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

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(Cω■H, γ) and give necessary and sufficient conditions for(Cω■H, γ) to be a Hom-bialgebra. We study the quasi-triangular structures over(Cω■H, γ)and show the necessary and sufficient conditions for(Cω■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)*.