发表机构
Hangzhou Normal University（杭州师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在辫子幺半范畴中，当$B$有左对偶$B^\ast$时构造$D\times B^\ast$，证明$B\times D$为其左smash积，并建立Yetter-Drinfeld模范畴的同构，涵盖代数、双代数、Hopf代数三个层次。
AI 中文摘要
设$B$和$D$均为辫子幺半范畴$\mathcal{C}$中的代数和余代数。文献中，$(B,D)$构成广义smash双积（或交叉积）$B\times D$的等价条件主要由Bespalov和Drabant于1999年以及Bulacu、Caenepeel和Torrecillas于2013年给出。本文致力于在$B$于$\mathcal{C}$中具有左对偶对象$B^\ast$时构造另一个双积$D\times B^\ast$。作为结果，我们证明$B\times D$是$D\times B^\ast$上的左smash积，并建立了范畴${}_{B\times D}\mathfrak{YD}(\mathcal{C})^{B\times D}\cong{}_{D\times B^\ast}\mathfrak{YD}(\mathcal{C})^{D\times B^\ast}$之间的具体同构，该同构涉及$\mathcal{C}$中（左右）Yetter-Drinfeld模。构造和结果在三个层次上给出：（1）$B\times D$是代数和余代数；（2）$B\times D$是双代数；（3）$B\times D$是Hopf代数。
英文摘要
Let $B$ and $D$ be both algebras and coalgebras in a braided monoidal category $\mathcal{C}$. In the literature, there are equivalent conditions for $(B,D)$ to form a generalized smash biproduct (or cross product) denoted by $B\times D$, which were mainly given by Bespalov and Drabant in 1999 as well as by Bulacu, Caenepeel and Torrecillas in 2013. This paper is devoted to constructing another biproduct $D\times B^\ast$ when $B$ has a left dual object $B^\ast$ in $\mathcal{C}$. As results, we show that $B\times D$ is left smash over $D\times B^\ast$, and establish a specific isomorphism ${}_{B\times D}\mathfrak{YD}(\mathcal{C})^{B\times D}\cong{}_{D\times B^\ast}\mathfrak{YD}(\mathcal{C})^{D\times B^\ast}$ between the categories of the (left-right) Yetter-Drinfeld modules in $\mathcal{C}$. The constructions and results are provided in three levels: (1) $B\times D$ is an algebra and a coalgebra; (2) $B\times D$ is a bialgebra; (3) $B\times D$ is a Hopf algebra.
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