Werner Greub Multilinear Igenra 2nd Edition a I Springer-Verlag New York Heidelberg Berlin Un iversitext Werner Greub Multiinear Algebra 2nd Edition Springer-Verlag New York Heidelberg Berlin Werner Greub Department of Mathematics University of Toronto Toronto M5S 1A1 Canada AMS Subject Classifications: 15-01, 15A75, 15A72 Library of Congress Cataloging in Publication Data Greub, Werner Hildbert, 1925- Multilinear algebra, (Universitext) Includes index. 1. Algebras, Linear. I. Title. QA 184.G74 1978 512'.5 78-949 ISBN-13:978-0-387-90284-5 e-ISBN-13:978-1-4613-9425-9 DO!: 10.1007/ 978-1-4613-9425-9 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag. © 1967 by Springer-Verlag Berlin Heidelberg © 1978 by Springer-Verlag New York Inc. 987654321 Preface This book is a revised version of the first edition and is intended as a sequel and companion volume to the fourth edition of Linear Algebra (Graduate Texts in Mathematics 23). As before, the terminology and basic results of Linear Algebra are frequently used without reference. In particular, the reader should be familiar with Chapters 1-5 and the first part of Chapter b of that book, although other sections are occasionally used. In this new version of Multilinear Algebra, Chapters 1-5 remain essen- tially unchanged from the previous edition. Chapter b has been completely rewritten and split into three (Chapters b, 7, and 8). Some of the proofs have been simplified and a substantial amount of new material has been added. This applies particularly to the study of characteristic coefficients and the Pf of f ian. The old Chapter 7 remains as it stood, except that it is now Chapter 9. The old Chapter 8 has been suppressed and the material which it con- tained (multilinear functions) has been relocated at the end of Chapters 3, S, and 9. The last two chapters on Clifford algebras and their representations are completely new. In view of the growing importance of Clifford algebras and the relatively few references available, it was felt that these chapters would be useful to both mathematicians and physicists. In Chapter 10 Clifford algebras are introduced via universal properties and treated in a fashion analogous to exterior algebra. After the basic isomorphism theorems for these algebras (over an arbitrary inner product space) have been established the chapter proceeds to a discussion of finite-dimensional Clifford algebras. The treatment culminates in the com- plete classification of Clifford algebras over finite-dimensional complex and real inner product spaces. v vi Preface The book concludes with Chapter 11 on representations of Clifford algebras. The twisted adjoint representation which leads to the definition of the spin-groups is an important example. A version of Wedderburn's theorem is the key to the classification of all representations of the Clifford algebra over an 8-dimensional real vector space with a negative definite inner product. The results are applied in the last section of this chapter to study orthogonal multiplications between Euclidean spaces and the ex- istence of orthonormal frames on the sphere. In particular, it is shown that the (n -1)-sphere admits an orthonormal k-frame where k is the Radon-Hurwitz number corresponding to n. A deep theorem of F. Adams states that this result can not be improved. The problems at the end of Chapter 11 include a basis-free definition of the Cayley algebra via the complex cross-product analogous to the defini- tion of quaternions in Section 7.23 of the fourth edition of Linear Algebra. Finally, the Cayley multiplication is used to obtain concrete forms of some of the isomorphisms in the table at the end of Chapter 10. I should like to express my deep thanks to Professor J. R. Vanstone who worked closely with me through each stage of this revision and who made numerous and valuable contributions to both content and presentation. I should also like to thank Mr. M. S. Swanson who assisted Professor Vanstone and myself with the proof reading. Toronto, April 1978 W. H. Greub Table of Contents Chapter 1 Tensor Products 1 Chapter 2 Tensor Products of Vector Spaces with Additional Structure 41 Chapter 3 Tensor Algebra 60 Chapter 4 Skew-Symmetry and Symmetry in the Tensor Algebra 84 Chapter 5 Exterior Algebra 96 Chapter 6 Mixed Exterior Algebra 148 Chapter 7 Applications to Linear Transformations 174 Chapter 8 Skew and Skew-Hermitian Transformations 193 Chapter 9 Symmetric Tensor Algebra 209 Chapter 10 Clifford Algebras 227 Chapter 11 Representations of Clifford Algebras 260 Index 291 vii Tensor Products Throughout this chapter except where noted otherwise all vector spaces will be defined over a fixed, but arbitrarily chosen, field T. Multilinear Mappings 1.1. Bilinear Mappings Suppose E, F and G are any three vector spaces, and consider a mapping p : E x F -+ G. 'p is called bilinear if it satisfies the conditions p(t x 1 + µx 2 , y) = p(x1,y) + µup(x2 , y) x1, x2EE, yeF,A,µeF, xeE,y1,y2eF. 'p(x, Ay 1 + µY2) _ A p(x, Y 1) + µup(x, Y2) Recall that if G = F, then 'p is called a bilinear function. The set S of all vectors in G of the form '(x, y), x e E, y e F is not in general a vector subspace of G. As an example, let E = F and G be re- spectively 2- and 4-dimensional vector spaces. Select a basis a 1, a2 in E and a basis c (v = 1, ... , 4) in G and define the bilinear mapping 'p by co(x, Y) = 'i 'c1 + 1i2C2 + 2i 'c3 + 2ii2C4 where x = c 1a 1 + 2a2 and y = ri' a 1 + Then it is easy to see that r12a2. a vector z = VC of G is contained in S if and only if the components satisfy the relation - = o. 1