On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation
On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation
复制标题
杨-巴克斯特方程集合论解的结构群
DOI：
10.1017/s0013091518000548
复制
发表时间：
2017
影响因子：
0.7
通讯作者：
L. Vendramin
中科院分区：
文献类型：
--
作者：
V. Lebed;L. Vendramin
研究Yang-Baxter方程有限非退化集论解（X,r）的结构群G（X,r）。即构造了G（X,r）的有限商$\overline {G}_{(X,r)}$，推广了Dehornoy关于对合解的类coxet群。这产生了测试注入性的有限设置：如果X注入到G（X,r）中，那么它也注入到$\overline {G}_{(X,r)}$中。我们将每个解压缩为具有相同结构群的单射解，并计算G（X,r）的阿贝尔化的秩。我们证明了多置换解是唯一具有弥散结构群的对合解；只有自由阿贝尔结构群是双序的；并且对于自分配解的结构群，下列条件是等价的：双有序的、左有序的、阿贝尔的、自由阿贝尔的和无扭转的。
Abstract This paper explores the structure groups G(X,r) of finite non-degenerate set-theoretic solutions (X,r) to the Yang–Baxter equation. Namely, we construct a finite quotient $\overline {G}_{(X,r)}$ of G(X,r), generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if X injects into G(X,r), then it also injects into $\overline {G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of G(X,r). We show that multipermutation solutions are the only involutive solutions with diffuse structure groups; that only free abelian structure groups are bi-orderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: bi-orderable, left-orderable, abelian, free abelian and torsion free.