Braces and the Yang–Baxter Equation

Abstract

Several aspects of relations between braces and non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation are discussed and many consequences are derived. In particular, for each positive integer n a finite square-free multipermutation solution of the Yang-Baxter equation with multipermutation level n and an abelian involutive Yang-Baxter group is constructed. This answers a problem of Gateva-Ivanova and Cameron. It is also proved that finite non-degenerate involutive settheoretic solutions of the Yang-Baxter equation whose associated involutive Yang-Baxter group is abelian are retractable in the sense of Etingof, Schedler and Soloviev. Earlier the authors proved this with the additional square-free hypothesis on the solutions. Retractability of solutions is also proved for finite square-free non-degenerate involutive set-theoretic solutions associated to a left brace.

Skew braces and the Yang–Baxter equation

“…The following theorem generalizes a result of Rump to the non-commutative setting, see [12,Lemma 2]. Theorem 3.1.…”

“…1 1 221 1 4 1 4 1 1 1 22 Remark 5.5. For information on square-free two-sided braces, see [12]. These braces are defined by nilpotent groups of class ≤ 2.…”

“…In [12], Cedó, Jespers and Okniński, defined a left brace as an abelian group (A, +) with another group structure, defined via (a, b) → ab, such that the compatibility condition a(b + c) + a = ab + ac holds for all a, b, c ∈ A. This definition is equivalent to that of Rump. This work is partially supported by CONICET, PICT-2014-1376, MATH-AmSud and ICTP.…”

Hopf braces and Yang-Baxter operators

“…A (left) brace is an abelian group (A, +) with another group structure, defined via (a, b) → ab, such that the compatibility condition a(b + c) + a = ab + ac holds for all a, b, c ∈ A. The theory of braces is being developed quite intensively, see for example [2,3,7,10,19,20]. One advantage of the language of braces is that one can imitate ring theory to discuss braided groups and sets.…”

“…Our Hopf-theoretical generalization of the concept of a brace is based on the definition given by Cedó, Jespers and Okniński, see [7,Definition 1]. Definition 1.1.…”