Spinors and Calibrations (Perspectives in Mathematics, Vol 9)

Spinors and Calibrations F Reese Harvey Department of Mathematics Rice University Houston, Texas

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers Boston San Diego New York Berkeley London Sydney Tokyo Toronto

This book is printed on acid-free paper. O

Copyright ©1990 by Academic Press, Inc. AN rights reserved.

No part of this publication maybe reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA 92101

United Kingdom Edition published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NW17DX

Library of Congress Cataloging-in-Publication Data Harvey, F Reese.

Spinors and calibrations/F. Reese Harvey. cm. - (Perspectives In mathematics: vol. 9) p. BibUography: p.

Includes Index ISBN 0-12-329650-1 1 Spinor analysis. 2 Matrix groups. QA433.H3271990

515'.63-dcl9

Printed in the United States of America

909192 987654321

I. Title.

II. Series.

89-74 CIP

This book is dedicated to my wife, Linda

TABLE OF CONTENTS PREFACE

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xi

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PART I: CLASSICAL GROUPS AND NORMED ALGEBRAS . 1. CLASSICAL GROUPS I . . . . The General Linear Groups Groups Defined by Bilinear Forms Other Miscellaneous Groups . Isomorphisms . . . . . . . . Summary . . . . . .

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3

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4

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6 8 13 15

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19

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22 23

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29

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31

2. THE EIGHT TYPES OF INNER PRODUCT SPACES The Standard Models . . . . . . . . . . . . . . . . . . . . . . . . . . Orthogonality . . . . . . . A Canonical Form (The Basis Theorem) . . . . . . The Parts of an Inner Product . . . . . . . . .

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. 3. CLASSICAL GROUPS II . Group Representations and Orbits . . Generalized Spheres . . . The Basis Theorem Revisited . . . Adjoints . . . . . . . . . . . . Lie Algebras . . . . . . . . . .

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41

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41 42 46 47 48

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57

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4. EUCLIDEAN/LORENTZIAN VECTORS SPACES The Cauchy-Schwarz Equality . . . . . . . . . . . . . . . . . . Special Relativity . . . . The Cartan-Dieudonne Theorem . . . . . . . Grassmannians and SOT(p, q), the Reduced Special . . . . . . . . . . Orthogonal Groups . . .

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58 62 67

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75

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81

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81 82

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83

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85 86 87 87

5. DIFFERENTIAL GEOMETRY . . . . . . . . . . . . . Real n-Manifolds: The Group GL(n, R) . . . . . . . Oriented Real n-Manifolds: The Group GL+(n, R) . . . . Complex (and Almost Complex) n-Manifolds: The Group GL(n, C) . . . . . . . . . . . . . . . Quaternionic (and Almost Quaternionic) n-Manifolds: The Group GL(n, H) H* . . . . . . . . . . . . . . Manifolds with Volume: The Groups SL(n, R) and SL(n, C) Riemannian Manifolds (of Signature p, q): The Group O(p, q) Conformal Manifolds (of Signature p, q): The Group CO(p, q) .

vii

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Table of Contents

viii

5. DIFFERENTIAL GEOMETRY (continued) Real Symplectic Manifolds: The Group Sp(n, R) . . . . . . Complex Riemannian Manifolds: The Group O(n, C) . . . . . Complex Symplectic Manifolds: The Group Sp(n, C) . . . . . Kahler Manifolds (of Signature p, q): The Group U(p, q) . . . . Special Kahler Manifolds (of Signature p, q): The Group SU(p, q) HyperKiihler Manifolds (of Signature p, q): The Group HU(p, q) Quaternionic Kahler Manifolds (of Signature p, q): . . . . . . . . The Group HU(p,q) HU(1) . . . . . . Quaternionic Skew Hermitian Manifolds: The Group SK(n, H) . . . . . . . . . . . . . . . . . . . . . Coincidences of Geometries in Low Dimensions . . . . .

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6. NORMED ALGEBRAS . . . The Cayley-Dickson Process . The Hurwitz Theorem . . . . . . . . Cross Products . . The Exceptional Lie Group G2 .

87 88 88 89 90 91

92 93 94 101

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104 107 110 113

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125

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125 127 130 134 138 144 146 156 163

7. CALIBRATIONS . . . . . . . . . . . . . . . . . The Fundamental Theorem . . . . . . . . . . . . . The Kahler Case The Special Lagrangian Calibration . . . . . . . . . The Special Lagrangian Differential Equation . . . . . Examples of Special Lagrangian Submanifolds . . . . Associative Geometry . . . . . . . . . . . . . . . The Angle Theorem . . . . . . . . . . . . . . . . Generalized Nance Calibrations and Complex Structures

8. MATRIX ALGEBRAS . . . . . . . Representations . . . . . . . . . . Uniqueness of Intertwining Operators Automorphisms . . . . . . . . . . Inner Products . . . . . . . . . .

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163 167 169 170

PART II: SPINORS 9. THE CLIFFORD ALGEBRAS The Clifford Automorphisms . The Clifford Involutions . . . The Clifford Inner Product . .

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177

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181 182 184

Spinors and Calibrations

ix

9. THE CLIFFORD ALGEBRAS (continued) The Main Symmetry . . . . . . . . . . . . . The Clifford Center . . . . . . Self Duality . . . . . Trace . . . . . . . . . . . . . . . . The Complex Clifford Algebras . . . . . .

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185 187 188 189 190

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195

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. 10. THE GROUPS SPIN AND PIN . . . . The Grassmannians and Reflections . . . . Additional Groups for Nondefinite Signature The Conformal Pin (or Clifford) Group . . . . . . Determinants . . . . . . . . .

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11. THE CLIFFORD ALGEBRAS Cl(r, s) AS ALGEBRAS . The Pinor Representations . . . . . . . . . . . . . . The Spinor Representations . . . . . . . . . . . . . The First Proof . . . . . . . . . . . The Spinor Structure Map on P(r, s) . . . . . . . . . Even Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Odd Dimensions . . . . . .

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210 213 215 217 217 220

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207

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12. THE SPLIT CASE Cl(p, p) . . . . . . . . . . A Model for C1(p, p) . . . . . . . . . . . . . Pinor Inner Products for Cl(p, p) = EndR(P(p, p)) The Complex Clifford Algebras (Continued from Chapter 9) . . . . . . . . Cl(r, s)(r + s = 2p) as a Subalgebra of Clc (2p) = Cl(p, P) OR C . . . . . . . . . . The Pinor Reality Map . . . . . . . . . . . A Second Proof of the Classification Theorem . . . . . . . . . . . . Pure Spinors . . . .

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197 199 201 203

227 227 230 233

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235 237 240 241

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247

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247

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13. INNER PRODUCTS ON THE SPACES OF SPINORS AND PINORS . . . . . . . . . . . . . . . . . The Spinor Inner Product . . . . . . . . . . . . The Spin Representation and the (Reduced) Classical Companion Group Cp°(r, s) . . . . . . . . . . . The Pinor Inner Products t and e . . . . . . . . . Pinor Multiplication . . . . . . . . . . . . . . . Signature . . . . . . . . . . . . . . . . . . .

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250 252 266 268

x

Table of Contents

. . . 14. LOW DIMENSIONS . . . . . . . . . Cartan's Isomorphisms . . . . . . . . . . . . . . . . . . . Triality . . . . . . . . . Transitive Actions on Spheres . . . . . . . . The Cayley Plane and the Exceptional Group F4 Clifford Algebras in Low Dimensions . . . . . Squares of Spinors and Calibrations . . . . . . .

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REFERENCES INDEX

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271

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271 275 283 289 298 308

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315

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317

PREFACE This book is intended to be a collection of examples. The (simple) Lie groups, the spin groups for general signature, normed algebras for general

signature, the exceptional groups G2 and F4, the orbit structure of the simpler representations of these groups, and the special Lagrangian and associative calibrations are all discussed in some detail. The underlying and not always mentioned motivation for these examples is differential geometry: Riemannian, symplectic, Kahler, hyperKahler, as well as complex and quaternionic. The book is divided into two parts. Readers who are primarily inter-

ested in the spin groups are encouraged to start with Part II-Spinors. Part I begins with an introduction to certain specific matrix groups, with the entries real, complex, or quaternionic. Some of these groups are defined by requiring that the matrix fix one of several types of (generalized)

inner products. This leads to a discussion in Chapter 2 of eight types of inner products. The groups introduced in Chapter 1 are also the subjects of Chapter 3. A brief discussion of group representations and orbits is followed by the computation of orbit structures for some of the classical representations of Chapter 1. Lie Algebras are also introduced in this chapter.

Chapter 4 is devoted to a study of inner product spaces in the usual sense, i.e., a real vector space equipped with a real nondegenerate symmetric bilinear form. This is just one of the eight types discussed in Chapter 2. The topics vary in intensity, beginning with a very elementary but fairly complete discussion of the Cauchy-Schwarz "equality."

The reader is assumed to have Euclidean intuition. A discussion of Lorenzian intuition is presented based on both special relativity and the analogue between complex numbers C and Lorentz numbers L. Some readers will prefer to skip the long section on the Cartan-Dieudonne Theorem until after reading Chapter 9, where additional motivation is provided.

The next chapter, titled "Differential Geometry," is quite different from the first four chapters. Chapter 5 is intended to provide two things: first, motivation (but not prerequisites) for other material in this book; second, a skeleton or bare outline of some of the various types of geometry.

xii

Preface

Hopefully the reader will be inclined to build on this outline by further study outside the book. In contrast, Chapter 6, on normed algebras, provides a fairly complete discussion. In fact, some of the information presented here is not available elsewhere in the literature. The chapter naturally includes a discussion of the exceptional Lie group G2 as the automorphism group of the octonians. Various examples of calibrations are examined in Chapter 7. The group that fixes a particular calibration provides an important ingredient in discussing the geometry of submanifolds determined by the calibration. The beautiful Angle Theorem of Lawlor and Nance is discussed. Both halves of the proof involve introducing an interesting class of examples: a) the Nance calibrations; and b) the Lawlor special Lagrangian submanifolds. Chapter 8 is a brief, elementary, and somewhat tedious discussion of

matrix algebras, but the material is essential for understanding Clifford algebras. Algebraists already familiar with this material are encouraged to skip to Part II. Part II begins with a presentation of Clifford algebras that is independent of Part I. The reader may wish to start the book here. The exterior algebra is assumed to be a familiar object, although a brief definition is provided. Since the exterior algebra and the Clifford algebra are canonically isomorphic as vector spaces (but not as algebras), a considerable amount of intuition about the exterior algebra carries over to the Clifford algebra. Our presentation emphasizes this relationship between the Clifford algebra

and the exterior algebra. For example, the natural inner product induced on the exterior algebra immediately provides a natural inner product on the Clifford algebra. This norm on the Clifford algebra usually appears somewhat myteriously as a norm only defined on those Clifford elements in the Clifford group.

Chapter 10 describes the Spin and Pin groups. Again the discussion emphasizes the connection with the exterior algebra, via the grassmannians as subsets of the exterior algebra. Each plane through the origin, once it is oriented in one of the two possible ways, can be considered a Clifford

element. By utilizing Clifford multiplication, both of these two oriented planes are associated with the orthogonal transformation "reflection along the plane," providing the core of the double cover of the orthogonal group by the Pin group. For the reader willing to accept the Cartan-Dieudonne Theorem, this Chapter is also independent of Part I. As algebras-forgetting the more subtle extra structure-the Clifford algebras are just matrix algebras. The matrices act on the vector space of pinors. Although not canonical, this pinor space is unique up to a scalar multiple of the identity. Chapter 11 describes the spaces of pinors and spinors.

Spinors and Calibrations

xiii

Each space of pinors or of spinors can, in a natural way, be given the extra structure of one of the types of (generalized) inner products discussed in Part I. This inner product is unique up to a change of scale, and it plays an important role in understanding the spin groups and the space of spinors. Chapter 12 analyzes the case of split signature and the complex case in even dimensions. This provides a basis for determining the spinor inner product for arbitrary signature in Chapter 13. Moreover, this spinor inner product determines a classical (companion) group containing the spin group that is, probably, the smallest such group. Chapter 14 consists of just a few of the many interesting applications of, as well as some very explicit models for, the various Clifford algebras in low dimensions. The importance of spinors in geometry (including, of course, general

relativity) is not universally accepted. Two recent texts should play an important role in correcting any previous oversight. The two-volume book

by Penrose and Rindler on general relativity contains many interesting topics on general relativity and Twistor Theory. Spin Geometry, by Lawson and Michelson, is a very exciting presentation of some of the most beautiful topics in geometry.

I wish to thank Robert Bryant, Blaine Lawson, and Roger Penrose for very valuable discussions and frequent encouragement concerning this book. In addition, Jack Mealy deserves special appreciation for his careful reading of the manuscript. Finally, I wish to thank typists Janie McBane, Anita Poley, and Janice Want. F. Reese Harvey

Houston, Texas

PART I. CLASSICAL GROUPS AND

NORMED ALGEBRAS

1. Classical Groups I

The elements of the groups defined in this chapter are matrices with entries in one of the three fields: R the field of real numbers, C the field of complex numbers, H the field of quaternions. Note that H, the field of quaternions (or hamiltonians), is not commutative. The quaternions will be examined in great detail, along with the octonions O (or Cayley numbers), in Chapter 6 on normed algebras. For the purposes of this chapter, a rudimentary knowledge of H is all that is presupposed. Consult Problem 6 for the multiplication rules for quaternions. Let M, (R), M (C), and M,, (H) denote the algebras of n x n matrices with entries in It, C, and H respectively. Represent elements x of R", C',

and H" as column n-tuples. Then each matrix A determines a linear transformation or endomorphism x - Ax by letting the matrix A act on the left of the column vector x, at least in the real and complex case. Special consideration is necessary for the quaternionic case since H is not

commutative. In order for the map A : H" --> H' (defined by A acting on x on left) to be H-linear, we are forced to let the scalars H act on the H-vector space H' on the right! Although it will be convenient to consider both right H-vector spaces

(where the scalars H act on the right of the vectors) and left H-vector spaces (where the scalars H act on the left of the vectors), the space H" of 3

4

The General Linear Groups

column n-tuples will always be considered as a right H-vector space. Then we have (1.1)

M"(R) = EndR(R"),

(1.2)

M"(C) = Endc(C"),

(1.3)

M"(H) = EndH(H"). Here EndF(V) denotes the F-linear maps from a vector space V, with

scalar field F, into itself. If V is a real vector space, then EndR V is naturally a real algebra (associative and with unit). If V is a complex vector space, then EndcV is naturally a complex algebra (associative and with unit) but may also be considered as a real algebra, which is convenient for some immediate purposes. Finally, if V is a right quaternionic vector space, then EndH(V) is naturally a real algebra-in fact, a real subalgebra of the algebra EndF(V). There is no canonical way to make EndH(V) into even a quaternionic vector space (right or left), much less a "quaternionic" algebra (see Problem 7).

THE GENERAL LINEAR GROUPS The group of units, or invertible elements, in the matrix algebra M"(F) is called the F-general linear group for F =_ R, C, or H and is denoted by GL(n, R), GL(n, C), or GL(n, H), respectively. If the group of units, or invertible elements, in EndF(V) is denoted by GLF(V), then (1.1')

GL(n, R) = GLR(R),

(1.2')

GL(n, C)

(1.3')

GL(n, H) 25 GLH(H').

GLc(R"),

In the quaternion case, there is another important group, larger than the H-general linear group GL(n, H), which we will call the enhanced Hgeneral linear group. First note that the H-general linear group GL(n, H) (which acts on the left) consists entirely of H-linear maps. However, right multiplication by a scalar A E H, denoted RA,, is not necessarily H-linear. In fact, RA is H-linear if and only if A commutes with all scalars µ E H

Classical Groups I

5

because RA(xp) = xµ.1, while Ra(x)p = xA

.

The reader should confirm

that A commutes with all µ E H if and only if A E R C H. Thus, Ra is H-linear if and only if A E R C H. Let H* denote the group of right multiplications by nonzero scalars. Then H* is not a subgroup of GL(n, H). However, both are contained in the algebra EndR,(H") of R-linear maps.

As noted above, the intersection GL(n, H) fl H* equals R* the group of real nonzero multiples of the identity. The enhanced H-general linear group, denoted GL(n, H) - H*, is defined to be the image of GL(n, H) x H* in EndR(H") via the map sending the pair (A, A) to LA . RA, where denotes multiplication in the algebra EndR(H"), i.e., composition. Thus, the following sequence of groups is -

exact:

1 -> R* -} GL(n, H) x H* -+ GL(n, H) - H* -* 1

(1.4)

with GL(n, C EndR,(H'). Note that the larger group GL(n, as well as the smaller group GL(n, H), maps quaternion lines to quaternion lines.

Given A E M,, (R), the real determinant of A will be denoted detK,A. Similarly, detc A denotes the complex determinant of A E M,, (C). The lack of commutativity for H eliminates the possibility of any useful notion of "quaternionic determinant." Of course,

GL(n,R)_JA EM"(R):detR,A36 0}, and

(1.5)

GL(n,C)=JA EM"(C):detcA#0}. The group (1.6)

GL+(n, R) = {A E M,, (R) : detR, A > 0}

is called the orientation-preserving general linear group. In both the real and the complex case, we have a special linear group, defined by (1.7)

SL(n, R) _ {A E M" (R) : detR, A = 1),

(1.8)

SL(n, C) _ {A E M,, (C) : detc A = 1).

Since there is no quaternion determinant, if we proceed in exact analogy with the real or the complex case, the special quaternion linear group

Groups Defined by Bilinear Forms

6

does not exist. However, it is useful to retain the notation SL(n, H) by employing the real determinant. Let SL(n, H) _ {A E GL(n, H) : detR. A = 1}

(1.9)

denote the special quaternion linear group.

GROUPS DEFINED BY BILINEAR FORMS Some very interesting groups are best defined as subgroups of the groups defined above that fix a certain bilinear form. R-symmetric

The orthogonal group O(p, q) with signature p, q is defined to be the

subgroup of GL(n, R) (n = p + q) that fixes the standard R-symmetric form (1.10)

E(x,y)=xlyl+...+xpyp-xp+lyp+l-...-xnyn.

That is, O(p, q)

{A E GL(n, R) : E(Ax, Ay) = e(x, y) for all x, y E R"}.

R-skew (or symplectic)

The real symplectic group Sp(n, R) is defined to be the subgroup of GL(2n, R) that fixes the standard R-symplectic (or R-skew) form (1.11)

or (1.11')

e

e(x, y)

dxl A dx2 +

x1y2 - x2y1 +

+ dx2n_l A dx2n,

+ X2, _1y2n - x2ny2n-1.

That is, Sp(n, R) _ {A E GL(2n, R) : e(Ax, Ay) = e(x, y) for all z, y E R2n}

.

7

Classical Groups 1

C-symmetric

The complex orthogonal group 0(n, C) is defined to be the subgroup of GL(n, C) that fixes the standard C-symmetric form (1.12)

-(Z, W) - z1 w1 +

+ znwn.

C-skew (or symplectic)

The complex symplectic group Sp(n, C) is defined to be the subgroup of GL(2n, C) that fixes the standard C-symplectic (or C-skew) form E - dzl A dz2 +

(1.13)

+ dz2n-1 A dz2n,

or (1.13')

e(z, w) - Z1W2 - Z2W1 +

+ z2n-lW2n - z2nW2n-1

C-hermitian (symmetric) The complex unitary group U(p, q) with signature p, q is defined to be the subgroup of GL(n, C) (n - p + q) that fixes the standard C-hermitian symmetric form (1.14)

E(z,w)_z12i1+...+Zpwp-zp+1TWp+1-...-zn1iJ

.

Remark 1.15. ie(z, w) is called the standard C-hermitian skew form. Note that the group that fixes ie is just the same group U(p,q) that fixes E. This contrasts sharply with the quaternion case. H-hermitian symmetric The hyper-unitary group HU(p, q) with signature p, q is defined to be the subgroup of GL(n, H) (n =- p + q) that fixes the standard H-hermitian symmetric form (1.16)

e(x, y) = xiyi + ... + xpyp - xp+lyp+1 - ... - xnyn-

Note: E(x, y) is H-hermitian. This means that e is additive in both variables x and y, and E(xA, y) = )E(x, y), e(x, yA) = E(x, y)) for all scalars A E H. Also note that xy is not H-linear in x. In fact, there is no standard H-symmetric or H-skew form (see Problem 8).

Remark. The group HU(p, q) is usually denoted "Sp(p, q)" and called the "symplectic group."

H-hermitian skew

The skew H-unitary group SK(n,H), or SK(n), is defined to be the subgroup of GL(n, H) that fixes the standard H-hermitian skew form e(x, y) = 71iy1 -l.....+ aniyn.

(1.17)

Remark. This i is the quaternion i (see Problem 6). In Chapter 2, we shall see that if the i occurring in (1.17) is replaced by any unit imaginary quaternion u E Sz C Im H, then the new form e' differs from the old form e by a coordinate change, i.e., an element of GL(n, H). Table 1.18. The groups defined by bilinear forms symmetric a

skew a

O(p, q)

Sp(n, R)

O(n, C)

Sp(n, C)

C

H

hermitian symmetric a

hermitian

U(p, q)

U(p, q)

HU(p, q)

SK(n, H)

skew e

OTHER MISCELLANEOUS GROUPS The subgroups defined by requiring either deter or detc to be equal to one can also be defined by requiring that an n-form be fixed. The skew n-form (1.19)

dx=dx1A---Adxn

on Rn is called the standard volume form on R1, while the skew n-form (1.19')

dz=dz1A---Adze

on Cn is called the standard complex volume form on Cn. The volume form transforms, under a coordinate change, by multiplication by the determinant: A*dx = (deter A) dx for all A E EndRV

9

Classical Groups 1

and B* dz = (detc B) dz

for all B E Endc V.

Here A* denotes the dual (or pull back) map associated with A, which

is defined by (A*a)(u) = a(Au) if a is a form of degree one and by

ifa = al is the simple product of degree one forms. This provides the most elegant definition of the determinant. Frequently, this is also the most useful. For example, see Problem 4. This definition gives (1.20)

SL(n, R) = {A E GL(n, R) : A*dx = dx},

(1.20')

SL(n, C) = {A E GL(n, C) : A*dz = dz}.

The special orthogonal group with signature p, q is defined by (1.21)

SO(p, q) _ {A E O(p, q) : deter A = 1}.

The special complex orthogonal group is defined by (1.22)

SO(n, C) _ {A E O(n, C) : detc A = 1).

The special unitary group is defined by (1.23)

SU(p, q) - {A E U(p, q) : detc A = 1}.

The various other possibilities do not lead to new groups. This is a consequence of the facts presented below-see (1.24), (1.25), (1.26), Lemma 1.28, (1.29), and (1.30). Consult Problem 5 for proofs of the following: (1.24)

if A E Sp(n, R), then detEt A = 1;

and (1.25)

if A E Sp(n, C), then detc A = 1.

Forgetting the complex structure on C", the complex vector space C" becomes a real vector space of dimension 2n. This embeds the algebra Endc(C") of complex linear maps into the algebra Endit(C") of all real linear maps. Thus, for a E M" (C), the real determinant detRA has meaning as well as detc A. See Problem 4 for a proof of the result: (1.26)

if A E M"(C), then detR,A = detcAl2.

