The Invertible Coalgebras And Relative Hom-Hopf Modules

In this paper, we mainly study the invertible coalgebras and rational modules de-fined on it. We also study the Hom-module coalgebras and relative Hom-Hopf modules. The paper is mainly made up of four chapters:In chapter 1. we introduce the definitions of Hom-algebras, Hom-coalgebras, Hom-modules and Hom-comodules.In chapter 2. we define the invertible Hom-coalgebras. and study the relations be-tween rational left C*-Hom-modules and right C-Hom-comodules. Furthermore, we study the basic properties on rational C*-Hom-modules.In chapter 3, we define Hom-comodule algebras, (A,B)-Hom-Hopf-modules and give the sufficient condition about the Hom-Hopf module M to be a injective A-Hom-module. Furthermore, we prove the Mo(?)B0 B which is regarded as (A.B)-Hom-Hopf-module can be embedded in M.In chapter 4, we study the relations between the category of (A,B)-Hom-Hopf mod-ules and the category of B0-Hom-modules via category theory.