BiHom-four-angle Hopf modules and BiHom-Yetter-Drinfel'd modules

Abstract

In this paper, we introduce the notion of four-angle Hopf modules over a BiHom-Hopf algebra $H$. We show that the category ${}_{H}^{H}\mathfrak{M}_{H}^{H}$ of BiHom-four-angle Hopf modules admits a strict monoidal category structure with respect to either the BiHom-tensor product $\otimes_{H}$ or the BiHom-cotensor product $\square_{H}$ as its monoidal product. We prove that the category $\mathcal{YD}_{H}^{H}(m,n,p,q)$ of BiHom-$(m,n,p,q)$-Yetter-Drinfel'd modules with parameters $m,n,p,q\in\mathbb{Z}$ forms a strict braided monoidal category equipped with a new monoidal product. Furthermore, we establish monoidal equivalences between the monoidal categories $\mathcal{YD}_{H}^{H}(m,n,p,q)$ and ${}_{H}^{H}\mathfrak{M}_{H}^{H}$, where ${}_{H}^{H}\mathfrak{M}_{H}^{H}$ carries either $\otimes_{H}$ or $\square_{H}$ as its monoidal product. Finally, we construct braiding structures for the monoidal categories $({}_{H}^{H}\mathfrak{M}_{H}^{H},\otimes_{H})$ and $({}_{H}^{H}\mathfrak{M}_{H}^{H},\square_{H})$.