BiHom-L-R-smash biproduct and BiHom-Yetter-Drinfel'd-Long category

Abstract

In this article, we first introduce the notion of BiHom-L-R-$\left(\genfrac{}{}{0ex}{}{m,n,p,q}{s,t,u,v}\right)$-smash biproduct over a BiHom-Hopf algebra, denoted by $D♮H$, where $m,n,p,q,s,t,u,v\in \mathbb{Z}$, and give the sufficient condition for $D♮H$ to be a BiHom-bialgebra. Furthermore, we describe the concept of BiHom-$\left(\genfrac{}{}{0ex}{}{m,n,p,q}{s,t,u,v}\right)$-Yetter-Drinfel'd-Long bimodule via BiHom-L-R-$\left(\genfrac{}{}{0ex}{}{m,n,p,q}{s,t,u,v}\right)$-smash biproduct bialgebra, and prove that the category $\mathcal{L}\mathcal{R}(H)(m,n,p,q)$ of BiHom-$\left(\genfrac{}{}{0ex}{}{m,n,p,q}{s,t,u,v}\right)$-Yetter-Drinfel'd-Long bimodule is a strict braided monoidal category. Finally, for a finite-dimensional BiHom-Hopf algebra H, \(\mathcal{LR}(H)\binom{m,n,p,q}{s,t,u,v}\) is isomorphic to the BiHom-$\left(\genfrac{}{}{0ex}{}{s,t}{p,q}\right)$-Yetter-Drinfel'd category \({}_{H\otimes H^*}^{H\otimes H^*}\mathcal{YD}\binom{s,t}{p,q}\) as braided monoidal categories.