Structure and Geometry of Lie Groups
This text is designed as an introduction to Lie groups and their actions on manifolds, one that is accessible both to a broad range of mathematicians and to graduate students. Building on the authors' Lie-Gruppen und Lie-Algebren textbook from 1
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Joachim Hilgert r Karl-Hermann Neeb
Structure and Geometry of Lie Groups
Joachim Hilgert Mathematics Institute University of Paderborn Warburgerstr. 100 33095 Paderborn Germany [email protected]
Karl-Hermann Neeb Department of Mathematics Friedrich-Alexander Universität Erlangen-Nürnberg Cauerstrasse 11 91054 Erlangen Germany [email protected]
ISSN 1439-7382 Springer Monographs in Mathematics ISBN 978-0-387-84793-1 e-ISBN 978-0-387-84794-8 DOI 10.1007/978-0-387-84794-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011942060 Mathematics Subject Classification (2010): 7Bxx, 22Exx, 22Fxx © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
Nowadays there are plenty of textbooks on Lie groups to choose from, so we feel we should explain why we decided to add another one to the row. Most of the readily available books on Lie groups either aim at an elementary introduction mostly restricted to matrix groups, or else they try to provide the background on semisimple Lie groups needed in harmonic analysis and unitary representation theory with as little general theory as possible. In [HN91], we tried to exhibit the basic principles of Lie theory rather than specific material, stressing the exponential function as the means of translating problems and solutions between the global and the infinitesimal level. In that book, written in German for German students who typically do not know differential geometry but are well versed in advanced linear algebra, we avoided abstract differentiable manifolds by combining matrix groups with covering arguments. Having introduced the basic principles, we demonstrated their power by proving a number of standard and not so standard results on the structure of Lie groups. The choice of results included owed a lot to Hochschild’s book [Ho65], which even then was not so easy to come by. This book builds on [HN91], but after twenty years of teaching and research in Lie theory we found it indispensable to also have the differential geometry of Lie groups available. Even though this is not apparent from the text, the reason for this is the large number of applications and furthe
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