  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.
