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 [5] and answers related questions from [3, 44]. The case of an uncountable field also answers a recent question by Zelmanov. In [38], 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 of \(A^{o}\) of a finite left brace \(A\) is a nilpotent group if and only if \(A^{n}=0\) for some \(n\). Moreover, if \(A^{o}\) is a nilpotent group then \(A\) is the direct sum of braces whose cardinatities are powers of prime numbers. Notice that \(A^{o}\) is sometimes called the multiplicative group of a brace \(A\) (for example in [13]). 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 \(m, n\) (Theorems 2, 3). In Section 2 we describe connections between our results and braided groups and the Yang-Baxter equation. It is worth noticing that by a result by Gateva-Ivanova [17] braces are in one-to-one correspondence with braided groups with involutive braided operators.