Some Research Of Braided Hopf Algebras

This thesis is related to the theory of braided Hopf algebras. In this paper, westudy the Hom functor of the braided Yetter-Drinfeld category, give an example ofduality theorem of braided Hopf algebra, the double factorization of braided Hopfalgebra and braided (co-)ribbon algebra.This paper is divided into four parts.First, we give some basic definitions and lemmas from the theory of braided tensorcategory, and study duality theorem, Hom functors and factorization of braided Hopfalgebras in it and so on.Second, we obtain new Hopf algebras structure for fixed Hopf algebras throughbraided reconstruction, which generalize the conclusion of S.Majid.Then, we define the braided (co-)ribbon algebra, and prove that its module cate-gory is a ribbon category in the forth chapter, which generalize the definition and itsrelated theory of Kassel's. Then, we use an example to prove that the ribbon structureof a (co-)Quasitriangular braided Hopf algebra can be constructed.In this paper, we used braided diagrams more extensively: Prove the closed ofHom functors in braided Yetter-Drinfeld category(i.e. if V,Ware in this category, thenHom(V,W) is in this category too), the braided reconstruction theorems by it. Onthe other hand, we also define the braided (co-)ribbon algebras and prove the maintheorem by braided diagrams.