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Communications in Mathematical Physics

, Volume 327, Issue 1, pp 101–116 | Cite as

Braces and the Yang–Baxter Equation

  • Ferran CedóEmail author
  • Eric Jespers
  • Jan Okniński
Article

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 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.

Keywords

Normal Subgroup Quantum Group Nilpotent Group Semidirect Product Radical Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain
  2. 2.Department of MathematicsVrije Universiteit BrusselBrusselBelgium
  3. 3.Institute of MathematicsWarsaw UniversityWarsawPoland

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