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

Volume 149 / 2017

Tianshui Ma, Haiying Li, Shaoxian Xu Colloquium Mathematicum 149 (2017), 309-323 MSC: Primary 16T05, 16T25. DOI: 10.4064/cm6987-9-2016 Published online: 30 June 2017

Abstract

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.

Authors

  • Tianshui MaSchool of Mathematics and Information Science
    Henan Normal University
    Xinxiang 453007, China
    e-mail
  • Haiying LiSchool of Mathematics and Information Science
    Henan Normal University
    Xinxiang 453007, China
    e-mail
  • Shaoxian XuSchool of Mathematics and Statistics
    Nanyang Normal University
    Nanyang 473061, China
    e-mail

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