Isotropy of non-nilpotent Riemannian solvable Lie groups
type: Item
instance of: scholarly article
title: Isotropy of non-nilpotent Riemannian solvable Lie groups
zbMATH Open document ID: 0843.53034
DOI: 10.1007/BF00128195
author: Ignacio Bajo
published in: Annals of Global Analysis and Geometry
publication date: 1996-03-11T00:00:00Z
review text: Let \((G, g)\) be a solvable Lie group endowed with a left-invariant Riemannian metric. It is known that if \(G\) is unimodular and all roots of its Lie algebra \({\mathfrak k}\) are real, then its isometry group \(I(G, g)\) is isomorphic to the semidirect product \(GK\) of \(G\) and the isotropy group at the identity \(K\), this being isomorphic to the group \(\Aut (G,g)\) of isometric automorphisms of \((G,g)\). In this paper, the author proves that for every compact Lie algebra \({\mathfrak k}\) and for every integer \(q\geq 3\) there exists a non-nilpotent \(q\)-step solvable Lie group \(G\) and a left-invariant Riemannian metric \(g\) on \(G\) such that \({\mathfrak k}\) is isomorphic to the Lie algebra of the isotropy group \(K\) of isometries fixing the identity of \((G, g)\).
reviewed by: Zixin Hou
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 854358
zbMATH Keywords: solvable Lie group
zbMATH Keywords: Riemannian metric
zbMATH Keywords: isometry group
zbMATH Keywords: compact Lie algebra
MaRDI profile type: Publication
Wikidata QID: Q115395228
cites work: Skew-symmetric derivations of a complex Lie algebra
cites work: Homogeneous nilmanifolds with prescribed isotropy
cites work: Q3815453
cites work: Isometry Groups of Riemannian Solvmanifolds
description: scientific article; zbMATH DE number 854358