt, Grundlehren der Mathematischen Wissenschaften, Band 220, review=MR # 54 #447,

Kirillov99article author=Kirillov, A. A., title=Merits and demerits of the orbit method, date=1999, ISSN=0273-0979, journal=Bull. Amer. Math. Soc. (N.S.), volume=36, number=4, pages=433488, review=MR # 2000h:22001,

KirGvi82book author=Kirillov, Alexander A., author=Gvishiani, Alexei D., title=Theorems and problems in functional analysis, series=Problem Books in Mathematics, publisher=Springer-Verlag, address=New York, date=1982,

Kisil08aarticle author=Kisil, Anastasia V., title=Isometric action of SL₂(ℝ) on homogeneous spaces, date=2010, journal=Advances in Applied Clifford Algebras, volume=20, number=2, pages=299312, note=E-print: arXiv:0810.0368,

Kisil93cincollection author=Kisil, Vladimir V., title=Clifford valued convolution operator algebras on the Heisenberg group. A quantum field theory model, date=1993, booktitle=Clifford algebras and their applications in mathematical physics, proceedings of the Third international conference held in Deinze, editor=Brackx, F., editor=Delanghe, R., editor=Serras, H., series=Fundamental Theories of Physics, volume=55, publisher=Kluwer Academic Publishers Group, address=Dordrecht, pages=287294, note=MR # 1266878,

Kisil95iarticle author=Kisil, Vladimir V., title=Möbius transformations and monogenic functional calculus, date=1996, ISSN=1079-6762, journal=Electron. Res. Announc. Amer. Math. Soc., volume=2, number=1, pages=2633, note=On-line, review=MR # MR1405966 (98a:47018),

Kisil96aarticle author=Kisil, Vladimir V., title=Plain mechanics: classical and quantum, date=1996, ISSN=0963-2654, journal=J. Natur. Geom., volume=9, number=1, pages=114, note=E-print: arXiv:funct-an/9405002, review=MR # MR1374912 (96m:81112),

Kisil97carticle author=Kisil, Vladimir V., title=Analysis in R^1,1 or the principal function theory, date=1999, ISSN=0278-1077, journal=Complex Variables Theory Appl., volume=40, number=2, pages=93118, note=E-print: arXiv:funct-an/9712003, review=MR # MR1744876 (2000k:30078),

Kisil97aincollection author=Kisil, Vladimir V., title=Two approaches to non-commutative geometry, date=1999, booktitle=Complex methods for partial differential equations (Ankara, 1998), series=Int. Soc. Anal. Appl. Comput., volume=6, publisher=Kluwer Acad. Publ., address=Dordrecht, pages=215244, note=E-print: arXiv:funct-an/9703001, review=MR # MR1744440 (2001a:01002),

Kisil98aarticle author=Kisil, Vladimir V., title=Wavelets in Banach spaces, date=1999, ISSN=0167-8019, journal=Acta Appl. Math., volume=59, number=1, pages=79109, note=E-print: arXiv:math/9807141, On-line, review=MR # MR1740458 (2001c:43013),

Kisil97barticle author=Kisil, Vladimir V., title=Umbral calculus and cancellative semigroup algebras, date=2000, ISSN=0232-2064, journal=Z. Anal. Anwendungen, volume=19, number=2, pages=315338, note=E-print: arXiv:funct-an/9704001. Zbl # 0959.43004, review=MR # MR1768995 (2001g:05017),

Kisil01amisc author=Kisil, Vladimir V., title=Spaces of analytical functions and wavelets—Lecture notes, date=2000–2002, note=92 pp. E-print: arXiv:math.CV/0204018,

Kisil02cincollection author=Kisil, Vladimir V., title=Meeting Descartes and Klein somewhere in a noncommutative space, date=2002, booktitle=Highlights of mathematical physics (London, 2000), publisher=Amer. Math. Soc., address=Providence, RI, pages=165189, note=E-print: arXiv:math-ph/0112059, review=MR # MR2001578 (2005b:43015),

Kisil01bincollection author=Kisil, Vladimir V., title=Tokens: an algebraic construction common in combinatorics, analysis, and physics, date=2002, booktitle=Ukrainian mathematics congress—2001 (ukrainian), publisher=Natsīonal. Akad. Nauk Ukraïni Īnst. Mat., address=Kiev, pages=146155, note=E-print: arXiv:math.FA/0201012, review=MR # MR2228860 (2007d:05010),

