The Construction of Braided T-Categories via Yetter–Drinfeld–Long Bimodules

Abstract

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