Meeting Descartes and Klein Somewhere in a Noncommutative Space

Abstract

We combine the coordinate method and Erlangen program in the framework of noncommutative geometry through an investigation of symmetries of noncommutative coordinate algebras. As the model we use the coherent states construction and the wavelet transform in functional spaces. New examples are a three dimensional spectrum of a non-normal matrix and a quantisation procedure from symplectomorphisms. Keywords: Heisenberg group, special linear group, symplectic group, Hardy space, Segal-Bargmann space, Clifford algebra, Cauchy-Riemann-Dirac operator, Möbius transformations, functional calculus, Weyl calculus (quantization), quantum mechanics, Schrödinger representation, metaplectic representation.