Other Miscellaneous Groups

10

The quaternion vector space H" can be considered as a complex vector space in a variety of natural ways (more precisely, a 2-sphere S2 of natural ways). Let ImH denote the real hyperplane in H with normal 1 E H. Let

S2 denote the unit sphere in Im H. Then, for each u E S2, u2 -JUJ2 = -1. Therefore, right multiplication by u, defined by

= -u

Rux - xu forallxEH", is a complex structure on Hn; that is, R,2, _ -1. This property enables one to define a complex scalar multiplication on H" by (a + bi)x = (a + bRu)(x) for all a, b E R and all x E Hn, where i2 = -1. Note that EndH(Hn) C Endc(Hn) for each of the complex structures Ru on H", where u E S2 C Im H. Choosing a complex basis for H" provides a complex linear isomorphism H" = Ctn. Sometimes it is convenient to select this complex basis as follows. Let C(u) denote the complex line containing 1 in each of the axis subspaces H C

Hn. Thus, C(u) is the real

span of 1 and u. Let C(u)1 denote the complex line orthogonal to C(u) a in H C H'. Then (1.27)

H" - [C(u) ® C(u)1]" = C2n

Assume the complex structure on H" has been fixed, say Ri, then as noted above EndH(H") C Endc(C2n). Moreover, given A E Endc(C2n), one can show that

A E Endf(H') if and only if ARj = R?A. This is a useful characterization of the subspace EndH(H') of Endc(C2n).

Lemma 1.28. For each complex structure R, on Hn (determined by right multiplication by a unit imaginary quaternion u E S2 C Im H) and for each A E Mn(H) the complex determinant detc A is the positive square root of detR, A, independent of the complex structure R,,.

Proof: First, we show that the complex determinant of A E MM(H) is real for all A E Mn(H). We will give the proof for the particular complex structure R,. Consider the case n = 1. Let e° = 1, el = i, e2 - j, and e3 k denote the standard real basis for the quaternions H. Let w°, w1, w2, w3 denote the standard dual basis. Then

dzl =w°+i 1,

dz2 =w2-iw3

is a basis for the complex forms of type 1, 0 on H = C2 (with complex structure Ri). Note that R (dz') = -dz2 and RR (dz2) = d 71, so that Rj(dz' Adze) =d71 Adx2.

Classical Groups I

11

Now

Rt A* (dzl A dz2) = Rj*(detc A dz' A dz2) = detc A d T' A dz2, while

A*Rj* (dzl A dz2) = A* (d zl A dz2) = detc A d zl A dz2.

Therefore, detc A E R is real since ARC = RJA. The proof, for n > 1, that detc A E R for all A E M"(H) is similar and omitted. Because of (1.26), it remains to show that detc A > 0 if A E GL(n, H).

Since detc I = 1 and GL(n, H) is connected (Problem 3), the set {detc A : A E GL(n, H)} is a connected subset of R - {0} containing 1, and hence it is contained in R+.

For elements of the subgroups HU(p, q) and SK(n, H) of GL(n, H), the real determinant is already equal to one (and hence by Lemma 1.28 all the various complex determinants are also equal to one). That is, (1.29)

detR A = 1 if A E HU(p, q);

(1.30)

detR A = 1 if A E SK(n, H).

Both of these facts follow from (1.24), since both HU(p, q) and SK(n, H) are contained in Sp(2n, R) for a suitable choice of coordinates. For example,

if A fixes the e defined by (1.16), i.e., A E HU(p, q), then A fixes the real valued skew form Re ie(x, y), which under a coordinate change is the

symplectic form given by (1.11'). The details are provided in the next chapter-see Lemma 2.80 and Equation (2.91). In the quaternion case, there is always the option of enlarging the group by utilizing right scalar multiplications. Recall (1.4) how the group GL(n, H) H* is an enhancement of the quaternionic general linear group GL(n, H). For another example, consider the enhanced hyper-unitary group (perhaps a better name is the quaternionic unitary group). This group is denoted HU(p, q) HU(1) and defined to be the subgroup of EndR(H") generated by letting HU(p, q) act on Hn on the left and the unit scalars HU(1) - S3 act on H" on the right. Since (1.31)

1 --> Z2 -p HU(p, q) x HU(1) -'-+ EndR(Hn)

Other Miscellaneous Groups

12

is exact, where Z2 = {1, -1} and where HU(1) = {Ry : y E S3 C H}, it follows that

HU(p, q) . HU(1) = (HU(p, q) x HU(1))/Z2.

(1.32)

See Problem 3.15 for more information about the quaternionic unitary group HU(p, q) HU(1). For example, this group fixes a 4-form 0 E A4(H")*.

Remark 1.33. In the special case of n = p = 1, and q = 0, HU(1) acting on the left equals {La : Jal = 1}, while HU(1) acting on the right equals {Rb : IN = 1}. In fact, the quaternionic unitary group is just the special orthogonal group. That is, (1.34)

HU(1)

HU(1) = SO(4),

or equivalently, (1.34')

X : HU(1) x HU(1) --+ SO(4),

is a surjective group homomorphism with kernel Z2 = {-1, 1}, where the map X is defined by

Xa,b(x)=axb forallxEH. To prove (1.34'), first note that by (1.29), or more directly, by Problem 6(b), detR La = 1 if lal = 1 (similarly detf Rb = 1 if IbI = 1). Second, one can show that Jax) _ Jal JxJ under quaternion multiplication. Thus, La E 0(4) if Jal = 1 (similarly Rb E 0(4) if IbI = 1). This proves that HU(1) HU(1) = x(HU(1) x HU(1)) C SO(4). The surjectivity of X can be demonstrated with a topological argument based on dimension, once it is known that SO(4) is connected (see Corollary 3.31). A nontopological proof that x is surjective is provided by Problem 4.9. If HU(1) denotes the diagonal copy of S3 = {a E H : Jal = 1} embedded in HU(1) x HU(1) and x is restricted to HU(1), then (1.35)

HU(1) Z2

25SO(3).

To prove (1.35), it suffices to note that the subgroup of SO(H) that fixes

1 E H is just SO(ImH) and that the subgroup of HU(1) x HU(1) that maps into SO(Im H) equals {(a, b) E HU(1) x HU(1) : a b = 1} = HU(1). The quaternionic enhancements are summarized as follows.

Classical Groups I

13

Enhanced Group

Group general linear special linear hyper-unitary skew-unitary

GL(n, H) SL(n, H) HU(p, q)

SK(n, H)

GL(n, H) H* SL(n, H) HU(1) HU(p, q) HU(1) SK(n, H) HU(1)

enhanced general linear enhanced special linear enhanced hyperunitary enhanced skew unitary

Of course, one can always enhance a group G with R+, or R* R - {0}, if the nonzero multiples of the identity do not already belong to G. The groups G R+ are usually referred to as conformal groups. For example, (1.36)

CO(p, q) = O(p, q) - R+ = O(p, q) x R+

is called the conformal (orthogonal) group of signature p, q. This group, perhaps the most important conformal group, is usually defined by requiring that the inner product s (see (1.10)) be fixed up to a positive scalar multiple (or conformal factor): (1.36')

CO(p, q) = {A E GL(n, R) : for some A E R+, e(Ax, Ay)

=.e(x,y)forallx,yER"}.

Similarly, (1.37)

CSO(p, q)

SO(p, q) R+

{AEGL+(n,R):A*e=as forsome AER+}

is called the special conformal group of signature p, q. If both p, q > 1, then (see Chapter 3) SO(p, q) has two connected components. The connected component of the identity, denoted by SOT(p, q), is, of course, a subgroup of SO(p, q). This subgroup SOT (p, q) of SO (p, q) is called the reduced special orthogonal group. Later, in Chapter 4, additional subgroups of O(p, q), denoted 0+ (p, q), and 0- (p, q) will be discussed in some detail. Briefly, if p, q > 1, then O(p, q) has four connected components. Adding any one of the remaining three components to SOT (p, q) yields three additional subgroups of O(p, q), denoted SO(p, q), O+(p, q), and 0-(p, q) . Thus, the intersection of any two of these three is always SOT(p, q). See Chapter 4 for the details.

ISOMORPHISMS The unit circle (1.38)

Sl-{zEC:Izi=1}

Isomorphisms

14

in the complex plane C is a group under complex multiplication. By definition, the groups U(1) and S' are the same. Of course, Sl - {en° : 0 E R} = R/2zrZ. The group of nonzero complex numbers under complex multiplication is denoted by C*, and by definition, GL(1, C) - C*. The set of unit quaternions (1.39)

S3-Ix EH:IxI =1}

also forms a group under quaternionic multiplication. Again, by definition, the groups HU(1) and S3 are the same. Also, by definitions (1.20) and (1.20'), we have SL(2, R) = Sp(1, R)

and SL(2, C) = Sp(1, C). The more difficult equality HU(1) - HU(1) = SO(4) has already been discussed. These and other coincidences are listed in the next proposition.

Proposition 1.40. The following isomorphisms hold (1.41)

SO(2) - U(1) = SK(1) = S1,

(1.42)

CSO(2) = GL(1, C) - C* = SO(2, C),

(1.43)

SU(2)

HU(1) - SL(1, H) t--- S3,

(1.44)

Sp(1, R) = SL(2, R) - SU(1, 1),

(1.45)

Sp(1, C) = SL(2, C),

(1.46)

(1.47)

HU(1) HU(1) - SO(4) and GL(1,H) H* - CSO(4), SOT (3, 1) c--- SO(3, C).

The last isomorphism (1.47) will be verified in the section on special relativity in Chapter 3. The proofs of all of the other isomorphisms in Proposition 1.40 are left as an exercise (see Problems 9, 10, and 11). One of these isomorphisms, SU(2) - HU(1), warrants the following discussion. Let H have the complex structure R;x =_ xi (right multiplication by i). Thus, H - C2, where each p E H can be expressed as p = z + jw with

15

Classical Groups I

z,w E C C H. Now each A E M1(H) = EndH(H) can be considered as acting on H on the left, hence A E Endc(C2) = M2(C) is a complex linear transformation of H = C2. Using the coordinates p = z + jw = (z, w) for p E H = C2, the complex linear map A expressed as a complex matrix is given by /

A= Iba

a6

,

where A = a + jb, a, b E C C H. This is because

Aj = (A - 1)j = (a-1- jb)j = -b+ja. This proves

M1(H)-

(1.48)

l\b

a) EM2(C):a,bEC}.

The isomorphism HU(1) = SU(2) is derived from (1.48) (see Problem 10).

Remark. In the standard reference (Helgason [10]), SL(n, H) is denoted by SU*(2n), SK(n, H) is denoted by SO*(2n), and HU(p, q) is denoted by Sp(p, q)

-

SUMMARY The three general linear groups GL(n, R), GL(n, C), and GL(n, H) and the seven groups described in Table 1.18 can be changed by imposing restrictions on determinants and/or by enhancing with scalar multiplication. The connected component of the identity in SO(p, q) with p, q > 1 is also a group. All the groups introduced in this chapter can be obtained in this way.

In low dimension, some of these groups coincide. One of the most interesting isomorphisms is SU(2) = HU(1). The topic of special isomorphisms in low dimensions will be discussed again in Chapter 14.

PROBLEMS 1. Establish R, C, H. 2. If A E

EndF(F") and GL(n, F) - GLF(F'") for F EndH(H") is injective, then A-1 is H-linear.

Problems

16

3. (a) Let e denote the column vector (1, 0, ..., 0) E F". Show that for

each x E F" - {0} there exist A E GL(n, F) with Ae = x. Let K denote the subgroup of GL(n, F) that fixes e. Show that K has the same number of connected components as GL(n - 1, F). (b) Use part (a) and induction to show that GL(n, C) and GL(n, H) are connected, while GL(n, R) has exactly two components. (c) Show that SL(n, R), SL(n, C), and SL(n, H) are all connected.

(d) Show that O(n), U(n), and HU(n) are compact. (e) Show that each of the groups defined in Table 1.18 is the level set of a vector-valued polynomial, and hence is a closed set. 4. Suppose A E Endc(C") C Enda,(R2n ). Let dt denote the standard volume form on R2' . Use the facts that: (a) A` dt = (detR, A)dt,

(b) A*dz = (detc A)dz, (c) dt = k dz A d x for some constant k, to show that detg A = I detc Ale. 5. Let e denote the standard symplectic form on R2s, and dx the standard volume form on R2n. Show that they are related by

1

E A ... AE = dx.

6. The quaternions H can be defined as the vector space R4 with 1 - ( 1 , 0, 0, 0),

i = (0, 1, 0, 0),

j

(0, 0, 1, 0),

k = (0, 0, 0,1),

and multiplication defined by

i2=.92=k2=-1 ij = k and all cyclic permutations of this equation are valid, and i, j, k skew commute. Given x = xo + xl i + x2 j + x3k, define conjugation by x - xo - x1i Z2j - x3k. (a) Show that ry = y x, x x = 1x2, 2(x, y) = xy + yx.

(b) Given a E H, left multiplication by a, denoted La, belongs to EndH(H) C Endit(H). In terms of the standard basis for H = R4, compute the 4 x 4 matrix for La, and then show that detR, La = Jal4.

Classical Groups 1

17

7. Suppose V is a right H-vector space. Let HomH(V, H) denote the real vector space of right H-linear maps from V to H. Given f E HomH (V, H) and A E H, define f A E HomH (V, H) by

(fA)(v) = af(v)

(1.49)

for all v E V.

(a) Show that (1.49) exhibits a right H-structure on HomH(V, H). The H-dual of V, denoted V*, is defined to be the vector space HomH (V, H) equipped with this right H-structure (1.49). (b) Exhibit a (canonical) right H-linear isomorphism (V*)* = V. (c) Given f E HomH(V, W), let f* (the dual map) be defined by (1.50)

(f * (w*))(v) = w* (f (v)) for all w* E W* and v E V.

Show that f* is right H-linear, i.e., f E HomH(W*, V*). (The real vector space HomH(V, W) cannot be (canonically) given the extra structure of either a right or a left H-vector space.) 8. Suppose : V x V -i H is H symmetric, i.e., e(x, y) = -(y, x) and s(x, y.A) = E(x, y)A. Prove that e = 0. 9. Show that

(a) U(1) - Sl, (b) SO(2) - S1, (c) SK(1) = S1, (d) Sp(1, R) = SL(2, R), (e) Sp(1, C) = SL(2, C), (f) SO (2, C) = C*, (g) SL(1, H) = S3. 10. Show that (a) HU(1) S3,

(b) U(2) =

a

-e'ab l

b

(c) SU(2) - (I b (d) SU(2) - HU(1).

ei9

I

l : a, b E C, (a12+ Ibj2 = 1, and 9 E R },

a f : a, b E C and 1a12+ 1b12 = 1

Problems

18

11. Prove the following.

(a) U(1, 1) = {

I

e'Ba) : a, c E C, B E R, and ja12 - Id12 = 1

a c

a

c ):acEC and

I-

Jal2-Ic12=1}.

a

J

(c) SU(1, 1) maps N into itself and M into itself, where N = {(z, z) z E C} and M = {(z, -x) E C2 : z E C} are both two-dimensional real subspaces of C2.

(d) C-'SU(1,1)C - SL(2, R), where C = I Transform.

\

1

_E) is the Cayley

/

2. The Eight Types of Inner Product Spaces The "inner products" (or "bilinear forms"), which we denoted by c in the previous chapter, will be examined in more detail in this chapter. Even if one is only interested in the orthogonal groups O(p, q), all types appear (in a natural way), as will be seen in a later chapter on spinor inner products. Suppose V is a vector space over F = R, C, or H and E is a biadditive map (2.1)

e:VxV--+F.

That is, c(x + y, z) = e(x, z) + e(y, z) and e(z, x -{- y) = E(z, x) -I- e(z, y). The biadditive map is said to be (pure) bilinear if and -(x, Ay) = A,-(x, y) for all scalars A = F.

e(Ax, y) = A,-(x, y)

The biadditive map e is said to be (hermitian) bilinear if e(xA, y) = aE(x, y) and e(x, yA) = E(x, y)A for all scalars A E F. 19

Standard Models

20

Frequently, when it is clear from the context, the adjective "pure" or the

adjective "hermitian" will be dropped. For example, if F = R there is only one kind of bilinear; the two notions, pure bilinear and hermitian bilinear, agree. If F = H and e is pure bilinear, then e = 0 because E(x, y)Ap = E(xA, yµ) = e(x, y) jA, while µ, A E H may be chosen so that [p, A] = uA - Ap 0 0. Thus, for F = H there is only one kind of bilinear, namely, hermitian bilinear. BlLhei.,n(V)-or just BIL(V) when the correct adjecLet

tive pure or hermitian is clear from the context-denote the space of bilinear forms on V. Note that these spaces are always real vector spaces. Suppose E E BlLpure(V). The form E is said to be F-symmetric, or F-pure symmetric if

for all x and y E V.

s(x, y) = E(y, x)

(2.4)

The form E is said to be F-skew or F-pure skew if

e(x, y) = -e(y, x)

(2.5)

for all x, y E V.

Suppose E E BlLherm(V). The form E is said to be F-(hermitian) symmetric if

E(x, y) = s(y, x)

(2.6)

for all x, y E V.

Finally, the form e is said to be F-(hermitian) skew if

s(x, y) = -e(y, x) for all x, y E V.

(2.7)

Note that for F = C (but not for F = H) a hermitian bilinear form e E BIL(V) is skew if and only if is is symmetric. The notations SYMpure(V), SYMherm(V), SKpure(V), and SKhe,.,,,(V)

should be self-explanatory. Each E E BIL(V) has a unique decomposition E = El + E2 into the sum of a symmetric part el and a skew part E2. If E is pure, then Ei (x, y) =

2

E(x, y) +

s(y, x)

and

E2(x, y) = 2 E(x, y) -

and

62(2

2

E(y, x)

while ifs is hermitian then 1

Cl (x, y)

E(x, y) +

2

e(y, x)

,

y) _ 2 E(x, y)

-

A bilinear form e E BIL(V) is said to be nondegenerate if (2.8)

E(x, y) = 0 for all y E V implies x = 0,

2

E(y, X).

21

Inner Product Spaces

and

e(x, y) = 0 for all x E V implies y = 0.

(2.8')

Definition 2.9 (Inner Product Space). Suppose V is a finite dimensional vector space of F, with F one of the three fields F = R, C, or H. An inner product e on V is a nondegenerate bilinear form on V that is either symmetric or skew. If F - C, then there are two types of symmetric and two types of skew: pure and hermitian. The eight types of inner products are

1. R--symmetric e is R-bilinear and e(x, y) = E(y, x), 2. R-skew or R-symplectic

is R-bilinear and e(x, y) = -E(y, x), 3. C-symmetric E is C-bilinear and e(x, y) = e(y, x), 4. C-skew or C-symplectic

E is C-bilinear and e(x, y) _ -e(y, x), 5. C-hermitian symmetric e is C-hermitian bilinear and e(x, y) = e(y, x), 6. C-hermitian skew

E is C-hermitian bilinear and e(x, y) _ -E(y, x), 7. H-hermitian symmetric e is H-hermitian symmetric and E(x, y) = e(y, x), 8. H-hermitian skew

e is H-hermitian and e(z, y) = -c(y, x).

Standard Models

22

In each of the eight cases the pair V, e is then called an inner product space (of one of the eight types).

THE STANDARD MODELS The standard models of inner product spaces are listed below. The reader may recall the names of the groups that fix these bilinear forms from Chapter 1. Please note that the conjugation in the C-hermitian symmetric case has been changed from the second variable to (as in (1.14)) to the first variable z in order to be consistent with the quaternion case. 1. R-symmetric: The vector space is R, denoted by R(p, q)(n = p + q), with (2.10)

e(x, y) = xlyl + ... + xpyp - ... - xnyn

2. R-skew or R-symplectic: The vector space is R2n with (2.11)

E(z,y) = zly2 - z2y1 +

+ z2n_1y2n - x2ny2n-1.

3. C-symmetric: The vector space is Cn with (2.12)

E(z, w) = ziwl + ... + znwn.

4. C-skew or C-symplectic: The vector space is C2n with (2.13)

E(z, w) = z1w2 - z2w1 + ...+. z2n-lw2n - z2nw2n-1

5. C-hermitian (symmetric): The vector space is Cn, denoted by C(p, q) (n = p + q) with (2.14)

e(z, w) - 71w1 + ... + zpwp - ... - znwn.

6. C-hermitian (skew): The vector space is Cn(n = p+ q) with (2.14')

E(z, w) = iz1W1 + ..+ izpwp

- ..

iznwn.

7. H-hermitian symmetric: The vector space is Hn, denoted by H(p, q) (n = p + q) with (2.15)

e(x, y) _ Tlyi + ... + x

zn yn

Inner Product Spaces

23

8. H-hermitian skew: The vector space is H'" with (2.16)

E(x, y) = xiiy1 + .....F. xniyn.

Some of the R-symmetric special cases are very important. The positive definite case R(n, 0) is called euclidean space, while the case of a single

minus, R(n - 1, 1), is called Lorentzian space. If n = 4, then Lorentzian space is also called Minkowski space or Minokowski space-time. The special

case R(p,p) will be referred to as the split case.

ORTHOGONALITY Definition 2.17. Suppose f : V --> V is a F-linear map from an inner product space V, e to another inner product space V, E of the same type. The map f is said to preserve inner products if (2.18)

E(f (x), f (y)) = e(x, y) for all x, y E V.

Since e is non degenerate such a map must be one to one. If, in addition, such a map is onto then f is called an isometry. The inner product spaces V, E and V, E are said to be isometric if there exists an isometry between them.

Remark 2.19. Suppose f is an isometry from V, e to V, E. Let G denote the e-isometry subgroup of GL(V, F), and let G denote the E-isometry subgroup of GL(V, F). Then the groups G and G are isomorphic: (2.20)

G= f o G o f-1

For example, the model C2, E with E(z, w) - zlw2 + z2w1 is isometric to the standard model C2, e with -(C, rl) 67]1 +6192. Using the model C2, E it is almost immediate that SO(2, C) = C* (cf. Problem 1.9(f)). Given a vector subspace W of the inner product space V,,-, let e1w denote the restriction of e to W. The restriction eJw is positive definite if e(x, x) > 0 for all nonzero x E W. Similarly, one defines negative definite. A subspace W of V is said to be (2.21) (2.22) (2.23)

positive (or spacelike) null (or degenerate or lightlike) negative

if c 1w is positive definite, if e w is degenerate, if -Iw is negative definite.

These three possibilities are by no means exhaustive. Positive (negative) subspaces can only occur for three types of inner product spaces:

Orthogonality

24

R-symmetric, C-hermitian (symmetric), and H-hermitian symmetric. In these three cases, another numerical invariant (in addition to the dimension), called the signature, is required in order to determine when two inner product spaces are isometric.

Definition 2.24. Suppose V, e is an inner product space of one of the three speciAl types mentioned above. A positive subspace W of V is said to be maximal positive if

dimZ < dim W for all positive Z C V. A negative subspace W of V is said to be maximal negative if

dim Z < dim W for all negati ve Z C V. The signature p, q of V is defined by

p equals the dimension of a maximal positive subspace, q equals the dimension of a maximal negative subspace. If V, e is positive definite (on V), then the signature is n, 0, and if V, e is negative definite (on V), then the signature is 0, n. If V, e has signature p = q, the inner product space is said to be split, or of split signature. The case of signature p = 2, q = 1 is pictured in Figure 2.27. Obviously, if V, e and V, a are isometric, then they have the same signature. In fact, as we will prove below, if V, c and V, E have the same dimension and signature, then they are isometric. First we must examine the important concept of orthogonality.

Two vectors u, v E V are said to be orthogonal, written u 1 v, if e(u, v) = 0. Note that (2.25)

u .1. v if and only if v .1. u

for all eight types of inner products. Suppose W is a subset of V. Then

uIW means uIvfor all vE W. The perp or orthogonal to W is defined by (2.26)

Wl-fu EV:uIW}.

The perp of the line [v] spanned by v E V will be denoted vl.

