The Duality Of π-comodules And Anti-Yetter-Drinfeld Module For T-coalgebras

The notions of Hopfπ-coalgebras, whereπis a discrete group, generalize those of Hopf algebras. Hopfπ-coalgebras were used by V. G. Turaev to construct Hennings-like and Kuperberg-like invariants of principalπ-bundles over link complements and over three-manifolds. This paper mainly studies the duality of anotherπ-comodules. The anti-Yetter-Drinfeld module will be extended on T-coalgebras. And we dicuss the properties of anti-Yetter-Drinfeld modules and the isomorphism on T-coalgebras. At last, we prove that Hom(M, N) is anti-Yetter-Drinfeld module, too.The thesis consists of three chapters as follows:In chapter 1, we review the basic definitions and notions used in this paper, For ex-ample,π-coalgebras,π-algebras,7r-comodule, Hopfπ-coalgebras, T-coalgebras and so on.In chapter 2, This chaper give another definition of 7r-comodule, and generalize the duality properties of comodules to thisπ-comodules. We prove that the duality of rightπ-C-comodules are the rightπ-C*-module.Suppose M be a rightπ-C-comodule, N is the subspace of M, we discuss the duality properties betweenπ-subcomodule N and 7r-submodule N1.In chapter 3, Firstly, on the basis of introducing 7r-comodules and T-coalgebras, we generalize the anti-Yetter-Drinfeld module on T-coalgebras. We define the left-left a-anti-Yetter-Drinfeld module on T-coalgebras, and give the condition of equivalence of consistency condition. Likewise, we give the definition of left-right a-anti-Yetter-Drinfeld module on T-coalgebras and its condition of compatibility.Particularly, we prove that M(?)N areαβ-anti-Yetter-Drinfeld module where M isα-anti-Yetter-Drinfeld module and N isβ-anti-Yetter-Drinfeld module. Moreover,βM isβαβ-1-anti-Yetter-Drinfeld module and the important isomorphism theorem(3.2.7).At last, we prove that Hom(M, N) is left-left a-anti-Yetter-Drinfeld module where M, N are left-left a-anti-Yetter-Drinfeld module.