Braided T-categories Over Hom-Hopf Algebras And Related Action Research

The main research objects of this paper are braided Turaev categories, monoidal Hom-Hopf algebras, twisted Yetter-Drinfeld Hom-modules and Hom-Hopf group coalge-bras. The main contents are arranged as follows:Chapter 2, we introduce the notions of twisted Yetter-Drinfeld Hom-modules for monoidal Hom-Hopf algebras and monoidal Hom-entwining structures, present the related examples of the twisted Yetter-Drinfeld Hom-modules. In addition, we takes advantage of such twisted Yetter-Drinfeld Hom-module categories to construct a new class of braided Turaev categories.Chapter 3, we propose the notion of the generalzied diagonal crossed Hom-product over monoidal Hom-Hopf algebras, and prove the left module category of suach generalzied diagonal crossed Hom-product is isomorphic to the twisted Yetter-Drinfeld Hom-module category. Furthermore, we study the isomorphism theory between the twisted Yetter-Drinfeld Hom-module category and the general Yetter-Drinfeld Hom-module category, and construct a braided Turaev category over additive group (Z,+).Chapter 4, we introduce the definitions of twisted Yetter-Drinfeld Hom-modules over Hom-Hopf algebras and Hom-Hopf group coalgebras, and construct a new braided Turaev categories using the twisted Yetter-Drinfeld Hom-module categories. In addition, we show that this new braided Turaev category is isomorphic to the representation category of Hom-Hopf group algebras.Chapter 5, we propose the notion of the generalized double crossproduct for monoidal Hom-Hopf algebras, present the condition for this double crossproduct to be a monoidal Hom-Hopf algebra, and construct the coquasitriangular structure. Moreover, we construct a monoidal Hom-category over the generalized double crossproduct, and establish the quantum Yang-Baxter Hom-operators.