On the Yang-Baxter equation and left nilpotent left braces

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• 作者：Ferran Cedó ; ^a ; ^{cedo@mat.uab.cat} ; Tatiana Gateva-Ivanova^b ; ^c ; ^{Tatyana@aubg.edu} ; Agata Smoktunowicz^d ; ^{A.Smoktunowicz@ed.ac.uk}
• 关键词：Primary ; 16T25 ; 16T20 ; 16N80 ; 20F45 ; 16N40 ; 81R50
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：221
• 期：4
• 页码：751-756
• 全文大小：266 K
• 卷排序：221

文摘

We study non-degenerate involutive set-theoretic solutions (X,r)(X,r) of the Yang–Baxter equation, we call them solutions  . We prove that the structure group G(X,r)G(X,r) of a finite non-trivial solution (X,r)(X,r) cannot be an Engel group. It is known that the structure group G(X,r)G(X,r) of a finite multipermutation solution (X,r)(X,r) is a poly-ZZ group, thus our result gives a rich source of examples of braided groups and left braces G(X,r)G(X,r) which are poly-ZZ 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)⁎AA(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.