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On the Yang-Baxter equation and left nilpotent left braces

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Original languageEnglish
Pages (from-to)751-756
Number of pages6
JournalJournal of pure and applied algebra
Volume221
Issue number4
Early online date29 Jul 2016
DOIs
Publication statusPublished - Apr 2017

Abstract

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.

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