Braces and the Yang–Baxter Equation

By Ferran Cedó, Eric Jespers, Jan Okninski. 2012

274 citations recorded by the source corpus.

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 proved that finite non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation whose associated involutive Yang–Baxter group is abelian are multipermutation solutions. Earlier the authors proved this with the additional square-free hypothesis on the solutions. It is also proved that finite square-free non-degenerate involutive set-theoretic solutions associated to a left brace are multipermutation solutions.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/64a8a07cb3cfa2e72258008d0c81ec0e85e67070

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Citations in the corpus

Foundations of Garside Theory (2013, methodology): “because of their connection with set-theoretic solutions o f the Yang–Baxter equation, received much attention, see for instance Rump [200, 199] or Ce dó–Jespers–Okniński [52].”
Classification of braces of order p3 (2014, result): “quation. This is one of the fundamental reasons for desiring a classiﬁcation of left braces, but there are other results relating this structure and the Yang-Baxter equation in a fundamental way; see [2] for a good introduction to braces and their relation to the Yang-Baxter equation. Another import”
Counterexample to a conjecture about braces (2015, background): “One of these classes, the non-degenerate involutive set-theoretical solutions, has received a lot of attention recently (see the introduction of [8], and the references there).”
Problems on skew left braces (2018, methodology): “Examples of skew left braces can be found in [3, 26, 43, 62].”
On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation (2015, background): “Cedó, Jespers and Okninski [18, 16] asked which groups are multiplicative groups of braces.”
Extensions, matched products, and simple braces (2015, background): “One class of solutions that can be studied with techniques from group theory has received a lot of attention recently, the non-degenerate involutive set-theoretic solutions (see [7], and the references there).”
Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation (2016, background): “Every left brace B admits a left action λ : ( B, · ) −→ Aut( B, +) deﬁned by λ ( b ) = λ b for all b ∈ B (see [13, Lemma 1]).”
On skew braces (with an appendix by N. Byott and L. Vendramin) (2017, background): “n [32], Rump introduced braces, a new algebraic structure that turns out to be equivalent to bijective 1-cocycles; see [8, 19, 34]. According to the deﬁnition given by Cedo´, Jespers and Oknin´ski in [10], a brace is a triple (A,·,+), where (A,·) is a group, (A,+) is an abelian group and a(b +c) +a ”
Solutions of the Yang-Baxter equation associated with a left brace (2015, background): “A subclass of this type of solutions, the non-degenerate involutive ones, has received a lot of attention in the last years [6, 7, 8, 11, 13, 14, 15, 16, 18, 19, 21, 22, 23].”
Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation (2023, unknown): “Further studies were carried out in [11, 12, 27, 50].”