On solvability of Engel equations in the group of triangular matrices over a field.
type: Item
instance of: scholarly article
title: On solvability of Engel equations in the group of triangular matrices over a field.
zbMATH Open document ID: 1269.20042
author: Agnieszka Bier
published in: Linear Algebra and its Applications
publication date: 2013-02-22T00:00:00Z
review text: Let \(G\) be any group and \(k\geq 0\) an integer. The (left normed) \(k\)-Engel commutator \([x,{_ky}]\) for any two elements \(x,y\in G\) is defined inductively as follows: \([x,y]=[x,{_1y}]=x^{-1}y^{-1}xy\), \([x,{_ky}]=[[x,{_{k-1}y}],y]\) for all \(k>1\). In the paper under review the author denotes \([x,{_ky}]\) by \(e_k(x,y)\). Let \(K\) be a field of arbitrary characteristic. By \(UT_n(K)\), \(D_n(K)\) and \(T_n(K)\) we denote the subgroups of the general linear group \(GL_n(K)\), which are respectively the groups of upper unitriangular, diagonal and triangular matrices of size \(n\times n\) with all entries from the field \(K\). In the group \(UT_n(K)\), the author distinguishes subgroups \[ UT_n^m(K)=\{\mathbf 1_n+\sum_{i<j-m\leq n}a_{i,j}e_{i,j},\;a_{i,j}\in K\},\quad 0\leq m\leq n-1, \] where \(\mathbf 1_n\) denotes the unity matrix of size \(n\times n\) and \(e_{i,j}\) denotes the matrix with unity in the place \((i,j)\) and zeros elsewhere. Let \(\varphi_n\colon T_n(K)\hookrightarrow T_{n+1}(K)\) denote the natural embedding, i.e., \[ \varphi_n(A)=\left(\begin{smallmatrix} A &\underline 0^T\\ \underline 0&\mathbf 1\end{smallmatrix}\right)\quad\text{for all }A\in T_n(K), \] where \(\underline 0\) denotes a zero vector from \(K^n\). Then \(\varphi(UT_n(K))\subseteq UT_{n+1}(K)\) and \(\varphi_n(D_n(K))\subseteq D_{n+1}(K)\). Let \[ UT_\infty(K)=\varinjlim_n(UT_n(K),\varphi_n)\quad\text{and}\quad T_\infty(K)=\varinjlim_n(T_n(K),\varphi_n) \] denote finitary unitriangular and triangular matrices over the field \(K\). The main results of the paper under review are the following: Theorem 1. Let \(K\) be an arbitrary field. Then for every matrix \(C\) contained in \(UT^m_n(K)\) the Engel equation \(e_m(x,y)=C\) has a solution in \(UT_n(K)\). Theorem 2. Let \(K\) be a field containing at least \(n+1\) elements. Then for every matrix \(C\) contained in \(UT_n(K)\) the Engel equation \(e_k(x,y)=C\) has a solution in \(T_n(K)\), \(k\in\mathbb N\). Theorem 3. Let \…
reviewed by: Alireza Abdollahi
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F45
Mathematics Subject Classification ID: 20F70
Mathematics Subject Classification ID: 20H20
zbMATH DE Number: 6138415
zbMATH Keywords: groups of triangular matrices
zbMATH Keywords: commutator equalities
zbMATH Keywords: commutator equations
zbMATH Keywords: Engel equations
zbMATH Keywords: Engel subgroup
zbMATH Keywords: Engel widths
MaRDI profile type: Publication
full work available at URL: j.laa.2012.10.009
OpenAlex ID: W2006618363
cites work: Verbal subgroups in the group of triangular matrices over field of characteristic 0.
cites work: The width of verbal subgroups in the group of unitriangular matrices over a field.
cites work: Engel subgroups of triangular matrices over local rings.
cites work: Q3849942
cites work: Commutator subgroup of Vershik-Kerov group.
cites work: The lower central series of subgroups of the Vershik-Kerov group.
DOI: 10.1016/J.LAA.2012.10.009
description: scientific article; zbMATH DE number 6138415