DG Poisson algebra and its universal enveloping algebra

• DOI: 10.1007/s11425-016-5127-4
• Publication Date: 2016-01-22T13:25:47Z
• Authors: JiaFeng Lü; XingTing Wang; GuangBin Zhuang
• Keywords: Rings and Algebras (math.RA); FOS: Mathematics; Mathematics - Rings and Algebras; 0101 mathematics; 01 natural sciences

Accepted by Science China Mathematics<br/>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.<br/>