The question of how the category of entwined modules can be made into a braided monoidal category is studied. First, the sufficient and necessary conditions making the category into a monoidal category are obtained by using the fact that if （A, C, ψ） is an entwining structure, then A × C can be made into an entwined module. The conditions are that the algebra and coalgebra in question are both bialgebras with some extra compatibility relations. Then given a monodial category of entwined modules, the braiding is constructed by means of a twisted convolution invertible map Q, and the conditions making the category form into a braided monoidal category are obtained similarly. Finally, the construction is applied to the category of Doi-Hopf modules and （α, β ）-Yetter-Drinfeld modules as examples.
Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of the crossed product in the classic Hopf algebra theory, first, let A be a k-linear category and H be a Hopf algebra, and the two crossed products A#_σH and A#＇_σH are isomorphic under some conditions. Then, let A#_σH be a crossed product category for a finite dimensional and semisimple Hopf algebra H. If V is a left A#σH-module and WC V is a submodule such that W has a complement as a left A-module, then W has a complement as a A#_σH-module.
Let H be a Hopf algebra and HYD the Yetter- Drinfeld category over H. First, the enveloping algebra of generalized H-Hom-Lie algebra L, i.e., Hom-Lie algebra L H in the category HYD, is constructed. Secondly, it is obtained that U（L） = T（ L）/L where I is the Hom-ideal of T（L） generated by {ll＇-l_（（-1））·l＇l_0-[l,l＇]｜l,l＇∈L}, and u： L,T（L）/I is the canonical map. Finally, as the applications of the result, the enveloping algebras of generalized H-Lie algebras, i.e., the Lie algebras in the category MyDn and the Hom-Lie algebras in the category of left H-comodules are presented, respectively.
Let H be a commutative, noetherian, semisimple and cosemisimple Hopf algebra with a bijective antipode over a field k. Then the semisimplicity of YD（H） is considered, where YD （H） means the disjoint union of the category of generalized Yetter-Drinfeld modules nYD^H（ α, β） for any α, β E Aut Hopf（H）. First, the fact that YD（H） is closed under Mor is proved. Secondly, based on the properties of finitely generated projective modules and semisimplicity of H, YD（H） satisfies the exact condition. Thus each object in YD（H） can be decomposed into simple ones since H is noetherian and cosemisimple. Finally, it is proved that YD （H） is a sernisimple category.
