Erlangen Program at Large-1: Geometry of Invariants

[摘要] 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.

[关键词] analytic function theory;semisimple groups;elliptic;parabolic;hyperbolic;Clif ford algebras;complex numbers;dual numbers;double numbers;split-complex numbers;M¨obius transformations