Clifford Algebras And Lie Theory

Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge A Series of Modern Surveys in Mathematics Editorial Board L. Ambrosio, Pisa G.-M. Greuel, Kaiserslautern M. Gromov, Bures-sur-Yvette J. Jost, Leipzig J. Kollár, Princeton S. S. Kudla, Toronto G. Laumon, Orsay H. W. Lenstra, Jr., Leiden J. Tits, Paris D. B. Zagier, Bonn Managing Editor R. Remmert, Münster For further volumes: www.springer.com/series/728 Volume 58 Eckhard Meinrenken Clifford Algebras and Lie Theory Eckhard Meinrenken Department of Mathematics University of Toronto Toronto, ON, Canada ISSN 0071-1136 Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics ISBN 978-3-642-36215-6 ISBN 978-3-642-36216-3 (eBook) DOI 10.1007/978-3-642-36216-3 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013933367 Mathematics Subject Classification: 15A66, 17B20, 17B35, 17B55 © Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Dedicated to my family Nozomi and Emma Preface Given a symmetric bilinear form B on a vector space V , one defines the Clifford algebra Cl(V ; B) to be the associative algebra generated by the elements of V , with relations v1 v2 + v2 v1 = 2B(v1 , v2 ), v1 , v2 ∈ V . If B = 0 this is just the exterior algebra ∧(V ), and for general B there is an isomorphism (the quantization map) q: ∧(V ) → Cl(V ; B). Hence, the Clifford algebra may be regarded as ∧(V ) with a new, “deformed” product. Clifford algebras enter the world of Lie groups and Lie algebras from multiple directions. For example, they are used to give constructions of the spin groups Spin(n), the simply connected double coverings of SO(n) for n ≥ 3. Going a little further, one then obtains the spin representations of Spin(n), which is an irreducible representation if n is odd and breaks up into two inequivalent irreducible representations if n is even. The “accidental” isomorphisms of Lie groups in low dimensions, such as Spin(6) ∼ = SU(4), all find natural expl