Hopf $\pi$-Crossed Biproduct and Related Coquasitriangular Structures

Abstract

Let $\pi$ be a group and $H=(\{H_{\alpha }\},\Delta ,\epsilon ,S)$ a Hopf $\pi$ -coalgebra (not nec essarily associative), $\alpha \in \pi$ . Let $A$ be an algebra and a coalgebra. We find the necessary and sufficient conditions on the $\pi$ -crossed product $A{\#}_{\sigma }^{\pi }H$ with suitable comultiplication and counit to be a Hopf $\pi$ -coalgebra. Moreover, the necessary and sufficient conditions for a Hopf $\pi$ -crossed product $A{♮}_{\sigma }^{\pi }H$ to be a coquasitriangular Hopf $\pi$ -coalgebra are given. In this case the category ${}^{A{♮}_{\sigma }^{\pi }H}\mathcal{M}$ of the left $\pi$ -comodules over $A{♮}_{\sigma }^{\pi }H$ is a braided monoidal category.