On Hom-Yetter-Drinfeld Category

Abstract

Let $(H, \beta )$ be a Hom-Hopf algebra. Recently we introduced the Hom-Yetter-Drinfeld category $_H^H{\mathcal {YD}}$ via Radford biproduct Hom-Hopf algebra, and proved that $_H^H{\mathcal {YD}}$ is a braided tensor category. Let $(H, \beta , \mathrm {R} (\hbox {or~}\sigma ))$ be a quasitriangular (or cobraided) Hom-Hopf algebra. In this paper, we prove that the category $_H{\mathcal {M}}$ (or $^H{\mathcal {M}}$) of left $(H, \beta )$-Hom-modules comodules) is a braided tensor subcategory of $_H^H{\mathcal {YD}}$. As a generalization of Radford biproduct Hom-Hopf algebra, we derive necessary and sufficient conditions for R-smash product Hom-algebra $(A\natural _R H, \alpha \otimes \beta )$ and T-smash coproduct Hom-coalgebra $(A\diamond _T H,\alpha \otimes \beta )$ to be a Hom-Hopf algebra. At last, two nontrivial examples are given.