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arXiv · 1205.3587

Braces and the Yang-Baxter equation

Ferran Cedo·Eric Jespers·Jan Okninski
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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 set-theoretic 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.

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Ferran Cedo, Eric Jespers, Jan Okninski. 2012-05-16. Braces and the Yang-Baxter equation. https://arxiv.org/abs/1205.3587

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