(Quasi-)Poisson enveloping algebras

TL;DR

This paper introduces quasi-Poisson and Poisson enveloping algebras for non-commutative Poisson algebras and establishes categorical equivalences with module categories, advancing the algebraic framework of non-commutative Poisson structures.

Contribution

It defines new enveloping algebras for non-commutative Poisson algebras and proves their module categories are equivalent to categories of modules over these enveloping algebras.

Findings

Categories of quasi-Poisson modules are equivalent to modules over the quasi-Poisson enveloping algebra.

Categories of Poisson modules are equivalent to modules over the Poisson enveloping algebra.

The framework extends the algebraic understanding of non-commutative Poisson structures.

Abstract

We introduce the quasi-Poisson enveloping algebra and Poisson enveloping algebra for a non-commutative Poisson algebra. We prove that for a non-commutative Poisson algebra, the category of quasi-Poisson modules is equivalent to the category of left modules over its quasi-Poisson enveloping algebra, and the category of Poisson modules is equivalent to the category of left modules over its Poisson enveloping algebra.