Twisted Modules, Twisted Comodules And Relative Yetter-Drinfel'd Modules On Hopf Algebra

Walter [1] first introduced the concept of twisted product in the way which he used the coaction of algebra of paired algebras. Dually, the author in [3] introduced the concept of twisted coproduct. They used the method of twisting to construct a new bialgebra from an old bialgebra and proved that using the way, we could naturally get smash product, Drinfel'd double, smash coproduct and generalized double crossproduct. Since modules and comodules are generalizations of algebras and coalgebras, in this paper, the method of twisting is used into modules and we can obtain a few of nice results.In section one, we first introduce skew paired Hopf modules and skew copaired Hopf modules which generalize skew paired bialgebras and skew copaired bialgebras. Then, we respectively give the neccessary and sufficient conditions which make them become skew paired Hopf modules or skew copaired Hopf modules. At last, we discuss their duality. Namely, if (M, N) is skew copaired Hopf modules, (M~°, N~°) is skew paired Hopf modules. On the contrary, if (M,N) is skew paired Hopf modules and satisfies σ*(1) ∈ H~° (?) K~° and γ*(1) ∈ M~° (?) N~°, (M~°, N~°) is skew copaired Hopf modules.In section two, under skew paired bialgebras, we use the method of twisting to construct a new module from an old module. Similarly, under skew paired bialgebras, we can get a new comodule from an old comodule by using the same method. Dually, under skew copaired bialgebras, we can get the similar results. In the end of the section, we obtain two important generalizations. We can get the conclusion that under a family of paired bialgebras or copaire bialgebras (H_i; K_i), i = 1,... , n, the result of twisting as a unit is the same as that of twisting step by step (theorem 2.3.1 and theorem 2.3.3).In section three, the author in [13] discussed the relation between Yetter-Drinfel'd modules and braided Hopf algebra. Dually, in the section, we discuss the relation between Yetter-Drinfel'd modules and quasitriangular Hopf algebra. In the end we give the duality between H and H~°.In section four, firstly, we introduce the concept of α-paired Hopf algebra and discuss some of its properties. We mainly show the fundamental structure theorem of twisted Hopf modules(theorem 4.1.5). Secondly, we also discuss relative Yetter-Drinfel'd modules and point out the double crossed product is the very relative Yetter- Drinfel'd modules. Thirdly, we introduce the concept of H-Hopf YD modules. Lastly, we consider the fact...