Braces and the Yang-Baxter Equation

Ferran Cedó, Eric Jespers, Jan Okniński

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97 Citations (Scopus)


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. © 2014 Springer-Verlag Berlin Heidelberg.
Original languageEnglish
Pages (from-to)101-116
JournalCommunications in Mathematical Physics
Publication statusPublished - 1 Jan 2014


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