Translations in simply transitive affine actions of Heisenberg type Lie groups
type: Item
instance of: scholarly article
title: Translations in simply transitive affine actions of Heisenberg type Lie groups
zbMATH Open document ID: 1023.17003
DOI: 10.1016/S0024-3795(02)00435-4
author: Tine De Cat
author: Karel Dekimpe
author: Paul Igodt
published in: Linear Algebra and its Applications
publication date: 2003-03-23T00:00:00Z
review text: Let \((x,y)\mapsto x\cdot y\) be a complete left-symmetric structure on a nilpotent Lie algebra \({\mathfrak g}\). Let \(T({\mathfrak g})\) be the set of \(x\in {\mathfrak g}\) satisfying \(x\cdot y=0\) for all \(y\in {\mathfrak g}\). The left-symmetric structure induces a simply transitive action of a nilpotent Lie group \(G\) such that \(\exp (T({\mathfrak g}))\) is precisely the subset of \(G\) consisting of those elements acting as pure translations. The authors prove the following theorem. If \({\mathfrak g}\) is a \(2\)-step nilpotent Lie algebra with \(1\)-dimensional commutator algebra, then for any complete left-symmetric structure on \({\mathfrak g}\) we have \(T({\mathfrak g})\neq 0\).
reviewed by: Dietrich Burde
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 1886830
zbMATH Keywords: left-symmetric structure
zbMATH Keywords: nilpotent Lie algebra
zbMATH Keywords: simply transitive action
Wikidata QID: Q115340027
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1886830