H-dimodule Algebras And Crossed Products, Yetter Drinfel'd Modules And α-H-Hopf Modules

In section 1, we first introduce the crossed product in the category of H-dimodule algebras and give the sufficient and necessary conditions for the crossed product of two H-dimodulc algebras to be an H-dimodule algebra (Theorem 1.8), as a result, we generalize the results in [1].In section 2, firstly, from different perspectives we introduce two notions, Hopf Ycttcr Drinfel'd (C, H)-modulcs and (C-D,H)-bicomodules, then we show that the two categories above both form monoidal categories and discuss the equivalence between these categories. Secondly, we study the dual of (α,β)-Ycttcr Drinfel'd modules and give the sufficient and necessary conditions for H to be an (α,β)-Yetter Drinfcl'd module, generalizing the results in [2].Using for the reference of the thinking in [3], we first introduce the notion of α-H-Hopf modules, and introduce two-sided α-H-Hopf modules, two-cosided α-H-Hopf modules, two sided two cosided (α,β)-H-Hopf modules in succession. We mainly verify that both the category of two-sided α-H-Hopf modules and the category of two-cosided α-H-Hopf modules form T-categorics over the group G (Theorem 3.12 and Theorem 3.19). Finally, we show that the category of two-sided two-cosided (α,β)-H-Hopf modules and the category of (α,β)-Ycttcr Drinfel'd modules are equivalent (Theorem 3.23).