25

Inner Product Spaces

Figure 2.27

Since "u .1. v" just involves a pair of vectors, the key to understanding orthogonality lies in understanding two dimensions-more specifically,

Elw where W = span{u,v}. In order to sharpen our intuition, we focus attention on inner product spaces of the R-symmetric type. In the euclidean plane R(2, 0), two orthogonal vectors are pictured as being at right angles to each other.

In the Lorentzian plane R(1, 1), first picture the null or light cone, consisting of all a = (x, t) E R(1, 1) which satisfy E(a, a) = x2 - t2 = 0. Such a vector a E R(1, 1) is said to be null or lightlike. Note that a 1 a for each lightlike vector a. Given a vector b = (x, t) E R(1, 1), the vector

b' = (t, x) E R(1,1) is orthogonal to b. The vectors ±b' are obtained pictorially by (euclidean) reflecting b through either one of the two lines making up the null cone (see Figure 2.28). The perp bl is just the span of ±b'.

Orthogonality

26

timelike t t

spacelike

x

null

Figure 2.28

This example shows that a subspace W and its perp W1 need not be complementary subspaces. Recall that a subspace W is null (or degenerate or lightlike) if eIw is degenerate. If eIW - 0, then W is said to be totally null or isotropic. Note that (2.29)

W is totally null if and only if W C W.

Totally null subspaces are particularly important in the split case R(p, p) and the symplectic cases (R-skew and C-skew). Despite these strong counterexamples to our euclidean intuition, which says W1 should be complementary to W, it is still true that the dimension of W1 is complementary to the dimension of W. The next lemma is basic for all eight types of inner products.

The Orthogonality Lemma 2.30. Suppose V, e is a n-dimensional inner product space, and W is a linear subspace of V. Then (2.31)

dim W+dim Wl=n,

(2.32)

(W1)1 = W,

Inner Product Spaces

27

and the following are equivalent:

(2.33)

(a)

W is nondegenerate,

(b)

W n W1 = {0},

(c)

W + W-L = V,

(d)

Wl is nondegenerate.

Proof: Let F denote the field of scalars (either R, C, or H). The inner product e : V x V --> F determines a map b : V -* V* from V to the dual space V* as follows. Consult Problem 1.7(b) for the definition of V* if V is a right H-space. For each z E V fixed, let b(x) E V* be defined by (2.34)

(b(x))(y) - E(x,y)

for all y E V.

Note that for all eight types of inner products b(x) E V* since e(x, y) is F-linear in y. The nondegeneracy hypothesis, (2.35)

E(x, y) = 0 for all y E V implies x = 0,

is equivalent to (2.35')

b : V -+ V* is one-to-one.

Since V and V* have the same real dimension and b is always R-linear, must be a real linear isomorphism. Given a subspace W of V, the annihilator of W, denoted W°, is the subspace of V* defined by (2.36)

W°- {0EV*:0(y)=0forallyEW}.

Suppose X E V and 0 E V* correspond under the isomorphism b. Then

E W° if and only if x E W. That is, (2.37)

Wl -- W° is an R-linear isomorphism.

In particular, (2.38)

dimF Wl = dimF W°.

Orthogonality

28

Choose a basis e1,.. ., e,n, ..., en for V with e1,.. ., e,,, a basis for W. Let e;, is a basis for e*,,. . ., en denote the dual basis for V*. Then W°. Therefore, dimF W° = n dimRW, which, by (2.38), proves the first part of the Lemma, (2.31). Obviously, W C (W')', and since the dimensions are the same, they must be equal. The proof that (a)-(d) are equivalent is left as an exercise.

Remark 2.39. (a) If V is the tangent space T to a (semi)Riemannian manifold at a point, the flat isomorphism b, defined above, between vectors and convectors is referred to as metric equivalence and corresponds to "lowering indices." The inverse map = b-1, called sharp, is referred to as "raising indices."

(b) If V is the tangent space to a symplectic manifold (e.g., phase space - the cotangent bundle T*M to a manifold M), then b allows one to start with a function (Hamiltonian function) f and associate a vector field (the associated Hamiltonian vector field) by first taking df, the exterior derivative of f, and then applying sharp 0 = b-1. The reader unfamiliar with these concepts may wish to glance at Chapter 5, "Differential Geometry." Corollary 2.40. If W is a non degenerate subspace, then each vector u E V has a unique orthogonal decomposition: (2.41)

u=w+z with wE W andzEWJ'.

The linear map a that is zero on W1 and the identity on W is called orthogonal projection onto W. In the three cases R-symmetric, C-hermitian, and H-hermitian symmetric, where positive/negative subspaces exist, the inner product space is said to have a signature.

The Signature Lemma 2.42. Suppose V, a is an inner product space that has a signature, say p, q. Let W denote a subspace.

(a) W is maximal positive if and only if Wl is maximal negative. (b) If W is positive and W' is negative, then both are maximal.

Corollary 2.43. If V has signature p, q, then p + q = n the dimension of V.

The obvious guess for the signatures of the standard models can be rigorously verified using the Signature Lemma (see Problem 2).

Inner Product Spaces

29

Corollary 2.44. The standard models R(p, q), C(q, p), and H(p, q) have signature p, q.

Proof of 2.42(a): Suppose W is maximal positive. It suffices to show that W1 is negative, because the maximality of W1 then follows from part (b). First, we establish that W1 is negative semidefinite; that is, E(x, x) < 0 for all x E W1. Otherwise, there exists a nonzero, spacelike vector u E W1, i.e., E(u, u) > 0. Since E restricted to either of the orthogonal subspaces W or [u] - span u is positive definite, an easy calculation shows that E restricted to W + [u] is also positive definite, contradicting the maximality of the dimension of W. Finally, we must rule out E(x, x) = 0 for x E W1 unless x = 0. If W1- contains a nonzero null vector u, then there exists a vector y E W1 with E(u, y) $ 0 because W1 is nondegenerate. We may assume that E(u, y) is real and greater than zero. Now 0 > e(u + ty, u -Ety)/t = 2E(u, y) +tE(y, y), for all t > 0, which is impossible. The hypothesis that V, e has a signature was used in this last equation to conclude that E(y, u) = E(u, y) from the assumption that e(u, y) is real. Can you find the other use of the signature hypothesis in this proof? Proof of 2.42(b): We shall prove that W is maximal (positive). Suppose P is another positive subspace. Since W1 is negative, {0} = P fl W1 = P fl 7r-1(0), where Tr is orthogonal projection onto W. Thus, a : P --> W is one-to-one, so that dim P < dim W. In a similar manner one can show:

Proposition 2.45. If V has a signature (say p, q), then each totally null subspace N must have dimension < min{p, q}.

A CANONICAL FORM (THE BASIS THEOREM) Using the Orthogonality Lemma and the Signature Lemma any inner product space may be put in canonical form.

The Basis Theorem 2.46. Suppose V, e is an inner product space of one of the eight types. Then V is isometric to the standard model of the same type that has the same dimension and signature. Corollary 2.47. Suppose V, e and V, 9 are two inner product spaces of the same type. The dimension and signature are the same if and only if V and V are isometric. For each of the eight types of inner product spaces, certain bases are of particular importance and will be used to prove Theorem 2.46.

A Canonical Form (The Basis Theorem)

30

Definition 2.48. Suppose V, a is an inner product space of dimension N and signature p, q (if V has a signature). A basis lei,. . -, eN} for V is said to be orthonormal if the linear map f from V to the standard model of the same type, dimension, and signature, defined by sending ej to the jch standard basis vector for the standard model, is an isometry. Thus, a basis {e! , ... , err} for V,,- is orthonormal if R-symmetric (dimR. V = n): e(ej , ej) = 1 (2.49)

for j = 1, .

E(ej,ej) -1 for p+ e(ei, ei) = 0 fori # j;

.

. , p,

1,.. ,n,

R-skew (dimR, V = 2n): (2.50)

e(e2i-1, e2j) = 1 for j = 1, ... , n, e(ei, e1) = 0 for all other pairs ei, ej,

(orthonormal basis = symplectic basis);

C-symmetric (dimc V = n):

e(e,,ej)=1 for (2.51)

e(ei,e1)=0 for i

j,

(orthonormal basis = C-orthonormal basis); C-skew (dimc V = 2n): (2.52)

e(e2j-l, e2j) = 1 for j = 1, ..., n, e(ei, eJ) = 0 for all other pairs ei, e

(orthonormal basis = C-symplectic basis);

C-hermitian (symmetric) (dimc V = n):

e(ei,ej)=1 forj=1,...,p, (2.53)

e(ej,ej)=-1 forj=p+1,...,n, e(ei, ei) = 0

fori # j,

(orthonormal basis = unitary basis);

Inner Product Spaces

31

H-hermitian symmetric (dimes V = n): 1

(2.54)

for

j

e(ej,ej)=-1 forj=p+1,...,n, e(ei,e?)=0 for i#j,

(orthonormal basis = H-unitary or hyperunitary basis); H-hermitian skew (dimH V = n): (2.55)

e(ej,ej) = i for j = 1,...,n

e(es,ej)=0fori#j,

(orthonormal basis = H-skew basis).

Proof of Theorem 2.46: It suffices to show that V,,- has an orthonormal basis. The proof is by induction on the dimension. R-symplectic, C-symplectic: Choose el any nonzero vector. Since e is nondegenerate, there exists u with e(ei, u) $ 0. Let e2 denote u rescaled so that

e(ei,e2) = 1. Let W = span {ei,e2}. Since W is nondegenerate, Lemma 2.30 implies that V = W +W1, W n W1 = {0}, and that W1, e is a lower dimensional (symplectic) inner product space. By the induction hypothesis W1 has a symplectic basis e3i ..., e2n. Since W 1 W1, el, e2i e3, ..., e2,a

is a symplectic basis for V = W + W. Signature: Assume V, e has a signature. Choose P maximal positive and N maximal negative with N = P1 by Lemma 2.42. Since P and N are orthogonal, it suffices to show that P and N have an orthonormal basis. Thus, we may assume that V is positive if V has a signature. The proof is completed, in the positive definite cases as well as in all remaining cases without signature, as follows. Choose a nonzero vector u E V. Let ei denote u properly normalized so that e(ei, ei) = 1, unless e is H-hermitian skew, in which case require e(ei, ei) = i (see Problem 6). Let W = span el. Then W, and hence W1, is nondegenerate. Thus, by the induction hypothesis W1 has an orthonormal basis. Since W and W1 are orthogonal, el combined with the orthonormal basis for W1 is an orthonormal basis for V = W + W1. I

THE PARTS OF AN INNER PRODUCT Suppose e is one of the eight types of inner products. If e is either C-valued or H-valued, then 6 has various parts.

32

The Parts of an Inner Product

The simplest case is when a is a complex-valued inner product. Then e has a real part a and an imaginary part /3 defined by the equation

e = a + i/3

(2.56)

(while requiring a and )3 to be real-valued).

Suppose e is a quaternion valued inner product. There are several

options for analyzing the parts of E. First, e has a real part a and a pure imaginary part Q defined by

e=a-}-/3,

(2.57)

where a = Re c is real-valued and /3 = Imc takes on values in ImH span{i, j, k}. The imaginary part 11 has three components defined by (2.57')

0 - i/3, + j/33 + k/3k,

where the parts /3;, /3? , /3k are real-valued.

Second, using the complex structure R; (right multiplication by i) on H, each quaternion x E H has a unique decomposition x = z + jw where z, w E C C H. Therefore,

e = y + jb with -y and b complex-valued

(2.58)

uniquely defines y (the first complex part of e) and 6 (the second complex part of e). Of course,

(2.58')

7=a+i,8 and b=/3j-i8k.

This section is devoted to computing the types of these parts of E. As an application, alternate definitions of the isometry group G for e are deduced. However, this material is better motivated by geometric considerations and should be read in conjunction with Chapter 5 on geometry. In fact, readers may wish to examine Chapters 3-5 before returning to this section. Because of its importance in geometry, each type of inner product e is decomposed into parts using notation from geometry.

C-Hermitian Symmetric Consider the standard C-hermitian (symmetric) form (2.59)

e(z, w) = zlwl + - . + zpwp - zp+lWp+l - ... - znwn

Inner Product Spaces

33

with signature p, q on C11. Since a is complex-valued, it has a real and imaginary part given bye = g - iw. For z = x + iy and to = l; + iq E Cn

R21, the real and imaginary parts g = Ree and w = -Ime are given by (2.60)

9(z, w) = x1Si + y17)1 + ... +

yprip - ... - xn. n - ynrln

and (2.61)

w(z, w) = x11

-

xnrln -1-

Thus, g is the standard R-symmetric form on R2n with signature 2p, 2q. Modulo some sign changes, w is the standard symplectic form on R21. In this context, when q = 0, w is exactly the standard Kahler form on Cn and is usually written as (2.62)

w = 2 dzl Adz, +

dzn A dzn.

Lemma 2.63. Suppose e is C-hermitian symmetric (signature p, q) on a complex vector space V with complex structure i. Then g - Ree is Rsymmetric (signature 2p, 2q) and w = -Ime is R-skew. Moreover, each determines the other by (2.64)

g(z, w) = w(iz, w)

and w(z, w) = g(iz, w).

Also, i is an isometry for both g and w: (2.65)

g(iz, iw) = g(z, w),

and w(iz, iw) = w(z, w).

Conversely, given g R-symmetric with i an isometry, if w is defined by (2.64), then e - g - iw is C-hermitian. Also, given w R-skew with i an isometry, if g is defined by (2.64), then e = g - iw is C-hermitian.

Proof: It suffices to prove that Ree and Ime have the desired properties when Cn, a is the standard model; this has already been carried out (see (2.60) and (2.61)). Alternatively, the properties of Ree and Ime can be easily derived from the fact that a is C-hermitian symmetric with signature p, q, providing a second proof, this one without coordinates. Reconstructing a from g (or from w), with i an isometry is Problem 8a.

Remark 2.66. If the signature is positive definite, then Lemma 2.63 may be cryptically summarized by saying that "the confluence of any two of

The Parts of an Inner Product

34

(a) complex geometry, (b) symplectic geometry, (c) Riemannian geometry is Kahler geometry." See Lemma 5.17 for more details. This decomposition e = g - iw also provides several alternative defintions of the unitary group U(p, q). Suppose V, e is a C-hermitian symmetric inner product space of signature p, q. Let GL(n, C) denote Endc(V), let O(2p, 2q) denote the subgroup of GL(V, R) fixing g - Re c, and let Sp(n, R) denote the subgroup of GL(V, R) fixing w. Corollary 2.67. The intersection of any two of the three groups

GL(n, C), Sp(n, C),

and 0(2p, 2q)

is the group U(p, q). Similar results are valid in the C-symmetric and C-skew cases, but perhaps not so interesting since in either of these cases the group that fixes a = Re e and the group that fixes /3 = Ims are just different versions of the same group under a coordinate change. These cases are left to the reader.

H-Hermitian Symmetric Consider the standard H-hermitian symmetric form E(x, y) _=71Y1 -{- ... + xp yp - ... - xnYn

(2.68)

on Hn with signature p, q. Note that a is H-valued. As noted earlier, it is natural to consider H as two copies of C, H - C ® jC, or x = z + jw. In particular, e = h + jo- with h and o complex-valued. Note that if x =- z -}- jw and y

1; -I- jq with z, w,

77 E C, then

xy = (z - jw)( + j'7) = z - jwjr7 + zjr7 - jwC

(2.69)

= x + 017 +?(zi - w. ). Therefore, the first complex part of h of e is given by (2.70)

h(x, y) = 91b1 + w1771 + ... +

wp?7p - ... - xnSn - wn77n.

Thus h is the standard C-hermitian symmetric form on C2n = H. Because of (2.69) the second complex part o- is given by v(x, y) = zi771 - w16 ± ... ± znr7n f

(2.71)

Thus, modulo some possible sign changes, a is the standard C-skew form on

Ctn.

Inner Product Spaces

35

Lemma 2.72. Suppose e = h + jo- is H-hermitian symmetric on a right H-space V. Then h, the first complex part of e, is C-hermitian symmetric and v, the second complex part of c, is C-skew. Moreover, each determines the other by (2.73)

h(x, y) = o(x, yj)

and

v(x, y) = -h(x, yj).

Also, (2.74)

h(xj, yj) = h(x, y)

and u(xj, yj) = o(x, y).

Conversely, given h C-hermitian symmetric and h(xj, yj) = h(x, y), if o is defined by (2.73), then e = h + jo- is H-hermitian symmetric. Also, given v C-skew and o(xj, yj) = o-(x, y), if h is defined by (2.73), then e = h+ ja H-hermitian symmetric.

Proof: The equation e(x, yj) = e(x, y)j can be used to prove (2.73), while e(xj, yj) = -je(x, y)j can be used to prove (2.74). The remainder of the proof is left to the reader (see Problem 8(b)). J Suppose V, a is an H-hermitian symmetric inner product space of signature p, q. Let GL(n, H) denote EndH(V), let U(2p, 2q) denote the subgroup of GL(n, H) that fixes the first complex part h of e, and let Sp(n, C) denote the subgroup of GL(n, H) that fixes the second complex part a of e.

Corollary 2.75. The intersection of any two of the three groups GL(n, H), U(2p, 2q),

and Sp(n, C)

is the group HU(p, q).

It is useful to try to construct the quaternionic structure from h and a.

Suppose V is a complex 2n-dimensional vector space, h is a C-hermitian symmetric inner product on V, and o- is a complex symplectic inner product on V. Then h and o define a complex antilinear map J by (2.76)

h(xJ, y) = o(x, y).

Now o-(xJ2, y) = -v(y, xJ2) = -h(yJ, xJ2) = -h(xJ2, yJ) = -o(xJ, yJ). Therefore, j2 = -1 if and only if (2.77)

o(xJ, yJ) = v(x, y).

In this case, h and o are said to be compatible.

The Parts of an Inner Product

36

Lemma 2.78. Suppose h is a C-hermitian symmetric inner product and a is a complex symplectic inner product on a complex 2n-vector space V. If h and a are compatible, then they determine a right H-structure on V and

E=h+ja

is an H-hermitian symmetric inner product on V.

Proof: It remains to verify that h + ja is H-hermitian symmetric. J There is yet another description of HU(p, q). First, recall (1.27) that for each unit vector u E Im H, right multiplication by u (denoted R,,) acting on H" determines a complex structure on Hn and hence an isomorphism Hn = C2n Second, recall that in the C-hermitian symmetric case the Kahler form w is minus the imaginary part of E and that w is determined by the formula

(2.64) in terms of g = Re E. Each of the complex structures R. on H' C2n determines a Kahler form in exactly the same manner. Let w,, (x, y) = Re E(xu, y),

(2.79)

and g(x, y) = Re e(x, y).

Lemma 2.80. For all x, y E Hn,

E=g+iw;-{-jwj +kwk. Proof: For u E ImH with Jul = u u = -u2 = 1, (u, e(x, y)) = (1, U6(x, y)) = Re e(xu, y) =

y).

For each complex structure u E Im H, Jul = 1, the complex C(u) valued form (2.81)

hu =g+uwu

is C(u)-hermitian symmetric. The group that fixes hu is a unitary group with signature 2p, 2q determined by the complex structure Ru. The next corollary justifies the name "hyper-unitary" for HU(p, q).

Corollary 2.82. The hyper-unitary group HU(p, q) is the intersection of the three unitary groups determined by the three complex structures R;, R, , Rk on H". Remark 2.83. Of course, HU(p, q) is also the intersection, over u E S2 C ImH, of the unitary groups determined by all the complex structures Ru.

Inner Product Spaces

37

The simplest case states that HU(1) is the intersection of all the unitary groups determined by the complex structures Ru on H = C2. Recall (1.43) the isomorphism SU(2) = HUU(1), where SU(2) denotes the special unitary group determined by the complex structure Ri on H = C2. Similarly, for each complex structure R, on H the corresponding special unitary group is again the same HU(1) and thus is independent of R.

H-Hermitian Skew Consider the standard H-hermitian skew form on H": (2.84)

e(x, y) = xliyl + ... + x"iy".

Since i is distinguished among all the unit imaginary quaternions u, it is natural to set H - C ®j C where each copy of C has the complex structure Ri induced from Ri acting on H. Let (2.85)

e = ih + jo,

where ih and o are the first and second complex parts of a respectively. Set x = z + jw and y =1; + jrt with z, w, , 71 E C. Then (2.86)

h(x, y) =

w1r11 +....f

U7"77"

is C-hermitian symmetric, and (2.87)

0'(X, Y) = x1711+w1S1 +...+xnrln+wnS,

is C-symmetric. In summary, if e = ih + jo is H-hermitian skew then (2.88)

h is C-hermitian symmetric and o is C-symmetric.

Since the form h is equivalent to the standard°C-hermitian symmetric form on C2n with signature n, n via a complex coordinate change, let U(n, n) denote the group fixing h. Since o is equivalent to the standard C-symmetric form on C2n via a complex coordinate change, let O(2n, C) denote the group that fixes v.

Problems

38

Proposition 2.89. The intersection of any two of the three groups GL(n, H),

U(n, n),

and

O(2n, C),

is the group SK(n, H).

Note that the H-skew sphere of radius i is the intersection of the unit sphere S(2p,2p), given by Iz12 - Iw12 = 1, with the null cone >z;w; = 0, where (z, w) E C2n = H' . For each u E S2 C ImH, (2.90)

g. (X, y) = 4X U, y) = (u, e(x, y))

defines a real symmetric inner product with split signature, while

w=ReE

(2.91)

is a real symplectic inner product. Because of (2.90), (2.92)

E = w + ig= + jgi + kgk.

Finally, note that SK(n, H) is also "hyper-unitary" (but in a different sense than HU(p, q)). The reader may wish to compute (in coordinates) w, g+, gi, and gk.

PROBLEMS 1. Give the proof of (2.33), that: (a) W is nondegenerate,

(b) WnWl={0),

(c)W+Wl=V, (d) Wl is nondegnerate are all equivalent. 2. (a) Prove that R(p, q), C(p, q), and H(p, q) have signature p, q. (b) If V, a has a signature, show that each positive subspace is contained in a maximal positive subspace. 3. Suppose V, E is an inner product space with a signature. A map f E EndF(V) is called an anti-isometry if e(f (x), f (y)) = --c (x, y) for all x, y E V.

39

Inner Product Spaces

A map f E EndF(V) is said to be anti-conformal if, for some negative constant A < 0, E(f(x),f(n)) = AE(w,y),

forallx,yEV.

Show that if there exists an anti-conformal map f E EndF(V), then the signature must be split. 4. Suppose V, E is an inner product space of R-symmetric type with signature p, q and p < q. Prove the following. (a) There exists a totally null subspace of dimension p. (b) Each totally null subspace is of dimension < p (i.e., prove Proposition 2.45).

(c) V is split (p = q) if and only if V = Ni ® N2 with N1, N2 totally null.

5. Suppose V is a real vector space and E : V x V -p R and b : V -> V* are related by the equation E(x,y) = (b(x))(y)

(Thus E(x, ) is linear since #(x) E V*.) Prove the following. (a) e(., y) is linear if and only if is linear. (b) e is symmetric if and only if (Here b* is the dual map not the adjoint.) 6. Suppose V, E is an H-hermitian skew inner product space. Show that there exists a vector el E V with e(e1, e1) = i. 7. A finite dimensional vector space V over F equipped with a bilinear form E satisfying all of the properties of an inner product except (possibly) the nondegeneracy condition is called a degenerate inner product space. The standard models for the degenerate inner product spaces are all of the form Fk e F'", e where F'", e is one of the standard models

for an inner product space. Show that each degenerate inner product space V, e is isometric to exactly one of the degenerate standard models.

