On the Yang–Baxter equation and left nilpotent left braces

Ferran Cedó, Tatiana Gateva-Ivanova, Agata Smoktunowicz

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21 Citations (Scopus)

Abstract

© 2016 Elsevier B.V. We study non-degenerate involutive set-theoretic solutions (X,r) of the Yang–Baxter equation, we call them solutions. We prove that the structure group G(X,r) of a finite non-trivial solution (X,r) cannot be an Engel group. It is known that the structure group G(X,r) of a finite multipermutation solution (X,r) is a poly-Z group, thus our result gives a rich source of examples of braided groups and left braces G(X,r) which are poly-Z groups but not Engel groups. We find an explicit relation between the multipermutation level of a left brace and the length of the radical chain A (n+1) =A (n) ⁎A introduced by Rump. We also show that a finite solution of the Yang–Baxter equation can be embedded in a convenient way into a finite left brace, or equivalently into a finite involutive braided group.
Original languageEnglish
Pages (from-to)751-756
JournalJournal of Pure and Applied Algebra
Volume221
Issue number4
DOIs
Publication statusPublished - 1 Apr 2017

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