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

In this article, we first introduce the notion of BiHom-L-R-$\binom{m,n,p,q}{s,t,u,v}$-smash biproduct over a BiHom-Hopf algebra, denoted by $D\natural H$, where $m,n,p,q,s,t,u,v\in \mathbb{Z}$, and give the sufficient condition for $D\natural H$ to be a BiHom-bialgebra.

Furthermore, we describe the concept of BiHom-$\binom{m,n,p,q}{s,t,u,v}$-Yetter-Drinfel'd-Long bimodule via BiHom-L-R-$\binom{m,n,p,q}{s,t,u,v}$-smash biproduct bialgebra, and prove that the category $\mathcal{LR}(H)(m,n,p,q)$ of BiHom-$\binom{m,n,p,q}{s,t,u,v}$-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-$\binom{s,t}{p,q}$-Yetter-Drinfel'd category \({}_{H\otimes H^*}^{H\otimes H^*}\mathcal{YD}\binom{s,t}{p,q}\) as braided monoidal categories.