The Duality Of π-Modules, Hopf π-algebras And Left,A-Hopf Modules In Right H-Hopf Module Categories

The notions of Hopf π-coalgebras. where π is a discrete group, generalize those of Hopf algebras. Hopf π-coalgebras were used by V. G. Turaev to construct Hennings-like and Kuperberg-like invariants of principal π-bundles over link complements and over three-manifolds. This paper mainly studies the duality of some algebraic structures. Firstly, it constructs the dual structures of π-algebras.π-modules and Hopf π-algebras, respectively, and discusses the relative properties. Then, when H is cocommutative, it studies the Hopf algebras and left A-Hopf modules in the right H-Hopf module category MHH.Finally. it shows the structure theorem of left A-Hopf modules in MHH.The thesis consists of three chapters as follows:In chapter1. we review the basic definitions and notions used in this paper, such as π-algebras.π-modules. Hopf π-algebras. Hopf π-ideals and right H-Hopf module categories. etc.In chapter2, we show that the existence of Hopf π-coalgebras dual to Hopf π-algebras (Theorem2.3.2). To prove this result, we construct a π-coalgebra A0dual to a π-algebra A(Theorem2.1.4). Also we show the existence of π-comodules dual to π-modules(Theorem2.2.1). Furthermore, we study the relationship between the π-submodule and the π-subcomodule. the π-ideal and the π-coideal, respectively. And we prove that one dual structure of π-module maps is a π-comodule map.In chapter3, let H be a cocommutative Hopf algebra over the ground field K with a bijective antipode SH, we discuss Hopf algebras and left A-Hopf modules in the right H-Hopf module category MHH.Firstly, we introduce the concept of the Hopf algebra in MHH. If A, C are finite dimensional Hopf algebras in MHH, we find that the duality A*and Hom(C, A) also are Hopf algebra. respectively(Theorem3.1.4.3.1.6). Meanwhile, we show that the tensor product of two Hopf algebras in MHH is a Hopf algebra(Theorem3.1.7).Secondly, we give the notion of a left A-Hopf module and two examples about it. It also be proved self-dual in the case of finite dimensional(Theorem3.2.6). at the end of the paper, we obtain the structure theorem of left A-Hopf modules in MHH(Theorem3.2.7).