The interior and the exterior of the image of the exponential map in classical Lie groups
type: Item
instance of: scholarly article
title: The interior and the exterior of the image of the exponential map in classical Lie groups
zbMATH Open document ID: 0638.22006
DOI: 10.1016/0021-8693(88)90134-2
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: In Lie group theory it is of essential importance to understand the exponential function exp: \({\mathfrak g}\to G\). Its global behavior is not as simple as it may seem at first glance. For instance, it is not an obvious matter to determine the image \(E=\exp {\mathfrak g}\) and its boundary \(\partial E\) in G. Previously, the author established that \(E=G\) if \(G=GL(n,{\mathbb{C}})\), GL(n,\({\mathbb{H}})\), U(k,n-k), O *(2n), Sp(k,n-k). In the present article he completely clarifies the situation for \(G=SL(n,{\mathbb{C}})\), GL(n,\({\mathbb{R}})\), Sl(n,\({\mathbb{R}})\), SU(k,n-k), O(n,\({\mathbb{C}})\), O(k,n-k), Sp(2n,\({\mathbb{C}})\), Sp(2n,\({\mathbb{R}}).\) It is beautifully demonstrated that, in the case of these linear Lie groups, the issue is a question of linear algebra. Since, for the purposes of this review it would require too much space to render a complete report of the results, their general flavor is well represented by the following sample: Theorem 1. If \(G=GL(n,{\mathbb{R}})\), Sl(n,\({\mathbb{R}})\) or Sp(2n,\({\mathbb{R}})\), then \(g\in G\) is in the interior of E iff Spec g contains no negative number, and \(g\not\in \bar E\) iff Spec g contains at least one negative eigenvalue of odd multiplicity. Theorem 2. If \(G=O(p,q)\) then g is in the interior of E iff Spec g contains no elements \(<-1\) and if the generalized eigenspace for the eigenvalue -1 is positive or negative definite. Further \(g\not\in \bar E\) iff g has at least one negative eigenvalue of odd multiplicity or else -1 is an eigenvalue of even multiplicity and the numbers p and q of the signature of the generalized eigenspace of the eigenvalue -1 are both odd.
reviewed by: Karl H. Hofmann
Mathematics Subject Classification ID: 22E15
zbMATH DE Number: 4040192
zbMATH Keywords: exponential function
zbMATH Keywords: linear Lie groups
zbMATH Keywords: interior
zbMATH Keywords: generalized eigenspace
zbMATH Keywords: eigenvalue
Wikidata QID: Q115366491
author: Dragomir Ž. Đoković
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90134-2
OpenAlex ID: W2093669232
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description: scientific article; zbMATH DE number 4040192