Yetter-Drinfeld Moduels And Relative Structures Over Weak Hopf π Coalgebra

This paper is to continue [25]'s work. The main result is as follows: First, we construct Yetter-Drinfeld moduels and weak center over weak hopf π coal-gebra. Based on the study of their properties, we get conclusion that they are isomoriphic. Morover, we study weak relative Hopf modules over weak hopf π coalgebra and build Maschke theorem. So we generalize the result of [4]and[22]. The whole paper consists of 5 parts.Chapter 1 is the introduction of this paper, which introduces the weak hopf algebra, hopf π coalgebra, and some properties of them.In chapter 2, we introduce weak hopf π coalgebra and two kinds of special subalgebra over it.Some basic properties are studied to make a basis for the following chapter.In chapter 3, we discuss how to establish Yetter-Drinfeld moduels, and classify it into four types of left-left, left-right, right-right, right-left types. Try to get equivalence conditions and get further realization.In chapter 4, we retroduce monoial category and center over it. moreover, we introduce weak center over weak hopf π coalgebra. After studying of the relationship between it and Yetter-Drinfeld moduels, we could found they are isomophic.In charpter 5, we study other type modules on weak hopf π coalgebra,that is weak relative hopf π comodules. In this charpter, we introduce the trace and the center to prove Maschke theorem under the given conditions.