A remark on the Ado-Iwasawa theorem
type: Item
instance of: scholarly article
title: A remark on the Ado-Iwasawa theorem
zbMATH Open document ID: 1043.17501
published in: Journal of Algebra
publication date: 1999-12-06T00:00:00Z
review text: The Ado-Iwasawa theorem states that every finite dimensional Lie algebra over a field \(K\) has a finite dimensional faithful representation. The paper under review shows that this theorem holds for Lie algebras over certain classes of rings instead of fields. Thus let \(L\) be Lie algebra over the commutative ring \(R\), \(1\in R\), and suppose \(L\) is finitely generated as an \(R\)-module. If \(R\) is either a principal ideal domain of characteristic 0, or a Noetherian ring with 1 of prime characteristic \(p\) then there exists a finitely generated free \(R\)-module \(V\) and a faithful representation of \(L\) in \(V\) (Theorem A). One refinement of this theorem is Theorem B which deals with the case \(R\) residually positive (i.e. R is a PID and for every nontrivial prime ideal \(I\) of \(R\), the characteristic of \(R/I\) is positive). Then there exists a faithful representation of \(L\) in some finitely generated \(R\)-module \(V\). As a corollary the author gives an application to pro-\(p\)-groups, namely Theorem C. Let \(\hat{G}\) be torsion-free finitely generated powerful pro-\(p\)-group. Then there exists a finitely generated \(Z_p\) module \(W\) and an injective morphism (of topological groups) \(\phi\colon \hat{G}\to PL(W)\). Here \(PL(W)=\{1_W+\text{\textbf{p}}\alpha\mid \alpha\in \hom_{Z_p}(W)\}\subseteq \hom_{Z_p}(W)\), \(\text{\textbf{p}}=p\) if \(p\) is odd, and 4 if \(p=2\).
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 1271823
zbMATH Keywords: Ado-Iwasawa theorem
zbMATH Keywords: Lie algebras over rings
zbMATH Keywords: Faithful representations of Lie algebras
author: Th. Weigel
reviewed by: Plamen Koshlukov
MaRDI profile type: Publication
full work available at URL: jabr.1998.7563
OpenAlex ID: W2041269182
cites work: Q5614712
cites work: Q3338360
cites work: Q3997907
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Q5809234
cites work: Observations on group rings
cites work: On the Representation of Lie Rings in Associative Rings
DOI: 10.1006/JABR.1998.7563
description: scientific article; zbMATH DE number 1271823