The local structure of groups of triangular automorphisms of relatively free algebras.
type: Item
instance of: scholarly article
title: The local structure of groups of triangular automorphisms of relatively free algebras.
zbMATH Open document ID: 1270.16031
published in: Algebra and Logic
publication date: 2013-02-21T00:00:00Z
review text: Let \(C_n\) be a relatively free algebra of rank \(n\) in a variety of algebras over an arbitrary field \(K\). In particular, \(C_n\) may be the polynomial algebra \(P_n\), the free associative algebra \(A_n\), or the absolutely free algebra \(F_n\). In the paper under review the author studies the local properties of the group \(UC_n\) of unitriangular automorphisms and the group \(TC_n\) of triangular automorphisms of \(C_n\), i.e., the properties of their finitely generated subgroups. When \(C_n=P_n,A_n,F_n\) the author establishes that every finitely generated subgroup \(G\) of \(TC_n\) has a faithful matrix representation over the field \(K\). Combined with the classical theorem of Mal'tsev that finitely generated matrix groups are residually finite, this implies that \(G\) is residually finite. As a consequence, for an arbitrary relatively free algebra \(C_n\) the group \(TC_n\) is locally solvable and the group \(UC_n\) is locally nilpotent; both groups are locally linear and locally residually finite. The solvability of \(UC_n\) and its nonlinearity (when the characteristic of \(K\) is 0 and \(n\geq 4\)) were established by \textit{V. A. Roman'kov, I. V. Chirkov} and \textit{M. A. Shevelin}, [Sib. Mat. Zh. 45, No. 5, 1184-1188 (2004); translation in Sib. Math. J. 45, No. 5, 974-977 (2004; Zbl 1059.17004)]. The nonlinearity for \(P_n\), \(n\geq 3\), over an arbitrary field \(K\) was given by \textit{Yu. V. Sosnovskij}, Sib. Mat. Zh. 48, No. 3, 689-693 (2007); translation in Sib. Math. J. 48, No. 3, 555-558 (2007; Zbl 1155.14315)]. The recent paper by \textit{V. G. Bardakov, M. V. Neshchadim} and \textit{Yu. V. Sosnovsky} [J. Algebra 362, 201-220 (2012; Zbl 1269.16033)] contains a detailed study of the groups of triangular automorphisms of \(P_n\) and \(A_n\). Finally, the present paper shows that the width of the commutator subgroup (the minimal number of commutators needed to write the elements of the commutator subgroup) of a finitely generated subgroup \(G…
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 14R10
Mathematics Subject Classification ID: 14R15
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 17A36
Mathematics Subject Classification ID: 17A50
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F12
zbMATH DE Number: 6138173
zbMATH Keywords: relatively free algebras
zbMATH Keywords: polynomial algebras
zbMATH Keywords: free associative algebras
zbMATH Keywords: absolutely free algebras
zbMATH Keywords: groups of unitriangular automorphisms
zbMATH Keywords: groups of triangular automorphisms
zbMATH Keywords: matrix representations
zbMATH Keywords: residual finiteness
zbMATH Keywords: width of commutator subgroup
author: Vitaliĭ Roman'kov
MaRDI profile type: Publication
full work available at URL: s10469-012-9202-9
OpenAlex ID: W1976601730
cites work: Automorphisms of a Free Associative Algebra of Rank 2.I
cites work: Automorphisms of a Free Associative Algebra of Rank 2. II
cites work: Über ganze birationale Transformationen der Ebene.
cites work: Q5818521
cites work: Q5622339
cites work: Subalgebras and automorphisms of polynomial rings.
cites work: The tame and the wild automorphisms of polynomial rings in three variables
cites work: Defining relations for tame automorphism groups of polynomial rings and wild automorphisms of free associative algebras.
cites work: The Anick automorphism of free associative algebras
cites work: Q4675478
cites work: Q3544752
cites work: Groups of triangular automorphisms of a free associative algebra and a polynomial algebra.
cites work: Q5193150
cites work: Commutator length of abelian-by-nilpotent groups
cites work: Faithful linear representations of certain free nilpotent groups
DOI: 10.1007/S10469-012-9202-9
description: scientific article; zbMATH DE number 6138173