Iteration Triangle Homogeneous Hom - Hopf Algebra

Coquasitriangular monoidal Hom-Hopf algebras is a twist generalization of coquasitriangular Hopf algebras. All kinds of these algebras, monoidal Hom-algebras morphisms and monoidal Hom-coalgebras morphisms are characterized in terms of changes of certain morphisms.In fact, the structure of coquasitriangular keeps when generalizing coquasitriangular Hopf algebras to coquasitriangular monoidal Hom-Hopf algebras. Therefore, we can find the examples of coquasitriangular monoidal Hom-Hopf algebras. And we will construct the example in the chapter 3.Monoidal Hom-Hopf algebras is an important algebras construction, it provids interpretation in the aspect of category. In order to check if the construction provids a solution for the generalized Hom-Yang-Baxter equations. In the chapter 3, we will study the coquasitriangular of monoidal Hom-Hopf algebras and construct the solution of the Hom-Yang-Baxter equations. Furthermore, we can induce a braiding on the category H(MH) of right H-Hom-comodules, find out a ralation of equivalence between the braiding and the coquasitriangular of monoidal Hom-Hopf algebras.In the chapter 4, we will discuss the Hom-2-cocycle of monoidal Hom-bialgebras.2-cocycle is an important method and tool to construct structure of new algebras. As an special structure, it has good characters. In order to prove another important conclusion,we check the inversion of Hom-2-cocycle has the similar characters. Take advantage of Hom-2-cocycle to induce a new multiplication, we can get new monoidal Hom-bialgebras and the construction keeps tensor structure of comodule category. The map in Hom(H ?H, κ) that is induced at the same time can be proved that it is a Hom-coquasitriangular of new monoidal Hom-bigebras.