Erlangen Program at Large-1: Geometry of Invariants

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 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.

Keywords

• analytic function theory
• semisimple groups
• elliptic
• parabolic
• hyperbolic
• Clifford algebras
• complex numbers
• dual numbers
• double numbers
• split-complex numbers
• Möbius transformations