The Hopf algebra structure of a crossed product in a braided monoidal category

Abstract

In this paper we define conditions under which a tensor product $A \otimes H$, in a braided monoidal category, together with a crossed product structure $A \mathbin{\sharp_{\sigma}} H$ and a smash coproduct structure $A \propto H$ is a Hopf algebra. When $\sigma = \varepsilon_{H} \otimes \varepsilon_{H} \otimes \eta_{A}$, Radford's theorems characterizing the biproduct are obtained and when the antipode of $H$ is a $\sigma$-antipode we find an analogous result with the one due to Wang, Jiao and Zhao.