Construction of a braided monoidal category for Brzeziński crossed coproducts of Hopf $\pi $-algebras

Tianshui Ma¹ ; Haiying Li¹ ; Shaoxian Xu²

¹ School of Mathematics and Information Science Henan Normal University Xinxiang 453007, China
² School of Mathematics and Statistics Nanyang Normal University Nanyang 473061, China

Colloquium Mathematicum, Tome 149 (2017) no. 2, pp. 309-323

Résumé

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.

Détail

DOI : 10.4064/cm6987-9-2016

Keywords: group hopf algebras alpha alpha otimes alpha rightarrow alpha otimes alpha alpha alpha otimes alpha rightarrow alpha otimes alpha families linear maps necessary sufficient conditions family brzezi ski crossed coproduct coalgebras alpha mathbin alpha alpha alpha alpha hopf algebra moreover necessary sufficient conditions brzezi ski crossed coproduct hopf algebra mathbin natural quasitriangular derived module category mathbin natural mathcal braided monoidal category