DG Poisson algebra and its universal enveloping algebra

TL;DR

This paper introduces DG Poisson algebras and their universal enveloping algebras, establishing their properties and categorical equivalences, with applications in differential geometry and homological algebra.

Contribution

It constructs the universal enveloping algebra for DG Poisson algebras and proves its universal property and categorical equivalences, extending the theory of Poisson algebras.

Findings

Universal enveloping algebra $A^{ue}$ has a natural DG algebra structure.

Category of DG Poisson modules over $A$ is equivalent to DG modules over $A^{ue}.

Universal enveloping algebra behaves well under opposite algebra and tensor product.

Abstract

In this paper, we introduce the notions of differential graded (DG) Poisson algebra and DG Poisson module. Let $A$ be any DG Poisson algebra. We construct the universal enveloping algebra of $A$ explicitly, which is denoted by $A^{ue}$. We show that $A^{ue}$ has a natural DG algebra structure and it satisfies certain universal property. As a consequence of the universal property, it is proved that the category of DG Poisson modules over $A$ is isomorphic to the category of DG modules over $A^{ue}$. Furthermore, we prove that the notion of universal enveloping algebra $A^{ue}$ is well-behaved under opposite algebra and tensor product of DG Poisson algebras. Practical examples of DG Poisson algebras are given throughout the paper including those arising from differential geometry and homological algebra.