Kisil04darticle author=Kisil, Vladimir V., title=Monogenic calculus as an intertwining operator, date=2004, ISSN=1370-1444, journal=Bull. Belg. Math. Soc. Simon Stevin, volume=11, number=5, pages=739757, note=E-print: arXiv:math.FA/0311285, On-line, review=MR # MR2130636 (2006a:47025),

Kisil02earticle author=Kisil, Vladimir V., title=p-Mechanics as a physical theory: an introduction, date=2004, ISSN=0305-4470, journal=J. Phys. A, volume=37, number=1, pages=183204, note=E-print: arXiv:quant-ph/0212101, On-line. Zbl # 1045.81032, review=MR # MR2044764 (2005c:81078),

Kisil02ainproceedings author=Kisil, Vladimir V., title=Spectrum as the support of functional calculus, date=2004, booktitle=Functional analysis and its applications, series=North-Holland Math. Stud., volume=197, publisher=Elsevier, address=Amsterdam, pages=133141, note=E-print: arXiv:math.FA/0208249, review=MR # MR2098877,

Kisil04aarticle author=Kisil, Vladimir V., title=p-mechanics and field theory, date=2005, ISSN=0034-4877, journal=Rep. Math. Phys., volume=56, number=2, pages=161174, note=E-print: arXiv:quant-ph/0402035, On-line, review=MR # MR2176789 (2006h:53104),

Kisil05carticle author=Kisil, Vladimir V., title=A quantum-classical bracket from p-mechanics, date=2005, ISSN=0295-5075, journal=Europhys. Lett., volume=72, number=6, pages=873879, note=E-print: arXiv:quant-ph/0506122, On-line, review=MR # MR2213328 (2006k:81134),

Kisil06aarticle author=Kisil, Vladimir V., title=Erlangen program at large–0: Starting with the group SL₂(R), date=2007, ISSN=0002-9920, journal=Notices Amer. Math. Soc., volume=54, number=11, pages=14581465, note=E-print: arXiv:math/0607387, On-line, review=MR # MR2361159,

Kisil05barticle author=Kisil, Vladimir V., title=Fillmore-Springer-Cnops construction implemented in GiNaC, date=2007, ISSN=0188-7009, journal=Adv. Appl. Clifford Algebr., volume=17, number=1, pages=5970, note=Updated full text and source files: E-print: arXiv:cs.MS/0512073, On-line, review=MR # MR2303056,

Kisil09carticle author=Kisil, Vladimir V., title=Erlangen program at large—2 1/2: Induced representations and hypercomplex numbers, date=2009, journal=Izvestiya Komi nauchnogo centra UrO RAN, volume=5, number=1, pages=410, note=E-print: arXiv:0909.4464,

Kisil09aarticle author=Kisil, Vladimir V., title=Comment on “Do we have a consistent non-adiabatic quantum-classical mechanics?” by Agostini F. et al, date=2010, journal=Europhys. Lett. EPL, volume=89, pages=50005, note=E-print: arXiv:0907.0855,

Kisil09barticle author=Kisil, Vladimir V., title=Computation and dynamics: Classical and quantum, date=2010, journal=AIP Conference Proceedings, volume=1232, number=1, pages=306312, url=http://link.aip.org/link/?APC/1232/306/1, note=E-print: arXiv:0909.1594,

Kisil07ainproceedings author=Kisil, Vladimir V., title=Erlangen program at large—2: Inventing a wheel. The parabolic one, date=2010, booktitle=Trans. Inst. Math. of the NAS of Ukraine, series=Trans. Inst. Math. of the NAS of Ukraine, volume=7, pages=8998, note=E-print: arXiv:0707.4024,

Kisil05aarticle author=Kisil, Vladimir V., title=Erlangen program at large–1: Geometry of invariants, date=2010, journal=SIGMA, Symmetry Integrability Geom. Methods Appl., volume=6, number=076, pages=45 pages, note=E-print: arXiv:math.CV/0512416,

Kisil10aarticle author=Kisil, Vladimir V., title=Erlangen Programme at Large 3.1: Hypercomplex representations of the Heisenberg group and mechanics, date=2010, journal=submitted, note=E-print: arXiv:1005.5057,

