Transitive actions of semigroups in semi-simple Lie groups
type: Item
instance of: scholarly article
title: Transitive actions of semigroups in semi-simple Lie groups
zbMATH Open document ID: 0915.22009
author: Luiz A. B. San Martin
author: Pedro A. Tonelli
published in: Semigroup Forum
publication date: 1999-06-28T00:00:00Z
review text: Given a connected semisimple Lie group \(G\) with finite center and a subsemigroup \(S\) with interior points, the authors point out a natural way of associating to \(S\) a parabolic subgroup \(P\) of \(G\) and an associated flag manifold \(B=G/P\). The notion of a contractive sequence in \(G\) with respect to \(B\) is introduced, and it is shown that for a closed subgroup \(L\) of \(G\), the action of \(S\) on \(G/L\) is transitive if and only if the action of \(L\) on \(B\) is minimal (no proper invariant closed subsets) and \(L\) contains a contractive sequence. For the case that \(G\not= L\) is simple, the theorem specializes to the result that if \(G/L\) is compact, then \(S\) is transitive on \(G/L\) if and only if \(S=G\). The restriction that \(G/L\) be compact corrects an earlier claim of the authors; they provide a counterexample to show that the theorem can fail if \(G/L\) is not compact.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22A20
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 1251164
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: semigroup
zbMATH Keywords: homogeneous space
zbMATH Keywords: control set
zbMATH Keywords: flag manifold
Wikidata QID: Q115388232
reviewed by: Jimmie D. Lawson
MaRDI profile type: Publication
full work available at URL: s002339900004
OpenAlex ID: W2068059759
DOI: 10.1007/S002339900004
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description: scientific article; zbMATH DE number 1251164