Algebras, Rings and Modules
As a natural continuation of the first volume of Algebras, Rings and Modules, this book provides both the classical aspects of the theory of groups and their representations as well as a general introduction to the modern theory of representations includi
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Mathematics and Its Applications
Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 586
Algebras, Rings and Modules Volume 2
by
Michiel Hazewinkel CWI, Amsterdam, The Netherlands
Nadiya Gubareni Technical University of Czestochowa, Poland and
V.V. Kirichenko Kiev Taras Shevchenko University, Kiev, Ukraine
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4020-5141-8 (HB) ISBN 978-1-4020-5140-1 (e-book)
Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com
Printed on acid-free paper
All Rights Reserved c 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.
Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Chapter 1. Groups and group representations . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Groups and subgroups. Deļ¬nitions and examples . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Symmetry. Symmetry groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Quotient groups, homomorphisms and normal subgroups . . . . . . . . . . . . . . 10 1.4 Sylow theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Solvable and nilpotent groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.6 Group rings and group representations. Maschke theorem . . . . . . . . . . . . . 26 1.7 Properties of irreducible representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.8 Characters of groups. Orthogonality relations and their applications . . . 38 1.9 Modular group representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 1.10 Notes and references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Chapter 2. Quivers and their representations . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.1 Certain important algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.2 Tensor algebra of a bimodule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.3 Quivers and path algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.4 Representations of quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.5 Dynkin and Euclidean diagrams. Quadratic f
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