Erlangen Program at Large-1: Geometry of Invariants

Erlangen Program at Large-1: Geometry of Invariants

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 Clif...

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Bibliographic Details
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
Description
Summary: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.
ISSN:1815-0659

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