Two Approaches to Non-Commutative Geometry

Abstract

Looking to the history of mathematics one could find out two outer approaches to Geometry. First one (algebraic) is due to Descartes and second one (group-theoretic)--to Klein. We will see that they are not rivalling but are tied (by Galois). We also examine their modern life as philosophies of non-commutative geometry. Connections between different objects (see keywords) are discussed. Keywords: Heisenberg group, Weyl commutation relation, Manin plain, quantum groups, SL(2, R), Hardy space, Bergman space, Segal-Bargmann space, Szeg"o projection, Bergman projection, Clifford analysis, Cauchy-Riemann-Dirac operator, Moebius transformations, functional calculus, Weyl calculus (quantization), Berezin quantization, Wick ordering, quantum mechanics.