Yetter-drinfeld Category Symmetry, Weak Hopf Algebras

This paper consists of two parts. The first part is devoted to study the symmetries of the categories of Yetter-Drinfeld module and quantum Yetter-Drinfeld module over weak Hopf algebras . It generalizes the theories of Yetter-Drinfeld modules on Hopf algebras, and associates the weak Doi-Hopf modules , quantum Yetter-Drinfeld modules with relative weak Hopf modules. In the second part, it generalizes two theorems on Hopf algebras and constructs a new way. Then we can obtain a class of coalgebras by this way in the representative categories of weak Hopf algebras.In section 1, prilementaries. It introduces the defination and the elementary properties of weak Hopf algebras. Then it gives the defintion of module algebra , comodule algebra , module coalgebra over weak Hopf algebras. At last, it introduces the representative categories of weak Hopf algebras. It paves the road for the further study.In section 2, it discusses the relative weak Hopf module and weak Doi-Hopf module. And it gives the relation between them. The results of Hopt algebra are generalized to weak Hopf algebra. The main result is as follows:Proposition 2.1.4 If if is weak Hopf algebra and the antipode S is bijective , then H^op is a weak Hopf algebra with S^-1 as an antipode and there exists an equivalence of categoriesIn section 3, it first gives some examples of Yetter-Drinfeld modules. Then it introduces the defintion of braided weak Hopf algebra, and gives a class of Yetter-Drinfeld modules. At last , it constructs the concept of u-condition and studies symmetaries of Yetter-Drinfeld module categories over weak Hopf algebras. The main results are as follows:Proposition 3.1.10 If (H, R) is quasitriangular, then for each M G h-M-, definePF.{m) = SJ?(2) ? i?(1) ? morp'R{m) = SS'(JR(1)) J?(2) ? m.Then < M, -,pr >, < M, -,p'R >G ^D, the familly of all < M, -,pR > or < M, ?, p'R > is a subcategory of #3^D- If (#, R) is triangular , then two subcategories are symmetric.Proposion 3.2.3 Let M,N G ^yD, and assume M and iV satisfies the u-condition , then M t N satisfies u-condition if and only if aM,Mls a symmetry.Corollary 3.2.5 Let M € j^yD, so that M satisfies the u-condition, and ffAf.Afis a symmetry, then for all n > l,M?nalso satisfies the u-condition.Proposition 3.2.6 Let if be a weak Hopf algebra with a bijective antipode , for all < H,m,p >G j^yD, where m is the left multiplication of H, if an./fis a symmetry , then there existsR E A°v(l)(H ?K H)A(l),Re withIn section 4, it introduces different types of Yetter-Drinfeld module over weak Hopf algebras. Then the quantum Yetter-Drinfeld modules are constructed. The relation among weak Hopf algebra , quantum Yetter-Drinfeld module and relative weak Hopf module are given out. The main results are as follows:Proposition 4.2.3 Let G = (H, A, C) is a weak Yetter-Drinfeld datum,?-module M is a left A—module and a right C—comodule, then M is a quantum Yetter-Drinfeld module if and only if for all m G M, a G A, we have...