Making the category of entwined modules into a braided monoidal category

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.

On the Braided Monoidal Categories of T-smash Products A ■_T H

Let A and H be Hopf algebra, T-smash product AT H generalizes twisted smash product A ＊ H. This paper shows a necessary and sufficient condition for T-smash product moduie category AT HM to be braided monoidal category.

On braided Lie algebras

Let （C, C） be a braided monoidal category. The relationship between the braided Lie algebra and the left Jacobi braided Lie algebra in the category （C, C） is investigated. First, a braided C2-commutative algebra in the category （C, C） is defined and three equations on the braiding in the category （C, C） are proved. Secondly, it is verified that （A, [, ] ） is a left （strict） Jacobi braided Lie algebra if and only if （A, [, ] ） is a braided Lie algebra, where A is an associative algebra in the category （C, C）. Finally, as an application, the structures of braided Lie algebras are given in the category of Yetter-Drinfel＇d modules and the category of Hopf bimodules.

The Braided Monoidal Structure on the Category of Comodules of Bimonads

We investigate how the category of comodules of bimonads can be made into a monoidal category.It suffices that the monad and comonad in question are bimonads,with some extra compatibility relation.On a monoidal category of comodules of bimonads,we cons true t a braiding and get the necessary and sufficien t conditions making it a braided monoidal category.As an application,we consider the category of comodules of corings and the category of entwined modules.

Sweedler's dual of Hopf algebras in HHYDQCM

Firstly,the notion of the left-left Yetter-Drinfeld quasicomodule M=(M,·,ρ)over a Hopf coquasigroup H is given,which generalizes the left-left Yetter-Drinfeld module over Hopf algebras.Secondly,the braided monoidal category HHYDQCM is introduced and the specific structure maps are given.Thirdly,Sweedler's dual of infinite-dimensional Hopf algebras in HHYDQCM is discussed.It proves that if(B,mB,μB,ΔB,εB)is a Hopf algebra in HHYDQCM with antipode SB,then(B^0,(mB0)^op,εB^*,(ΔB0)^op,μB^*)is a Hopf algebra in HHYDQCM with antipode SB^*,which generalizes the corresponding results over Hopf algebras.

Weak crossed biproducts and weak projections

In this paper, we present the general theory and universal properties of weak crossed biproducts. We prove that every weak projection of weak bialgebras induces one of these weak crossed structures. Finally, we compute explicitly the weak crossed biproduct associated with a groupoid that admits an exact factorization.

Hopf Quasimodules and Yetter-Drinfeld Modules over Hopf Quasigroups

We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup H in a,symmetric monoidal category C.li H possesses an adjoint quasiaction,we show that symmetric Yetter-Drinfeld categories are trivial,and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over H and the category of four-angle Hopf modules over H under some suitable conditions.