BiHom-L-R-smash 双积与 BiHom-Yetter-Drinfel'd-Long 范畴

中文摘要 AI 辅助

本文中，我们首先在 BiHom-Hopf 代数上引入 BiHom-L-R-$\binom{m,n,p,q}{s,t,u,v}$-smash 双积的概念，记为 $D\natural H$，其中 $m,n,p,q,s,t,u,v$ 均为整数，并给出 $D\natural H$ 成为 BiHom-双代数的充分条件。此外，我们通过 BiHom-L-R-$\binom{m,n,p,q}{s,t,u,v}$-smash 双积双代数描述 BiHom-$\binom{m,n,p,q}{s,t,u,v}$-Yetter-Drinfel'd-Long 双模的概念，证明 BiHom-$\binom{m,n,p,q}{s,t,u,v}$-Yetter-Drinfel'd-Long 双模范畴 $\boldsymbol{\text{LR}}(H)(m,n,p,q)$ 是严格辫子幺半范畴。最后，对于有限维 BiHom-Hopf 代数 $H$，$\boldsymbol{\text{LR}}(H)\binom{m,n,p,q}{s,t,u,v}$ 作为辫子幺半范畴同构于 BiHom-$\binom{s,t}{p,q}$-Yetter-Drinfel'd 范畴 ${}_{H\boxtimes H^*}^{H\boxtimes H^*}\boldsymbol{\text{YD}}\binom{s,t}{p,q}$。

英文摘要

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.

发表机构

• School of Mathematics and Physics, Nanjing Institute of Technology（南京工程学院数学与物理学院）

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