The Factorization For A Class Of Hom-algebras And Hom-coalgebras

In recent years,the research about Hom-type structures has become important in the theory of Hopf algebra.Hom-(co)algebra is actually the generalization of(co)algebra.Here the(co)associativity is replaced by the Hom-(co)associativityα(a)(bc)=(ab)α(c)(β(c1)(?)C21(?)C22 = C11(?)c12(?)β(C2)),where a:A → A(β C → C)is a linear map.In the sense of Hom-structure,on one hand we focus on what the structure of crossed coproduct and cleft coextension are and what the relation between them is,on the other hand we.discuss under what conditions a coalgebra can be factorized into the crossed coproduct.This thesis is organized as follows:In the first chapter,some preliminary definitions and basic results are given.Hom-(co)algebra,Hom-Hopf-algebra,Hom-Hopf-module and Hom-module coalgebra are includ-ed.In the second chapter,we first define a weak action on the Hom-comodule algebra with Hom-Hopf module structure,and obtain the Hom-crossed product.Then we discuss the relation between the cleft extension and Hom-crossed product,the cleft extension and Hom-Hopf module.In the third chapter,we introduce the definitions of weak coaction,Hom-crossed co-product and cleft coextension,discuss the necessary and sufficient conditions for C(?)λ H to be Hom-crossed coproduct and give some examples of Hom-crossed coproduct.Afterwards we discuss the relations between the cleft coextension and Hom-crossed coproduct,the cleft coextension and Hom-Hopf module.In the fourth chapter,we obtain the Hom-algebra(Hom-coalgebra)factorization for Hom-comodule algebra(Hom-module coalgebra)with Hom-Hopf module structure,that is B(?)A□ ρH as Hom-bialgebra.