Local splitting of locally compact groups and pro-Lie groups
type: Item
instance of: scholarly article
title: Local splitting of locally compact groups and pro-Lie groups
zbMATH Open document ID: 1246.22003
author: Sidney A. Morris
author: Karl H. Hofmann
publication date: 2012-05-07T00:00:00Z
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 22D05
zbMATH DE Number: 6031523
zbMATH Keywords: Iwasawa's Theorem
zbMATH Keywords: local splitting
zbMATH Keywords: locally compact group
zbMATH Keywords: pro-Lie group
zbMATH Keywords: local structure
zbMATH Keywords: projective limit
zbMATH Keywords: Lie group
zbMATH Keywords: connected group
zbMATH Keywords: almost connected group
zbMATH Keywords: coreductive radical
zbMATH Keywords: reductive Lie algebra
MaRDI profile type: Publication
full work available at URL: jgt.2011.090
OpenAlex ID: W2037625254
cites work: Transitive actions of compact groups and topological dimension
cites work: Iwasawa's Local Splitting Theorem for Pro-Lie Groups
cites work: On some types of topological groups
cites work: On the conjecture of Iwasawa and Gleason
cites work: A generalization of a theorem of Gleason
DOI: 10.1515/JGT.2011.090
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published in: Journal of Group Theory
review text: The note provides a new proof for a local splitting theorem for pro-Lie groups, i.e., (not necessarily locally compact) topological groups which are projective limits of (finite-dimensional, real) Lie groups. A profound theory of pro-Lie groups was developed in a recent monograph by the authors [The Lie theory of connected pro-Lie groups. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups. Zürich: European Mathematical Society (2007; Zbl 1153.22006)], (*). The book already contains a local splitting result (op. cit., Theorem 13.19) for connected pro-Lie groups, stimulated by the classical work of \textit{K. Iwasawa} [Ann.\ Math. (2) 50, 507--558 (1949; Zbl 0034.01803)] which applies to locally compact groups. The condition of connectedness was removed afterwards by \textit{A. A. George Michael} (see [J. Math. Sci. Univ. Tokyo 17, 123--133 (2010; Zbl 1254.22001)]). Based on the general theory developed in (*), the authors give a short new proof for George Michael's result.NEWLINENEWLINERecall that a pro-Lie group \(G\) is said to be almost connected if \(G/G_e\) is compact (where \(G_e\) is the connected component of the identity element). Let \({\mathfrak g}\) be the Lie algebra associated to \(G\) (as in (*)) and \({\mathfrak s}\) be its coreductive radical, i.e., the smallest closed ideal of \({\mathfrak g}\) such that \({\mathfrak g}/{\mathfrak s}\) is reductive (viz.\ a direct product of a family of simple finite-dimensional Lie algebras and copies of \({\mathbb R}\)). George Michael's result is established in the following form (Theorem 2): Let \(G\) be an almost connected pro-Lie group such that the coreductive radical of \({\mathfrak g}\) is finite-dimensional. Then there are arbitrarily small normal subgroups \(N\) such that there is a simply connected Lie group \(L\) and a morphism \(\alpha: L\to G\) such that \(N\times L\to G\), \((n,x)\mapsto n\alpha(x)\) is open and has discrete kernel. In particular, \(G\) and …
reviewed by: Helge Glöckner
description: scientific article; zbMATH DE number 6031523