Hint: Write V = N ® W with N = V. 8. (a) Suppose g is an R-symmetric inner product on R2n and i is a complex structure on R2n that is orthogonal with respect to g. Show

that E(z, w) = g(z, w) - ig(iz, w)

is C-hermitian symmetric on C" = RZ", i.

Problems

40

(b) Suppose o is a C-symmetric (or C-skew) inner product on CZ" H", R; that satisfies a(x.1, yj) = a(x, y)

for all x,yEH".

Show that E(x, y) = cr(x, y.9) +?cr(x, y)

is H-hermitian symmetric (or H-hermitian skew). 9. The Basis Theorem 2.46 may also be interpreted purely algebraically. Only the first case (R-symmetric) is discussed. Show that the two statements (a) and (b) are equivalent. (a) (See Problem 7.) Each degenerate R-symmetric inner product space is isometric to one of the standard models R(p, q) ® Rk.

(b) (Sylvester's Theorem.) Given a symmetric matrix A E M (R), there exists an invertible matrix B E so that BAB-1 is diagonal with each nonzero diagonal entry either +1 or -1. 10. State and prove the H-hermitian skew analogue of Lemma 2.78. 11. Show that the intersection of any two of the unitary groups U; (2p, 2q),

Uj(2p, 2q), Uk(2p, 2q) (based on the three complex structures R;, R Rk respectively) is equal to the hyper-unitary group HU(p, q).

3. Classical Groups II

The classical groups introduced in Chapter 1 are examined more carefully in this chapter. In particular, the "Lie algebra" of each of these groups is computed explicitly.

GROUP REPRESENTATIONS AND ORBITS Suppose G is a group and V is a real vector space.

Definition 3.1. (a) A representation p of G on V is a group homomorphism p : G GL(V). Sometimes, the notation p is suppressed, and we say that G acts on V, when the action p is understood from the context. (b) Two representations p : G -+ GL(V) and o : G --+ GL(W) are said to be equivalent if there exists a R-linear isomorphism L : V --> W with o(g) = L o p(g) o L-1 for all g E G. The linear map L is called an intertwining operator. Since the groups defined so far were defined as subgroups of GL(n, F), each such group comes equipped with a natural action on F". Suppose p : G -+ GL(V) is a action of G on V. A set of the form (3.2)

{p(g)v: G E G}

is called an orbit of G, or the orbit of G through v E V. The subgroup K - {g E G : p(g)v = v} is called the isotropy subgroup of G at v. The 41

Generalized Spheres

42

group G is said to act transitively on a subset X C V if X is the orbit of G through each point v E X. Two isotropy subgroups, K. and If,, at two points u, v in the same orbit X are isomorphic:

Ku=LK,, L'1,

(3.3)

where L E G is chosen so that Lv = u. Thus, if G acts transitively on X, we write

G/K - X

(3.4)

for the quotient of the group G by the equivalence relation gl - g2 defined by g2 = glk for some k E K, where K represents the isotropy subgroup at a point u E X.

Recall from Problem 1.3 that the connectivity of GL(n, F) and SL(n, F) (for F - R, C, or H) was derived from

GL(n, F)/K = F" - {0} SL(n, F) f K F" - {0}

(3.5) (3.6)

(n > 2).

Thus, GL(n, F) and SL(n, F) (n > 2) acting on F" have two orbits, namely {O} and F" - {0}.

GENERALIZED SPHERES For each of the seven types of inner products, the proof of the Basis Theorem can be used to compute the orbits of the isometry group G (that fixes the inner product e) acting on F". Of course, each orbit of G is contained in one of the level sets (3.7)

{x E F" : e(x, x) = c} for some constant c E F.

The most important cases occur when e is an R-symmetric inner prod-

uct.

Definition 3.8. The positive sphere of radius r > 0 in R(p, q) (p > 1) is: (3.9)

Sr -{xER(p,q):(x,x)=r2}.

The negative sphere of radius r < 0 in R(p, q) (q > 1) is: (3.10)

Sr -{xER(p,q):(x,x)_-r2}.

The unit spheres S: in R(p, q) are obtained by setting the radius r = E1.

43

Classical Groups II

The standard notation for the euclidean sphere Sr in R(n, 0) is ST -1 since this sphere has dimension n - I. We will use the same notation,

Sr'1, for the sphere Sr- in R(0, n). S° consist of two points. Note that the positive unit sphere S+ in R(p, q) is the same as the negative unit sphere S- in R(q, p). S- future timelike

S- past timelike Figure 3.11

Proposition 3.12. (a) (p > 1) the unit positive sphere S+ in R(p, q) is diffeomorphic to, SP-1 x R. (b) (q > 1) the unit negative sphere S- in R(p, q) is diffeomorphic to RP x S9-1.

(c) The null cone N diffeomorphic to

{z E R(p,q) : z # 0 but jjzjj = 0} in R(p,q) is SP-1

x SQ'1 x R+.

Generalized Spheres

44

Proof: Let (3.13)

llxl 2 = 1x12 - lyl2, for all z = (x; y) E R(p, q) = RP x R9

denote the square norm of z = (x, y) where IxI and Iyl denote the euclidean

norms ofx R" and yE R. Proof of (a): If z =- (x, y) E S+ then llzll = IxI2 - lyl2 =

1.

In

particular lxl # 0 so that (x/Ixl,y) E SP-1 x R4 is well defined. This map from S+ to SP-1 x R9 has inverse sending (v, y) E SP-1 x RQ to

z E( 1+ lyl2v, y) E S+.

The proof of (b) is similar and omitted. Proof of (c): Since IxI2 - IyI2 = 0,

(3.14)

for allz = (x, y) E N,

if z = (x, y) is null but non-zero then both x and y are non-zero. Thus the map from N to SP-1 X SQ-1 x R+ sending z - (x, y) to ( , YT, r), where r = IxI = lyl, is a well defined smooth map. The inverse of this map sends (x, y, r) E SP-1 x 59-1 x R+ to z = (rx, ry) E N.

The next result follows since S° consists of two points, S"(n > 1) is connected, and S" (n > 2) is simply connected.

Corollary 3.15. (a) The unit positive sphere S+ in n-dimensional Lorentzian space R(1, n - 1) has two connected components diffeomorphic to Ri-1. (These components are called the future and past spheres.) (b) The unit positive sphere S+ in R(p, q) for p > 2 is connected. (c) The unit positive sphere S+ in R(p, q) for p > 3 is simply connected. If C(p, q), E is the standard model for the C-hermitian symmetric case of signature p, q, then R(2p, 2q) S' C(p, q) with (, ) - Re a is the standard model for the R-symmetric case of signature 2p, 2q. Also note that E(x, x) only takes on real values. Therefore, (3.16)

{x E C(p, q) : E(x, x) = ±r2} = S, , the sphere in R(2p, 2q).

Similarly,

(3.17)

{x E H(p, q) : E(x, x) = ±r2} = S1, the sphere in R(4p, 4q).

If C', E is the standard model for the C-symmetric inner product space, then ST'1(C) _ {z E C' : e(z, z) = r} is called the complex (or hyperquadric) sphere of radius r E C.

45

Classical Groups II

If H'2, E is the standard model for the H-hermitian skew inner product

space, note that -(x, X) only takes values in ImH. Given r E ImH, the level set

Sr (H"-skew) _ {x E H' : E(x, x) = r}

(3.18)

,

fixed by the group SK(n), will be called the H-skew sphere of radius r. Note that if u E Im H is a given unit vector then the real linear isomorphism R,' :

H" - H" sends the H-skew sphere of radius r E Im H onto the H-skew sphere of radius uru E Im H. This fact can be used to yield a description of the general H-skew sphere from calculating the special case r = i, where S;(H'-skew) = {a+jb : a, b E C",1aI2-1612 = 1, and akbk+bkak = 0}.

Given an inner product, the spheres are the orbits of the associated isometry group.

Theorem 3.19 (Orbit Structure). Let n = p + q. (3.20)

(3.21)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

O(p, q)

SO(p, q)

O(p - 1, q)

SO(p - 1, q)

O(p , q) O(p, q 1)

SO(q p, q )

-

U(p, q)

U(p - 1, q)

SU(p, q)

=

SU(p, q - 1)

_

U(p, q)

- 1)

SO(p,

SU(p, q)

U(p, q - 1) HU(p,

SU(p, q - 1)

q)

HU(p - 1, q) HU(p, q)

_, S

,

r S:, the sphere in R(p, q).

,, S* the sphere in R(p, q).

- S,+, the sphere in R(2p, 2q).

, S,-, the sphere in R(2p 2q).

the sphere in R(4p, 4q).

,,, S,-, the sphere in R(4p, 4q).

HU(p, q - 1) O(n, C)

_

SO(n, C)

0(n - 1, C) - SO(n - 1, C)

-1(C) for r E C' and n > 2.

The Basis Theorem Revisited

46

R) R2" - {0}, Sp(n, = K x Sp(n - 1, R). where K is topologically the same as

(3.27)

R2n-1

Sp(n, C) = C2n - {0}, K where It is topologically the same as C2i-1 X Sp(n - 1, C). (3.28)

(3.29)

SK(n, H)

_ S,.(H " skew)

SK(n - 1, H) -

for each r E ImH.

Proof: Let G/If - X correspond to one of the cases in Theorem 3.19. First, the group G acts transitively on X. In each case, this is verified by examining the proof of the Basis Theorem 2.46. For example, O(p, q) acts transitively on Sr since, given a vector u E S* , the vector el = u/vfrmay be chosen as the first vector in an orthonormal basis for R(p, q). The isometry f E 0(p, q) that maps the standard orthonormal basis e1, ... , e for R(p, q) to this orthonormal basis sends el to e"1 and hence \el to u. Second, the isotropy subgroup K of G at el must be computed. In all cases, the fact that (3.30)

if f(el) = el then f : (span el)1 -i (span el)1

is of key importance. The individual cases are left to the reader.

-1

Corollary 3.31. Each of the groups Sp(n, R), U(p, q), SU(p, q), SO(n, C), Sp(n, C), HU(p, q), and SK(n, H) is connected. The groups SO(p, q), p, q > 1 and O(n, C) have two connected components, while the groups O(p, q), p, q > 1, have four connected components. This corollary can be deduced from the theorem by induction on the dimension.

THE BASIS THEOREM REVISITED The Basis Theorem may be interpreted as calculating an orbit of one of the groups GL(n, F). Let e denote the standard inner product on F". The group GL(n, F) acts on the space BIL(R") containing e as follows. Suppose rl E BIL(F'). For each A E GL(n, F), let A*ij denote the action of A on q where A* is defined by (3.32)

(A"rl)(x, y) = q(Ax, Ay)

for all x, y E F".

Let I(V) denote the subset of BIL(V) consisting of all inner products of the same type (and signature if applicable) as e. Let G(e) denote the classical group that fixes a (i.e., the isometry group of the given type).

Classical Groups II

47

Theorem 3.33. GL(n, F) acts transitively on I(V) with isotropy G(e) at E. That is,

GL(n, F)/G(e) - I(V).

(3.34)

Note that {77 E SYM(V) : 71 is nondegenerate} is an open subset of SYM(V) if e is symmetric; while ifs is skew, {77 E SKW(V) : r7 is nondegenerate} is an open subset of SKW(V). Since this theorem is a reinterpretation of the Basis Theorem, the proof is an exercise (see Problem 10).

ADJOINTS Recall from the proof of the Orthogonality Lemma 2.30 that the flat map b : V --+ V* is a real linear isomorphism for all seven types of inner product spaces V, E. In particular, if A E EndF(V) is given and y E V is fixed, then e(y, Ax) E V* is considered as a function of x. Thus, E(y, Ax) = (b(z))(x)

for a unique z E V. By definition of the flat map (b(z))(x) = e(z, x). Now consider z E V to be a function of the given point y E V and denote z by A* y. For each of the eight types of inner product spaces, it is easy to check that e(y, Ax) = e(z, x) if and only if e(Ax, y) = e(x, z). Thus the map A* : V -* V is characterized by (3.35) either e(y, Ax) = e(A* y, x) or e(Ax, y) = e(x, A* y) for all x, y E V.

This map A* is called the adjoint (not the dual) map of A, and definitely depends on the given inner product e. The proof of the next lemma is straightforward and hence omitted (i.e., should be checked once carefully enough to see that it is routine).

Lemma 3.36. The adjoint A --r A* is an anti-automorphism of the algebra EndFV that squares to the identity (an involution). That is, A* E EndF(V) for each A E EndF(V) and (3.37)

(AB)* = B*A*

and (A*)* = A.

Note that, given A E EndF(V), (3.38)

e(Ax, Ay) = e(x, y) for all

x, y E V

and (3.39)

A*A = Id

(i.e., A* = A-1),

(i.e., A fixes e),

Lie Algebras

48

are equivalent. Thus, each of the groups, defined by requiring that e be fixed, can also be described as {A E EndF(V) : A* = A-'}.

Depending on the type of the inner product space, the adjoint is referred to as the R-orthogonal adjoint, R-symplectic adjoint, C-orthogonal adjoint, C-symplectic adjoint, C-hermitian adjoint, H-hermitian symmetric adjoint, and H-hermitian skew adjoint. Given a matrix A = (aid) E Mn(F) the matrix transpose is denoted by At = (aj2), while the matrix conjugate is denoted by A = (a;j). Using the matrix transpose and the matrix conjugate, each of the various types of adjoints can be explicitly computed (Problem 3). For example, the C-hermitian (positive definite) adjoint of is given by A* = ;P the conjugate transpose. Combining the AE fact that A fixes e if and only if AA* = 1 with the explicit computations of the adjoints for A E Mn(F), it is easy to give alternative definitions of the groups: O(p, q), Sp(n, R), O(n, C), Sp(n, C), U(p, q), HU(p, q) and SK(n, H). For example, U(n) = {A E Mn(C) : AAt = 1}.

LIE ALGEBRAS Each of the groups G considered so far is a subset of M, (F), with F = R, C, or H, defined implicitly by a certain number of polynomial equations. The objective of this section is to show that G is a submanifold of Mn(F) and

to compute the tangent space to G at the identity 1 E G in Mn(F). This tangent space is denoted g and called the Lie algebra of G. Given two matrices A, B E Mn (F) the commutator (3.40)

[A, B]

AB - BA

is also called the Lie bracket of A and B. This product provides the natural algebraic structure on each of the Lie algebras g. The basic identity satisfied by a Lie algebra g, [ , ] is called the Jacobi identity: [A, [B, C]] = [[A, B], C] + [A, [B, C]], for all A, B, C E g. The reader is referred to the plethora of texts on Lie algebras for more information.

The Lie algebra of a particular group is usually denoted by using the same label used for the group-except in German lower case. For example, so(r, s) denotes the Lie algebra of SO(r, s). First note that (3.41)

gI(n, R) = MM(R)

Classical Groups II

49

(3.42)

g((n, C) = Mn(C)

(3.43)

g[(n, H) = M,,(H),

since GL(n, F) is just an open subset of Mn(F). Now suppose that G is one of the seven classical groups that fixes one of the standard inner products e. Then G is given implicitly as a subset of Mn(F) by

G = {X E M,,(F) : XX* = 1},

(3.44)

where F is R, C, or H and X* denotes the E-adjoint. Let SymE(n, F) denote the real vector subspace of Mn(F) consisting of the --symmetric matrices, i.e., (3.45)

SymE (n, F) _ {A E M (F) : A = A*}.

Let SkewE (n, F) denote the real vector subspace of Mn(F) consisting of the --skew matrices, i.e., (3.46)

SkewE (n, F) _ {A E Mn(F) : A* = -Al.

Proposition 3.47. Suppose e is one of the standard inner products. The e-orthogonal group

G={AEMn(F):AA*=1} is a real submanifold of the real vector space Mn(F) with (3.48)

dime. G = dims Skew£ (n, F).

The tangent space g at 1 E G is given by (3.49)

g = Skew, (n, F).

Proof. Consider X X * as a function from M (F) to SymE (n, F) (i.e., from RP to R9 for some p and g). The group G is defined to be the level set of this function at level 1 E SymE (n, F). To prove that G is a real submanifold of M (F) we invoke the implicit function theorem. Thus we must compute

the rank of the derivative of XX* at each point B E G and show that it equals dimrt SymE(n, F).

Lie Algebras

50

The derivative (i.e., the linearization) of the function XX* at B E G is given by (3.50)

d (B +tX)(B +tX)* It_o= XB* + BX*.

The linearized map XB* + BX* from Mn(F) to SymE (n, F) is surjective;

given A E SymE (n, F), take X = AB and note that X B* + BX * z ABB* + BB* A* = A. Thus thezrank of the linear map sending X E z comMn(F) to XB* + BX* E SymE(n,F) is equal to pleting the proof that G is an implicitly defined submanifold of M,(F). The tangent space to G at B E G is the kernel of the linear map XB* + BX * . Thus, the tangent space g to G at 1 E G is the kernel of X + X*, which equals F) C MM(F). Li The submanifold G of Mn (F) defined by (3.44) can also be described parametrically. Since, given B E G, right multiplication by B provides a diffeomorphism of Mn(F) that maps G to G and sends 1 E G to B E G, it suffices to give a parametric description of a neighborhood of 1 E G. First, we need a lemma about power series. The sup norm on M (F) is defined by (3.51)

sup norm A = sup

I

ICI

for A E Mn(F),

I

I denotes the standard positive definite norm on Fn(F = R, C, or H). The proof is an exercise. where I

Lemma 3.52. Suppose f (z) of convergence r. The series

n=0

a,Zn is a power series with radius

0" anAn f (A) = >2

n=0

converges in the sup norm on Mn(F) if the sup norm A < r.

Definition 3.53. Given A E M, (F), define

exp(A) = eA = 1+A+ 1 A2 + Proposition 3.54. Let G denote the group that fixes the inner product e. The exponential map (3.55)

exp : SkewE (n, F) --+ G

Classical Groups II

51

provides a parameterization of a neighborhood of the identity 1 for the group G.

Proof: If A E Skew, (n, F), then CA E G, since (eA)(eA)* = eAeA* _ eA . e-A = eA-A = eo = 1. The derivative of exp at A = 0 is given by = X.

etx

t.0

Wt

This proves that exp is injective near A = 0. Now the proposition follows since G has dimension equal to the dimension of the parameter space

Skew, (n,F). Li The special linear groups are defined implicitly by setting the polynomial detF X equal to one, where F =_ R or C. The derivative, or linearization, of this function at a point A is by definition (3.56)

dt detF (A + tX)

Lo

.

At the point A = 1 (the identity matrix), this formula provides a convenient coordinate free definition of the trace of a linear map X E EndF(V). That is, for F = R or C, we define

tracer X = dt detF(1 + tX)

(3.57)

Lo

Of course, it is important to compute that if X - (xii) E Mn(F) for F =- R or C, then

xii. i.1 To check this consider detF(I+ tX)ej A . . A en = (el +tXel) A (en + tXen), where e1, ... , e is an F-basis for V. If X = (xii) E MM(H), then it follows that traceF X =

(3.58)

A

n

(3.59)

tracer- X = 4 Re

xii i.1

Now suppose that A E SL(n, F). In particular, A is invertible. The derivative of detF X at the point A is equal to (3.60)

dt

detF(A+tX)

Lo

= dt detF(1 +tXA-1)A) t=o

Problems

52

Therefore, the linearization has rank 1 for detR, and rank 2 for detc. Because of the implicit function theorem, this provides a proof that the special linear groups are submanifolds. The tangent space at the identity 1 is the kernel of the linear map traceFX, obtained by setting A = 1 in (3.60). This proves the next proposition.

Proposition 3.61. (3.62)

st(n, R) = {A E A, (R) : traceR,A = 0},

(3.63)

st(n, C) = {A E Mn (C) : tracecA = 0} ,

(3.64)

sl(n, H) = {A E M (H) : traceRA = 0} .

PROBLEMS 1. (a) Prove that (AB)* = B*A* and (A*)' = A for each of the seven types of adjoints. (b) Show that e(Ax, Ay) = e(x, y) if and only if A* = A-1.

2. (a) Show that det A* = det A if A* is the adjoint with respect to e, where e is either of R-symmetric type or C-symmetric type. (b) Show that each A E O(p, q) has deter A = ±1. (c) Show that each A E O(n, C) has detc A = ±1. (d) Suppose A E CO(p, q) with conformal factor A. Show (detR A)2 = A", where n = p + q is the dimension. 3. (Adjoints) Compute the adjoint of a matrix A E M,,, (F) in the following cases. Note that a decomposition of the vector space F'° into the direct sum of two subspaces induces a 2 x 2 blocking of the matrices:

Case (a) R-positive definite or C-pure symmetric. Show A* = At. Case (b) Positive (or negative) definite, F - R, C, or H. Show

A* = q' (conjugate transpose). This case is a subcase of the next case, F-hermitian symmetric.

Case (c) F(p, q) = F" x F4, with F - R, C, or H. Show d\ * ( C c

-

_V

T/

Classical Groups 11

53

Case (d) H-hermitian skew: A* = iAt z', where i denotes i times the identity matrix. Case (e) R- or C-symplectic: R2i = RI (DR' or C21 = C"(BCn, written as the direct sum of the even coordinates and the odd coordinates. Show (-dtCt

(c

d)*=

at)

Hint: Reduce all cases to either Case (a) or Case (b) by finding a linear map L so that e(x, Ly) = (x, y), with (,) an inner product of the type in either case (a) or case (b). 4. Suppose F", e is one of the standard models for an inner product space.

Let G denote the isometry group of F", e, and g = SkewE (n, F) the Lie algebra of G.

(a) Suppose e is (either pure or hermitian) symmetric. Exhibit an isomorphism g = SKW(F") with the space SKW(F") of all skew bilinear forms on F" of the same type as e. (b) Formulate the analogue of part (a) for b skew. (c) The common notation for the space of all skew bilinear forms on RI is A2(R")*, while S2(Rn)* denotes the space of all symmetric bilinear forms on R". Thus, by (a) so(p, q) = A2(R")* (n = p+q), and by (b) zp(n, R) = S2(R2n)*. Compute that dimR SO (p, q) = 2n(n + 1) and that dimR Sp(n, R) = 2n2 + n. 5. Using the definition (3.57) for traceF X show that the formula (3.58)

is valid for X E M"(R) or X E M"(C). If X E M"(H), show that (3.59) can be used to compute traceR X.

6. (a) Show that lyu(1) = ImH and that exp : ImH -* S3 = HU(1) C H is surjective.

(b) Show that exp : s((2, R) -; SL(2, R) is not surjective. 7. Explicitly compute o(n), o(r, s); u(n), u(r, s); I u(n), lju(r, s); sp(n, R), .sp(n, C); o(n, C); and st(n, H). 8. The real manifold dimension of a Lie group G is the same number as the real vector space dimension of the Lie algebra S. Show that (a)

dimR GL(n, R) = n2, dimR SL(n, R) = n2 - 1, dimR GL(n, C) = 2n2, dimR SL(n, C) = 2n2 - 2, dimR GL(n, H) = 4n2, dimR SL(n, H) = 4n2 - 1.

Problems

54

(b) Using Problem 12, compute that (n = r + s)

dimR SO(r, s) = 1 n(n - 1), dimR U(r, s) = n2, dimR SU(r, s) = n2 - 1, dimR HU(r, s) = 2n2 + n, dimR O(n, C) = n(n - 1), dimR SK(n, H) = 2n2 - n, dimR Sp(n, R) = 2n2 + n, dimR Sp(n, C) = 4n2 + 2n. 9. Give the proof of Lemma (3.52) on power series.

10. Give the proof of the seven results contained in Theorem 3.33. For example, show that

GL(n, R)/O(r, s) - I(r, s; R), the set of R-symmetric inner products on R' with signature r, s. 11. Prove that SK(n, H) is the intersection of three distinct split unitary groups on H", corresponding to the three complex structures R;, RR, and Rk on H. 12. Show that (a) su(p, q) OR C = 5f(n, C), (b) hu(p, 4) OR C = sp(n, C) (c) st(n) OR C - so(2n, C),

(d) sf(n, H) 0 C - sf(2n, C). 13. Suppose G is one of the classical groups considered in this chapter. Given an element h E G, the mapping from G to G defined by sending g E G to hgh'1 E G is a diffeomorphism of G that maps the identity element 1 E G into itself. The linearization of this map, denoted Adh, is a linear map from g = T1 G into itself. (a) Show that Adh A = hAh-1, and that h Adh is a representation of the group G on the vector space g. This representation is called the adjoint representation of G. (b) Show that for B E g fixed, the map sending h E G to Adh B has the derivative at h = 1 E G given by the linear map sending A E g - T1G to [A, B] __ AB - BA.

55

Classical Groups II

14. Suppose G is a Lie group, that is, a group equipped with a compatible differentiable manifold structure. Let Xinv(G) denote the vector space of right invariant vector fields, that is, vector fields on G which are fixed by the diffeomorphisms R. (right multiplication by g) for each g E G. (a) Show that Xi,,,,(G) is closed under the binary operator [X, Y]

XY -YX. (b) Under the natural map sending X E Xi,,v(G) to X1 E T1(G) . show that X;,,,(G), [ ] and g, [ , ] are isomorphic. Hint: Consult a text on Lie groups. 15. (The quaternionic unitary group) The space of imaginary quaterions ImH has several realizations relating to H", E. First, ImH can be identified with the coefficient space {Ru : u E ImH} of R-linear maps of H. Second, using E, Im H can be identified with the space {wu : u E ImH} of Kahler forms on H", where each 2-form wu(x, y) is defined to be (u, E(x, y)) (cf. (2.79)). (a) Show that the quaternionic unitary group HU(p, q) HU(1) can be characterized as the subgroup of O(4p, 4q) that fixes R; A R? A Rk, where each element g E SO(4p, 4q) acts on the coefficient space {R,, u E Im H} by sending Ru to g o Ru o g-1. Hint: Recall (1.35) that HU(1)/Z2 SO(ImH) to reduce to the case ,

where g E O(4p,4q) leaves the coefficient space {Ru : u E ImH} pointwise fixed.

(b) Prove that the two representations of HU(1) given by

Xa(x)=axa for allxEImH and pa (wu) = Rawu for all u E Im H are equivalent.

(c) Prove that the following two representations of the group HU(1) Z2

= SO(ImH)

(see 1.35) are equivalent. First, pa(wu A representation on

R°a(wu A

defines a

span {wu A w : u, v E Im H} C A4 (H" )*

Second, oa(A) = X. o A o Xa defines a representation of Sym(ImH), the space of symmetric 3 x 3 matrices on R3 = Im H.

Problems

56

(d) Prove that the quaternionic unitary group HU(p, q) HU(1) fixes the quaternionic Kahler form 1 (w; A wi + wj Awl + wk A Wk) E

A4 (Hn)* .

(e)* (n > 2) Prove that if g E GLR.(H") and g*4(b _
(a) Suppose p = pl ®p2 : SO(n) -- O(Vi) ® O(V2) is a reducible representation of SO(n) on a positive definite inner product space V = Vl ® V2, with inner product (, ). Show that there exists an inner product on V different from c(, ), where c is a constant, which is fixed by SO(n). (b) Using part (a), show that the representation of SO(n) on AkRn is

irreducible unless k = n/2 and n = 0 mod 4. (In this case, the representations of SO(n) on A' R11* and Ak R2k are both irreducible, where A}R2k . {u E AkR2k *u = ±u} denotes the space of self/anti-self dual k-vectors, and * is the Hodge star operator.)

4. Euclidean/Lorentzian Vector Spaces

In this chapter, we will study R-symmetric inner product spaces. If the inner product is positive definite, then the inner product space is better known as a euclidean vector space. If the signature is n - 1, 1, then the inner product space is better known as a Lorentzian vector space. Rather than refer to a Lorentzian vector space as "pseudo" or "semi" euclidean or to a euclidean vector space as "pseudo" or "semi" Lorentzian, we shall refer to an R-symmetric inner product space with signature p, q simply as a euclidean vector space with signature p, q. Remark. For the reader familiar with manifolds, a Riemannian manifold is a smooth manifold whose tangent space at each point is equipped with a positive definite euclidean inner product, and the inner product varies smoothly. A Lorentzian manifold is a smooth manifold whose tangent space at each point is equipped with a Lorentzian inner product, varying smoothly. To be consistent with our inner product space terminology, we shall refer to a Lorentzian manifold as a Riemannian manifold with signature n - 1, 1 rather than as a "pseudo" or "semi" Riemannian manifold (cf. Chapter 5). The first topic in this chapter is very elementary: the Cauchy-Schwarz equality. The second topic is a brief introduction to special relativity, and

57

The Cauchy-Schwarz Equality

58

the third topic is the Cartan-Dieudonne theorem. This last result is crucial to our discussion of the spin groups.

THE CAUCHY-SCHWARZ EQUALITY Suppose V, e = (,) is a euclidean vector space with signature p, q and dimension n = p + q.

Definition 4.1. (a) The norm.of a vector v E V is defined to be IV l

I(TV)I.

(b) The square norm or quadratic form associated with the bilinear form is defined by; Hull

(v, v).

The inner product e = (,) is completely determined by the quadratic form 1111 by the process of polarization-namely, replace v by x + y in hull = (v, v) and use bilinearity to obtain (4.2)

(llx + yll - Ilxll - IIyIU Recall the musical isomorphism (metric equivalence or lowering indices) b : V -* V*. This linear map naturally induces a linear map, also denoted b, on tensors: b :O kv ®kV* = ((&kV)* (e.g., b(u ®v) = b(u) ®b(v)), (x, y) =

2

b : AkV --> AkV* S' (AkV)*

(e.g., b(u A v) = b(u) A b(v)).

Similarly, 0 = b!1 : V* -+ V naturally induces a linear map # on tensors and #b = b# = Id. Thus, ®kV and AkV naturally inherit a euclidean structure

V with the inner product e defined by e(x, y) _ (b(x))(y). This inner product on tensors will also be denoted by e - (, ). Suppose e1,. .. , e is an orthogonal basis for V with the sign o-j =_ (ej, ej) either ±1. Then, for example,

{ei Aej : i < j} is an orthonormal basis for A2V with (ei Aej,ei Aej) = aio-j, and more generally, (4.3)

lei,

:il < ...
is an orthonormal basis for APV.

Note: If dim V = 2, then the one-dimensional space A2V is positive in both the case V positive and the case V negative, while A2V is negative if V has signature 1, 1.

Euclidean/Lorentzian Vector Spaces

59

Theorem 4.4 (Cauchy-Schwarz Equality). Suppose V is a euclidean vector space of signature p, q. For all x, y E V, (x, y)2 + (x A y, x A y) = IIxII IIyII-

First Proof: The idea of this proof is that since the theorem only involves two vectors it is really a two-dimensional result. Let W denote the span

of x and y. If x and y are colinear, the proof is immediate. Thus, we may assume that W is a plane. Because of the Basis Theorem and its extension to the degenerate cases (Problem 2.7), the (possibly degenerate) inner product space W is isometric to one of the standard models:

R(2, 0), R(1, 1), R(0, 2) if W is nondegenerate,

(4.6) or

R(1, 0) x R, R(0,1) x R, R x R if W is degenerate.

(4.7)

Finally, it is a simple matter to verify directly (4.5) in each of these six cases.

For example, if x, y E R(1, 1), then

(x, y)2 = (xiy1 - x2y2)2 = xiyi + x2 y2 - 2xlyix2y2, (x A y, x A y) _ -(xiy2 - x2yi)2 = -xiy22 - x22 y1 + 2xiyix2y2,

and

IIxII IIyll = (xi - x2)

(Y2

- 14) = x2y2 +x2y2 -x2y2 -x2y2

Li

Second Proof: (x A y, x A y) = (b(x A y))(x A y) = (b(x) A b(y))(x A y) = (b(x)(x))(b(y)(y)) - (b(x)(y))(b(y)(x)) = (z,x)(y,y) - (x, y)2, where each of these equalities is an application of a definition. J

Third Proof: First, we verify the algebraic identity (Lagrange):

E

2

(4.8)

/ z=wi

`:_i

{

i<j

(ziwj

- zjwi)2 =

(2 z2

i=1

I

E WJ j_i

The Cauchy-Schwarz Equality

60

for all z, w E Cn, by noting that (ziwj - Zjwi)2 = i<j

=

n

2

(ziwj - wizj)2

9=1 n

2

E (ziwj2 -2ziwjzjwi+z?w;) i,j=1

n

= i-1

n

n

i=1

i_1

z; wj2 -

2

ziwi

Now, substituting

xj forzj and yi for wj

ifj=1,...,p,

and

ixj for zj and iyj for wj ifj = p+ 1, ..., n,

with xi, yi real, yields the C.-S. equality for R(p, q). Since V, (,) is isometric to R(p, q), this completes the proof. Proposition 4.9. Suppose x, y E V and W = span{x, y} is a two-dimensional plane. The plane W is degenerate if and only if (z A y, x A y) = 0 but x A y # 0. If W is nondegenerate, then W is either positive, negative, or Lorentzian.

Positive/Negative: The following are equivalent: (4.10)

(i)

W is a positive or a negative plane,

(ii)

A2 W is positive,

(iii)

(xAy,xAy) > 0,

(iv)

(x,y)2 < IIzII IIyII

Lorentzian: The following are equivalent: (4.11)

W is a Lorentzian plane,

(i) (ii) (iii)

A2 W is negative, (x A y, x A y) < 0,

(iv)

(x, y)2 > IIxII Ilyll

Euclidean/Lorentzian Vector Spaces

61

Proof: The fact that (i)-(iv) are all equivalent, in either (4.10) (positive/negative) or (4.11) (Lorentzian), follows from the note before Theorem 4.4 and Theorem 4.4 (C.-S. Equality). If W is a degenerate plane, then there exists a nonzero vector u E W

with (u, z) = 0 for all z E W. Pick v E W with u A v = x A y. Then, by the C.-S. Equality, (x A y, x A y) = (u A v, u A v) = (u, v)2 - IluJI Mvii = 0. Conversely, if x A y # 0 but (x A y, x A y) = 0, then W must be a degenerate plane, since W cannot have dimension < 2 (x A y i4 0) and W cannot be a positive, negative, or Lorentzian plane. _1

Corollary 4.12 (Cauchy-Schwarz Inequality). If V is positive euclidean, then

(u,v)<JulJvi forallu,vEV, with equality if and only if one of the vectors is a positive multiple of the other.

Remark. Since -Jul lvJ < (u, v) < Jul lvJ, (4.13)

cos 0 - (u, v)

Jul Ivi

uniquely defines an angle 0 between any two non-zero vectors u and v with 0 < 0 < ir. Note that sin 0 = lu A vi/luJ JvJ.

A curve M in a euclidean vector space V is, by definition, a onedimensional oriented submanifold with boundary. Since (by definition) a curve is oriented, each point on a curve possesses a well-defined tangent ray (or half-line) and, if this tangent line is non-null, a well-defined unit tangent vector. The length of a curve M is defined to be (4.14)

1(M)

Jx'(t)J dt,

where x : [a,#] -* M C V parameterizes M. The initial point a = x(a) and the terminal point b - x(3) form the boundary of M. If the tangent line to M is never null, then the parameterization can be chosen so that ix'(s)I = 1 and is called parameterization by arclength. The length of the line segment from a to b is I b - aI.

Proposition 4.15. Suppose V is positive definite. Ib - al < 1(N)

Special Relativity

62

for any curve N with initial point a and terminal point b. Equality occurs if and only if N = the line segment from a to b. The proof of this result is fundamental in the theory of "calibrations" (see Chapter 7). Proof: Let M denote the line segment from a to b. Let M denote the unit

tangent vector to M (i.e., M= b - a/lb - al). Let 0 = b(M) denote the corresponding 1-form tangent to M. Then lb - aI = 1(M) = fm 0, which equals JN 0 by the fundamental theorem of calculus for paths since do = 0.

Choose the parameterization x(s) of N by arclength. Then

f = j(Mz'(s)) N

By the C.-S. Inequality, this is < fo ds = 1(N).

Li

SPECIAL RELATIVITY The terminology and intuition provided by special relativity is very helpful in understanding Lorentzian vector spaces, and, in particular, in understanding dimension four (Minkowski space).

A Lorentzian vector space V (signature n - 1, 1) is said to be time oriented if one of the two components of the timecone {v E V : (v, v) < 01 is designated as the future timecone. Vectors in the future timecone are said to be future timelike vectors.

Corollary 4.16 (Backwards Cauchy-Schwarz Inequality). Suppose V is a time-oriented Lorentzian vector space. If u and v are future timelike vectors, then

-(u, v) > lul M, with equality if and only if u and v are positive multiples of each other.

Remark. For u, v future timelike (4.17)

cosh 0 =

-(u'v)

lul IvI

uniquely determines an angle 0 < 0 < +oo between u and v. Note that sinh 0 = juAvj/jul Ivj. Suppose V is a time-oriented Lorentzian vector space. A curve M in V is called a worldline of a particle if its tangent is future timelike at each

.Euclidean/Lorentzian Vector Spaces

63

point. The arclength parameter is called proper time and usually denoted -r. The length of M is called the proper time of the worldline or particle. If the worldline M is a line segment, then the particle is said to be in free fall. Of course, the proper time of a particle in free fall from a to b is lb - al.

Figure 4.18

Proposition 4.19 (Twin Paradox). The proper time of a particle is maximized by free fall.

Proof: See Problem 3.

LJ

Special Relativity

64

Corollary 4.20 (Backwards Triangle Inequality). If u and v are future timelike vectors, then

Iu+vI >_ Iul+ Ivh with equality if and only if u and v are positive multiples of one another.

The null lines in a Lorentzian vector space V are called light rays (or

the worldline of a photon). Thus, the union of all the light rays is the lightcone (or nullcone) {x E V : IxII = 0}.

In Minkowski space R(3, 1), the lightcone is given by {(x, t) E R(3, 1) : IIxII = t2}.

The speed of light has been normalized to be one. One of Einstein's axioms of special relativity, referred to as the Constancy of Light, states that If a linear coordinate change A E GL(R(3, 1)) on Minkowski space is physically admissible (or provides a new inertial coordinate system), then A must preserve the set of light rays. Any invertible linear map A on V, II ' II determines a new square norm by II for all z E V. IIxII=IIAzil If A preserves the nullcone for II II, then II - II' has the same nullcone as -

II Under this hypothesis, the next theorem states that II II' and II II differ by a constant c. That is, A must preserve II . II up to a constant. In II

-

-

summary, the next theorem has as a consequence: Einstein's axiom, the Constancy of Light, implies that each inertial coordinate system on R(3, 1) is obtained from the standard coordinate system by a coordinate change A that preserves II . II up to a constant. If additionally A preserves futuretime, then it is easy to show that the constant is positive, i.e., A must be conformally Lorentzian, A E CO(3, 1). (An additional axiom, called the Axiom of Relativity, implies that the constant is 1; that is, A E SOT(3, 1) the Lorentz group.)

Lemma 4.21. Suppose V, II . II has signature p, q with p, q > 1 and II II' is another quadratic form on V. If each null vector for II II is also null for 11', then for some constant c, 11

IIxII' = c11x11-

Euclidean/Lorenizian Vector Spaces

65

Proof: First assume that V has dimension 2. Then V, II II is isometric to the Lorentz plane R(1, 1). Let R ® R denote R2 with the square norm IIxII = xlx2. Then R(1, 1) and RED R are isometric. Using the model R ® R, the hypothesis that II II' vanishes on the null cone for II . II, says that IIxII' = Ax2+Bxlx2+Cx2 must vanish when xi = 0 or x2 = 0. Thus -

-

IIxII'=BIIxII. Now assume V has dimension > 2. It suffices to prove that for any pair of vectors u, v with one spacelike and one timelike the ratios IIuII'/IIuII and IIuII'/IIuII are the same. Repeated use of this will yield IIxII' = cIIxII

for all x. Since span{u, v} is a Lorentz plane, there exists a constant c (depending on u and v) such that IIxII' = cJIxII

for all x E span{u, v}.

A future timelike vector might also be called the instantaneous world-

line of a particle. This particle can also be thought of as an "observer." It is useful to define an observer to be a curve whose unit tangent is always future timelike. That is, "observer" and "worldline of a particle" are mathematically identical concepts. Finally, an instantaneous observer is another name for a (unit) future timelike vector. To say that an instantaneous observer u is provided or given should be thought of as the same as to say that an orthogonal decomposition of Minkowski space is provded or given. Namely,

R(3,1) = S ® span u, where S =_ (span u)1 is a spacelike 3-plane referred to as rest space. Armed

with this expanded definition of an instantaneous observer, it is usually possible to guess the classical concepts that are "observed" or "deduced" from a relativistic concept. Two examples are discussed. For both, suppose an instantaneous observer u is given at a point z in Minkowski space with associated rest space S. In the first example, suppose M is the worldline of a particle. Let M denote the unit (future timelike) tangent vector to M at the point z. H H Then M has the orthogonal decomposition M = to + to, and the observed velocity of the particle is v = w/t E S. As a second example, consider the relativistic concept of an electromagnetic field on Minkowski space. By definition, this is just a two-form F on R(3, 1); that is, for each point z E R(3, 1), F(z) E A2R(3,1). Now, given an instantaneous observer u at z E R(3, 1), what can be "observed"? The vector field E that is metrically equivalent to the one-form u L F is called the (observed) electric field.

Special Relativity

66

Recall that if a euclidean space V, (,) is oriented, then there exists a unique unit volume element A E An V defined by A = el A ... A e where e 1, ..., e is an oriented orthonormal basis for V. Moreover, the equation a A (*/3) = (a, /3)A

defines a linear isomorphism *: A'kV --+ A"-kV

called the Hodge * operator.

Now consider the unit volume element A = dx' A dx2 A dx3 A dt along with the standard inner product (, ) on R(3, 1). Note that * maps A211(3, 1) isomorphically on A2R(3, 1). Let b(u) denote the one-form metrically equivalent to u. The vector field B that is metrically equivalent to *(F Ab(u)) is called the (observed) magnetic field. If the instantaneous

observer u = e4 is taken to be the standard unit timelike basis vector in R(3, 1), then F = B3dx1 A dx2 + B1 dx2 A dx3 + B2dx3 A dx'

+E'dx1Adt+E2dx2Adt+E3dx3Adt. The classical Maxwell equations are most succinctly written

dF = 0

and

d(*F) = 0.

We conclude this section with a sketch of the proof of the isomorphism between the Lorentz group SO1(3,1) and the special complex orthogonal group SO(3, C). First note that SOT(3, 1) can be defined as the subgroup of SO(3, 1), which preserves the future timecone. In particular, -10 SO1(3,1). Later, the alternate description of SO1(3, 1), as the connected component of the identity in SO(3, 1), is used. The standard action of SO1(3,1) on R(3, 1) induces an action of SOT (3, 1) on A2R(3,1) by the usual "pull back" map on forms. The following facts are straightforward to verify. The square, *2, of the star operator * on A2R(3,1) is minus the identity. Consequently, * provides A2R(3, 1)

with a complex structure J - *. The real valued symmetric bilinear form *(aAQ) on A'R(3,1) is nondegenerate, and J is an anti-isometry with respect to this bilinear form. Consequently, e(a, 6) = *(a A /3) + i * ((*a) A fl)

is a C-symmetric inner product on the complex vector space A2R(3, 1), J *. Since SO1(3,1) commutes with J =_ *, we obtain a group homomorphism

67

Euclidean/Lorenizian Vector Spaces from SOT(3, 1) to SO(3, C), the subgroup of GLc(AZR(3,1)) (where J that fixes E.

THE CARTAN-DIEUDONNE THEOREM Given u E V nonnull, then ul is a nondegenerate hyperplane and by Corollary 2.40, each vector x E V has a unique orthogonal decomposition z w+Au with WE u1 and A ER.

w - Au is called the reflection through (the hyperplane) ul. or reflection along u. That is, R is the identity on ul and minus the identity on the line through u. Therefore,

Definition 4.22. The map Ru defined by

&x

-x- 2 (x, u) Ilull

Note that R. E O(V ), det R _ -1, and R,,', = I. By definition, the identity I E O(V) is said to be the product of r = 0 reflections along lines.

Theorem 4.23 (Cartan-Dieudonne). (a) (Weak Form) Each orthogonal transformation A E O(V) can be expressed as the product of a finite number r of reflections (4.24)

A=R,,o...oR,,,.

along none ull lines, so that I Iui 11 : 0 for all j = I,-, r. (b) (Strong Form) The number of reflections required is at most n; that is, r < n. (c) (Sherk Version) First note that the fixed set (or axis of revolution)

FA =- {x E V : Ax = x} = ker(A - I) is always orthogonal to image(A - I). Now suppose that (4.24) is any representation of A (as the product of r reflections) with r minimal. Then

(1) If A - I is not skew, then r = n - dim FA. In fact, ul, ... , ur is a basis for image(A - I). (2) If A - I is skew, then r = n - dim FA + 2, but FA must be totally null so dim FA < n/2. This case cannot occur if V is positive or negative definite, or positive or negative Lorentzian.

Remark 4.25. The part of this theorem that will be used later to study the Spin and Pin groups is simply the weak form: each A E O(V) can be expressed as a (finite) product of reflections. Consequently, some readers may wish to skip the proof of parts (b) and (c).

The Cartan-Dieudonne Theorem

68

The proof of the theorem begins with the case of dimension n = 2. As a prelude to Chapter 6 on normed algebras, this case provides us with an excuse to discuss the complex numbers C and the less familiar Lorentz numbers L. In order to emphasize the analogy between C and L, a brief discussion of the well-known complex numbers is included (for later comparison with L).

The Complex Numbers C The complex numbers C are defined to be R(2, 0) with the extra structure of multiplication, given by (a, b)(c, d) _ (ac - bd, ad + bc).

Let I =- (1, 0) and i (0,1), so that (a, b) = a + bi and i2 = -1. Conjugation is defined by 7 = a - ib for z = a -{- ib. Note that zw = T W, zz = 1IzI and hence JJzwJJ = JJzJi JJwJJ. If z # 0, then z-1 = x/JIzJJ, so that C is a (commutative) field. Also, note that (z, w) = Re zw

(zw -{- 7w)

.

Let eie = cos 0 + i sin 0 denote a point on the unit circle and note that Mere, multiplication by e19, is an orthogonal transformation since ese As a 2 x 2 real matrix, Me.e = (cosO

`sin 0

-sing) cos 0 J

so that M. E SO(2). Since Meiseio = Mei(e+o), the map 0 r-, Meie induces the group isomorphism,

R - SO(2). 27rZ Lemma 4.26.. Suppose A E 0(2) and define 0 by Al = e'B. If det A = 1, then

(4.27)

Rie;e/2 o A = Ri,

where Ri (reflection along i) is just conjugation Cz then (4.28)

Rieie/2 o A = Id.

= z. If det A = -1,

Euclidean/Lorentzian Vector Spaces

69

Proof: It suffices to show that Riei0/2 o A fixes 1, because Ri and Id are the only two orthogonal transformations fixing 1. The reflection of eie = Al along ieie/2 (or through span eie/2) is 1 since both unit vectors eie and 1 have the same inner product with ei9/2, namely, cos 0/2. J

Corollary 4.29. Suppose A E SO(2). Then A = Rie0/2 o Ri

(4.30)

is the product of two reflections. Moreover, A = Mere .

Suppose A E 0(2) and det A = -1. Then (4.31)

A = Rieie/2

is a reflection. Also, A = C o M,-;e.

Proof: Solving (4.27) and (4.28) for A, we obtain A as the product of reflections proving the first equalities in (4.30) and (4.31). To prove A = Mete, note that both satisfy det = 1 and map 1 to eie so that both satisfy (4.27). The proof of A = C o Me_;e is similar. J

The Lorentz Numbers L The Lorentz numbers L are defined to be the inner product space R(1,1), (, ) with the extra structure of multiplication given by (a,b)(c,d) _ (ac+bd,ad-{-bc).

Let 1 = (1, 0) and r =- (0,1), so that (a, b) = a+br and r2 = 1. Conjugation is defined by z = a - br for z = a + br. Note that FT = T U Y and z z = IIzII, so that IlzwJJ = IIzII JIwII. Thus, if IIzII :0 (z nonnull), then z-1 = x/IlzlI exists, while for IIzII = 0 (z null) z cannot have an inverse. Also note that (z, w) = Re zw = (zw + zw). Let ere = cosh2 B + r sinh B (calculate the formula power series for eTe to see that this definition is appropriate). Note that Me,e, multiplication by ere, is an orthogonal transformation since IIeT°II = 1. As a 2 x 2 real matrix, Mere = cosh 0 sink 0 ) sinh 9

cosh 0 J

so that det Mere = 1. Define a timelike vector z = a + br to be future timelike if b > 0. Since Me .e (r) = sinh 0 + r cosh 0, multiplication by

The Cart an-Dieudonn 6 Theorem

70

reserves the futuretime cone. Thus, Mere E SO1(1, 1). In fact, since ere preserves

ere erW = e`(8+) the map sending 0 i- Mere determines the group isomorphism (4.32)

R S--- SOT(l, 1).

-

Note that the set of unit spacelike vectors {z = a + br : IJzIJ = a2 b2 = 1} consists of two disjoint curves {ere 0 E R} and {-ere : 0 E R). Therefore, given A E 0(1, 1), the unit spacelike vector Al uniquely determines an angle 0 by Al = ±ere :

L

~\

Figure 4.33

Lemma 4.34. Suppose A E 0(1, 1) and Al = ±ere defines 0 E R. (a) If Al = ere and det A = 1, then RTer@12 OA = R.

(b) If Al = -ere and det A = 1, then RTere/2 o A = R1.

Euclidean/Lorenizian Vector Spaces

71

(c) If Al = ere and det A = -1, then RTere/2 o A = Id.

(d) If Al = -ere and det A = -1, then RTere/2 o A = -Id.

Proof: The reflection of ere along the line rere/2 (or through the line ere/2) is 1 since both ere and 1 have the same inner product with ere12 (since ere = e-re and (ere , ere/2) = Re eTee- re/2 = (ere/2 , 1)). Thus, (RTere/2 o A)(1) = ±1, where Al = ±er9 has the same ± sign. Since RTere/2 o A is orthogonal, it must also map the 7--axis into itself and hence

be ± Id on the r-axis. The four cases now follow.

Corollary 4.35. Suppose A E 0(1,1). Ifdet A = 1, then A is the product of two reflections. Either A = RTere/2 c Rr

or

A = RTere/2 o R1.

If det A = -1, then A is a reflection. Either A = RTere/2

or A = Rere/a.

This completes the proof of Theorem 4.23 in the special case of dimension two using the complex numbers C and the Lorentz numbers L.

General Proof of Theorem 4.23: The general proof proceeds by induction on the dimension n of V. Given A E O(V), there are cases to consider.

Case 1: A is said to have a nonnull axis of rotation if there exists a nonnull vector x with Ax = x.

Proof of Case 1: In this case, the fixed set FA can be written as FA = N ® H, where N is totally null and H is nondegenerate. Since A is the identity on H, A maps H1 to H-L. By the induction hypothesis, the theorem is true for AIH-L E O(H1). In particular, if (4.24) is valid for AIHI E O(H -), then the same representation is valid for A E 0(V). Note that R,,, reflection along a line u in Hl, naturally extends to reflection along the same line u, but now considered a line in V. Case 2: Suppose u = Av - v and v are nonnull vectors. Then v.

fixes

The Cart an-Die udonne Theorem

72

Proof of Case 2: Note that u = Av - v and to = Av + v are orthogonal since A E O(V). Therefore, Av has the orthogonal decomposition (4.36)

Av = 2 u + 2 w

-1 u + iw = v with respect to V span u ® (span u)-L. Thus, can be expressed as the fixes v. By the induction hypothesis, product of < n-1 reflections. Therefore A can be expressed as the product of < n reflections. Now the proof of the weak form (i.e., each A E O(V) is the product of a finite number of reflections) can be completed. Pick any nonnull vector and hence can be v E V. If u = Av - v is nonnull, then

expressed as a product of a finite number of reflections. If to = Av + v = -((-A)v - v) is nonnull, then R. (-A) Ispaz, vim, and hence Rv, (-A), can b e expressed as a product of a finite number of reflections. Finally, note that both u and w cannot be null, since they are orthogonal and v = - au+ 1w . Before preceeding with Case 3, we need to make several observationsRecall that for any linear map L E End (V), (4.37)

image L = (ker L* )1

and (4.38)

dim image L + dim ker L = n.

Since A* = A-1, the equations Ax - x and A*x = x are equivalent. Thus, ker(A - I) = ker(A* - I). Combined with (4.35) this proves that (4.39)

image(A - I) = FA

for A E O(V).

Note that if A E O(V) does not belong to Case 2, then u - Av - v is null for each nonnull vector v. By continuity, this implies that each u = Av - v is null. Therefore, if A E O(V) does not belong to Case 2, then image(A-I) is totally null. If Case 1 is not applicable, then FA = ker(A- I) is totally null. Case 3: Suppose both image(A - I) and ker(A - I) are totally null.

Proof of Case 3: Since by (4.38) each is the orthogonal compliment of the other, they must be equal, i.e.,

N =- image(A - I) = ker(A - I). By (4.38) this implies that dim V = 2p, where p = dim N.

Euclidean/Lorentzian Vector Spaces

73

Since N is totally null, Proposition 2.45 implies that V has split signature p, p. Note that A is the identity on N. Pick M complimentary to N. Given

Y E M, Ay = (Ay - y) + y is the direct sum decomposition for Ay E V =

N ®M since Ay - y E N. Let b : M - N denote A - I restricted to M. Then V = N® M determines the 2 x 2 blocking of A as (4.40)

A=

(oI b). I

Therefore det A = 1. This proves that if A belongs to Case 3, then detA = 1. Choose any reflection R through a nondegenerate hyperplane. Then det RA = -1. Thus, RA must be in Case 1 or Case 2. Therefore RA =

Rl . Rkk < n. Now det RA = -1 implies k is odd. Since n = 2p is even, we must have k < n - 1. Therefore A = RR, ... Rk with k < n - 1. This completes the proof of what is usually called the Cartan-Dieudonne Theorem, i.e., part (b). Remark. By choosing a null basis for V = N ® M, i.e., n1, ..., np a basis for N and M = span{ml,..., mp} totally null with the planes span{ni, mi} nondegenerate and orthogonal, one can show that A E O(V) belongs to Case 3 if and only if A is of the form (4.40) with b a skew matrix. See Problem 10 for an example of an orthogonal transformation A E O(V) that belongs to Case 3 (dimension of V equals four). Now we complete the proof of the theorem. Since A satisfies part (c) of the theorem if and only AIH1 satisfies part (c), we may assume that the fixed point set FA is totally null. Several statements equivalent to

"L = A - I is skew" are provided in Problem 11. In particular, A - I is skew if and only if image(A - I) C ker(A - K) _- FA. Therefore, because of the assumption that FA is totally null, the cases where A - I is skew fall under Case 3. This case is discussed in Problem 11. Now assume that A -.1 is not skew, FA is totally null, and hence Case

2 is applicable. Suppose j = Lv is nonnull with L = A - I and IIvil # 0. Then fixes v. We must show that, for some choice of u, R,,A - I is not skew: (4.41)

If RA - I is skew then, for all v E V, IILvII(L + L*)(y) = -4(Lv, y) (L*v, y) for all y E V.

Proof of (4.41): Assume u = Lv is nonnull. Note HUM

The Cartan-Dieudonne Theorem

74 Therefore,

L-2(u,A( ))n

RUA - I =

Mull

IIuIi This gives

R A-I=L+2 (v,

(442)

U

(lull

since L*A = (A-' - I)A = I - A = -L. Thus, RUA- I is skew if and only if

(L + L*)(y) + 4jjujj-1(L*v, y) (Lv, y) = 0.

(4.43)

Since (4.41) is valid for all v E with jjLvjj # 0, by continuity it is valid for all v

V.

Note that {v E V : Lv = v} _ {v E V : L*v = V} = FA. Consequently, if RUA is skew, then

Lv null implies Lv = 0,

(4.44)

or each nonzero u E image L is also nonnull. Consequently, the subspace image L = image(A - I) is either positive or negative definite. In this case, the theorem is valid for A. Therefore, we may choose u = Av - v with u and v nonnull so that RUA - I is not skew. It remains to show that (4.45)

rank(RUA - I) < rank(A - I).

Now RUA still fixes FA since u E Im(A-I) = FA,.. In addition, RUA fixes v and v 0 FA. Thus dim dim FA,

which proves (4.45). This completes the proof of Theorem 4.23. 'J

Definition 4.46. Given a non degenerate subspace W of V, reflection along W (or through W1), denoted Rw, is defined to be -1 on W and +1 on Wl. I f u1, ... , uk is a basis for W, then reflection along W will also be denoted by RU,n...AU,,.

Just as O(V) is generated by reflections along lines, SO(V) is generated by reflections along planes.

Euclidean/Lorentzian Vector Spaces

75

Theorem 4.47. If n = dimR V > 3, then each f E SO(V) can be expressed as the product of an even number k < n of reflections along nondegenerate planes: f = Ru1Ao1 0... o RukAvk,

k even and < n

Proof: By the Cartan-Dieudonne Theorem, f = Ru, o . o R, with each uk E V nonnull and with k even. If n = 3, then either k = 0 and f = 1, or, k = 2 and f = Rut o Ru2. Now -Ru = R,AW, where span(v A w) = (span u)1. Thus f = Ru, o Rue = (-R+1,) o (-RT1z) = Rv1AW, o

R.,A.,.

If n > 4, then given a pair of nonnull vectors u1, u2 there exists a nonnull vector v orthogonal to both ul and u2 (see Problem 4.10). Therefore, f = Rut o Rut = R,,, o R o Ru o Ru2 = Ru1Aa o RuaAo as desired.

GRASSMANIANS AND SOt(p,q), THE REDUCED SPECIAL ORTHOGONAL GROUPS Suppose V, (,) is an R-symmetric inner product space with signature p, q that is not definite, i.e., p, q > 1. Let G+(p, V) denote the grassmanian of all oriented p-planes through the origin that are maximally positive. Given A ep E ApV by choosing an oriented 1 E G+(p, V), represent ty =_ el A span 4 denote the p-plane without orthonormal basis for V. Let P orientation. If l E G+(p, V) is given, then each ij E G+(P, V) can be graphed over P = span 4 because 7r, the orthogonal projection onto P along Pl, is injective when restricted to Q = span 77 (ker a nQ = {0}). Consequently, there is a well-defined notion of when , q are compatibly oriented. If el, ..., ep

is an oriented orthonormal basis for 4 and Q = span n is the graph of a linear map A : P - Pl, then either {el + Ae + 1, ... , ep + Aep} or {-el - Ae1, e2 + Ae2,.. . , ep + Aep } is an oriented basis for r). In the first case, we say that 4 and n have the same (spacelike) orientation. Note that in terms of the natural inner product (,) on ApV, f and rt have the same orientation if and only if (l;, 17) is positive and the opposite orientation if and only if (4, 77) is negative. Thus, G+(p, V) is naturally divided into two sets.

76

Grassmanians and The Reduced Special Orthogonal Groups

It is also easy to see that G+(p, V) has two connected components. Shrink the graphing map A to zero. That is, if l; and rl are compatibly span ., then rlt oriented with Q = span r7 the graph of A over P graph to defines a path in G+(p, V) connecting q to e. Similarly, the grassmannian, G_(q, V), of oriented maximal negative subspaces of V has two connected components. Selecting G+ (p, V) as one of the two components of G+(p, V) is called the choice of spacelike orientation, while selecting Gt (q, V) as one of the two components of G_(q, V) is called the choice of timelike orientation (or future orientation). An orthogonal transformation A E 0(V) is said to preserve the space-

like orientation if A preserves the two components of the grassmannian G+(p, V) of oriented maximal positive subspaces. An orthogonal transformation A E O(V) is said to preserve the timelike orientation if A preserves the two components of the grassmannian G_ (q, V) of oriented maximal negative subspaces. In this context, A E O(V) with det A = 1 is said to preserve the total orientation of V.

Definition 4.48. The orthogonal group O(V) in the cases that are not definite (p, q > 1) has subgroups defined by

SO(V) _ {A E O(V) : A preserves the total orientation}, O+(V) _ {A E O(V) : A preserves the spacelike orientation}, (4.51) 0-(V) _ {A E O(V) : A preserves the timelike orientation}. (The notation Ot (V) is also used for 0-(V). The arrow indicates that the direction of time is preserved.) The intersection of any two of these three subgroups is the same group and is denoted by SO1(V) __ {A E O(V) : A preserves the total, spacelike, (4.52) and timelike orientation}. (4.49) (4.50)

O+(V) is called the spacelike reduced orthogonal group, 0-(V) is called the timelike reduced orthogonal group, and SO1(V) is called the reduced special orthogonal group.

Proposition 4.53. 0+(p, q)/SO(p) x O(q) - GI (p, V). (b) O- (p, q)/0(p) x SO(q) - GL (q, V) The straightforward proof is omitted. In summary, the group O(p, q) (with signature that is not definite) has four connected components given by the four possibilities for A E O(V) : A preserves (or reverses) space orientation and A preserves (or reverses) time (a)

orientation.

Euclidean/Lorenizian Vector Spaces

77

Lemma 4.54. Suppose A E O(V) is expressed as the product of reflections

A=Ru,o...oRur.

(a) The number of spacelike vectors in {ul,..., u,.} is even if and only if A E 0+(V). (b) The number of timelike vectors in {ul, ..., u,.} is even if and only if

A E 0-(V). In particular, the parity of the number of reflections along spacelike lines (or along timelike lines) is independent of the representation of A as the product of reflections.

Definition 4.55. Given A E O(V), let IIAIIs denote the parity of the number of reflections along timelike vectors when A is expressed as the product of reflections. Note that

II'11s:0(V)- Z2

(4.56)

is a group homomorphism. The number IIAIIs is called the spinorial norm of A.

The two group homomorphisms det : O(V) --+ Z2 and II I Is : 0(V) --3 Z2 distinguish the four components of O(V ). In addition, each of the three groups SO(V), Ot(V), and 0+(V) 21 -

consist of the identity component SOT(V) plus one additional component given by (4.57)

det =

(4.58) (4.59)

det = -1 and det = -1 and

1

and

for SO(V), for 0-(V) = OT (V), for 0+(V).

PROBLEMS 1. (a) Verify (4.3) directly from the definition of the inner product on A2V.

(b) Show that A2 V has signature (2) + (2 ), pq. In particular, A2 V has signature 3, 3 if V has signature 3, 1 or 1, 3. 2. (A canonical form for 0(n) acting on o(n)) Given a E A2R*(n), find an orthonormal basis w', ..., w" for R*(n) and al > )2 > > h, > 0,

so that a = Alwl A w2 + ... + A,w2r-1 A w2r.

Problems

8

Hint: If the maximum value al of a(x A y) taken over all Ix A yl = 1 is attained at el A e2, then show that a(el A u) = 0 for all u E R(n) with u 1 e2 by considering the function a(el A (cos 0 e2 + sin 0 u)) of 0.

3. Give the proof of the twin paradox, Proposition 4.19. 4. Suppose V is a time-oriented Lorentzian vector space. The backwards triangle inequality says that lu + vl > Jul + Ivl for all future timelike vectors, and equality holds if and only if u and v are positive multiples of one another. Deduce this result (a) algebraically from the backwards C.-S. Inequality; (b) as a special case of the twin paradox (for piecewise smooth curves). Given a, b with b - a future timelike, show that there exist (piecewise smooth) particles from a to b with arbitrarily small proper time. 5. Suppose V, II II is a Lorentzian vector space. If II II' is a square norm

on V with Ilxll' < M for all unit timelike vectors, then show that II

II' = cll . II for some constant c.

6. (a) Show thatzw=zwandthatz'z=zz=llzllforallz,wEL. (b) Show that

SO(1,1)={±MM,e :OER}. (c) Let Cz = z denote conjugation on L. Show that C E 0(1, 1) and that

0(1,1)_{±Me,e :0ER}U{±CMe,e :O E R}. 7. (Double Numbers) Consider the algebra R ® R of diagonal 2 x 2 matrices

ul 0

0

u2)'

with matrix multiplication, and with the inner product determined by the quadratic form u1 u2. Define conjugation by

C (0

vv

)

- (0

uu)

.

Exhibit an isomorphism between L and R e R preserving the multiplication, the inner products, and the conjugations. Exhibit eTa as a 2 x 2 matrix. 8. Suppose V is a euclidean vector space of signature p, q. Recall that the unit sphere is defined to be {u E V : Jul = 1}. Show that if x is a point on the unit sphere, then x is normal to the unit sphere at x.

Euclidean/Lorenizian Vector Spaces

79

9. Consider R(4, 0) = H. (a) Show that reflection along a line spanned by a E H with Ial = 1 is

given by Rax = -aia. (b) Use part (a) to give a proof that X : S3 x S3 -+ SO(4) defined by Xa,b(x) = axb for all x E H, as in Remark 1.33, is a surjection. 10. (a) Exhibit a pair of nonnull vectors u1i u2 E R(2, 1) that do not have a nonnull vector v E R(2, 1) orthogonal to both of them. (b) If n - dim V > 4, show that each pair u1i u2 of nonnull vectors has a nonnull vector v E V orthogonal to both of them. (c) Find A E SO(2, 1) with the fixed axis FA =- {x E R(2, 1) : Ax = x} a null line.

11. Let V - M2(R) with JJvll = detv. Let A E End(V) denote left multiplication by the matrix C

1

11

0

1

(a) Show that A E SO(V). (b) Show that image (A - Id) is totally null of dimension 2. (c) Show that FA = {x E V : Ax = x} is totally null of dimension 2.

(d) Show that A cannot be written as the product of less than four reflections along (nonnull) lines. 12. Given A E O(V), show that the following are equivalent:

(a) A - I is skew, (b)

2(A + A*) = I,

(c) A'1 is the identity on image(A - I), (d) image(A - I) C FA - ker(A - I). (e) FA C FA. 13. Suppose A, B E O(V) both leave the subspace F C V fixed and agree

on F.

(a) Show that A = B if V is positive definite or Lorentzian. (b) Find a counterexample to A = B if V = R(2,2). 14. Suppose V, (, ) is positive definite euclidean space of dimension n. Given a subspace W of V, let Rw denote reflection along W through Wl and let Pw denote orthogonal projection from V onto W. Show

that Pw = 2 (I - Rw) or Rw = I - 2Pm. 15. Let G(k, V) denote the grassmanian of (not necessarily oriented) kplanes in V through the origin. (a) Show that G(k, R"), the grassmannian of non-oriented k-planes in R', can be identified with a subset (actually a submanifold) of the

Problems

80

vector space of self-adjoint linear maps Sym(R') _ {A E EndR,(R") A' = A}. More precisely, identify G(k, R") with

{ A E Sym(R') : A2 = A and traces A = k)

.

(b) Similarly, identify G(k, C"), the grassmannian of complex k-planes in C", with

{A E Herm(C") : A2 = A and traces A = k} . (c) Identify the grassmannian of quaternionic k-planes in H", denoted G(k, H") with the set of projection operators

{ A E Herm(H") : A2 = A and trace, A = 4k).

In the special case k = 1, G(k, F") is called projective space and denoted P"(F) _ G(1, F")-see Chapter 5 and Problem 6.16 for more information. 16. Suppose V, II . II has split signature p, p. Given N, totally null of di-

mension p, show that there exist M complimentary to N, also totally null of dimension p, and an isometry from V to RP x RP, with square +xpyp, which sends N to RP x {0} and M norm II(x, y) 11 = x1y1 + to {0} x RP. 17. Show that each A E O(n, C) can be expressed as the product A=

o . . . o R,,,,

with r < n,

where each uj is nonnull. The complex reflection Ru along a nonnull line u is defined by exactly the same formula as in the real case.

5. Differential Geometry

This chapter provides a brief description of several types of geometries. Its purpose is to add motivation for the other chapters. As such, it may be either browsed over quickly or studied carefully in conjunction with some of the standard texts in geometry. The chapter should only be considered as introductory and motivational, despite the fact that the reader is assumed to be familiar with the notion of a manifold. While the novice in geometry may wish to use this chapter as a reference point for further study, the more experienced student may be disappointed in its introductory nature, especially after reading the following table of geometries. Each of the geometries discussed is labeled with a particular group (see Table 5.1). However, a particular group may be associated with one, two, or even three different but closely related geometries. The connection between

the group and the geometry is meant only to be suggestive. A deeper relationship is provided by the notions of principle G-bundles, reduction of the structure group, and integrability. The reader is encouraged to pursue these topics.

REAL n-MANIFOLDS: THE GROUP GL(n, R) Briefly, for the sake of completeness, the general notion of a (smooth) real n-dimensional manifold M is defined as follows. 81

Oriented Real n-Manifolds

82

Definition 5.2. M must be a topological space equipped with a countable atlas that is smoothly compatible. An atlas is a collection of charts, where

in turn each chart consists of two things: an open subset of M and a homeomorphism of this open set with an open subset of R". Of course,

each point of M is required to belong to at least one chart. Two overlapping

charts induce a homeomorphism from one open set in R' onto another, called a transition function. The atlas is said to be smoothly compatible if each transition function is a diffeomorphism. This completes the definition of a real n-manifold. The group GL(n, R) is associated with this geometry, since the linearization, at a point, of each transition function belongs to GL(n, R). In later discussions in this chapter, the reader is assumed to be familiar with the notion of the tangent space

TyMtoMatapoint xEM.

Table 5.1 Manifold

Group

Real n-manifolds GL(n, R) Oriented real n-manifolds GL+(n, R) Complex (and almost-complex) n-manifolds GL(n, C) Quaternionic (and almost-quaternionic) n-manifolds GL(n, H) H* Manifolds with volume SL(n, R) and SL(n, C) Riemnnian manifolds (of signature p, q) O(p, q) Conformal manifolds (of signature p, q) CO(p, q) Real symplectic manifolds Sp(n, R) Complex Riemannian n-manifolds O(n, C) Complex symplectic manifolds Sp(n, C) Kahler manifolds (of signature p, q) U(p, q) Special Kahler manifolds (signature p, q) SU(p, q) HyperKahler manifolds (of signature p, q) HU(p, q) Quaternionic Kahler manifolds (of signature p, q) HU(p, q) HU(1) Quaternionic skew hermitian manifolds Sk(H)

ORIENTED REAL n-MANIFOLDS: THE GROUP GL+(n, R) Suppose M is a real n-manifold equipped with an atlas so that each transition function preserves orientation; that is, the linearization of each transition function, at each point, belongs to the subgroup GL+(n, R), consisting of all invertible matrices with positive determinant. Then M is called an

Differential Geometry

83

oriented real n-manifold. Note that each tangent space TxM carries a natural orientation as follows. Choose any chart with open set containing x. Then the associated homeomorphism between this open set and an open subset of R" induces an orientation on TxM, and this orientation on TxM is independent of the choice of chart since the transition functions have derivatives in GL+(n, R). An oriented real n-manifold may also be (equivalently) defined as follows. Assume M is a real n-manifold. Suppose there exists a nevervanishing n-form 0 on M (but Q is not necessarily distinguished), then M is said to be orientable. In this case, any other never-vanishing n-form S2' is a multiple f of S2 where f is a never-vanishing function. If f is positive, then we say 52' and S2 are equivalent. Thus, on an orientable manifold there exist precisely two equivalence classes of never-vanishing n-forms. If a choice of one of these two equivalence classes is made, then M is called an oriented manifold.

