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arXiv · 2006.15267

Four-angle Hopf modules for Hom-Hopf algebras

Xiaoqian Liu·Dongdong Yan·Xuchen Deng·Danhua Wang
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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 $\Box_{H}$ 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 $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\Box_{H})$ 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 $(~\!^{H}_{H}\mathfrak{M}^{H}_{H},\Box_{H})$).

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Xiaoqian Liu, Dongdong Yan, Xuchen Deng, Danhua Wang. 2020-06-27. Four-angle Hopf modules for Hom-Hopf algebras. https://arxiv.org/abs/2006.15267

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