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.
 Publication:

arXiv eprints
 Pub Date:
 March 2014
 arXiv:
 arXiv:1403.3130
 Bibcode:
 2014arXiv1403.3130L
 Keywords:

 Mathematics  Rings and Algebras;
 16E45;
 16S10;
 17B35;
 17B63
 EPrint:
 37 pages, the abstract is rewritten, another construction of the universal enveloping algebra is given and several typos are fixed