Two Approaches to Non-Commutative Geometry

Details Two Approaches to Non-Commutative Geometry Two Approaches to Non-Commutative Geometry

1997-03-04

arxiv.org/abs/funct-an/9703001v3

H.~Begehr et al eds., Complex Methods for Partial Differential Equations, Ch~14, pp 219--248, Kluwer, 1999

Mathematics

Operator Algebras

05.02.98: minor corrections

Scientific paper

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.

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