The Structure Of Comodule Over Hopf Algebra And Weak Hopf Algebra

The notion of a Hopf algebra was developed by algebraic topologists abstracting the work of H.Hopf on manifolds in 1941. Starting with issue of the article "On the structure of Hopf algebras", written by J.Milnor and J.Moore in 1965, Hopf algebras became a subject of study from a strictly algebraic point of view. By the end of the 1990s, research in this field was given a strong boost by the appearance of quantum groups and the development of the action theory of Hopf algebras which unifies earlier results known for group actions, actions of Lie algebras, and graded rings. Since the 1990s, there appear many Hopf algebraic constructions, such as weak Hopf algebra, quasi-Hopf algebra, Hopf coalgebra and so on.Weak Hopf algebras have been proposed by G.Bohm, K.Szlachanyi first, it is the promotion of Hopf algebra. Weak Hopf algebra has strict mathematical and physical background, Such as surface algebra, Quantum group and so on are all the example of weak Hopf algebra.In recent years, weak Hopf algebra obtained great achievements which attracted more attention of mathematicians. This makes the development of weak Hopf algebra is particularly fast, compared with other Hopf algebra is concerned, weak Hopf algebra is a coassociative algebra. This enables us to define coaction as Hopf algebra does.This paper are divided into three chapters.The first chapter is preliminaries, including the background of Hopf algebra and weak Hopf algebra, and preliminary notions and lemmas.In the second part, we introduce co-convolution comodule and Hopf module, first, we discuss how to construct comodule with co-convolution, and then we extend this construction to Hopf-module.In the third part, firstly, We give some basic properties of weak Hopf algebra and its dual algebra, then we introduce some formulas of two special kinds of subalgebras over weak Hopf algebra, Ht and Hs, Finally we focuses on studying the structure of module and comodule over weak Hopf algebra.