On the Yang–Baxter equation and left nilpotent left braces

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.