Dense Lie group homomorphisms
type: Item
instance of: scholarly article
title: Dense Lie group homomorphisms
zbMATH Open document ID: 0847.22005
published in: Journal of Algebra
publication date: 1994-11-06T00:00:00Z
full work available at URL: https://pub.uni-bielefeld.de/record/1775497
review text: This article gives a survey of the basic facts on the structure and properties of dense Lie group homomorphisms, continuous homomorphisms between connected Lie groups with dense images. Dense images of vector spaces in tori are widely known examples. The treatment leads to the description of all dense embeddings of a given Lie group into other Lie groups, thus solving a problem of Goto. Among results that the author obtains, it is shown that if \(\varphi : H \to G\) is a dense continuous homomorphism of connected Lie groups, then \(G = \varphi (H)T\) for any maximal torus \(T\) in \(G\). In particular, the different maximal tori in \(G\) are conjugate by elements in \(\varphi (H)\). The classification of dense embeddings of \(H\) is reduced to an abelian problem concerning a given dense homomorphism \(\alpha : U \to T\), where \(T\) is a maximal torus in the closure of \(\text{Ad} (H)\), the image of adjoint representation, and \(U\) is its inverse (under Ad) in \(H\).
Mathematics Subject Classification ID: 22E20
zbMATH DE Number: 682587
zbMATH Keywords: dense Lie group homomorphisms
zbMATH Keywords: vector spaces
zbMATH Keywords: tori
zbMATH Keywords: dense embeddings
zbMATH Keywords: adjoint representation
author: D. Poguntke
reviewed by: Jimmie D. Lawson
MaRDI profile type: Publication
OpenAlex ID: W2031228322
Wikidata QID: Q115396229
DOI: 10.1006/JABR.1994.1300
Recommended article: Dense Subgroups of Lie Groups
Recommended article: Lie groups with dense exponential image
Recommended article: On dense free subgroups of Lie groups
Recommended article: Zariski density in Lie groups
Recommended article: Homomorphisms of Lie groups
Recommended article: Density properties of complex Lie groups
Recommended article: Dense subgroups with property \((T)\) in Lie groups
Recommended article: Homomorphic images of Lie groups
Recommended article: Q5340976
Recommended article: Q4008043
description: scientific article; zbMATH DE number 682587