Four-angle Hopf modules for Hom-Hopf algebras

Abstract

In this paper, we introduce the notion of a four-angle Hopf module for a Hom-Hopf algebra $(H,\beta )$ and show that the category ${}_H^H{\mathfrak{M}}_H^H$ of four-angle Hopf modules is a monoidal category with either a Hom-tensor product ${\otimes }_H$ or a Hom-cotensor product ${\square }_H$ as a monoidal product. We study the category ${\mathcal{YD}}_H^H$ of Yetter-Drinfel'd modules with bijective structure map can be organized as a braided monoidal category, in which we use a new monoidal structure. Finally, We prove an equivalence between the monoidal category $({}_H^H{\mathfrak{M}}_H^H,{\otimes }_H)$ or $({}_H^H{\mathfrak{M}}_H^H,{\square }_H)$ of four-angle Hopf modules, and the monoidal category ${\mathcal{YD}}_H^H$ of Yetter-Drinfel'd modules, and furthermore, we give a braiding structure of the monoidal categorys $({}_H^H{\mathfrak{M}}_H^H,{\otimes }_H)$ (and $({}_H^H{\mathfrak{M}}_H^H,{\square }_H)$).

References in corpus (5)

• Deformations of Lie Algebras using $\sigma$-derivations
• Representations of hom-Lie algebras
• Hom-bialgebras and comodule Hom-algebras
• Module Hom-algebras
• Hom-quantum groups III: Representations and module Hom-algebras