Kisil09dinproceedings author=Kisil, Vladimir V., title=Wavelets beyond admissibility, date=2010, booktitle=Proceedings of the 10^th ISAAC Congress, London 2009, publisher=Imperial College Press, pages=6 pp., note=E-print: arXiv:0911.4701,

Kisil10carticle author=Kisil, Vladimir V., title=Covariant transform, date=2011, journal=Journal of Physics: Conference Series, volume=284, number=1, pages=012038, url=http://stacks.iop.org/1742-6596/284/i=1/a=012038, note=E-print: arXiv:1011.3947,

Kisil11barticle author=Kisil, Vladimir V., title=Erlangen Programme at Large 1.1: Integral transforms and differential operators, date=2011, journal=in preparation,

Kisil11aarticle author=Kisil, Vladimir V., title=Erlangen Programme at Large 3.2: Ladder operators in hypercomplex mechanics, date=2011, journal=Acta Polytechnica, note=(accepted). E-print: arXiv:1103.1120,

Kisil11cincollection author=Kisil, Vladimir V., title=Erlangen programme at large: an Overview, date=2012, booktitle=Advances in applied analysis, editor=Rogosin, S.V., editor=Koroleva, A.A., pages=165, note=E-print: arXiv:1106.1686,

Klauder96aarticle author=Klauder, John R., title=Coherent states for the hydrogen atom, date=1996, ISSN=0305-4470, journal=J. Phys. A, volume=29, number=12, pages=L293L298, url=http://dx.doi.org/10.1088/0305-4470/29/12/002, review=MR # 1398598 (97f:81052),

KlaSkag85book editor=Klauder, John R., editor=Skagerstam, Bo-Sture, title=Coherent states. applications in physics and mathematical physics., publisher=World Scientific Publishing Co., address=Singapur, date=1985,

KleinIbook author=Klein, Felix, title=Elementary mathematics from an advanced standpoint, publisher=Dover Publications Inc., address=Mineola, NY, date=2004, ISBN=0-486-43480-X, note=Arithmetic, algebra, analysis, Translated from the third German edition by E. R. Hedrick and C. A. Noble, Reprint of the 1932 translation, review=MR # 2098410,

KleinIIbook author=Klein, Felix, title=Elementary mathematics from an advanced standpoint, publisher=Dover Publications Inc., address=Mineola, NY, date=2004, ISBN=0-486-43481-8, note=Geometry, Translated from the third German edition and with a preface by E. R. Hendrik and C. A. Noble, Reprint of the 1949 translation, review=MR # 2078728 (2005c:01029),

KnappWallach76article author=Knapp, A.W., author=Wallach, N.R., title=Szegö kernels associated with discrete series, date=1976, journal=Invent. Math., volume=34, number=3, pages=163200,

KollarMori08book author=Kollár, János, author=Mori, Shigefumi, title=Birational geometry of algebraic varieties. With the collaboration of C. H. Clemens and A. Corti. Paperback reprint of the hardback edition 1998., language=English, publisher=Cambridge Tracts in Mathematics 134. Cambridge: Cambridge University Press. viii, 254 p., date=2008, note=Zbl # 1143.14014,

Konovenko08aarticle author=Konovenko, Nadiya G., title=Algebras of differential invariants for geometrical quantities on affine line., language=Ukrainian. English summary, date=2008, journal=Visn., Ser. Fiz.-Mat. Nauky, Kyïv. Univ. Im. Tarasa Shevchenka, volume=2008, number=2, pages=915,

KonovenkoLychagin08aarticle author=Konovenko, N.G., author=Lychagin, V.V., title=Differential invariants of nonstandard projective structures., language=Russian. English summary, date=2008, journal=Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky, volume=2008, number=11, pages=1013, note=Zbl # 1164.53313, review=MR # MR2529116,

Krein48aarticle author=Kreĭn, M. G., title=On Hermitian operators with directed functionals, date=1948, journal=Akad. Nauk Ukrain. RSR. Zbirnik Prac’ Inst. Mat., volume=1948, number=10, pages=83106, note=MR # 14:56c, reprinted in [],

KreinIIbook author=Kreĭn, M. G., title=Izbrannye Trudy. II, publisher=Akad. Nauk Ukrainy Inst. Mat., address=Kiev, date=1997, ISBN=5-7702-0681-0, note=MR # 96m:01030,

