The Construction Of Poisson Algebra

Poisson algebra first appears in Hamiltonian mechanics.It is is an important part of research on quantum groups.It is both an associative algebra and a Lie algebra.The Lie bracket and associative multiplication satisfy Leibniz rule.With the development of noncommutative geometry,noncommutative Poisson algebra has become one of the focuses of Poisson research.This paper mainly explores the construction of some noncommutative Poisson algebra.The construction of Poisson algebra can be considered from two aspects:In the first aspect,we define the Lie bracket satisfying Leibniz rule on associative algebra.Secondly,we define the associative algebra structure satisfying Leibniz rule on Lie algebra.From these two perspectives,the main results are obtained in this paper.The first chapter is the preparatory knowledge,and also introduces the background knowledge and literature review of Poisson algebra.In the second chapter,we introduce the construction of Poisson algebra structure on associative algebras.First of all,we introduce the standard Poisson structure and the construction method by defining Poisson brackets on the generator and then extending to this algebra.Then,we recall some research results of Poisson polynomial rings.On the basis of previous work,we give Poisson structure on ore extension.In the third chapter,we recall the construction of Poisson algebra structure on Virasoro Lie algebra.The Poisson structure on this kind of Lie algebras is mainly constructed by the root hierarchical method.By this method,we further generalize the structure of Poisson algebra on Lie algebra W(a,b).On the basis of previous work,we also obtain Poisson algebra structure on Block Lie algebra and its central extension.