The quasi-triangular structures over Hom-ω-smash coproduct Hopf algebras

Abstract

Let (C, α) and (H, β) be Hom-bialgebras and ω: C ⊗H → H ⊗C a linear map. We introduce a Hom-ω-smash coproduct (C ω ⋈ H, γ) and give necessary and sufficient conditions for (C ω ⋈ H, γ) to be a Hom-bialgebra. We study the quasi-triangular structures over (C ω ⋈ H, γ) and show the necessary and sufficient conditions for (C ω ⋈ H, γ, R) to be a quasi-triangular Hom-Hopf algebra. As applications of our results, we introduce the concept of D(H)* and construct quasi-triangular structures over D(H)*.