Rota95book editor=Kung, Joseph P.S., title=Gian-carlo rota on combinatorics: Introductory papers and commentaries, series=Contemporary Mathematicians, publisher=Birkhäuser Verlag, address=Boston, date=1995, volume=1,

Lang69book author=Lang, Serge, title=Algebra, publisher=Addison-Wesley, address=New York, date=1969,

Lang85book author=Lang, Serge, title=SL₂(R), series=Graduate Texts in Mathematics, publisher=Springer-Verlag, address=New York, date=1985, volume=105, ISBN=0-387-96198-4, note=Reprint of the 1975 edition, review=MR # 803508 (86j:22018),

LavrentShabat77book author=Lavrent’ev, M. A., author=Shabat, B. V., title=Problemy gidrodinamiki i ikh matematicheskie modeli. (Russian) [Problems of hydrodynamics and their mathematical models], edition=Second, publisher=Izdat. “Nauka”, Moscow, date=1977, review=MR # 56 #17392,

Lidskii66aarticle author=Lidskiĭ, V. B., title=On the theory of perturbations of nonselfadjoint operators, date=1966, ISSN=0044-4669, journal=Z. Vyčisl. Mat. i Mat. Fiz., volume=6, number=1, pages=5260, review=MR # 0196930 (33 #5114),

Litvinov05article author=Litvinov, G. L., title=The Maslov dequantization, and idempotent and tropical mathematics: a brief introduction, date=2005, ISSN=0373-2703, journal=Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), volume=326, number=Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 13, pages=145182, 282, note=E-print: arXiv:math/0507014, review=MR # MR2183219 (2006i:46104),

Mazorchuk09abook author=Mazorchuk, Volodymyr, title=Lectures on sl₂-modules, publisher=World Scientific, date=2009,

Miller68book author=Miller, Willard, Jr., title=Lie theory and special functions, publisher=Academic Press, address=New York, date=1968, note=Mathematics in Science and Engineering, Vol. 43, review=MR # 41 #8736,

Mirman01abook author=Mirman, R., title=Quantum field theory, conformal group theory, conformal field theory, publisher=Nova Science Publishers Inc., address=Huntington, NY, date=2001, ISBN=1-56072-992-9, note=Mathematical and conceptual foundations, physical and geometrical applications, review=MR # MR2145706 (2006j:81001),

MityushevRogosin00abook author=Mityushev, Vladimir V., author=Rogosin, Sergei V., title=Constructive methods for linear and nonlinear boundary value problems for analytic functions, series=Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, publisher=Chapman & Hall/CRC, Boca Raton, FL, date=2000, volume=108, ISBN=1-58488-057-0, note=Theory and applications, review=MR # 1739063 (2001d:30075),

MoroBurkeOverton97aarticle author=Moro, Julio, author=Burke, James V., author=Overton, Michael L., title=On the Lidskii-Vishik-Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure, date=1997, ISSN=0895-4798, journal=SIAM J. Matrix Anal. Appl., volume=18, number=4, pages=793817, url=http://dx.doi.org/10.1137/S0895479895294666, review=MR # 1471994 (98k:15014),

MotterRosa98article author=Motter, A. E., author=Rosa, M. A. F., title=Hyperbolic calculus, date=1998, ISSN=0188-7009, journal=Adv. Appl. Clifford Algebras, volume=8, number=1, pages=109128, review=MR # MR1648837 (99m:30099),

Niederer73aarticle author=Niederer, U., title=The maximal kinematical invariance group of the free Schrödinger equation, date=1972/73, ISSN=0018-0238, journal=Helv. Phys. Acta, volume=45, number=5, pages=802810, review=MR # 0400948 (53 #4778),

Nikolskii86book author=Nikol’skiĭ, N. K., title=Treatise on the shift operator, publisher=Springer-Verlag, address=Berlin, date=1986, ISBN=3-540-15021-8, note=Spectral function theory, With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller, Translated from the Russian by Jaak Peetre., review=MR # 87i:47042,

Olver93book author=Olver, Peter J., title=Applications of Lie groups to differential equations, edition=Second, publisher=Springer-Verlag, address=New York, date=1993, ISBN=0-387-94007-3; 0-387-95000-1, review=MR # 94g:58260,

Olver95book author=Olver, Peter J., title=Equivalence, invariants, and symmetry, publisher=Cambridge University Press, address=Cambridge, date=1995, ISBN=0-521-47811-1, review=MR # 96i:58005,

