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

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.