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

Abstract

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