Semiclassical limits of Ore extensions and a Poisson generalized Weyl algebra

TL;DR

This paper explores the connection between Ore extensions and Poisson polynomial extensions, constructing a Poisson generalized Weyl algebra and analyzing its structure and symmetries in the semiclassical limit.

Contribution

It introduces a Poisson version of the quantum generalized Weyl algebra and characterizes its simplicity and endomorphisms, linking quantum and Poisson algebra structures.

Findings

Poisson polynomial extensions are semiclassical limits of Ore extensions

Constructed a Poisson generalized Weyl algebra $A_1$

Established conditions for $A_1$ to be Poisson simple

Abstract

We observe \cite[Proposition 4.1]{LaLe} that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra $A_1$ considered as a Poisson version of the quantum generalized Weyl algebra is constructed and its Poisson structures are studied. In particular, it is obtained a necessary and sufficient condition such that $A_1$ is Poisson simple and established that the Poisson endomorphisms of $A_1$ are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra.