Tensor Space s and Exterio r Algebra Recent Titles in This Series 108 Take o Yokonuma, Tenso r space s and exterior algebra , 199 2 107 B . M. Makarov, M. G. Goluzina, A. A. Lodkin, and A. N. Podkorytov, Selecte d problem s in real analysis, 199 2 106 G.-C . Wen, Conforma l mapping s and boundary value problems, 199 2 105 D . R. Yafaev, Mathematica l scatterin g theory: Genera l theory, 199 2 104 R . L. Dobrushin, R. Kotecky, an d S. Shlosman, Wulf f construction : A global shape fro m local interaction, 199 2 103 A . K. Tsikh, Multidimensiona l residue s and their applications, 199 2 102 A . M. Il'in, Matchin g o f asymptotic expansions of solution s of boundary valu e problems, 199 2 101 Zhan g Zhi-fen, Ding Tong-ren, Huang Wen-zao, and Don g Zhen-xi, Qualitativ e theory of differential equations , 199 2 100 V . L. Popov, Groups , generators, syzygies, and orbits i n invariant theory, 199 2 99 Nori o Shimakura, Partia l differential operator s of elliptic type, 199 2 98 V . A. Vassiliev, Complement s o f discriminants o f smooth maps: Topolog y and applications, 199 2 97 Itir o Tamura, Topolog y of foliations: A n introduction, 1 992 96 A . I. Markushevich, Introductio n to the classical theory of Abelian functions, 199 2 95 Guangchan g Dong, Nonlinea r partial differential equation s of second order, 199 1 94 Yu . S. Il'yashenko, Finitenes s theorems for limit cycles , 199 1 93 A . T. Fomenko and A. A. Tuzhilin, Element s of the geometry and topology of minima l surfaces i n three-dimensional space , 199 1 92 E . M. Nikishin and V. N. Sorokin, Rationa l approximations and orthogonality, 199 1 91 Mamon i Mimura and Hirosi Toda, Topolog y of Li e groups, I and II, 199 1 90 S . L. Sobolev, Som e applications o f functiona l analysi s in mathematical physics, third edition, 199 1 89 Valeri i Y. Kozlov and Dmitrii V. Treshchev, Billiards : A genetic introduction t o the dynamics of systems with impacts, 199 1 88 A . G. Khovanskii, Fewnomials , 199 1 87 Aleksand r Robertovic h Kemer, Ideal s of identities of associative algebras, 199 1 86 V . M. Kadets and M. I. Kadets, Rearrangement s o f series in Banach spaces , 199 1 85 Miki o Ise and Masaru Takeuchi, Li e groups I, II, 199 1 84 Da o Trong Thi an d A. T. Fomenko, Minima l surfaces, stratifie d multivarifolds , an d th e Plateau problem, 199 1 83 N . I. Portenko, Generalize d diffusio n processes , 199 0 82 Yasutak a Sibuya, Linea r differential equation s in the complex domain: Problem s of analytic continuation, 199 0 81 I . M. Gelfand an d S. G . Gindikin, Editors, Mathematica l problems o f tomography, 199 0 80 Junjir o Noguchi an d Takushiro Ochiai, Geometri c function theor y i n several comple x variables, 199 0 79 N . I. Akhiezer, Element s of the theory o f elliptic functions, 199 0 78 A . V. Skorokhod, Asymptoti c methods of the theory of stochastic differential equations , 1989 77 V . M. Filippov, Variationa l principles for nonpotentia l operators , 198 9 76 Philli p A. Griffiths, Introductio n to algebraic curves, 198 9 75 B . S. Kashin and A. A. Saakyan, Orthogona l series, 198 9 74 V . I. Yudovich, Th e linearization metho d i n hydrodynamical stabilit y theory, 198 9 (Continued in the back of this publication) 10.1090/mmono/108 Translations o f MATHEMATICAL MONOGRAPHS Volume 10 8 Tensor Space s and Exterio r Algebr a Takeo Yokonuma Translated b y Takeo Yokonum a o America n Mathematical Societ y i? Providence , Rhod e Islan d T-vw&mttmft TENSORU KUUKA N T O GAISEKIDAISU U (Tensor Spaces and Exterior Algebra) by Takeo Yokonuma Copyright © 197 7 by Takeo Yokonuma Originally published in Japanese by Iwanami Shoten, Publishers, Tokyo in 1977 Translated fro m th e Japanese by Takeo Yokonuma 2000 Mathematics Subject Classification. Primar y 15A69 , 47-XX ; Secondary 15A75 , 81-XX . ABSTRACT. Thi s book provides an introduction to tensors and related topics. Th e book begins with definitions o f th e basic concepts of th e theory; tensor product s of vecto r spaces, tensors, tensor algebras, and exterior algebra. Thei r properties are then studied and applications given. Algebraic systems with bilinear multiplication are introduced i n the final chapter. I n partic- ular, the theory of replica s of Chevalley and several properties of Li e algebras that follow fro m this theory are presented. Library of Congress Cataloging-in-Publication Data Yokonuma, Takeo, 1939- [Tensoru kuukan to gaisekidaisuu. English ] Tensor spaces and exterior algebra/Takeo Yokonuma; translated by Takeo Yokonuma. p. cm.—(Translations of mathematical monographs, ISSN 0065-9282; 108) Translation of: Tensor u kuukan to gaisekidaisuu. Includes bibliographical reference s and index. ISBN 0-8218-4564-0 1. Multilinear algebra. 2 . Tensor products. I . Title. II . Series. QA199.5.Y6513 199 2 92-1672 1 512'.57-dc20 CI P AMS softcover ISBN 978-0-8218-2796-3 Copyright ©1992 b y the American Mathematical Society . Al l rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America Information on Copying and Reprinting can be found at the back of this volume. The paper used in this book is arid-free and falls within the guidelines established to ensure permanence and durability. © This publication was typeset using Aj^S-T^i, the American Mathematical Society's T^ X macr o system. Visit the AM S home page at URL : http://www.ams.org/ 10 9 8 7 6 5 4 3 2 0 6 05 04 03 02 01 Contents Preface t o the Englis h Edition vi i Preface i x Chapter I. Definition o f Tensor Products 1 §1. Preliminaries 1 §2. Definition o f bilinear mappings 2 §3. Linearization o f bilinear mapping s 4 §4. Definition o f tensor products 7 §5. Properties o f tensor products 8 §6. Multilinear mapping s and tensor products o f more than tw o vector spaces 1 1 §7. Tensor products o f linear mapping s 1 4 §8. Examples of tensor product space s 1 8 §9. Construction o f tensor products with generators and relations 2 4 §10. Tensor products o f i?-module s 2 6 Exercises 3 0 Chapter II. Tensors an d Tensor Algebras 3 3 §1. Definition an d examples o f tensors 3 3 §2. Properties o f tensor spaces 3 6 §3. Symmetric tensors and alternating tensors 4 3 §4. Tensor algebras and their properties 5 4 §5. Symmetric algebras and their properties 6 2 §6. Definition o f relativ e tensors (pseudotensor , tenso r density) 6 8 Exercises 7 3 Chapter III. Exterior Algebr a and it s Applications 7 7 §1. Definition o f exterior algebra and it s properties 7 7 §2. Applications to determinants 8 3 §3. Inner (interior ) products o f exterior algebras 8 8 §4. Applications to geometry 9 1 Exercises 9 7 VI CONTENTS Chapter IV . Algebraic Systems with Bilinear Multiplication. Li e Algebras 10 1 §1. Algebraic system s with bilinear multiplication 10 1 §2. Replicas of matrices 10 5 §3. Properties of Lie algebras 11 4 Exercises 12 5 References fo r th e Englis h Edition 12 7 Subject Inde x 129 Preface t o the English Editio n This is a translation o f m y work originally published b y Iwanami Shoten , as a volume in their Lecture Serie s on linear algebra . We assume, therefore, tha t reader s are familiar wit h severa l fundamenta l concepts of linear algebra such as vector spaces, matrices, determinants, etc., though w e review som e o f these concepts i n the text. W e have made som e changes in the references . On thi s occasion , I woul d lik e to expres s m y gratitude t o Professo r Na - gayoshi Iwahori for giving me encouragement and many valuable suggestions during the preparatio n o f th e origina l wor k an d als o to Mr. Hide o Ara i o f Iwanami Shote n fo r continue d support . I am grateful t o the American Mathematical Societ y and the staff for thei r effort i n publishin g thi s Englis h edition . I als o than k Kazunar i Nod a an d Nami Yokonuma fo r their excellent typing. Takeo Yokonum a December 199 1 This page intentionally left blank Preface The subject matter of the present book is generally called multilinear alge- bra; w e shall discus s mainl y tensors an d relate d concepts . Th e readers ma y recall severa l importan t tensor s tha t ar e use d i n differentia l geometry , me - chanics, electromagnetics, and s o on. O n the other hand, i t is generally sai d that tensor s ar e difficul t t o understand . On e o f th e reason s fo r thi s i s tha t formerly tensor s and tensor fields (mappings whose values are tensors) wer e not distinguished , an d tenso r fields were discusse d withou t definin g tensor s in advance. ( ) I n fact , reader s should be aware that sometime s tensor fields are simpl y calle d tensor s i n th e literature .