Relative Hopf Modules In Yetter-Drinfeld Category

Relative Hopf modules are generalizations of Hopf modules. In this paper, we discuss the structures of two kinds of relative Hopf modules in Yetter-Drinfeld category, and we will give the fundamental structure theorem.In the introduction, we state some background and present situation of relative Hopf modules in Yetter-Drinfeld category.In the third section, we give the definition and some propositions of (A, B )-Hopf module in Yetter-Drinfeld category, and show the fundamental structure theorem: In Yetter-Drinfeld category, if B is a right ,4-comodule algebra, and there exists a right A -comodule map,which is a algebra map. Then M as (A, B )-Hopf module for all (A, B )-Hopf module M in Yetter-Drinfeld category. In addition, we also discuss the relationship between YD and YD, presenting the conditions of the both being equivalent categories.The section 4 is the dual of the third section. We give some propositions of [C, A]-Hopf module in Yetter-Drinfeld category. For example, we get equivalent conditions about [C,A]-Hopf modules being projective A -modules and the fundamental structure theorem in Yetter-Drinfeld category YD.In section 5, we change the conditions of important theorems which are in section 3 and 4. We prove that the fundamental structure theorem still hold if A is a Hopf algebra, and B a cleft comodule algebra or if A is a Hopf algebra, and C a cocleft module coalgebra.In section 6, we study some propositions of antipode and distinguished grouplike elements on braided Hopf algebra and we get the following results: if A is a braided Hopf algebra with antipode s, then s4 is inner. Besides we also get the sufficient and necessary conditions that s2is inner in a finite dimensional Hopfalgebra. That is, s2is inner is equivalent to there being such that where t is a integral in A.