Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras

In this paper, we investigate a more general category of Θ \Theta -Yetter-Drinfeld modules ( Θ ∈ Aut H ( H ) \Theta \in {\rm{Aut}}\hspace{0.33em}H\left(H) ) over a Hom-Hopf algebra, which unifies two different definitions of Hom-Yetter-Drinfeld category introduced by Makhlouf and Panaite, Li and Ma, respectively. We show that the category of Θ \Theta -Yetter-Drinfeld modules with a bijective antipode S S is a braided tensor category and some solutions of the Hom-Yang-Baxter equation and the Yang-Baxter equation can be constructed by this category. Also by the method of symmetric pairs, we prove that if a Θ \Theta -Yetter-Drinfeld category over a Hom-Hopf algebra H H is symmetric, then H H is trivial. Finally, we find a sufficient and necessary condition for a Θ \Theta -Yetter-Drinfeld category to be pseudosymmetric.