Two-step nilpotent Lie groups and homogeneous fiber bundles
type: Item
instance of: scholarly article
title: Two-step nilpotent Lie groups and homogeneous fiber bundles
zbMATH Open document ID: 1038.53049
DOI: 10.1023/A:1024241806100
author: Hiroshi Tamaru
published in: Annals of Global Analysis and Geometry
publication date: 2003-08-20T00:00:00Z
review text: A new subclass of two-step nilpotent Lie groups is introduced. This class is attached to homogeneous fiber bundles \(H\slash K\to G\slash K\to G\slash H,\) where \((G,H)\) is a compact symmetric pair and \(K\) is a closed subgroup of \(H.\) If \(\mathfrak{g},\) \(\mathfrak{h}\) and \(\mathfrak{k}\) are the Lie algebras of \(G,\) \(H,\) and \(K,\) respectively, then the author constructs the two-step nilpotent Lie algebras \(\mathfrak{n}(\mathfrak{g},\mathfrak{h},\mathfrak{k})\) and investigates its properties. In this paper the author considers the case when \((G,H)\) is irreducible and shows close relations between the isotropy representations of \(G\slash K\) and a simply connected nilpotent Lie group \(N(\mathfrak{g},\mathfrak{h},\mathfrak{k})\) corresponding to \(\mathfrak{n}(\mathfrak{g},\mathfrak{h},\mathfrak{k}).\) The relation between weak symmetry on the total space \(G\slash K\) and the nilpotent groups \(N(\mathfrak{g},\mathfrak{h},\mathfrak{k}),\) found by the author, provides new examples of weakly symmetric nilpotent groups. The author has also found the relation between the geodesic orbit property of \(G\slash K\) and \(N(\mathfrak{g},\mathfrak{h},\mathfrak{k})\). (Recall, that a connected Riemannian manifold has geodesic orbit property if every geodesic is an orbit of a one parameter subgroup of the isometry group.)
reviewed by: Roman Urban
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1968157
zbMATH Keywords: two-step nilpotent groups
zbMATH Keywords: symmetric spaces
zbMATH Keywords: weakly symmetric spaces
zbMATH Keywords: Riemannian geodesic orbit spaces
MaRDI profile type: Publication
full work available at URL: a:1024241806100
OpenAlex ID: W24932585
description: scientific article; zbMATH DE number 1968157