Olver99book author=Olver, Peter J., title=Classical invariant theory, series=London Mathematical Society Student Texts, publisher=Cambridge University Press, address=Cambridge, date=1999, volume=44, ISBN=0-521-55821-2, review=MR # MR1694364 (2001g:13009),

Perelomov86book author=Perelomov, A., title=Generalized coherent states and their applications, series=Texts and Monographs in Physics, publisher=Springer-Verlag, address=Berlin, date=1986, ISBN=3-540-15912-6, review=MR # 87m:22035,

Pimenov65aarticle author=Pimenov, R.I., title=Unified axiomatics of spaces with maximal movement group, language=Russian, date=1965, journal=Litov. Mat. Sb., volume=5, pages=457486, note=Zbl # 0139.37806,

PolyaSzegoIbook author=Pólya, George, author=Szegő, Gabor, title=Problems and theorems in analysis. I, series=Classics in Mathematics, publisher=Springer-Verlag, address=Berlin, date=1998, ISBN=3-540-63640-4, note=Series, integral calculus, theory of functions, Translated from the German by Dorothee Aeppli, Reprint of the 1978 English translation, review=MR # 1492447,

Pontryagin86abook author=Pontryagin, L. S., title=Obobshcheniya chisel, series=Bibliotechka “Kvant” [Library “Kvant”], publisher=“Nauka”, address=Moscow, date=1986, volume=54, review=MR # MR886479 (88c:00005),

Porteous95book author=Porteous, Ian R., title=Clifford algebras and the classical groups, series=Cambridge Studies in Advanced Mathematics, publisher=Cambridge University Press, address=Cambridge, date=1995, volume=50, ISBN=0-521-55177-3, review=MR # MR1369094 (97c:15046),

Sawyer82book author=Sawyer, W.W., title=Prelude to mathematics, series=Popular Science Series, publisher=Dover Publications, date=1982, ISBN=9780486244013,

Segal60book author=Segal, Irving E., title=Mathematical problems of relativistic physics, series=Proceedings of the Summer Seminar (Boulder, Colorado, 1960), publisher=American Mathematical Society, address=Providence, R.I., date=1963, volume=II,

Segal76book author=Segal, Irving Ezra, title=Mathematical cosmology and extragalactic astronomy, publisher=Academic Press [Harcourt Brace Jovanovich Publishers], address=New York, date=1976, note=Pure and Applied Mathematics, Vol. 68, review=MR # 58 #14894,

Shubin87book author=Shubin, M. A., title=Pseudodifferential operators and spectral theory, edition=Second, publisher=Springer-Verlag, address=Berlin, date=2001, ISBN=3-540-41195-X, note=Translated from the 1978 Russian original by Stig I. Andersson, review=MR # 2002d:47073,

SrivastavaTuanYakubovich00aarticle author=Srivastava, H. M., author=Tuan, Vu Kim, author=Yakubovich, S. B., title=The Cherry transform and its relationship with a singular Sturm-Liouville problem, date=2000, ISSN=0033-5606, journal=Q. J. Math., volume=51, number=3, pages=371383, url=http://dx.doi.org/10.1093/qjmath/51.3.371, review=MR # 1782100 (2001g:44010),

NagyFoias70book author=Sz.-Nagy, Béla, author=Foiaş, Ciprian, title=Harmonic analysis of operators on Hilbert space, publisher=North-Holland Publishing Company, address=Amsterdam, date=1970,

JTaylor72article author=Taylor, Joseph L., title=A general framework for a multi-operator functional calculus, date=1972, journal=Advances in Math., volume=9, pages=183252, review=MR # 0328625 (48 #6967),

MTaylor86book author=Taylor, Michael E., title=Noncommutative harmonic analysis, series=Mathematical Surveys and Monographs, publisher=American Mathematical Society, address=Providence, RI, date=1986, volume=22, ISBN=0-8218-1523-7, review=MR # 88a:22021,

ATorre08aarticle author=Torre, A, title=A note on the general solution of the paraxial wave equation: a Lie algebra view, date=2008, journal=Journal of Optics A: Pure and Applied Optics, volume=10, number=5, pages=055006 (14pp), url=http://dx.doi.org/10.1088/1464-4258/10/5/055006,

