Yetter-Drinfeld modules for Hom-bialgebras

Abstract

The aim of this paper is to define and study Yetter-Drinfeld modules over Hom-bialgebras, a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism. Yetter-Drinfeld modules over a Hom-bialgebra with bijective structure map provide solutions of the Hom-Yang-Baxter equation. The category \documentclass[12pt]{minimal}\begin{document}$_H^H{\mathcal {Y}D}$\end{document}YHHD of Yetter-Drinfeld modules with bijective structure maps over a Hom-bialgebra H with bijective structure map can be organized, in two different ways, as a quasi-braided pre-tensor category. If H is quasitriangular (respectively, coquasitriangular) the first (respectively, second) quasi-braided pre-tensor category \documentclass[12pt]{minimal}\begin{document}$_H^H{\mathcal {Y}D}$\end{document}YHHD contains, as a quasi-braided pre-tensor subcategory, the category of modules (respectively, comodules) with bijective structure maps over H.