On Noncommutative Poisson Algebras

Poisson algebras appears naturally from Poisson geometry. A vector space A over a field K is called a Poisson algebra, if A is both an associative algebra and a Lie algebra, and the Leibniz rule holds. In Poisson geometry, A is commutative in associative sense, because A is the algebra of smooth functions on a Poisson manifold. With the development of noncommutative geometry in recent two decades, noncommutative Poisson algebras were widely investigated by some mathematicians. Naturally, there are two ways to investigate the structure of Poisson algebra. One is to discuss the Lie bracket on an associative algebra such that the Leibniz rule holds. The other is to determine the associative multiplication on a Lie algebra satisfying the Leibniz rule. In the thesis, our idea is from the first respective, and mainly investigate the Poisson algebra without identity in associative sense and Poisson structure on a class of quiver algebras. It consists of the following three parts.In the first chapter, we recall some notion, notations, researching history and some background of Poisson algebras. Furthermore, we recall the quiver theory of associative algebras, and some results on Poisson structures over quiver algebras.Chapter 2 deals with Poisson algebra without identity in associative sense. By the standard technique, we construct a new Poisson algebra with identity such that the first Poisson can be viewed as an ideal of the second one. Moreover, we discuss the relation between the categories of Poisson modules over these Poisson algebras.In the third chapter, we mainly discuss the Poisson structure on truncated basic cycles. By the quiver technique, we show that all Poisson structure on truncated basic cycles are inner, and obtain all Poisson structure.