Erlangen program at large:
SL₂(ℝ) case study

The Erlangen program of F. Klein (influenced by S. Lie) defines geometry as a study of invariants under a certain 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.

In this course we consider in detail the group SL₂(ℝ) and the corresponding geometrical and analytical invariants with their interrelations. Consequently the course has a multi-subject nature touching algebra, geometry and analysis. No special knowledge beyond a standard undergraduate curriculum is required.

The considered topics are:

• the group SL₂(ℝ) and Möbius transformations of the real line.
• Complex, dual and double numbers and Clifford algebras with two generators.
• Iwasawa decomposition of SL₂(ℝ).
• Möbius transformations in the upper half-plane.
• Cycles (quadrics) as geometric SL₂(ℝ)-invariants.
• Fillmore–Springer–Cnops construction and algebraic invariants of cycles.
• Linearisation of the Möbius transformations.
• Linearised Möbius actions in the spaces of functions: the Hardy and Bergman spaces.
• The Cauchy integral formula as a wavelet transform.
• The Cauchy–Riemann and Laplace equations from invariant vector fields.
• The Laurent and Taylor expansions over eigenvectors of rotations.
• A functional calculus as an intertwining operator.
• Prolongation of representations and functional calculus of non-selfadjoint operators.

Lectures notes for this course are in the process of writing now. The current version can be browsed on-line as Web pages split per chapter, or as the huge single Web page, or downloaded as a PDF file.

There is an interactive Computer Algebra System (CAS) [3] associated to this work, to get started read information about installation and usage.

References

[1]

Vladimir V. Kisil. Meeting Descartes and Klein somewhere in a noncommutative space. In Highlights of mathematical physics (London, 2000), pages 165–189. Amer. Math. Soc., Providence, RI, 2002. E-print: arXiv:math-ph/0112059.

[2]

Vladimir V. Kisil. Erlangen program at large–0: Starting with the group SL₂(R). Notices Amer. Math. Soc., 54(11):1458–1465, 2007. E-print: arXiv:math/0607387, On-line.

[3]

Vladimir V. Kisil. Fillmore-Springer-Cnops construction implemented in GiNaC. Adv. Appl. Clifford Algebr., 17(1):59–70, 2007. Updated full text and source files: E-print: arXiv:cs.MS/0512073, On-line.

[4]

Vladimir V. Kisil. Erlangen programme at large: an Overview. In S.V. Rogosin and A.A. Koroleva, editors, Advances in Applied Analysis, chapter 1, pages 1–65. 2012. E-print: arXiv:1106.1686.

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Figure 1: Unification of elliptic, parabolic and hyperbolic cycles

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