COMPLEX (AND ALMOST COMPLEX) n-MANIFOLDS: THE GROUP GL(n, C) The general notion of a complex n-manifold is defined by mimicking the definition of a real n-manifold, except R° is replaced by C' and the transition functions are required to be biholomorphisms (from an open subset of C" to an open subset of C'). There are several useful, and of course equivalent, definitions of a holomorphic map. The simplest, from our point of view, requires that the linearization or derivative of the map, at each point, should be complex linear. Thus, a complex n-manifold M is, first of all, a real 2n-dimensional manifold, but with the additional property that

each transition function (from an open subset of C" - R2" to another open subset of C" = R2n) has its derivative (_ differential) in the subgroup GL(n, C) of GL(2n, R). The obvious choice is to label the geometry of all complex n-manifolds with the, group GL(n, C). On a complex vector space T, complex scalar multiplication by i is a real linear map (denoted I) that squares to minus the identity. Conversely, given any real linear map I (on a real vector space) that squares to minus the identity, it is easy to convert the real vector space to a complex vector space by setting the given linear map I equal to multiplication by i. Such a map I is called a complex structure. Now, on a complex manifold M each

tangent space TM (considering M as a real 2n-manifold) is naturally equipped with a complex structure due to the fact that the linearization of a transition function commutes with the complex structure on C".

84

Complex (and Almost Complex) n-Manifolds

An almost-complex n-manifold is a real 2n-manifold that is equipped with a complex structure I on each tangent space TrM (that varies smoothly with the point x E M). The differential forms (with complex coefficients) on an almost complex manifold are particularly interesting. The space of degree one forms (at a point) is naturally the direct sum of two vector spaces: the space T*1,0 of forms of type 1, 0 and T*O,l the space of forms of type 0, 1 (see Problem 3). More generally the space of degree k forms, denoted AkT*, is the direct sum EP+q-k ®AP,QT*, where AP,9T* is the space of forms of bidegree p,q. There are convenient nontrivial ways of testing whether a given almost complex n-manifold is a complex n-manifold, which are provided by the "Newlander-Nirenberg Theorem." There is a further weakening of the concept of an almost-complex nmanifold that is not of much interest itself but does provide a basis for a useful quaternion analogue. A "weak" almost-complex n-manifold is a real 2n-manifold that is equipped with two complex structures dI, one the negative of the other, on each tangent space TrM (and they should locally vary smoothly with x). The point is (see Problem 2) that there may not be any way globally to make a choice of I versus -I. Remark 5.3. For the reader comfortable with the notion of a vector bundle on a manifold, the distinction between almost-complex and the "weaker form" of almost-complex can be described as follows: Let End(TM) denote the endomorphism vector bundle on M, i.e., the vector bundle with fiber End(TrM) at each point x E M. A two-dimensional subbundle C of End(TM) is said to be a bundle of C-algebras on M if, for each point x, the fiber Cx is real algebra isomorphic to C. Now a "weak" almost-complex n-manifold is a real 2n-manifold equipped with a subbundle C C End(TM) of C-algebras. Since z --*Y is a real algebra automorphism of C sending i to -i and the isomorphism Cr C is not canonical, the set of operators

{I, -I} C C., corresponding to {i, -i} C C is distinguished, but not the individual operators I and -I. If, in addition, a global choice of I is possible, then by identifying I E Cz with i E C (and by identifying the identity in each Cr with 1 E C) we have a global trivialization of the bundle C. Thus, we have the following equivalent definition of an almost complex manifold.

Definition 5.4. An almost-complex n-manifold is a real 2n-manifold equipped with a trivial subbundle M x C C End(TM) of C-algebras.

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85

QUATERNIONIC (AND ALMOST-QUATERNIONIC) n-MANIFOLDS: THE GROUP GL(n, H) H* Now the situation is more delicate, and the concept that is analogous in a straightforward manner to the concept of a complex n-manifold is barren, i.e., there are essentially no examples. The example of quaternionic projective space Pn(H) provides some guidance. Let [x] denote the quaternionic line [x] =_ {xA A E H} through the origin and the point x = (xo, ..., xn) E Hn+1 - {0}. By definition, P"(H) E {[x] : x E Hn}1 - {0}}. The standard atlas for P'(H) consists of n + 1 charts Uk, Ok, k = 0,.. ., n, defined as follows. Let Uk = {[x] E Pn(H) : xk # 0},1,and define , 1, maps 4k : Uk --+ Hn by sending [x] E Uk to Ok([x]) = (xoxk functions are all of the same form. xnxk 1) E H'. Now the transition : 0o(Uo) --> 01(U1) is given by For example, the transition function 1,. 1, 1). The linearization 0'(y) (or deriv. ., ynyl ti(yl, yn) = (yl 2y1 ative) of z/i at y sends u E Hn to :

,

(Y1 lulyl

u'2y11.... ,yny11th y11 -unyl1).

In particular, 0'(y) __ Ay o Rye where Ayu = -O(y)ul + (0, u2, ..., un) is H-linear and Rye u = uyi 1 is simply right multiplication by yi 1

In summary, the linearizations ik'(y) belong to GL(n, H) H* but not GL(n, H). Thus, P'(H) is a "quaternionic n-manifold" in the sense that the transition functions for the standard atlas have linearizations in the enhanced quaternionic linear group GL(n, H) H*. Unfortunately, even adopting this more general notion of "quaternionic n-manifold," it follows that each such manifold is locally equivalent to P"(H) (see Besse [3], p. 411). Consequently, it is most useful to reserve the terminology "quaternionic n-manifold" for yet another notion due to Salamon. Since

the precise definition involves the concept of a torsion-free connection, we only state the definition and refer the reader to the literature (see Salamon [16] or Besse [3]).

A quaternionic n-manifold M is an almost-quaternionic n-manifold

whose GL(n, H).H* reduced frame bundle admits a torsion-free connection. If n = 1, this condition is automatic. The useful notion of "almost-quaternionic" is simpler to define. First consider Pn(H) again. Suppose a point [x] E Pn(H) is fixed. Right multiplication on the tangent space T1x1Pn(H) by a quaternion A can be defined

in several ways using different charts. Using Uk, 4, it is natural to define uA by representing the tangent vector u E Tx3Pn(H) as a vector in Ok(Uk) C Hn and multiplying by A on the right. Let this multiplication

Manifolds with Volume

86

on T1t1P"(H), for x E Uk, be denoted by R. Then (5.5)

Ra = Rk

3atkSJ 1

(see Problem 5). Therefore, right multiplication by a scalar A E H is not well-defined on T[ ]P"(H). However, the vector space H C End(T[ ]M) defined by H = {Ra E End(T[,-]M) : A E H} for each k with xk $ 0 is independent of k and (non-canonically) isomorphic as a real algebra to

the algebra H, since A i aAa-1, with a = xjxk1 fixed, is an algebra automorphism of H. This example makes it clear, since a productive definition of "almost-

quaternionic n-manifold" should include P"(H), that one should elect to utilize the quaternionic analogue of the notion of "weak" almost-complex. Definition 5.6. An almost-quaternionic n-manifold M is a real 4n-manifold equipped with a subbundle C of End(TM) with C5 = H (as real algebras) for each point x E M. The subbundle C is called the coefficient bundle or the almost-quaternionic structure bundle. Suppose this subbundle C is trivial and M is equipped with a particular trivilization (a global bundle isomorphism)

C=MxH

so that operators I, J, K equal to right multiplication by i, j, k E H are globally defined. Then M will be referred to as an almost- quaternionic manifold with trivialized coefficient bundle. This stronger motion of an almost-quaternionic manifold with trivialized coefficient bundle is the exact analogue of almost-complex.

MANIFOLDS WITH VOLUME: THE GROUPS SL(n, R) AND SL(n, C) A real n-manifold M equipped with a never-vanishing n-form S2 is called a (real) manifold with volume. Of course, such a manifold is automatically oriented. A interesting result of Moser says that if a compact n-manifold M is a manifold with volume in two different ways (i.e., two different nevervanishing n-forms Q and ' are prescribed), then M, 0 and M, SZ' are equiv-

alent (i.e., there exists a diffeomorphism f of M with f' (Q') = S2) if and only if the l volume of M is the same as the S2' volume of M' (see Problem 10).

Similarly, a complex n-manifold M equipped with a never-vanishing n, 0-form Sl is called a complex manifold with (complex) volume. If, in addition, 0 is closed under exterior differentiation d, then M is said to be a complex manifold with trivialized canonical bundle.

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87

Remark 5.7. If an almost-complex manifold M of complex dimension n is equipped with a never-vanishing d-closed form SZ of bidegree n, 0, then the hypothesis of the Newlander-Nirenberg Theorem can be verified so that M must, in fact, be a complex manifold. (And hence S2 must be a holomorphic n-form.)

RIEMANNIAN MANIFOLDS (OF SIGNATURE p, q): THE GROUP O(p, q) A Riemannian manifold M of signature p, q is a real n-manifold (with n - p+q) equipped with a real symmetric inner product of signature p, q on each tangent space TTM (and varying smoothly with the point x E M). In particular, this includes the usual concept of Riemannian manifold (q = 0) and the concept of Lorentzian manifold (q = 1) as particular cases. Refer to Chapter 4 for additional information.

CONFORMAL MANIFOLDS (OF SIGNATURE p, q): THE GROUP CO(p, q) A conformal manifold of signature p, q is a real n-manifold with n = p + q equipped with a (smoothly varying) ray [g] of R-symmetric inner products of signature p, q, where g is a Riemannian inner product and [g] denotes the ray {.\g : A E R+}. The group associated with an oriented conformal manifold is CSO(p, q).

REAL SYMPLECTIC MANIFOLDS: THE GROUP Sp(n, R) We shall give two definitions of a symplectic manifold. A theorem of Dar-

boux states that the two definitions are equivalent. First, a symplectic manifold is a real 2n-manifold -hose atlas has the property that each transition function is a symplectomorphism from an open subset of RZ", w (the standard symplectic vector space) to another such open subset of R`, w. A symplectic map is a map whose linearization at each point belongs to the subgroup Sp(n, R) of GL(2n, R), or equivalently a map that fixes the degree 2 form w under "pull back." Thus, the global degree 2-form it obtained by pulling back the standard symplectic form w on RZ",w (for each chart) is well-defined. Moreover, the 2-form S2 on M has the properties

that (5.8)

At each point x E M, S2 defines a real skew inner product on TTM, the tangent space to M.

88

(5.9)

Complex Symplectic Manifolds

S2 is closed under exterior differentiation, i.e., dQ = 0 on M.

The second definition of a symplectic manifold M requires that M be a real 2n-manifold equipped with a (smoothly varying) real skew inner product 0 on T.M at each point x E M (i.e., a nondegenerate 2-form on M) with dS2 = 0 on M. Symplectic geometry is important for two reasons. First, it provides a framework for classical mechanics. As noted in Remark 2.39 (b), a function h (the Hamiltonian) can be converted to a vector field by first taking the exterior derivative dh and then (using the symplectic form) converting dh to a vector field Vh. Many important flows are the solutions to such Hamiltonian vector fields. The main point is this fact: Another function f is constant on the flow lines of Vh (i.e., a first integral or conserved quantity) if and only if the original Hamiltonian function h is constant in the directions Vj, i.e., Vj(h) = 0. There are far-reaching global consequences. See the beautiful book by Arnold entitled Classical Mechanics [1] for more information. The second reason symplectic geometry is important is hinted at in Problem 8. In particular, symplectic manifolds provide a geometric framework for solving first order nonlinear partial differential equations. As one might suspect, these two reasons for the importance of symplectic geometry are closely related.

COMPLEX RIEMANNIAN MANIFOLDS: THE GROUP O(n, C) A complex Riemannian n-manifold is a complex manifold equipped with a complex symmetric inner product on each tangent space TM (and varying

smoothly with the point x E M). These manifolds arise naturally by "complexifying" a (real analytic) Riemannian manifold.

COMPLEX SYMPLECTIC MANIFOLDS: THE GROUP Sp(n, C) A complex symplectic manifold is a complex 2n-manifold equipped with a complex symplectic inner product o- on each tangent space (and varying smoothly), where the bidegree 2, 0 form o- is required to be closed under exterior differentiation. These manifolds arise naturally by "complexifying" a (real analytic) symplectic manifold. Compact examples occur in algebraic

geometry.

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89

Remark 5.10. (a) On a complex symplectic manifold M, o the 2, 0-form o is automatically 8-closed (i.e., holomorphic) so that M, o is also called a holomorphic symplectic manifold. (b) An almost-complex symplectic manifold M, v is automatically a complex manifold because of Remark 5.7: since an is d-closed, nevervanishing and of bidegree n, 0.

KA.HLER MANIFOLDS (OF SIGNATURE p,q): THE GROUP U(p, q) A Kahler manifold M is a complex n-manifold equipped with a complex hermitian inner product h on each tangent space TrM (and varying smoothly), with the property that the 2-form (5.11)

w = -Im h (the Kahler form) is d-closed.

Note that g=_Reh

(5.12)

is a Riemannian structure with signature 2p, 2q on M. The hermitian structure is given by (5.13)

h = g- iw.

Given a Riemannian structure g, a real symplectic structure w, and an almost-complex structure J on a manifold M: (5.14)

w and J are compatible if J is an w isometry,

(5.15)

g and J are compatible if J is a g isometry,

(5.16)

w and g are compatible if the linear operator J defined by w(x, y) = g(Jx, y) is an isometry for either (or both) w and g.

Lemma 5.17. Given any two of the following three structures-(1) a Riemannian structure, (2) a real symplectic structure, and (3) an (almost)

Special Kahler Manifolds (of Signature p, q)

90

complex structure, that are compatible-they determine the third structure. In fact, they determine an (almost) Kahler structure.

Proof: Apply Lemma 2.63. Kahler manifolds have been studied extensively.

SPECIAL KAHLER MANIFOLDS (OF SIGNATURE p,q): THE GROUP SU(p, q) Suppose M is a Kiher manifold with signature p, q and complex dimension n. If, in addition, in a neighborhood of each point, there exists a form a of bidegree n, 0 that is both

(a) d-closed, and (5 18)

(b) of constant size dal = c,

then M is called a special Kahler manifold of signature p, q. Note that do = 0 and ao = 0 (i.e., a holomorphic) are equivalent since there are no forms of bidegree n + 1, 0. If M is a simply connected Kihler manifold, then the definition of special Kahler-can be simplified. In this case, M is a special Kihler manifold if and only if there exists a global holomorphic n, 0 form a of constant size

lo'l = 1. In particular, note that in this case the canonical bundle A"'0 is holomorphically trivialized (by a). Remark 5.19. Suppose M is a Kihler manifold of complex dimension n

and that a is an n, 0 form on M. One can show that either of the two conditions:

(a) a is parallel, (b) the first chern form of A"'° vanishes (i.e., Ricci flat), are equivalent to the condition (5.18) required in the definition of special Kahler manifolds.

Remark 5.20 (Calabi-Yau). A beautiful theorem of Yau provides a means of constructing special Kihler manifolds in the important positive definite case. Suppose M is a (positive definite) compact Kihler manifold with Kahler form w. Assume that the canonical bundle A"'0 is trivial. Let a denote the global, never-vanishing holomorphic n-form (unique up to a constant). Yau proved that there exists a new Kihler form Co that is homologous tow (i.e., w - w = da for some global 1-form a) such that M, w is a special Kahler manifold, i.e., a has constant size with respect to the new Kahler metric based on (Z. (Yau's theorem can be restated in a stronger form if M is not simply connected.)

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Differeniial Geometry

HYPERKAHLER MANIFOLDS (OF SIGNATURE p, q): THE GROUP HU(p, q) There are quite a few useful ways of defining a hyperKahler manifold. The next definition contains in a certain sense the maximal amount of information.

Definition 5.21. A hyperKahler manifold M is a real 4n-manifold equipped with the following extra structure. First, assume that the quaternions H act (smoothly) on the tangent bundle on the right, giving each tangent space TT the structure of a right quaternionic vector space (i.e., M is equipped with an almost quaternionic structure with trivial coefficient bundle, cf. Problem 7). Second, supposes is an H-hermitian inner product of signature p, q on each tangent space (which varies smoothly with

the point x E M). Thus, g = Rer provides M with the structure of a Riemannian manifold of signature 4p, 4q; and each operator R,+ (right multiplication by u) with u E S2 C Im H a unit imaginary quaternion, provides

M with the structure of an almost-complex manifold. Third, require that each associated (Kihler) form w,,, defined by (5.22)

z) - Rer(wu, z) for all w, z E TM,

be closed under exterior differentiation. Fourth, require that the almostcomplex manifold M, R,,, with u E S2 C Im H, be, in fact, a complex manifold. This completes the definition of a hyperKahler manifold.

Remark 5.23. The fourth and final requirement can be proved to be a consequence of the first three requirements, by utilizing Remark 5.10b-see the definition of au given by (5.27) below. Recall from the section in Chapter 2 on "The Parts of an Inner Prod-

uct" that an H-hermitian symmetric inner product s has first and second complex parts h and v defined by (5.24)

e=h+jo.

Further, with respect to the complex structure R;, h is a C-hermitian symmetric inner product and o is a C-skew (complex symplectic) inner product. In particular, (5.25)

h = g + iwl

provides a natural Kahler structure on M, and (5.26)

0'= wJ - iwg

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Quaternionic Kahler Manifolds (of Signature p, q)

provides a complex symplectic structure (with almost-complex structure

I). More generally, for each u E S2 C ImH, h = g + iwu provides a Kahler structure on M (thus the name hyperKahler), while (5.27)

E = hu + uou

defines a complex symplectic structure ou (with complex structure Ru). Also note that (5.28)

c=g+iwi+jwj +kwk.

On a hyperKahler manifold, with the complex structure I chosen, i.e., distinguished, the complex hermitian structure h and the complex symplectic structure o defined by (5.26) are compatible in the sense that (5.29)

h(xJ, y) = o(x, y)

and (5.30)

o(xJ, yJ) = o(x, y).

Conversely, suppose a complex hermitian structure h and a complex symplectic structure o are given. Define a linear map J by requiring (5.29) to be valid. In Chapter 2, it was shown that (5.30) is valid if and only if J2 = -1 (see Equation (2.77)). Now once (5.29) and (5.30) are satisfied one can define c to be h + jo and show that c is H-hermitian symmetric. This proves that the next definition of a hyperKahler manifold is equivalent to the earlier definition. A hyperKahler manifold M is a Kahler manifold equipped with a complex symplectic structure o that is compatible with the hermitian structure

h=g+iw.

QUATERNIONIC KAHLER MANIFOLDS (OF SIGNATURE p, q): THE GROUP HU(p, q) HU(1) A quaternionic Kahler manifold of signature p, q is, by definition,

(a) an almost-quaternionic manifold (with coefficient bundle C), (b) equipped with a Riemannian structure of signature 4p, 4q, with each coefficient u E Cs of unit length an orthogonal map, (c) such that the coefficient bundle C is parallel with respect to the riemannian structure.

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93

Recall by Problem 3.15 (a) that the quaternionic unitary group, IIU(p, q) HU(1) can be defined to be the subgroup of SO(4p, 4q) that fixes H c--- C,.

By choosing a particular algebra isomorphism Cy = H between the fiber of the coefficient bundle C and the quaternions H, we may define the quaternionic 4-form 4D by

FE

6 (W1 -{- w -- wk) ,

where the two forms w,, with u E ImH are defined by

wu(x, y) _ (xu y) for all tangent vectors x, y, using the Riemannian metric (, ). Since any other isomorphism CC = H differs by an element a E HU(1) acting on H by Xa(x) = axa, Problem 3.15 (c) can be used to show that does not depend on the choice of isomorphism Q, c--- H. Thus, the quaternionic 4-form is a well-defined global 4-form on any is a nonzero volume form. quaternionic Kahler manifold. Also note that Moreover, is parallel since the coefficient bundle is parallel. Based on Problem 3.15, one can show that (n > 2) an almost-quaternionic n-manifold equipped with a 4-form 4i, that (at each point) can be expressed as (w; +w? +w,) for some choice of metric (, ), in fact, uniquely s metric. If -1 is parallel, then the manifold is quaternionic determines this Kahler. In fact, for n > 3, the weaker condition d(k = 0 implies that P is parallel (see Salamon [16]). The case n = 2 is unknown. Note: Since HU(p, q) . HU(1) is the quaternionic analogue of the unitary group U(p, q), quaternionic Kahler geometry is the analogue of Kahler geometry. (Similarly, since HU(p, q) is the quaternionic analogue of SU(p, q), hyper-Kahler geometry is the quaternionic analogue of special Kahler geometry.) However, a quaternionic Kahler manifold need not be Kahler manifold.

QUATERNIONIC SKEW HERMITIAN MANIFOLDS: THE GROUP SK(n, H) An H-hermitian skew manifold M is a real 4n-manifold equipped with the following extra structure. First, assume that the quaternions H act (smoothly) on the tangent bundle on the right giving each tangent space T.,M the structure of a right quaternion vector space (cf. Problem 7). Second, suppose c is an H-hermitian skew inner product on each tangent space

94

Coincidences of Geometries in Low Dimensions

(that varies smoothly). Third, require that the 2-form defined by (5.31)

w=Ree

be closed under exterior differentiation. Fourth, and last, require that each of the almost-complex structures R,,, right multiplication by a unit imaginary quaternion, be, in fact, a complex structure on M. For each u E S2 C Im H, (5.32)

g. (z, w) = Re -(zu, w)

for all z, w E TxM

defines a split Riemannian structure on M. In fact, gu-uw is a C-hermitian symmetric inner product on the complex manifold M, Ru with Kahler form w = Re e independent of u. Note that (5.33)

E =w + igt -f jgg +kgk.

Consult the H-hermitian skew case in the section The Parts of an Inner Product in Chapter 2.

COINCIDENCES OF GEOMETRIES IN LOW DIMENSIONS In this section, we discuss the coincidences of certain geometries in low dimensions. These coincidences can be labeled with group coincidences (cf. Proposition 1.40, see Problem 9). Sp(1, R) 25 SL(2, R) and Sp(1, C)

SL(2, C)

Suppose M is a real (respectively complex) manifold of real (complex) dimension 2. Then the notion of a real (complex) symplectic manifold M is exactly the same as the notion of a real (complex) manifold with volume. That is, M must be equipped with a particular never-vanishing 2-form. Suppose M is a complex manifold of complex dimension 2. Then the notion of an almost-complex symplectic manifold (or equivalently a holomorphic symplectic manifold-see Remark 5.10b) is exactly the same as the notion of a complex surface with trivialized canonical bundle. That is, M must be a complex surface equipped with a never-vanishing holomorphic 2, 0-form.

