DG Poisson algebra and its universal enveloping algebra  

作　　者：LU JiaFeng WANG XingTing ZHUANG GuangBin 

机构地区：[1]Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China [2]Department of Mathematics, Temple University, Philadelphia 19122, USA [3]Department of Mathematics, University of Southern California, Los Angeles 90089-2532, USA

出　　处：《Science China Mathematics》2016年第5期849-860,共12页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11571316 and 11001245);Natural Science Foundation of Zhejiang Province(Grant No.LY16A010003)

摘　　要：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.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.

关 键 词：differential graded algebras differential graded Hopf algebras differential graded Lie algebras differential graded Poisson algebras universal enveloping algebras monoidal category 

分 类 号：O152.5[理学—数学] O141.4[理学—基础数学]