Density properties for certain subgroups of semi-simple groups without compact components
type: Item
instance of: scholarly article
title: Density properties for certain subgroups of semi-simple groups without compact components
zbMATH Open document ID: 0094.24901
DOI: 10.2307/1970150
published in: Annals of Mathematics. Second Series
publication date: 1960-01-01T00:00:00Z
review text: Suppose \(G\) is a connected, real, semisimple Lie group without compact components. A subgroup \(H\) of \(G\) may be such that (i) \(H\) is discrete and the measure of \(G/H\) is finite, or (ii) for every neighbourhood \(U\) of the identity in \(G\) and every element \(g\in G\) there is an integer \(n\) so that \(g^n\in UHU\). (i) implies (ii), and is, of course, the more interesting property. For the author's purposes, however, it is convenient to assume (ii) alone. The author then shows that if a subalgebra of the Lie algebra of \(G\) is stable under \(H\) it is an ideal. Among other corollaries he deduces that (i) the centralizer of \(H\) is the centre of \(G\) and (ii) if \(\rho\) is a representation of \(G\) in a real (complex) vector space then every element of \(\rho(G)\) is a real (complex) linear combination of elements of \(\rho(H)\).
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 3154010
zbMATH Keywords: connected, real, semisimple Lie group without compact components
zbMATH Keywords: stable subalgebra
zbMATH Keywords: centralizer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1970150
OpenAlex ID: W2321597728
author: Armand Borel
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description: scientific article; zbMATH DE number 3154010