Structure monoids of set-theoretic solutions of the Yang-Baxter equation
Cedó, Ferran (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Jespers, Eric (Vrije Universiteit Brussel. Department of Mathematics)
Vermwimp, Charlotte (Vrije Universiteit Brussel. Department of Mathematics)

Data:	2021
Resum:	Given a set-theoretic solution (X, r) of the Yang-Baxter equation, we denote by M = M(X, r) the structure monoid and by A = A(X, r), respectively A0 = A0 (X, r), the left, respectively right, derived structure monoid of (X, r). It is shown that there exist a left action of M on A and a right action of M on A0 and 1-cocycles π and π 0 of M with coefficients in A and in A0 with respect to these actions, respectively. We investigate when the 1-cocycles are injective, surjective, or bijective. In case X is finite, it turns out that π is bijective if and only if (X, r) is left non-degenerate, and π 0 is bijective if and only if (X, r) is right non-degenerate. In case (X, r) is left non-degenerate, in particular π is bijective, we define a semi-truss structure on M(X, r) and then we show that this naturally induces a set-theoretic solution (M, r) on the least cancellative image M = M(X, r)/η of M(X, r). In case X is naturally embedded in M(X, r)/η, for example when (X, r) is irretractable, then r is an extension of r. It is also shown that non-degenerate irretractable solutions necessarily are bijective.
Ajuts:	Ministerio de Economía y Competitividad MTM2017-83487-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1725
Nota:	The first author was partially supported by the grants MINECO-FEDER MTM2017-83487-P and AGAUR 2017SGR1725 (Spain). The second author is supported in part by Onderzoeksraad of Vrije Universiteit Brussel and Fonds voor Wetenschappelijk Onderzoek (Belgium). The third author is supported by Fonds voor Wetenschappelijk Onderzoek (Flanders), via an FWO Aspirant-mandate.
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Yang-baxter equation ; Set-theoretic solution ; Structure monoid ; 1-cocycle ; Semi-truss
Publicat a:	Publicacions matemàtiques, Vol. 65 Núm. 2 (2021) , p. 499-528 (Articles) , ISSN 2014-4350

Adreça alternativa: https://raco.cat/index.php/PublicacionsMatematiques/article/view/390240
DOI: 10.5565/PUBLMAT6522104

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