Abstract: The Erlangen programme of F. Klein (influenced by S. Lie) defines geometry as a study of invariants under a certain transitive group action. This approach proved to be fruitful much beyond the traditional geometry. For example, special relativity is the study of invariants of Minkowski space-time under the Lorentz group action. Another example is complex analysis as study of objects invariant under the conformal maps.

These notes systematically apply the Erlangen approach to various areas of mathematics. In the first instance we consider the group SL₂(ℝ) in details as well as the corresponding geometrical and analytical invariants with their interrelations. Consequently the course has a multi-subject nature touching algebra, geometry and analysis.

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.

• Contents
• Preface

• Erlangen Programme: Preview
  • Make a Guess in Three Attempts
  • Covariance of FSCc
  • Invariants: Algebraic and Geometric
  • Joint Invariants: Orthogonality
  • Higher Order Joint Invariants: Focal Orthogonality
  • Distance, Length and Perpendicularity
  • Erlangen Programme at Large
• Groups and Homogeneous Spaces
  • Groups and Transformations
  • Subgroups and Homogeneous Spaces
  • Differentiation on Lie Groups and Lie Algebras
  • Integration on Groups
• Homogeneous Spaces from the Group SL(2,R)
  • The Affine Group and the Real Line
  • One-dimensional Subgroups of SL₂(ℝ)
  • Two-dimensional Homogeneous Spaces
  • Elliptic, Parabolic and Hyperbolic
  • Orbits of the Subgroup Actions
  • Unifying EPH Cases: The First Attempt
  • Isotropy Subgroups
• Extended Fillmore–Springer–Cnops Construction
  • Invariance of Cycles
  • Projective Spaces of Cycles
  • Covariance of FSCc
  • Origins of FSCc
• Indefinite Product Space of Cycles
  • Cycles: an Appearance and the Essence
  • Cycles as Vectors
  • Invariant Cycle Product
  • Zero-radius Cycles
  • Cauchy–Schwartz Inequality and Tangent Cycles
• Joint Invariants of Cycles: Orthogonality
  • Orthogonality of Cycles
  • Orthogonality Miscellanea
  • Ghost Cycles and Orthogonality
  • Actions of FSCc Matrices
  • Inversions and Reflections in Cycles
  • Higher Order Joint Invariants: Focal Orthogonality
• Metric Invariants in Upper Half-Planes
  • Distances
  • Lengths
  • Conformal Properties of Moebius Maps
  • Perpendicularity and Orthogonality
  • Infinitesimal Radius Cycles
  • Infinitesimal Conformality
• Global Geometry of Upper Half-Planes
  • Compactification of the Point Space
  • (Non)-Invariance of The Upper Half-Plane
  • Optics and Mechanics
  • Relativity of the Space-Time
• Conformal Unit Disk
  • Elliptic Cayley Transforms
  • Hyperbolic Cayley transform
  • Parabolic Cayley Transforms
  • Cayley Transforms of Cycles
• Geodesics
  • Metric, Curves’ Length and Extrema
  • Invariant Metrics
  • Geodesics: Additivity of Metric
  • Geometric Invariants
• Unitary Rotations
  • Unitary Rotations—an Algebraic Approach
  • Unitary Rotations—a Geometrical Viewpoint
  • Rebuilding Algebraic Structures from Geometry
  • Invariant Linear Algebra
  • Linearisation of the exotic form

• Representation Theory
  • Representations
  • Decomposition of Representations
  • Schur’s Lemma
  • Induced Representations
• Wavelets on Groups
  • Wavelet Transform on Groups
  • Square Integrable Representations
  • Fundamentals of Wavelets on Homogeneous Spaces
• Linear Representations
  • Hypercomplex Characters
  • Induced Representations
  • Similarity and Correspondence: Ladder Operators
• Covariant Transform
  • Extending Wavelet Transform
  • Examples of Covariant Transform
  • Symbolic Calculi
  • Inverse Covariant Transform
• Analytic Functions
  • Induced Covariant Transform
  • Induced Wavelet Transform and Cauchy Integral
  • The Cauchy-Riemann (Dirac) and Laplace Operators
  • The Taylor Expansion
  • Wavelet Transform in the Unit Disk and Other Domains

• Covariant and Contravariant Calculi
  • Functional Calculus as an Algebraic Homomorphism
  • Intertwining Group Actions on Functions and Operators
  • Jet Bundles and Prolongations
  • Spectrum and Spectral Mapping Theorem
  • Functional Model and Spectral Distance
  • Covariant Pencils of Operators

• Harmonic Oscillator and Ladder Operators
  • Harmonic Oscillator
  • Heisenberg Group and Its Automorphisms
  • Ladder Operators in Quantum Mechanics
  • Ladder Operators for the Hyperbolic Subgroup
  • Ladder Operator for the Nilpotent Subgroup
  • Similarity and Correspondence
• Open Problems
  • Geometry
  • Analytic Functions
  • Functional Calculus
  • Quantum Mechanics
• Supplementary Material
  • Dual and Double Numbers
  • Classical Properties of Conic Sections
  • Comparison with Yaglom’s Book
  • Other Approaches and Results
  • FSCc with Clifford Algebras
• How to Use Software
  • Viewing Colour Graphics
  • Installation of CAS
  • Using the CAS and Computer Exercises
  • Library for Cycles
  • Predefined Object on Initialisation
• Index

This document was translated from L^AT_EX by H^EV^EA.