On structure groups of set-theoretic solutions to the Yang–Baxter equation
| Autor: | Victoria Lebed, Leandro Vendramin |
|---|---|
| Přispěvatelé: | Laboratoire de Mathématiques Nicolas Oresme (LMNO), Centre National de la Recherche Scientifique (CNRS)-Université de Caen Normandie (UNICAEN), Normandie Université (NU)-Normandie Université (NU), Departamento de Matemática [Buenos Aires], Facultad de Ciencias Exactas y Naturales [Buenos Aires] (FCEyN), Universidad de Buenos Aires [Buenos Aires] (UBA)-Universidad de Buenos Aires [Buenos Aires] (UBA), Mathematics |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Mathematics(all) Rank (linear algebra) General Mathematics Structure (category theory) multipermutation solution structure group 010103 numerical & computational mathematics Group Theory (math.GR) 01 natural sciences biquandle [MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] abelianization orderable group Mathematics - Quantum Algebra FOS: Mathematics Quantum Algebra (math.QA) bijective 1-cocycle 0101 mathematics Abelian group Quotient ComputingMilieux_MISCELLANEOUS Mathematics quandle Group (mathematics) Yang–Baxter equation 010102 general mathematics birack 16. Peace & justice Injective function yang-baxter equation diffuse group structure rack Torsion (algebra) Mathematics - Group Theory injective solution |
| Zdroj: | Proceedings of the Edinburgh Mathematical Society Proceedings of the Edinburgh Mathematical Society, Cambridge University Press (CUP), 2019, pp.1-35. ⟨10.1017/S0013091518000548⟩ |
| ISSN: | 0013-0915 1464-3839 |
| Popis: | 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 group; that only free abelian structure groups are biorderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: biorderable, left-orderable, abelian, free abelian, torsion free. 32 pages. Final version. Accepted for publication in Proc. Edinburgh Math. Soc |
| Databáze: | OpenAIRE |
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