The PBW Theorem and simplicity criteria for the Poisson enveloping algebra and the algebra of Poisson differential operators

TL;DR

This paper establishes a Poisson analogue of the Poincaré-Birkhoff-Witt theorem, providing presentations, simplicity criteria, and explicit formulas for the Poisson enveloping algebra and Poisson differential operators, with applications to finite type algebras.

Contribution

It introduces new presentations and criteria for Poisson enveloping algebras and Poisson differential operators, extending classical theorems to Poisson algebra context.

Findings

Proves a Poisson analogue of the Poincaré-Birkhoff-Witt theorem.

Provides explicit generators and relations for Poisson enveloping algebras.

Establishes simplicity and isomorphism criteria for these algebras.

Abstract

For an arbitrary Poisson algebra $\CP$ over an arbitrary field, an (analogue of) the Poincar\'{e}-Birkhof-Witt Theorem is proven and several presentations/constructions for its Poisson enveloping algebra $\CU (\CP )$ are given. As a result, explicit sets of generators and defining relations are given for $\CU (\CP )$ and the algebra $P\CD (\CP)$ of Poisson differential operators on $\CP$. Simplicity criteria for the algebras $\CU (\CP )$ and $P\CD (\CP )$ are given. In the case when the algebra $\CP$ is of essentially finite type, a criterion for the algebra $\CU (\CP )$ to be a domain is presented and a criterion for a natural epimorphism $\CU (\CP )\ra P\CD (\CP )$ to be an isomorphism is given. The kernel of the epimorphism is described and for large classes of Poisson algebras an explicit set of generators is given. Explicit formulae for the Gelfand-Kirillov dimension of the algebras $\CU (\CP )$ and $P\CD (\CP)$ are given. In the case when the Poisson algebra $\CP$ is a regular domain of essentially finite type an explicit simplecticity criterion for $\CP$ is found and a criterion is presented for the algebra $\CU (\CP )$ to be isomorphic to the algebra $\CD (\CP )$ of differential operators on $\CP$.