LAUSR

Let (H,αH) be a Hom-bialgebra. In this paper, we firstly introduce the notion of Hom-L-R smash coproduct (C♮H,αC ⊗ αH), where (C,αC) is a Hom-coalgebra. Then for a Hom-algebra and Hom-coalgebra (D,αD), we introduce the notion of Hom-L-R-admissible pair (H,D). We prove that (D♮H,αD ⊗ αH) becomes a Hom-bialgebra under Hom-L-R smash product and Hom-L-R smash coproduct. Next, we will introduce a prebraided monoidal category ℋℒℛ(H) of Hom–Yetter–Drinfel’d–Long bimodules and show that Hom-L-R-admissible pair (H,D) actually corresponds to a bialgebra in the category ℋℒℛ(H), when αH and αD are involutions. Finally, we prove that when H is finite dimensional Hom-Hopf algebra, ℋℒℛ(H) is isomorphic to the Yetter–Drinfel’d category H⊗H∗H⊗H∗???????? as braid monoidal categories where (H ⊗ H∗,α H ⊗ αH−1∗) is the tensor product Hom–Hopf algebra.