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

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