The Structure Of Poisson Algebras On Sweedler 4-Dimensional Hopf Algebras

Hopf algebras are involved in many areas of mathematical research,and are actually quantum groups.Since the concept of Hopf algebras was introduced,a famous mathematician Sweedler,elaborated and studied the theories and methods of Hopf algebras,and published a lot of original research results.This laid a foundation for the later studies of Hopf algebras.As is well known that group algebra and the universal enveloping algebra of Lie algebra are the common examples of Hopf algebras.In addition,Sweedler 4-dimensional Hopf algebra,as the lowest dimension Hopf algebra which is neither commutative nor cocommutative,is also concerned by people.Poisson geometry was originated from the studies of Hamiltonian mechanics,and then Poisson structures were endowed with algebraic properties,this is,so-called Poisson algebras.Many studies have shown that Poisson algebra plays an important role in the studies of mathematics and mathematical physics.In recent years,many experts began to pay attention to the studies of Poisson Hopf algebras,and obtained a number of meaningful and valuable results.In this paper,Poisson algebras and Poisson Hopf algebras are constructed in view of Sweedler 4-dimensional Hopf algebra.This thesis is divided into three chapters:In Chapter 1,we give the background of this paper,and introduce the current development of Poisson algebras and Poisson Hopf algebras.Then we give the main results of this paper as well as preliminary knowledge.In Chapter 2,we give the construction of Poisson algebras on dendriform algebras,Rota-Baxter system and Sweedler algebras,respectively.In Chapter 3,we construct Poisson Hopf algebra on Sweedler 4-dimensional Hopf algebra.