Hom-entwined Modules And Hom-Doi-Hopf Modules

In recent years, Hom-algebras as a class of deformation algebras have attracted the attention of many mathematical scholars. Hom-(co)algebra is a weakening concept of (co)algebra. The (co)associativity is replaced by the Hom-(co)associativity: α(a)(bc)= (ab)α(c), (α(a1)(?) a21 (?) a22= a11 (?)a12 (?) α(a2)). In this paper, the relationships between the Hom-entwining structures and Hom-smash (co)product、Hom-Doi-Hopf module and crossed H-Hom-module are studied. The main contents are as follows:(1) Firstly, the Hom-entwining structures and Hom-Doi-Hopf structures are introduced and the concepts of Hom-entwined module and Hom-Doi-Hopf-module are given. Then, the relationship between Hom-entwined module and Hom-smash (co)product is studied: if A is Hom-algebra, C is a finitely generated and projective Hom-coalgebra, then there exists a bijective correspondence between left-right Hom-entwining structures·HE·(κ) of (A, C, ψ) and Hom-smash product structure C*#sA of (A, C*, R). It is also showed that left-right Hom-entwined module categories AHM(ψ)C and right C*#sA-Hom-module categories MC*#sA are isomorphic.(2) The relationships between Hom-Doi-Hopf module and crossed H-Hom-module are discussed:Hom-Doi-Hopf module category AHD(H)C and cross H-Hom-module category AHM(Hop (?) H)C are isomorphic (Theorem 4.1.3).(3) The relationship between Hom-Doi-Hopf module and Hom-smash (co)product are obtained:if (C,αC) is a right H-Hom-comodule coalgebra, (D,α.D) is a right H-Hom-module coalgebra, then the category HM(H)CD and HMD(?)C are isomorphic(Theorem4.2.3). Let (B, αB) be right H-Hom-comodule algebra, (D, αD) left H-Hom-module coalgebra, un-der certain conditions,HM.(H)D is the left B#D*-Hom-module category for (?)M ∈B. If D is finitely generated and projective, then BHM(H)D is equal to B#D*M.