The construction of braided $T$-category via Yetter-Drinfeld-Long bimodules

Abstract

Let $H_1$ and $H_2$ be Hopf algebras which are not necessarily finite dimensional and $\alpha ,\beta \in Au{t}_{Hopf}(H_1),\gamma ,\delta \in Au{t}_{Hopf}(H_2)$. In this paper, we introduce a category ${}_{H_1}{\mathcal{L}\mathcal{R}}_{H_2}(\alpha ,\beta ,\gamma ,\delta )$, generalizing Yetter-Drinfeld-Long bimodules and construct a braided $T$-category $\mathcal{L}\mathcal{R}(H_1,H_2)$ containing all the categories ${}_{H_1}{\mathcal{L}\mathcal{R}}_{H_2}(\alpha ,\beta ,\gamma ,\delta )$ as components. We also prove that if $(\alpha ,\beta ,\gamma ,\delta )$ admits a quadruple in involution, then ${}_{H_1}{\mathcal{L}\mathcal{R}}_{H_2}(\alpha ,\beta ,\gamma ,\delta )$ is isomorphic to the usual category ${}_{H_1}{\mathcal{L}\mathcal{R}}_{H_2}$ of Yetter-Drinfeld-Long bimodules.

References in corpus (2)

• On $G$--equivariant modular categories
• Constructing New Braided $T$-categories over Monoidal Hom-Hopf Algebras