与辫群的Artin表示相关的动力系统和算子代数
Dynamical systems and operator algebras associated to Artin's representation of braid groups
摘要：
Artin表示是从$n$链上的辫子群$B_n$到$\operatorname{Aut}\mathbb{F}_n$的单射同态，$n$生成元上的自由群$\mathbb{F}_n$的自同构群。该表示将映射$B_n\to\operatorname{Aut}C^*_r(\mathbb{F}_n)$和$B_n\to\operatorname{Aut}C^*(\mathbb{F}_n)$引入到$\mathbb{F}_n$的相应群$C^*$ -代数的自同构群中.这些映射对纯辫群$P_n$也有自然的限制。本文考虑了值在圆上的余圈作用的扭曲形式，并讨论了相应的交叉积的理想结构。此外，我们利用Artin表示证明了无穷多条链上的辫群$B_\infty$和$P_\infty$都是$C^*$ -单的.
英文摘要：
Artin's representation is an injective homomorphism from the braid group $B_n$ on $n$ strands into $\operatorname{Aut}\mathbb{F}_n$, the automorphism group of the free group $\mathbb{F}_n$ on $n$ generators. The representation induces maps $B_n\to\operatorname{Aut}C^*_r(\mathbb{F}_n)$ and $B_n\to\operatorname{Aut}C^*(\mathbb{F}_n)$ into the automorphism groups of the corresponding group $C^*$-algebras of $\mathbb{F}_n$. These maps also have natural restrictions to the pure braid group $P_n$. In this paper, we consider twisted versions of the actions by cocycles with values in the circle, and discuss the ideal structure of the associated crossed products. Additionally, we make use of Artin's representation to show that the braid groups $B_\infty$ and $P_\infty$ on infinitely many strands are both $C^*$-simple.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
