A Method of Constructing Braided Hopf Algebras

Tianshui Ma, Shuanhong Wang, Shaoxian Xu

Let $A$ and $B$ be two Hopf algebras and $R\in Hom(B\otimes A, A\otimes B)$, the twisted tensor product. Hopf algebra $A\#_RB$ was introduced by S. Caenepeel et al.\ in [3] and further studied in our recent work [6]. In this paper we give the necessary and sufficient conditions for $A\#_RB$ to be a Hopf algebra with a projection. Furthermore, a braided Hopf algebra A is constructed by twisting the multiplication of $A$ through a $(\gamma,R)$-pair $(A,B)$. Finally we give a method to construct Radford's biproduct directly by defining the module action and comodule action from the twisted tensor biproduct.