Noncommutative Poisson Algebras

Poisson algebra plays an important role in the study of Poisson geome-try and quantum groups. Poisson manifolds are known as smooth manifolds equipped with a Poisson structure. And then different versions of commu-tative and noncommutative Poisson algebras are introduced and studied by many authors from different perspectives.In this paper,we begin our work at the commutative Poisson algebras, and then we present some different kinds of Poisson algebras and the modules over noncommutative Poisson algebras. Our main result is to show the universal property of the Poisson enveloping algebra in the noncommutative case.This paper is divided into three parts.In the first chapter,we introduce the background, the main work and the concerning basic concepts in the paper.In the second chapter, on the basis of commutative Poisson algebras,we present some different generalized versions of Poisson algebras and then we show the relationships between them and commutative Poisson algebras.In the third chapter,we show the definition of modules over noncommu-tative Poisson algebras and the definition of quasi-Poisson enveloping algebra. And then we pay our main attention on the universal property of the quasi-Poisson enveloping algebra. In the end, we present a definition of one-sided Poisson modules of noncommutative Poisson algebras and show some proper-ties of it.