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Kishore Marathe

Topics in Physical Mathematics

ABC

Prof. Kishore Marathe City University of New York Brooklyn College Bedford Avenue 2900 11210-2889 Brooklyn New York USA [email protected]

Whilst we have made considerable efforts to contact all holders of copyright material contained in this book, we have failed to locate some of them. Should holders wish to contact the Publisher, we will make every effort to come to some arrangement with them. ISBN 978-1-84882-938-1 e-ISBN 978-1-84882-939-8 DOI 10.1007/978-1-84882-939-8 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2010933531 Mathematics Subject Classification (2010): 53C25, 57M25, 57N10, 57R57, 58J60, 81T13, 81T30 c Springer-Verlag London Limited 2010 ° Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: deblik Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Dedicated to the memory of my mother, Indumati (1920 – 2005), who passed on to me her love of learning. ========================== Memories Your voice is silent now. But the sound of your soft Music will always resonate In my heart. The wisp of morning incense Floats in the air no more. It now resides only in my Childhood memories.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 1

Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Graded Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Kac–Moody Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .