Retractability of set theoretic solutions of the Yang–Baxter equation

Abstract

It is shown that square free set theoretic involutive non-degenerate solutions of the Yang-Baxter equation whose associated permutation group (referred to as an involutive Yang-Baxter group) is abelian are retractable in the sense of Etingof, Schedler and Soloviev. This solves a problem of Gateva-Ivanova in the case of abelian IYB groups. It also implies that the corresponding finitely presented abelian-by-finite groups (called the structure groups) are poly-Z groups. Secondly, an example of a solution with an abelian involutive Yang-Baxter group which is not a generalized twisted union is constructed. This answers in the negative another problem of Gateva-Ivanova. The constructed solution is of multipermutation level 3. Retractability of solutions is also proved in the case where the natural generators of the IYB group are cyclic permutations. Moreover, it is shown that such solutions are generalized twisted unions.

Skew braces and the Yang–Baxter equation

“…The seminal works of Etingof, Schedler and Soloviev [15], and Gateva-Ivanova and Van den Bergh [24], discussed algebraic and geometrical interpretations and introduced several structures associated with the class of non-degenerate involutive solutions. Such solutions have been intensively studied, see for example [17,18,19], [21,22,23], [25,26], [20], [32,34], [6], [10], [11], [13], [28], and [40].…”

Retractability of solutions to the Yang–Baxter equation and p-nilpotency of skew braces

“…As an application of Theorem 2.3 we obtain the following particular case of a theorem proved by Cedó, Jespers and Okniński in [13] and by Cameron and Gateva-Ivanova in [29]. For a direct proof (without the finiteness assumption), see [44,Proposition 10].…”

“…This means that multipermutation solutions generalize those solutions of Lyubashensko. Several papers study multipermutation involutive solutions, see for example [4,7,13,28,29,47,48,50].…”

Finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids

“…For some special classes of such semigroups (called monoids of I-type), it turns out that their algebras are noetherian domains of finite global dimension, satisfy a polynomial identity, are Koszul, Auslander-Gorenstein and Cohen-Macaulay [16]. Furthermore, the monoids of I-type yield set-theoretic solutions of the Yang-Baxter equation, and the properties of the associated algebraic structures have been extensively investigated in [3][4][5]8,[19][20][21]. In [15], Jespers, Okniński and Gateva-Ivanova investigated the algebraic structure of semigroup algebras K[S] for the wider class of monoids S of skew type.…”