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

### Tianshui Ma

Henan Normal University, Xinxiang, China### Yanan Song

Henan Normal University, Xinxiang, China

## 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.

## Cite this article

Tianshui Ma, Yanan Song, Hopf $\pi$-Crossed Biproduct and Related Coquasitriangular Structures. Rend. Sem. Mat. Univ. Padova 130 (2013), pp. 127–145

DOI 10.4171/RSMUP/130-3