On Structure Groups of Set-Theoretic Solutions to the Yang–Baxter Equation

By Victoria Lebed, Leandro Vendramin. Proceedings of the Edinburgh Mathematical Society, 2017

76 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/667e0d15a2eaee1d1c42d232a9bb18064f3d94c5

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In Algebra and Number Theory

1. Chengming Bai
2. Li Guo
3. Y. Sheng
4. Rong Tang
5. Abdenacer Makhlouf

In Discrete Mathematics and Combinatorics

1. Chengming Bai
2. Y. Sheng

In Geometry and Topology

1. Chengming Bai
2. Li Guo
3. Y. Sheng
4. Rong Tang
5. Abdenacer Makhlouf

Citations in the corpus

• Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation (2023, background): “Recently, the finite quotients of these structure groups were studied in [35].”