Lie automorphisms of the algebra of two generic \(2\times 2\) matrices
type: Item
instance of: scholarly article
title: Lie automorphisms of the algebra of two generic \(2\times 2\) matrices
zbMATH Open document ID: 0961.16012
author: Vesselin Drensky
author: Chander K. Gupta
published in: Journal of Algebra
publication date: 2000-11-20T00:00:00Z
review text: Let \(R\) be the (associative) algebra generated by two generic \(2\times 2\) matrices \(x_1\) and \(x_2\) over a field \(K\) of characteristic 0. The authors study the Lie automorphisms of \(R\). These are the automorphisms of \(R\) that send \(x_1\) and \(x_2\) to linear combinations of commutators of \(x_1\) and \(x_2\). They prove that if \(\phi_{kl}(x_1)=x_1\), \(\phi_{kl}(x_2)=x_2+x_2\text{ ad}^{2k}x_1\text{ ad}^{2l+1}[x_1,x_2]\), \(k\geq 1\), \(l\geq 0\), then the subgroup of the Lie automorphisms of \(R\) generated by \(\text{GL}_2(K)\) and by \(\phi_{kl}\) is dense. Here the density is meant with respect to the formal power series topology.
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 17B40
zbMATH DE Number: 1428245
zbMATH Keywords: generic matrices
zbMATH Keywords: tame automorphisms
zbMATH Keywords: wild automorphisms
zbMATH Keywords: Lie automorphisms
reviewed by: Plamen Koshlukov
MaRDI profile type: Publication
full work available at URL: jabr.1999.8055
OpenAlex ID: W2094493528
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Wikidata QID: Q115395793
DOI: 10.1006/JABR.1999.8055
description: scientific article; zbMATH DE number 1428245