论文标题
比较Grothendieck-Teichmüller和Kashiwara-Vergne组的结论方法
A knot-theoretic approach to comparing the Grothendieck-Teichmüller and Kashiwara-Vergne groups
论文作者
论文摘要
同构膨胀是打结物体的组合不变的，从某种意义上说，所有有限型（Vassiliev）不变性因素通过它们来考虑。同构膨胀也很重要，因为在低维拓扑和量子代数之间桥接对象。例如，括号的编织物的同形扩展与Drinfel'd员工（Bar-Natan 1998）一对一，并且$ W $ -FOAMS的同型扩展与与Kashiwara-Vergne（KVARA-VERGNE（KV）方程式（Bar-Natanan and Bar-Natan和First First Furse）的解决方案是一对一的对应。 Drinfel'd协会和KV解决方案的集合都是双重的，分别是联合无能的Grothendieck-Teichmüller和Kashiwara-Vergne组的行动。上面的对应关系实际上是Bi-Torsors的地图（Bar-Natan 1998，以及Halacheva 2022的第一和第三作者）。 Drinfel'd的同伴和KV方程之间存在着深厚的关系 - 由Alekseev，Enriquez和Torossian在2010年代发现 - 包括用协会人来构建KV解决方案的明确公式，以及注射性的MAP $ρ：\ mathsf {grt} _1 _1 _1 _1 \ to \ mathssf}本文是对Kashiwara-Vergne对称组中Grothendieck-Teichmüller组的图像的拓扑/示意研究，使用括号内的辫子和$ W $ -FOAMS允许相应的有限演示作为operad和Tensor类别（电路Algebra或Per）。
Homomorphic expansions are combinatorial invariants of knotted objects, which are universal in the sense that all finite-type (Vassiliev) invariants factor through them. Homomorphic expansions are also important as bridging objects between low-dimensional topology and quantum algebra. For example, homomorphic expansions of parenthesised braids are in one-to-one correspondence with Drinfel'd associators (Bar-Natan 1998), and homomorphic expansions of $w$-foams are in one-to-one correspondence with solutions to the Kashiwara-Vergne (KV) equations (Bar-Natan and the first author, 2017). The sets of Drinfel'd associators and KV solutions are both bi-torsors, with actions by the pro-unipotent Grothendieck-Teichmüller and Kashiwara-Vergne groups, respectively. The above correspondences are in fact maps of bi-torsors (Bar-Natan 1998, and the first and third authors with Halacheva 2022). There is a deep relationship between Drinfel'd associators and KV equations--discovered by Alekseev, Enriquez and Torossian in the 2010s--including an explicit formula constructing KV solutions in terms of associators, and an injective map $ρ:\mathsf{GRT}_1 \to \mathsf{KRV}$. This paper is a topological/diagrammatic study of the image of the Grothendieck-Teichmüller groups in the Kashiwara-Vergne symmetry groups, using the fact that both parenthesised braids and $w$-foams admit respective finite presentations as an operad and as a tensor category (circuit algebra or prop).
