Structure monoids of set-theoretic solutions of the Yang-Baxter equation

• Cedó Giné, Ferran ^[1] ; Jespers, Eric ^[2] ; Vermwimp, Charlotte ^[2]
  1. [1] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
  2. [2] Vrije Universiteit Brussel

     Vrije Universiteit Brussel

     Arrondissement Brussel-Hoofdstad, Bélgica

     
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 65, Nº 2, 2021, págs. 499-528
• Idioma: inglés
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• Resumen

  • 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.