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 $({}_H^H{\mathfrak{M}}_H^H,{\square }_H)$.

24pages