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∈AutH(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 SS 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 HH is symmetric, then HH is trivial. Finally, we find a sufficient and necessary condition for a Θ\Theta -Yetter-Drinfeld category to be pseudosymmetric.

Abstract licence: CC BY

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