On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation

Abstract:

It is shown that over an arbitrary field there exists a nil algebra $R$ whose adjoint group $R^{o}$ is not an Engel group. This answers a question by Amberg and Sysak from 1997. The case of an uncountable field also answers a recent question by Zelmanov.

In 2007, Rump introduced braces and radical chains $A^{n+1}=A\cdot A^{n}$ and $A^{(n+1)}=A^{(n)}\cdot A$ of a brace $A$. We show that the adjoint group $A^{o}$ of a finite right brace is a nilpotent group if and only if $A^{(n)}=0$ for some $n$. We also show that the adjoint group $A^{o}$ of a finite left brace $A$ is a nilpotent group if and only if $A^{n}=0$ for some $n$. Moreover, if $A$ is a finite brace whose adjoint group $A^{o}$ is nilpotent, then $A$ is the direct sum of braces whose cardinalities are powers of prime numbers. Notice that $A^{o}$ is sometimes called the multiplicative group of a brace $A$. We also introduce a chain of ideals $A^{[n]}$ of a left brace $A$ and then use it to investigate braces which satisfy $A^{n}=0$ and $A^{(m)}=0$ for some $m, n$.

We also describe connections between our results and braided groups and the non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation. It is worth noticing that by a result of Gateva-Ivanova braces are in one-to-one correspondence with braided groups with involutive braiding operators.