On the exponential function of real splittable and real semisimple Lie groups
type: Item
instance of: scholarly article
title: On the exponential function of real splittable and real semisimple Lie groups
zbMATH Open document ID: 0902.22005
author: Michael Wüstner
published in: Beiträge zur Algebra und Geometrie
publication date: 1998-04-07T00:00:00Z
full work available at URL: https://eudml.org/doc/222825
full work available at URL: 5.html
review text: A Lie group \(G\) is called exponential if \(G= \exp{\mathfrak g}\), and weakly exponential if \(G\) is equal to the closure of \(\exp{\mathfrak g}\). As a generalization of his previous paper [J. Algebra 184, 1082-1092 (1996; Zbl 0870.22004)], the author gives a characterization of the exponential real analytic splittable groups and their exponential covering groups. Let \(G\subset GL(V)\) be a splittable analytic group, that is, for all \(g\in G\), the Jordan decomposition \(g=g_ug_s\) satisfies \(g_u,g_s\in G\). Then \(G\) is exponential if and only if for each nilpotent \(x\in{\mathfrak g}\) the centralizer \(Z_G(x)\) is weakly exponential. Furthermore, this characterization is even true for exponential covering groups \(\widetilde G\) of \(G\). As a corollary, we see that, if \(G\) is a real semisimple Lie group and \({\mathfrak g}\subset{\mathfrak g}{\mathfrak l}(V)\), then \(G\) is exponential if and only if for each nilpotent \(x\in{\mathfrak g}\), \(Z_G(x)\) is weakly exponential.
Mathematics Subject Classification ID: 22E15
zbMATH DE Number: 1127637
zbMATH Keywords: exponential Lie groups
zbMATH Keywords: exponential covering groups
zbMATH Keywords: splittable analytic group
zbMATH Keywords: semisimple Lie group
reviewed by: Takeshi Kawazoe
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1127637