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.

LaTeX 25p, amsproc style, 5 PS pics

Cited by in corpus (10)

• Erlangen Programme at Large: An Overview
• Erlangen Program at Large-1: Geometry of Invariants
• Erlangen Program at Large-0: Starting with the group SL(2,R)
• Elliptic, Parabolic and Hyperbolic Analytic Function Theory--0: Geometry of Domains
• Tokens: An Algebraic Construction Common in Combinatorics, Analysis, and Physics
• Spaces of Analytical Functions and Wavelets--Lecture Notes
• Erlangen Program at Large: Outline
• Relationships Between Quantum and Classical Mechanics using the Representation Theory of the Heisenberg Group
• Spectrum as the Support of Functional Calculus
• Monogenic Calculus as an Intertwining Operator