Groups and Symmetries From Finite Groups to Lie Groups
Unlike many other texts, this book deals with the theory of representations of finite groups, compact groups, linear Lie groups and their Lie algebras, concisely and in one volume. Key Topics: • Brisk review of the basic definitions of group theory, with
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S. Axler K.A. Ribet
For other titles published in this series, go to http://www.springer.com/series/223
Yvette Kosmann-Schwarzbach
Groups and Symmetries From Finite Groups to Lie Groups Translated by Stephanie Frank Singer
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Yvette Kosmann-Schwarzbach Centre de Math´ematiques Laurent Schwartz ´ Ecole Polytechnique 91128 Palaiseau France [email protected] Translated by:
Stephanie Frank Singer Philadelphia, PA USA
Editorial Board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Universit`a degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley ´ Claude Sabbah, CNRS, Ecole Polytechnique Endre S¨uli, University of Oxford Wojbor Woyczy´nski, Case Western Reserve University
ISBN 978-0-387-78865-4 e-ISBN 978-0-387-78866-1 DOI 10.1007/978-0-387-78866-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009937461 Mathematics Subject Classification (2000): 20-01, 20G05, 20G45, 17B45, 33C55 c Springer Science+Business Media, LLC 2010 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. Cover Illustration: The figure on the front cover represents weights for the fundamental representation of sl(4). Courtesy of Adrian Ocneanu. Printed on acid-free paper Springer is a part of Springer Science+Business Media(www.springer.com).
Sophus Lie (1842–1899 ), around 1865, at the end of his studies at the University of Christiana (Oslo), approximately seven years before his first work on continuous groups, later known as “Lie groups.” (Photo Frederik Klem/Joronn Vogt, with the kind permission of Joronn Vogt and Arild Stubhaug)
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1
General Facts About Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Review of Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Examples of Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Cyclic Group of Order n . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Symmetric Group Sn . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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