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

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Let H₁ and H₂ be Hopf algebras which are not necessarily finite dimensional and α,β∈ AutHopf(H₁), γ,δ∈ AutHopf(H₂). In this paper, we introduce a category ₕ₁LRₕ₂(α, β, γ, δ), generalizing Yetter-Drinfeld-Long bimodules and construct a braided T-category LR(H₁,H₂) containing all the categories ₕ₁LRₕ₂(α, β, γ, δ) as components. We also prove that if (α, β, γ, δ) admits a quadruple in involution, then ₕ₁LRₕ₂(α, β, γ, δ) is isomorphic to the usual category ₕ₁LRₕ₂ of Yetter-Drinfeld-Long bimodules.