ATorre10aarticle author=Torre, A, title=Linear and quadratic exponential modulation of the solutions of the paraxial wave equation, date=2010, journal=Journal of Optics A: Pure and Applied Optics, volume=12, number=3, pages=035701 (11pp), url=http://stacks.iop.org/2040-8986/12/035701,

Tyrtyshnikov97abook author=Tyrtyshnikov, Eugene E., title=A brief introduction to numerical analysis, publisher=Birkhäuser Boston Inc., address=Boston, MA, date=1997, ISBN=0-8176-3916-0, review=MR # 1442956 (97m:65005),

Uspenskii88book author=Uspenskiĭ, V. A., title=CHto takoe nestandartnyi0 analiz? (Russian) [What is non-standard analysis?], publisher=“Nauka”, address=Moscow, date=1987, note=With an appendix by V. G. Kanoveĭ, review=MR # MR913941 (88m:26028),

VignauxDuranona35aarticle author=Vignaux, J. C., author=Durañona y Vedia, A., title=Sobre la teoría de las funciones de una variable compleja hiperbólica., language=Spanish, date=1935, journal=Univ. nac. La Plata. Publ. Fac. Ci. fis. mat., volume=104, pages=139183, note=Zbl # 62.1122.03,

Vilenkin68book author=Vilenkin, N. Ja., title=Special functions and the theory of group representations, publisher=American Mathematical Society, address=Providence, R. I., date=1968, note=Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22, review=MR # 37 #5429,

Wilson08abook author=Wilson, P. M. H., title=Curved spaces, publisher=Cambridge University Press, address=Cambridge, date=2008, ISBN=978-0-521-71390-0, note=From classical geometries to elementary differential geometry, review=MR # MR2376701,

Wulfman10abook author=Wulfman, Carl E., title=Dynamical Symmetry, publisher=World Scientific, date=2010, ISBN=978-981-4291-36-1, url=http://www.worldscibooks.com/physics/7548.html,

Yaglom79book author=Yaglom, I. M., title=A simple non-Euclidean geometry and its physical basis, publisher=Springer-Verlag, address=New York, date=1979, ISBN=0-387-90332-1, note=An elementary account of Galilean geometry and the Galilean principle of relativity, Heidelberg Science Library, Translated from the Russian by Abe Shenitzer, With the editorial assistance of Basil Gordon, review=MR # MR520230 (80c:51007),

Zejliger34book author=Zejliger, D.N., title=Kompleksnaya linei0chataya geometriya. Poverhnosti i kongruencii. (Russian) [Complex lined geometry. Surfaces and congruency], publisher=GTTI, address=Leningrad, date=1934, note=195 pp.,

	elliptic	parabolic	hyperbolic
cosine	[−1,1]	(−∞,1]	[1,∞)
sine	[−1,1]	(−∞,∞)	(−∞,∞)

Notion	Yaglom’s usage	This work
Circle	Defined as a locus of equidistant points in metric d(u,v; u′,v′)=\| u−u′ \|. Effectively is a pair of vertical lines.	A limiting case of p-cycles with n=0. Form a Möbius invariant subfamily of self-adjoint p-cycles (Definition 7). In this case all three centres coincide. We use term “circle” only to describe a drawing of cycle in the elliptic point space ℝ^e.
Cycle	Defined as locus of point having fixed angle view to a segment. Effectively is a non-degenerate parabola with a vertical axis.	We use this word for a point of the projective cycle space ℙ³ . Its drawing in various point spaces can be a circle, parabola or hyperbola, single or pair of lines, single point or an empty set.
Centre	Absent, Yaglom’s cycles are “centreless”.	A cycle has there EPH centres.
Diameter	A quarter of the focal length of the parabola.	The distance between real roots.
Special lines	Vertical lines, special role reflects absolute time in Galilean mechanics.	The intersection of invariant sets of self-adjoint and zero radius p-cycles, i.e. having the form (1,l,0,l²).
Orthogonal, perpendicular	The relation between two lines, if one of them is special. Delivers the shortest distance.	We have a variety of various orthogonality and perpendicularity relations, which are not necessary local, symmetric.
Inversion in circles .	Defined through the degenerated p-metric	Conjugation with a degenerate parabola (n=0).
Reflection in cycles	Defined as a reflection in the parabola along the special lines.	Composition of conjugation with three parabolas, see Exercise 7.

