Erlangen Program at Large-1: Geometry of Invariants
arXiv:math/0512416 · doi:10.3842/SIGMA.2010.076
Abstract
This paper presents geometrical foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theories based on the representation theory of SL(2,R) group. We describe here geometries of corresponding domains. The principal role is played by Clifford algebras of matching types. In this paper we also generalise the Fillmore-Springer-Cnops construction which describes cycles as points in the extended space. This allows to consider many algebraic and geometric invariants of cycles within the Erlangen program approach. For an easy-reading introduction see arXiv:math/0607387. An outline of the whole approach is given in arXiv:1006.2115.
AMS-LaTeX, 47 p, 80 PS graphics in 19 figures; v2: minor corrections v3: a substantial revision; v4 & v5: small improvements; v6: revised sections on lengths, infinitesimal cycles, parabolic Cayley transform; v7, v8 & v9: numerous minor improvements and updates; v10: the final version published in SIGMA; v11: the reference to Schwerdtfeger's book is added
References in corpus (13)
- Conformal compactification and cycle-preserving symmetries of spacetimes
- Erlangen Program at Large-1: Geometry of Invariants
- Meeting Descartes and Klein Somewhere in a Noncommutative Space
- Erlangen Program at Large-0: Starting with the group SL(2,R)
- Erlangen Program at Large--2: Inventing a wheel. The parabolic one
- How Many Essentially Different Functional Theories Exist?
- Isometric action of SL(2,R) on homogeneous spaces
- Elliptic, Parabolic and Hyperbolic Analytic Function Theory--0: Geometry of Domains
- Schwerdtfeger-Fillmore-Springer-Cnops Construction Implemented in GiNaC
- Clifford Algebras and Possible Kinematics
- Spaces of Analytical Functions and Wavelets--Lecture Notes
- Noncommutative space-time models
- Erlangen Program at Large: Outline
Cited by in corpus (6)
- Erlangen Programme at Large: An Overview
- Erlangen Program at Large-1: Geometry of Invariants
- Schwerdtfeger-Fillmore-Springer-Cnops Construction Implemented in GiNaC
- MoebInv: C++ libraries for manipulations in non-Euclidean geometry
- Towards a Quantum Erlangen Program
- The Vlasov-Poisson equation, the Moebius Geometry and then-body problem in a negative space form