AI 中文总结
本文定义BiHom四角Hopf模范畴，证明其在两种BiHom积下为严格幺半范畴，建立其与特定BiHom-Yetter-Drinfel'd模范畴的幺半等价，并为前者构造辫结构。
AI 中文摘要
本文引入BiHom-Hopf代数$H$上的四角Hopf模概念，证明BiHom四角Hopf模构成的范畴${}_{H}^{H}\boldsymbol{\frak{M}}_{H}^{H}$以BiHom张量积$\boxtimes_{H}$或BiHom余张量积$\boxtimes_{H}$为幺积时，均具有严格幺半范畴结构；证明带参数$m,n,p,q\in\boldsymbol{\frak{Z}}$的BiHom-$(m,n,p,q)$-Yetter-Drinfel'd模构成的范畴$\boldsymbol{\frak{YD}}_{H}^{H}(m,n,p,q)$形成带新幺积的严格辫幺半范畴；建立$\boldsymbol{\frak{YD}}_{H}^{H}(m,n,p,q)$与${}_{H}^{H}\boldsymbol{\frak{M}}_{H}^{H}$之间的幺半等价，其中${}_{H}^{H}\boldsymbol{\frak{M}}_{H}^{H}$分别以$\boxtimes_{H}$或$\boxtimes_{H}$为幺积；最后为幺半范畴$({}_{H}^{H}\boldsymbol{\frak{M}}_{H}^{H},\boxtimes_{H})$和$({}_{H}^{H}\boldsymbol{\frak{M}}_{H}^{H},\boxtimes_{H})$构造辫结构。
英文摘要
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 $({}_{H}^{H}\mathfrak{M}_{H}^{H},\square_{H})$.
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