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Erlangen program at large:
SL2(ℝ) case study

Vladimir V. Kisil

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:

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.

YouTube playlist with lectures on the group SL2(ℝ):

YouTube playlist with lectures on the Heisenberg group::

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

Download moebinv


Vladimir V. Kisil. Meeting Descartes and Klein somewhere in a noncommutative space. In A. Fokas, J. Halliwell, T. Kibble, and B. Zegarlinski, editors, Highlights of mathematical physics (London, 2000), pages 165–189. Amer. Math. Soc., Providence, RI, 2002. arXiv:math-ph/0112059.
Vladimir V. Kisil. Erlangen program at large–0: Starting with the group SL2(R). Notices Amer. Math. Soc., 54(11):1458–1465, 2007. arXiv:math/0607387, On-line. Zbl # 1137.22006.
Vladimir V. Kisil. Schwerdtfeger–Fillmore-Springer-Cnops construction implemented in GiNaC. Adv. Appl. Clifford Algebr., 17(1):59–70, 2007. On-line. Updated full text, source files, and live ISO image: arXiv:cs.MS/0512073. Project page Zbl # 05134765.
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 1–94. Birkhäuser Verlag, Basel, 2012. arXiv:1106.1686.
Vladimir V. Kisil. Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL2(R). Imperial College Press, London, 2012. Includes a live DVD. Zbl # 1254.30001.

Figure 1: Unification of elliptic, parabolic and hyperbolic cycles

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Last modified: January 21, 2019.
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