Notes on a particular Weyl Algebra

TL;DR

This paper explores the structure of Weyl algebras using quantum group theory concepts, revealing new examples with physical relevance derived from groupoid models inspired by quantum mechanics.

Contribution

It introduces a novel approach to understanding Weyl algebras through cross product algebras in quantum groups, connecting algebraic structures with physical models.

Findings

Identifies Weyl algebra structures as quantum groups via cross product algebras.

Provides examples like the Drinfeld quantum double and the generalized Schrödinger representation.

Links algebraic structures to physical models in quantum mechanics.

Abstract

By means of the notions of cross product algebras of the theory of quantum groups, in the context of classical Hopf algebra structures, we deduce some known structures of Weyl algebras type (as the Drinfeld quantum double, the restricted Heisenberg double, the generalized Schroedinger representation, and so on) that may be considered as a non-trivial examples of quantum groups having physical meaning, starting from a particular example of groupoid motivated by elementary quantum mechanics.