论文标题
编织的霍夫夫通过简单结构越过模块
Braided Hopf Crossed Modules Through Simplicial Structures
论文作者
论文摘要
任何简单的HOPF代数都涉及$ 2N $不同的预测，而Hopf代数$ H_N，H_ {N-1} $对于每个$ n \ geq 1 $。一个单词投影，在这里表示元组$ \ partial \ colon h_ {n} \ to h_ {n-1} $和$ i \ i \ colon h_ {n-1} \ to hopf algebra形态的h_ {n} $给定一个编织的单体类别$ \ mathfrak {c} $中的Hopf代数投影$（\ partial \ colon i \ to H，i）$，由于Radford的TheoRem，人们可以获得生活在$ H $的Yetter-Drinfeld模块类别中的新的Hopf代数结构。此HOPF代数的基础集是通过均衡器获得的，该均衡器仅定义$ \ Mathfrak {c} $中的$ i $ $ i $的子代数（不是次核）。实际上，这是一个编织的Hopf代数，因为具有可逆的反座的Hopf代数上的Yetter-Drinfeld模块类别是编织的。要连续地将Radford的定理应用于Simplicial Hopf代数，我们需要一些Yetter-Drinfeld模块的功能性属性。此外，这使我们能够从简单结构的角度对Majid的编织的Hopf交叉模块概念进行建模。
Any simplicial Hopf algebra involves $2n$ different projections between the Hopf algebras $H_n,H_{n-1}$ for each $n \geq 1$. The word projection, here meaning a tuple $\partial \colon H_{n} \to H_{n-1}$ and $i \colon H_{n-1} \to H_{n}$ of Hopf algebra morphisms, such that $\partial \, i = \mathrm{id}$. Given a Hopf algebra projection $(\partial \colon I \to H,i)$ in a braided monoidal category $\mathfrak{C}$, one can obtain a new Hopf algebra structure living in the category of Yetter-Drinfeld modules over $H$, due to Radford's theorem. The underlying set of this Hopf algebra is obtained by an equalizer which only defines a sub-algebra (not a sub-coalgebra) of $I$ in $\mathfrak{C}$. In fact, this is a braided Hopf algebra since the category of Yetter-Drinfeld modules over a Hopf algebra with an invertible antipode is braided monoidal. To apply Radford's theorem in a simplicial Hopf algebra successively, we require some extra functorial properties of Yetter-Drinfeld modules. Furthermore, this allows us to model Majid's braided Hopf crossed module notion from the perspective of a simplicial structure.
