Construction of a braided monoidal category for Brzeziński crossed coproducts of Hopf $\pi $-algebras
Tom 149 / 2017
Streszczenie
Let $\pi $ be a group, $C, H$ Hopf $\pi $-algebras, and $g_{\alpha }: C_{\alpha }\otimes H_{\alpha }\rightarrow H_{\alpha }\otimes H_{\alpha }$ and $T_{\alpha }: C_{\alpha }\otimes H_{\alpha }\rightarrow H_{\alpha }\otimes C_{\alpha }$ families of linear maps. We give necessary and sufficient conditions for the family of Brzeziński crossed coproduct coalgebras $\{C_{\alpha }\mathbin {\#^{g_{\alpha }}_{T_{\alpha }}} H_{\alpha }\}_{\alpha \in \pi }$ to be a Hopf $\pi $-algebra. Moreover, necessary and sufficient conditions for the Brzeziński crossed coproduct Hopf $\pi $-algebra $C\mathbin {{\natural ^{g}_{T}}^{\pi }} H$ to be quasitriangular are derived, and in this case, the left $\pi $-module category ${}_{C\mathbin {{\natural ^{g}_{T}}^{\pi }} H}{\mathcal M}$ is a braided monoidal category.
