On structure groups of set-theoretic solutions to the yang-baxter equation

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 of G(X,r), generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting f...

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Detalles Bibliográficos
Autores principales:	Lebed, V., Vendramin, L.
Formato:	INPR
Materias:	abelianization bijective 1-cocycle biquandle birack diffuse group injective solution multipermutation solution orderable group quandle structure group structure rack Yang-Baxter equation
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_00130915_v_n_p_Lebed
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	todo:paper_00130915_v_n_p_Lebed
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spelling	todo:paper_00130915_v_n_p_Lebed2023-10-03T14:10:41Z On structure groups of set-theoretic solutions to the yang-baxter equation Lebed, V. Vendramin, L. abelianization bijective 1-cocycle biquandle birack diffuse group injective solution multipermutation solution orderable group quandle structure group structure rack Yang-Baxter equation 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 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. 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. © Edinburgh Mathematical Society 2019. INPR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00130915_v_n_p_Lebed
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	abelianization bijective 1-cocycle biquandle birack diffuse group injective solution multipermutation solution orderable group quandle structure group structure rack Yang-Baxter equation
spellingShingle	abelianization bijective 1-cocycle biquandle birack diffuse group injective solution multipermutation solution orderable group quandle structure group structure rack Yang-Baxter equation Lebed, V. Vendramin, L. On structure groups of set-theoretic solutions to the yang-baxter equation
topic_facet	abelianization bijective 1-cocycle biquandle birack diffuse group injective solution multipermutation solution orderable group quandle structure group structure rack Yang-Baxter equation
description	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 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. 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. © Edinburgh Mathematical Society 2019.
format	INPR
author	Lebed, V. Vendramin, L.
author_facet	Lebed, V. Vendramin, L.
author_sort	Lebed, V.
title	On structure groups of set-theoretic solutions to the yang-baxter equation
title_short	On structure groups of set-theoretic solutions to the yang-baxter equation
title_full	On structure groups of set-theoretic solutions to the yang-baxter equation
title_fullStr	On structure groups of set-theoretic solutions to the yang-baxter equation
title_full_unstemmed	On structure groups of set-theoretic solutions to the yang-baxter equation
title_sort	on structure groups of set-theoretic solutions to the yang-baxter equation
url	http://hdl.handle.net/20.500.12110/paper_00130915_v_n_p_Lebed
work_keys_str_mv	AT lebedv onstructuregroupsofsettheoreticsolutionstotheyangbaxterequation AT vendraminl onstructuregroupsofsettheoreticsolutionstotheyangbaxterequation
_version_	1807317962243178496

On structure groups of set-theoretic solutions to the yang-baxter equation

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