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Let $H_1$ and $H_2$ be Hopf algebras which are not necessarily finitedimensional and $\alpha,\beta \in Aut_{Hopf}(H_1), \gamma,\delta \inAut_{Hopf}(H_2)$. In this paper, we introduce a category${}_{H_1}\mathcal{LR}_{H_2}(\alpha, \beta, \gamma, \delta)$, generalizingYetter-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. Wealso prove that if $(\alpha, \beta, \gamma, \delta)$ admits a quadruple ininvolution, then ${}_{H_1}\mathcal{LR}_{H_2}(\alpha, \beta, \gamma, \delta)$ isisomorphic to the usual category ${}_{H_1}\mathcal{LR}_{H_2}$ ofYetter-Drinfeld-Long bimodules.

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