Radford's biproducts and Yetter-Drinfeld modules for monoidal Hom-Hopf algebras

Abstract

Let (H, α) be a monoidal Hom-bialgebra and (B, β) be a left (H, α)-Hom-module algebra and also a left (H, α)-Hom-comodule coalgebra. Then in this paper, we first introduce the notion of a Hom-smash coproduct, which is a monoidal Hom-coalgebra. Second, we find sufficient and necessary conditions for the Hom-smash product algebra structure and the Hom-smash coproduct coalgebra structure on B ⊗ H to afford B ⊗ H a monoidal Hom-bialgebra structure, generalizing the well-known Radford's biproduct, where the conditions are equivalent to that (B, β) is a bialgebra in the category of Hom-Yetter-Drinfeld modules $^{H} _{H}\mathcal {HYD}$HYDHH. Finally, we illustrate the category of Hom-Yetter-Drinfeld modules $^{H} _{H}\mathcal {HYD}$HYDHH and prove that the category $^{H} _{H}\mathcal {HYD}$HYDHH is a braided monoidal category.