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 $\alpha,\beta \in Aut_{Hopf}(H_1), \gamma,\delta \in Aut_{Hopf}(H_2)$. 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.