| The aim of this dissertation is to carry out a series of researches on the construction of Yetter-Drinfel’d modules,four-angle Hopf modules and Yetter-Drinfel’d-Long bimoduls of Hopf type algebras and the related braided monoidal categories from the following parts:First,we introduce the notion of a four-angle Hopf module over Hom-Hopf algebra(H,β)and show that the categoryHHMHH of four-angle Hopf modules is a monoidal category with either a Hom-tensor product (?)H or a Hom-cotensor product □H as a monoidal structure.We then prove the category yDHH of Yetter-Drinfel’d modules with bijective structure map can be organized as a braided monoidal category,in which we use a new monoidal structure,and describe an equivalence between the monoidal category(HHMHH,(?)H)or(HHMHH,□H)of four-angle Hopf modules,and the monoidal category(yDHH,(?)H)of Yetter-Drinfel’d modules.Furthermore,we give a braiding structure of the monoidal categories(HHMHH,(?)H)and(HHMHH,□H),respectively.Second,let H be a Hom-Hopf T-coalgebra over a group π.We introduce and study the left-right α-Yetter-Drinfel’d category yD(H)α over H,with α∈G,and construct a class of new braided T-categories.Next,we define the quasitriangular structure of H and give some basic properties and the related results.Furthermore,the Drinfel’d structure D(H)of H is constructed,and an equivalent relation between the category yD(H)and the representation category of D(H)is obtained.Third,let H be a Hopf algebra and LR(H)the category of Yetter-Drinfel’d-Long bimodules over H.We give sufficient and necessary conditions for LR(H)to be symmetry and pseudosymmetry,respectively.We then introduce the definition of the u-condition in LR(H)and discuss the relation between the u-condition and the symmetry of LR(H).Furthermore,we show that LR(H)over a triangular(cotriangular,resp.)Hopf algebra contains a symmetric subcategory.Finally,let H be a weak Hopf quasigroup.We give more interested properties of H,introduce the notion of Yetter-Drinfel’d weak quasimodule over H and prove that the category W2yDHH of Yetter-Drinfel’d weak quasimodules is braided monoidal category under suitable conditions.Furthermore,we describe the notion of coquasitriangular weak Hopf quasigroup(H,σ),and study a relation between Yetter-Drinfel’d weak quasimodule and the coinvariant space of right H-comodule over(H,σ). |
