Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 076, 45 pages      math.CV/0512416     https://doi.org/10.3842/SIGMA.2010.076

Erlangen Program at Large-1: Geometry of Invariants

Vladimir V. Kisil
School of Mathematics, University of Leeds, Leeds LS29JT, UK

Received April 20, 2010, in final form September 10, 2010; Published online September 26, 2010

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 SL2(R) group. We describe here geometries of corresponding domains. The principal rôle 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.

Key words: analytic function theory; semisimple groups; elliptic; parabolic; hyperbolic; Clifford algebras; complex numbers; dual numbers; double numbers; split-complex numbers; Möbius transformations.

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