Representations	—	Function Theory	—	Type of Spectrum


discrete series	—	elliptic	—	discrete spectrum
principal series	—	hyperbolic	—	continuous spectrum
complementary series	—	parabolic	—	residual spectrum

Erlangen program at large

Vladimir V. Kisil School of Mathematics, University of Leeds, Leeds LS2 9JT, UK email: kisilv@maths.leeds.ac.uk Web: http://www.maths.leeds.ac.uk/~kisilv/

September 29, 2011

Contents

Preface

Part I Geometry

Lecture 1 Erlangen Programme: Preview

1.1 Make a Guess in Three Attempts

1.2 Covariance of FSCc

1.3 Invariants: Algebraic and Geometric

1.4 Joint Invariants: Orthogonality

1.5 Higher Order Joint Invariants: Focal Orthogonality

1.6 Distance, Length and Perpendicularity

1.7 Erlangen Programme at Large

Lecture 2 Groups and Homogeneous Spaces

2.1 Groups and Transformations

2.2 Subgroups and Homogeneous Spaces

2.2.1 From a Homogeneous Space to the Isotropy Subgroup

2.2.2 From a Subgroup to the Homogeneous Space

2.3 Differentiation on Lie Groups and Lie Algebras

2.3.1 One-parameter Subgroups and Lie Algebras

2.3.2 Invariant Vector Fields and Lie Algebras

2.3.3 Commutator in Lie Algebras

2.4 Integration on Groups

Lecture 3 Homogeneous Spaces from the Group SL(2,R)

3.1 The Affine Group and the Real Line

3.2 One-dimensional Subgroups of SL2(ℝ)

3.3 Two-dimensional Homogeneous Spaces

3.3.1 From the Subgroup K

3.3.2 From the Subgroup N′

3.3.3 From the Subgroup

3.3.4 Unifying All Three Cases

3.4 Elliptic, Parabolic and Hyperbolic

3.5 Orbits of the Subgroup Actions

3.6 Unifying EPH Cases: The First Attempt

3.7 Isotropy Subgroups

Lecture 4 Extended Fillmore–Springer–Cnops Construction

4.1 Invariance of Cycles

4.2 Projective Spaces of Cycles

4.3 Covariance of FSCc

4.4 Origins of FSCc

4.4.1 Projective Coordiantes and Homegeneous Polynomials

4.4.2 Coadjoint Representation

Lecture 5 Indefinite Product Space of Cycles

5.1 Cycles: an Appearance and the Essence

5.2 Cycles as Vectors

5.3 Invariant Cycle Product

5.4 Zero-radius Cycles

5.5 Cauchy–Schwartz Inequality and Tangent Cycles

Lecture 6 Joint Invariants of Cycles: Orthogonality

6.1 Orthogonality of Cycles

6.2 Orthogonality Miscellanea

6.3 Ghost Cycles and Orthogonality

6.4 Actions of FSCc Matrices

6.5 Inversions and Reflections in Cycles

6.6 Higher Order Joint Invariants: Focal Orthogonality

Lecture 7 Metric Invariants in Upper Half-Planes

7.1 Distances

7.2 Lengths

7.3 Conformal Properties of Moebius Maps

7.4 Perpendicularity and Orthogonality

7.5 Infinitesimal Radius Cycles

7.6 Infinitesimal Conformality

Lecture 8 Global Geometry of Upper Half-Planes

8.1 Compactification of the Point Space

8.2 (Non)-Invariance of The Upper Half-Plane

8.3 Optics and Mechanics

8.3.1 Optics

8.3.2 Classical Mechanics

8.3.3 Quantum Mechanics

8.4 Relativity of the Space-Time

Lecture 9 Conformal Unit Disk

9.1 Elliptic Cayley Transforms

9.2 Hyperbolic Cayley transform

9.3 Parabolic Cayley Transforms

9.4 Cayley Transforms of Cycles

Lecture 10 Geodesics

10.1 Metric, Curves’ Length and Extrema

10.2 Invariant Metrics

10.3 Geodesics: Additivity of Metric

Vladimir V. Kisil
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
email: kisilv@maths.leeds.ac.uk
Web: `http://www.maths.leeds.ac.uk/~kisilv/`

Part I
Geometry

3.2 One-dimensional Subgroups of SL₂(ℝ)

Part II
Analytic Functions

Part III
Functional Calculi and Spectra

Part IV
Quantum Mechanics