The Poisson enveloping algebra and the algebra of Poisson differential operators of a generalized Weyl Poisson algebra

TL;DR

This paper studies the structure and properties of the Poisson enveloping algebra and Poisson differential operators of a generalized Weyl Poisson algebra, providing generators, relations, simplicity criteria, and dimension calculations.

Contribution

It introduces explicit generators and relations for the Poisson enveloping algebra and establishes criteria for simplicity, minimal primes, and regularity of these algebras.

Findings

Explicit generators and relations for $ ext{CU}(A)$ are provided.

Simplicity criteria for $ ext{CU}(A)$ and $P ext{CD}(A)$ are established.

Gelfand-Kirillov dimensions are calculated for these algebras.

Abstract

For a generalized Weyl Poisson algebra $A$, explicit sets of generators and defining relations are presented for its Poisson enveloping algebra $\CU (A)$. Simplicity criteria are given for the algebra $\CU (A)$ and algebra of Poisson differential operators $P\CD (A)$ on $A$. The Gelfand-Kirillov dimensions of the algebras $\CU (A)$ and $P\CD (A)$ are calculated. It is proven that the algebra $\CU (A)$ is a domain provided that the coefficient ring $D$ of the generalized Weyl Poisson algebra $A$ is a domain of essentially finite type over a perfect field. For the algebra $A$, the set of its minimal primes and the prime radical are described and an equidimensionality criterion is given. For the equidimensional algebra $A$ of essentially finite type, two regularity criteria are presented.