Symmetry and pseudosymmetry of v-Yetter-Drinfeld categories for Hom-Hopf algebras

The purpose of this paper is to introduce the category of v-Yetter-Drinfeld modules (v is an element of Z) over a Hom-Hopf algebra. We first prove that every category of v-Yetter-Drinfeld modules over a Hom-Hopf algebra with a bijective antipode S is a braided tensor category and that every v-Yetter-Drinfeld module can provide the solution of the Hom-Yang-Baxter equation. Secondly, we find sufficient and necessary conditions for H-H HYDv to be symmetric and pseudosymmetric, respectively. Finally, we construct examples of v-Yetter-Drinfeld modules by a quasitriangular Hom-Hopf algebra and study their relationship.