# The generalized Weyl Poisson algebras and their Poisson simplicity criterion

## TL;DR

This paper introduces generalized Weyl Poisson algebras, provides a criterion for their Poisson simplicity, and describes their Poisson centers, expanding the understanding of Poisson algebra structures.

## Contribution

It defines a new class of Poisson algebras and establishes a Poisson simplicity criterion, linking them to generalized Weyl algebras.

## Key findings

- Poisson simplicity criterion established
- Explicit descriptions of Poisson centre provided
- Numerous examples analyzed

## Abstract

A new class of Poisson algebras, the class of {\em generalized Weyl Poisson algebras}, is introduced. It can be seen as Poisson algebra analogue of generalized Weyl algebras or as giving a Poisson structure to (certain) generalized Weyl algebras. A Poisson simplicity criterion is given for generalized Weyl Poisson algebras and explicit descriptions of the Poisson centre and the absolute Poisson centre are obtained. Many examples are considered.