Erlangen Program at Large-1: Geometry of Invariants

• Main Author: Vladimir V. Kisil
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2010-09-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: analytic function theory; semisimple groups; elliptic; parabolic; hyperbolic; Clifford algebras; complex numbers; dual numbers; double numbers; split-complex numbers; Möbius transformations
• Online Access: http://dx.doi.org/10.3842/SIGMA.2010.076

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.