Generalized Drinfel’d Double And Relative Research Of The Category Of Yetter-Drinfel’d Modules

The aim of this dissertation is to carry out a series of further research on Hom-Hopf algbebras and the category of Hom-Yetter Drinfel’d modules from the following aspects.In chapter 2, we will introduce the notion of smash coproduct on Hom-Hopf algebra and construct the Drinfel’d double in the setting of Hom-Hopf algebras by generalizing Majid’s bicrossproduct for Hopf algebras and by introducing the notion of dual pairs. As an application we will consider the relation between the Drinfel’d double and Heisenberg double.In chapter 3, we will introduce the notion of Hom-type Doi-Hopf module and monoidal Doi-Hopf datum such that the category of Doi-Hopf modules becomes monoidal. Then we introduce the braiding in this category and give the necessary and sufficient conditions to make the category braided. Moreover we show that when the category of Hom-type Doi-Hopf modules is braided monoidal, it corresponds to a module category over a quasitriangular Hom-bialgebra.In chapter 4, we study the symmetry and pseudosymmetry in the Hom Yetter-Drinfel’d category over (H,α). We also introduce the notion of u-condition. Assume that (H,α) itself is an object in HHYD, and we prove that if the braiding cH,H is symmetric, H is forced to be triangular and cotriangular.In chapter 5, we introduce the notion of Hom-L-R compatible pair, and show that under the compatible pair, Hom-L-R smash product and Hom-L-R smash coproduct make a Hom-bialgebra. Moreover we prove that Hom-L-R compatible pair in fact corresponds to a bialgebra in some prebraided monoidal category. Then we point out that this category is actually a Hom Yetter-Drinfel’d category.