Related Researches On Yetter-drinfeld Category

Yetter-Drinfeld category is one of the celebrated objects, and it is widely used in math-ematical physics, topology and other fields. Majid realized a categorical interpretation of Radford’s biproduct: B is a Hopf algebra in the Yetter-Drinfeld category HHyD if and only if BX# H is a Radford’s biproduct Hopf algebra. Recently, Yetter-Drinfeld category has attracted many researchers in the theory of Hopf algebras.In this paper, the Yetter-Drinfeld category is studied. The main contents are as fol-lows:(1) We extend the construction of two-sided smash coproduct to two-sided crossed coproduct C×αHβ× D. Then we derive the necessary and sufficient conditions for the two-sided smash product algebra C#H#D and the two-sided crossed coproduct coalgebra C×αHβ×D to be a bialgebra, which generalizes the Majid’s double biproduct. Taking either C =K or D=K of the two-sided crossed coproduct coalgebra C×α Hβ× D, we obtain right or left crossed coproduct.(2) We give constructions of Rota-Baxter monoidal Hom-(co)algebras from Hom-Hopf module (co)algebras, then introduce the concept of Rota-Baxter monoidal Hom-bialgebras,and also provided example from Radford biproduct monoidal Hom-Hopf algebras. Further-more, we consider the relations between Rota-Baxter monoidal Hom-systems and monoidal Hom-dendriform algebras, and also derive the structures of pre-Lie Hom-(co)algebras via Rota-Baxter monoidal Hom-(co)algebras of different weight.(3) We will study Lie algebras in the categoryH(HHyD(Z)). Firstly, we introduce the notion of (m, n)-Hom-Lie algebra and then prove that (A, a) can give rise to an (m, n)-Hom-Lie algebra with suitable Lie bracket when the braiding τ in (m, n)-Hom-Yetter-Drinfeld category H(HHyD(Z)) is symmetric on (A, α). We also show that if also (A, α) is a sum of two (H,β)-commutative Hom-subalgebras,then [A, A][A,A] = 0.(4) Firstly, we introduce the notion of (lazy) Hom-2-cocycles on Hom-Hopf algebras,study their properties, and extend (lazy) Hom-2-cocycles in Hom-Yetter-Drinfeld category to a Radford biproduct Hom-Hopf algebra.(5) Firstly, we introduce the notions of a smash product monoidal BiHom-algebra(B#H, αB(?)αH,,βB(?)βH) and a smash coproduct monoidal BiHom-coalgebra (B×H,αB(?)αH, βB(?)βH). Furthermore, we get sufficient and necessary conditions for(B#H, αB(?)αH,βB(?)βH) and (B×H,αB(?)αH,βB(?)βH) to be a Radford biproduct monoidal BiHom-Hopf algebra. It also provides conditions for constructing the new braided tensor category(i.e. a generalized form of Yetter-Drinfeld category).