Universal enveloping algebras of differential graded Poisson algebras

Abstract

In this paper, we introduce the notion of differential graded Poisson algebra and study its universal enveloping algebra. From any differential graded Poisson algebra $A$, we construct two isomorphic differential graded algebras: $A^e$ and $A^E$. It is proved that the category of differential graded Poisson modules over $A$ is isomorphic to the category of differential graded modules over $A^e$, and $A^e$ is the unique universal enveloping algebra of $A$ up to isomorphisms. As applications of the universal property of $A^e$, we prove that $(A^e)^{op}\cong (A^{op})^e$ and $(A\otimes_{\Bbbk}B)^e\cong A^e\otimes_{\Bbbk}B^e$ as differential graded algebras. As consequences, we obtain that ``$e$'' is a monoidal functor and establish links among the universal enveloping algebras of differential graded Poisson algebras, differential graded Lie algebras and associative algebras.