TY - JOUR

T1 - On the Yang–Baxter equation and left nilpotent left braces

AU - Cedó, Ferran

AU - Gateva-Ivanova, Tatiana

AU - Smoktunowicz, Agata

PY - 2017/4/1

Y1 - 2017/4/1

N2 - © 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.

AB - © 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.

U2 - 10.1016/j.jpaa.2016.07.014

DO - 10.1016/j.jpaa.2016.07.014

M3 - Article

VL - 221

SP - 751

EP - 756

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 4

ER -