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95

SO(2) - U(1) and CSO(2) - GL(1, C)

Theorem 5.34. (a) The notion of an oriented conformal manifold of real dimension 2 is the same as the notion of a complex manifold of complex dimension 1. (b) The notion of an oriented Riemannian manifold of real dimension 2 is the same as the notion of a Kahler manifold of complex dimension 1.

Proof: First, assume that M is a one-dimensional complex manifold with complex structure J. Then M is oriented (apply (1.26) to the linearization of a holomorphic transition function). Let w denote a choice of orientation preserving volume form. All other choices must be of the form qw where ¢ is a smooth positive function on M. Each skew 2-form w determines a Riemannian structure g by (5.35)

g(x, y) - w(x, Jy) for all tangent vectors x, y.

Since Ow determines fig, the orientation class [w]

{4w : 0 > 0} determines a conformal structure [g] _ log : 0 > 0} on M. If M is Kahler, then M is equipped with a particular w that, by (5.35), determines a particular Riemannian structure g on M. Conversely, assume that M is equipped with an orientation class [w] and a conformal class [g]. Given a choice of Riemannian structure g E [g], renormalize w so that w is a unit volume form. Define a linear map J by (5.36)

w(x, Jy) = g(x, y) for all tangent vectors x, y.

If ¢g is any other representative of the conformal class [g], then ow is the unit volume form (because the dimension is 2) in the metric ¢,g. Thus, ¢g and g determine the same map J. To prove that J is an almost-complex

structure, i.e., j2 = -1, fix g and select an oriented orthonormal basis el, e2. Then w(el, e2) = 1 so that (5.36) implies Jel = e2,Je2 = -el. (Thus, J is counterclockwise rotation by 90°.) Therefore j2 = -1. It is a standard classical result that, for complex dimension one, each almostcomplex manifold is a complex manifold. This result is a special case of the Newlander-Nirenberg Theorem. The reader is referred to the literature. If M is an oriented Riemannian 2-manifold, then the metric g yields the almost-complex structure J by (5.36). Note that in addition to the property J2 = -1, J is an isometry with respect to both g and w. Therefore,

h_g - iw is a complex hermitian inner product. Of course, dw = 0 since dw is a 3-form on a 2-manifold. This proves that M is a Kahler manifold. U

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Coincidences of Geometries in Low Dimensions

SO(4) = HU(1) HU(1) and CSO(4) = GL(1, H) H*

Theorem 5.37. (a) The notion of an oriented conformal manifold of real dimension 4 is the same as the notion of an (almost) quaternionic manifold of quaternionic dimension 1. (b) The notion of an oriented Riemannian manifold of real dimension 4 is the same as the notion of a quaternionic Kahler manifold of quaternionic dimension one.

Proof: Suppose M is an almost quaternionic manifold of quaternionic dimension one. Let C C End(TM) denote the coefficient bundle (i.e., the almost-quaternionic structure) so that, at each point x E M, Cx = H as real algebras. This isomorphism is not canonical, but if fl : Cx -* H and f2 : Cx --* H are two such isomorphisms, then f2 o fi 1 E Aut(H) is an

automorphism of the algebra H. Later, in Chapter 6, in the context of normed algebras, it is easy to compute that Aut(H) - SO(ImH). Assuming this fact, it follows that the standard real inner product (,) and the standard orientation (on H) induce a real inner product and orientation

on C. Now pick any nonzero tangent vector u. Then any other nonzero tangent vector.v is of the form Rau = v for a unique choice of Ra E C. Thus, we may define (5.38)

9(V1, v2) = (Rai, Rae) = (al, a2),

where vj = R,,, u. The inner product depends on u, but if u is replaced

by u = uA, then v = ua = ua)'1a = uA-la, so that a is replaced by a - A-1a. Thus, the new inner product g equals JAI-2 times the old inner product g, so that we have determined a conformal structure [g] on the tangent space. Similarly, the orientation {1, i, j, k} on H determines an orientation on the tangent space. If, in addition, M is quaternionic Kiihler, then M is automatically equipped with a volume form ' = (w; + w + wk). Now, there exists a s g in the conformal class determined unique choice of Riemannian structure by the quaternionic structure, so that the given 4-form is of unit norm. Next suppose M is an oriented conformal manifold of real dimension 4 with conformal structure [g]. Given a choice of g E [g], let 4D denote the unit oriented volume form. The inner product g and the volume determine a linear map * on forms by (5.39)

a A*,3 = g(al3),

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97

where g also denotes the inner product induced by g on forms. If g on tangent vectors is rescaled by A so that g = Ag, then g on forms of degree

k rescales by g = A-2kg, so that the new unit volume form 1i = A41b. Therefore, if a, E A2 are of degree 2, g(a, Q)41 = g(a, Q)db is independent

of the rescaling A. This proves that the * operator on A2 is uniquely determined by the oriented conformal structure on M. Using an oriented orthonormal basis e1, . . . , e4, it is easy to prove that * : A2 -> A2 squares

to 1, and that the ±1 eigenspaces are both 3-dimensional. Thus, A2 = A+ ® A- decomposes into the space of self-dual forms and the anti-selfdual forms. Each self-dual form a E A+ determines a linear map Ja on the tangent space by (5.40)

a(u, v) = g(J, u, v) for all tangent vectors u, V.

The coefficient bundle C C End(TM) is defined by CI = span 1 ®tJa a E A+} C End TIM. Note that if g = )g is a rescaling of g then Ja = A-'J,,,. Therefore, the bundle C only depends on the conformal structure [g] and not the choice of g E [g]. In terms of an oriented orthonormal basis a1, a2, a3, a4 for the cotangent space (based on a choice g E [g]), it is easy

to compute that ,31 - a1 A a2 + a3 A a4, (32 = a1 A a3 - a2 A a4, and /33 =- a1 A a4 + a2 A a3 provide a basis for A+. The operators JJ11 Jp Jp,

satisfy Jp1 = J 3 = J 3 = -1 and Jp1Jp2 = Jp etc., so that CI = H as real algebras. This proves that M is naturally an almost quaternionic manifold. In this dimension n = 1, each almost-quaternionic manifold is a quaternionic manifold.

Now, if M is an oriented Riemannian manifold with metric g, the previous discussion applies yielding an almost-quaternionic structure on M. In addition, the 4-form (Qi + Q2 +,6,1)

is just the unit volume element and hence is independent of the choice of oriented orthonormal basis a1, a2, a3, a4. Combining the coefficient bundle C, the form -1), and the metric g provides the quaternionic Kahler structure on M.

Remark 5.41. The right H-structure determined by C C End(TM) determines EndH(TM), the space of quaternionic linear maps. This subbundle EndH(TM) of End(TM) is exactly the bundle span {1} ® {Ja : a E A-} of operators obtained from the space of anti-self-dual 2-forms, i.e., A-.

Problems

98

SU(2) t--- HU(I)

Theorem 5.42. (a) Suppose M is a hyperKahler manifold of quaternion dimension one. Then M is naturally a special Kahler (positive definite) manifold for a 2-sphere of different complex structures on M. (b) Suppose M is a (positive definite) special Kahler surface. Then M is naturally a hyperKahler manifold.

Proof: (a) Suppose M, e is a hyperKahler manifold of quaternionic dimension

one. For each complex structure R,, on M (with u E S2 C ImH a unit imaginary quaternion), M, h is a Kahler manifold with complex hermitian inner product h = g - uw and M, a,, is a complex holomorphic symplectic manifold with complex symplectic form a,, defined by e = h + uo-,,. (See Lemma 2.72.) Moreover, to is globally constant since it equals the norm of dzldz2 on the standard model when u = i. (b) Suppose M is a special Kahler surface with complex structure I, hermitian structure h = g + iwj, and unit holomorphic volume form a-. Define a linear map J on the tangent space at each point p by (5.43)

h(xJ, y) = Y (x, y)

for all tangent vectors x, y.

Choose any isomorphism TpM = C2 and note that a = e'Bdzl A dz2. If y is replaced by a - dz' A dz2 in (5.43), then (5.44)

h(xJ, y) = A(x, y)

obviously defines a standard quaternionic structure I, J, K = IJ on C2.

Since or = e"), J must equal e'BJ. Therefore, J satisfies the same

identities, namely IJ = -JI and j2 = -1, as J. This proves that I, J provides a quaternionic structure on the tangent space TpM. Now one can show that e = h + ja defines an H-hermitian symmetric inner product on each tangent space.

PROBLEMS 1. (a) Prove that the two definitions of oriented (real) manifold given in the text are equivalent.

Differential Geometry

99

(b) Prove that each complex n-manifold is automatically oriented as a real 2n-manifold. 2. Prove that each non-orientable Riemannian 2-manifold (of signature 2, 0) is a "weak" almost-complex manifold as described in the text. 3. Suppose a real 2n-dimensional vector space V with complex structure J is given. Let Vc = V OR C = V ® iV denote the complexification

of V. That is, Vc E {u + iv : u,v E V}. Vc is naturally equipped with a conjugation that fixes V C Vc, i.e., u + iv = u - iv. Define Re z = (z--z) and Imz = (z - x)/2i for all z E Vc. Extend J to be z linear on V. Let V1'0 denote the +i eigenspace of J and Vo,l, complex denote the -i eigenspace of Jon Vc, so that Vc = V1,o® Vo,l Let 9r denote the projection of Vc onto V',o along V°'1. Show that V with complex structure J and V1-° with complex structure i are complex isomorphic via the map 7rlv and that 2R.e provides the inverse. 4. When V, J is taken as TM, the tangent space to a complex manifold, then utilizing the coordinate functions z1,.. ., z' with z - x + iy, a real basis for V is given by: aa, , ... , a=n , ay l , ... , aa . Show that (a)

, ... , OZn , 11

where

a =

2/ ( - i ay l

\ is a complex basis for Vl,o (b) The isomorphism V, J = V1-1, i of complex vector spaces exhibited in Problem 3 does not preserve the bracket [ , ] of vector fields. 5. Verify (5.5), i.e., Rya = 1. Rkt=k

6. Prove that 0 =_ da, where a is a 1-form defined by a(V) E ¢(a*V) at each point Or E T*M, provides a symplectic structure on the cotangent bundle T*M to a manifold M. 7. Suppose M is an almost-quaternionic manifold equipped with an Hhermitian inner product a that is either symmetric or skew. Prove that the coefficient bundle C C End(TM) is trivial, i.e., exhibit r, J, K globally on M. 8. (Lagrangian submanifolds) Consider the standard model RZn, w of a symplectic vector space as a symplectic manifold, where w=_dxl

Suppose M is the graph of a smooth function y = f (x) over an open simply connected subset U of Rn, i.e., f : U -* R' and M {(x, f (x)) E R2n : x E U). Prove that the following are equivalent.

Problems

100

(a) f = V4 for some scalar-valued function 0 defined on U. (b) The 1-form E fi(x)dx' is d-closed on U. (c) The 1-form a - E y'dx' on R21 restricts to a d-closed form on M. (d) The 2-form w vanishes when restricted to M (i.e., M is an isotropic or totally null submanifold of R 2 '

,

).

In particular, note that 0 satisfies a nonlinear partial differential equation of the form P(x, VO(x)) = 0 if and only if both of the following geometric conditions on the graph M are satisfied:

(1) M is contained in {(x, y) E U x R" : P(x, y) = 0}, and (2) M is isotropic.

9. (a) Extract a sublist of the list of group isomorphisms provided by Proposition 1.40 with the stronger property that the corresponding lowest dimensional representations are isomorphic. For example, SO(2) acting on R(2) and U(1) acting on C(1) are isomorphic representations, but the representation Sk(1) on H is different, so that SO(2) L, U(1) should be included on the new list but SO(2) - Sk(1) should be deleted. (b) Compare this new list with the list of low dimensional coincidences of geometries presented in this chapter.

10. Prove Moser's Theorem on compact manifolds with volume.

Hint: (a) Show 1' = 11 + da for some global (n - 1) form a. (b) Define a vector field V by a = V .i 0. Make use of the corresponding flow Ot and the beautiful formula Lv (a) = d(V _j a) + V _j da

(valid for any form a), where Lv denotes the Lie derivative with respect to the vector field V.

6. Normed Algebras

Suppose V, (, ) is a euclidean vector space of signature p, q. The norm

of a vector Ivi . fj(v v)I is always positive. In this section, we shall deal with the square norm or quadratic form (6.1)

lixll = (x,x).

By a normed or euclidean algebra, we mean a (not necessarily associative) finite dimensional algebra over R with multiplicative unit 1, and equipped

with an inner product (, ) of general signature whose associated square norm I satisfies the multiplicative property I

(6.2)

I

I

lixyil = Ilxll Ilyll

for all x, y.

The inner product (x, y) can be determined from the square norm lull by polarization. That is, replace z by x + y in the formula ilzil = (z, z) and

obtain

Ilx+yll = (x+y,x+y) = Ilxll+2(x,y)+Ilyll or

2(x, y) = lix + yli - Ilxii - 11Y11-

Similarly, the identity Iizwjj = ilzli Ilwil can be polarized. 101

Normed Algebras

102

Lemma 6.3. The following identities are all equivalent: (6.2)

Iixyll = IIxII IIyII

(xw) yw) = (x, y)IIwII

(6.4) (6.5)

(wx, wy) = IIwII(x, y)

(xz, yw) + (yz, xw) = 2(x, y) (z, w)

(6.6)

Note that the final identity is linear in all of the variables x, y, z, and w.

Proof: Setting z = w in the last identity yields (6.4), and one obtains (6.5) similarly. Setting x = y in (6.4) yields the multiplicative identity (6.2). Thus, it remains to deduce all the identities from the multiplicative property lixyll = IIxII Ilyll via polarization. Replace x by x + y and y by w in (6.2). This proves that II(x + y)wII = ((x + y)w, (x + y)tv) = (xw, xw) + 2(xw, yw) + (yw, yw)

= IIxwII + Ilywll + 2(xw, yw) = IIxII IiwII + IIyII IIwII + 2(xw, yw)

must equal Ix + yll IiwII = (IIxII + 2(x, y) + IIyII)IIwII,

yielding (6.4). The proof of (6.5) is similar. The identity (6.4) is quadratic in w and hence may be further polarized by replacing w by z + w. Thus,

(x(z + w), y(z + w)) = (xz, yz) + (xz, yw) + (xw, yz) + (xw, yw) =I 1Z11(x, y) + IIwlI(x, y) + (xz, yw) + (xw, yz)

must equal

1k+wII(x,y) = I1zII(x,y)+2(z,w)(x,y)+IIwII(x,y), yielding (6.6). In summary, the defining property Ilxyll = IIxII IIyII, which is quadratic in x and y, may be re-expressed as an identity, (6.6), linear in x, y, z, and

w. L Given an algebra, let Rw denote the linear operator right multiplication by w. Similarly, let Lw denote left multiplication by w. The identity (6.4) may be rewritten (6.4')

(Rwx, Rwy) = Ilwll(x, y)

103

Normed Algebras

That is, the key axiom for a normed algebra may be stated as: Right multiplication by w is conformal with conformal factor IIwII. Similarly, (6.5')

(L. x, Lwy) = IIwII(x,y) In terms of the operators RZ and R,,, the fully polarized identity (6.6) becomes

(6.6')

(R-- x, Rwy) + (R. x, R. y) = 2(x, y)(z, w)

Note: If IIwII = -1, then R,,, V O(A). Interchanging x with z and y with w in (6.6) yields (6.6")

(L,2x, Lwy) + (Lwx, Lzy) = 2(x, y)(z, w)

We will adopt the following notational conventions. Let Re V denote

the span of 1 E V. Since IIxII = III . xII = 11111 IIxII, the multiplicative identity 1 cannot be null. Let Im V denote the orthogonal compliment of Re V. Then, by Lemma 2.30, Im V is a nondegenerate hyperplane, and each x E V has a unique orthogonal decomposition:

x=x1+x', with xiEReV, x'EImV. Occasionally, we let Rex denote x1 and Imx denote x'. Conjugation is defined by

x = xl - x'.

(6.7)

Thus (6.8)

xlRex= x'Imx= The adjoints of R,, and L. are

(6.9)

Ra, = R. , L*, = Lw. To prove (6.9), first note that these identities are linear in w, and obviously true when w E Re V. Thus, we may assume w 1 Re V or w E Im V. Now setting z = 1 in (6.6') yields (x, R. y) + (Rwx, y) = 0

or R*, _ -R, = Rw. Similarly, Lv, = L. The elementary facts concerning conjugation are contained in the next lemma.

The Cayley-Dickson Process

104

Lemma 6.10. (a) -'X= = x and (x, y) _ (x, y)

(b) (x, y) = Re xy = Re xy. (c) xy = y T. (d) xx = xx = JIxJJ.

Proof: (a) is true because conjugation is reflection through a hyperplane. Part (b) is a special case of Ry = Ry. Part (c) is true because

(xy, z) = (xy, z) = (y, x x) _ (yz, x) = (z' y x) Part (d) is now an easy direct calculation. The associator

[x, y, z] - (xy)z - x(yz)

measures the lack of associativity in an algebra. For a normed algebra, there is always a weak form of associativity, namely

Lemma 6.11. The associator [x, y, z] vanishes if any two of the arguments are set equal, or equivalently, the trilinear form [x, y, z] is alternating. An algebra on which the associator is alternating is called alternative. Lemma 6.11 states that any normed algebra is alternative.

Proof: Note that the associator vanishes if one of its variables is real. Hence, it suffices to show that the associator vanishes when two of its variables are set equal to w E Im V pure imaginary. Note, we show that [x, w,] = 0. Since ww = JJwJJ, [x, w, w] = (xw)w-xJJwJJ. Since Rw = R, and R. is conformal, ((xw)w, z) = (xw, zw) = IIwII(x, z),

which proves [x, w, w] = 0. Similarly, [w, w, z] = 0. Since [x, w, w] = 0, polarization yields [x, y, z] _ -[x, z, y]. Therefore [w, y, w] -[w, w, y] 0.

THE CAYLEY-DICKSON PROCESS Using the facts that Lw = Lo- and Rw = Ru, (as well as Lemma 6.10a), the identities (6.6') and (6.6") can be written as

105

Normed Algebras

Lemma 6.12. x(yw) + y(7w) = 2(x, y)w, (wy)x + (wx)y = 2(x, y)w.

In particular, we obtain the key identities that motivate the CayleyDickson process:

Corollary 6.13. If x 1 y, then xy = -yx and (6.14)

x(yw) = -y(xw) and (w'y)x = -(wx)y, for all w.

These crucial equations enable us to interchange a pair of orthogonal vectors x and y.

Lemma 6.15. Suppose A is a normed subalgebra (with 1 E A) of the normed algebra B and that E E A-' is a unit vector orthogonal to A with 11611 = ±1. Then Ae is orthogonal to A and (6.16)

(a + bE)(c + dE) = (ac+ db) + (da + bc)E for all a, b, c, d E A.

Note that if IIEII = 1, then -db occurs, while if Ilel I = -1, then +db occurs.

Proof: First, we prove A J. Ae. Since 1 E A, x E A if and only if F E A. Thus, if a, b E A, then (a, be) = (ba, e) = 0 because ba E A. Note that e2 = T1 if and only if IIehl = ±1 since e 1 1 is pure imaginary (i.e., e = -e and eE = IIEII implies e2 = -ee = -IIEII). Finally, to prove (6.16) we use the key identities (6.14) several times, applied to the terms in (a + be)(c + de) = ac + (be)(de)-I-a(de) + (be)c.

That is, (be)(de) = -d((be e) = d((eb)E) = -d(& )b) = -IIeIIdb a(de) = a(ed) = e(ad) = (ad) e = (da)E (be)c = (b'c)E.

Lemma 6.15 mandates the following doubling process.

Definition 6.17 (Cayley-Dickson). Suppose A is a nonmed algebra. Motivated by Lemma 6.15, we define two algebras A(+) and A(-). As vector spaces both A(±) = A ®A.

The Cayley-Dickson Process

106

Multiplication is defined by (a, b)(c, d) _ (ac + db, da + bc),

(6.18)

with the coefficient of db taken to be -1 for A(+) and +1 for A(-). Both A(+) and (A-) are easily seen to be algebras with multiplicative unit 1 = (1, 0). The algebra A is naturally a subalgebra of A(±). In fact, given a E A, we also let a denote (a, 0), embedding A as a subalgebra of both A(+) and A(-). Let e denote (0, 1). Then (a, b) = a + be.

Note that e2 = -1 for A(+) and e2 = +1 for A(-). There is a natural square norm or quadratic form to impose on A(+) and on A(-). If (a, b) = a + be E A(±), let (6.19)

II(a, b)II = IIail + IIbii.

Note that IIeII = 1 for A(+) and IIell = -1 for A(-). The associated inner product is given by (x, y) _ (a, c) f (b, c),

(6.19')

if x = a + be and y = c + de. Conjugation is defined by (a, b) = (d, -b) or a + be = i - be. Real and imaginary parts are defined in terms of conjugation or equivalently in terms of the orthogonal splitting of A(±) into R and RJ-. Next, we collect together some of the formulas valid in A(+). The proofs are straightforward calculations using the definition (6.18) of multiplication.

Lemma 6.20. Suppose A is a normed algebra. Let A(±) denote the algebra defined via the Cayley-Dickson process. Suppose x = a + ae, y =

b+/3e, z=c+yeEA(+). (6.21)

xy = y T.

(6.22)

xx = xx = II xII

(6.23)

(xy + yx) = Rexy = (x, y). [a, b] ± Im aQ + (Q Im a - a Im b) c.

[x, y] =

(6.24) 2

2

Normed Algebras (6.25)

107

[x, y, z] = ± [a, TO] ± [b, ay] ± [c, 8a]

+ a[b, C]e + Q[a, 4e + y[a, b]e ± (ofy - yQa)e,

assuming A is associative. [x, g, y] = ±[a, Q, a,] + [a, b, ale. (6.26)

jlxii Ilyil - Iixyll = ±2(a, (Q, a, Ll) -

Corollary 6.27. Suppose A(+) is the algebra defined via the CayleyDickson process from a normed algebra A. (6.28)

A(±) is commutative if and only if A = R.

(6.29)

A(±) is associative if and only if A is commutative and associative.

A(±) is alternative, A(±) is normed, and

(6.30)

A is associative are all equivalent.

THE HURWITZ THEOREM Corollary 6.27 can be used to deduce the properties of the euclidean algebras obtained via the Cayley-Dickson process. First, we list (some of) these algebras.

Definition 6.31. C = R(+)

H = C(+)

0=H(+) L = R(-)

M2(R) = C(-)

0 = H(-)

the complex numbers. the quaternions or Hamiltonians. the octonians or Cayley numbers. the Lorentz numbers. real 2 x 2 matrices.

the split octonians.

Remark. The normed algebra M2(R) of real 2 x 2 matrices (with IIAII detp. A defined to be the square norm or quadratic form) occurs in several different ways via the Cayley-Dickson process. By Corollary 6.38(a) below,

M2(R) = C(-) = L(+) = L(-).

The Hurwitz Theorem

108

To show M2(R) = C(-), identify C with the 2 x 2 reals of the form C b

= a + ib ab

and choose -1

to be the matrix corresponding to conjugation on R2 = C. Check that 1 C and E2 = 1. Then by Lemma 6.15, C(-) and M2(R) are algebra and norm isomorphic.

Corollary 6.32. C =R(+) and L = R(-) are commutative, associative, (6.33)

(6.34)

and normed.

H =C(+) and M2(R) are not commutative but associative and normed.

O =H(+) and O = H(-) are neither commutative (6.35)

nor associative but are alternative and normed.

Proof: Apply Corollary 6.27.

Remark 6.36. O(+) and O(-) also fail to be alternative or normed. Theorem 6.37 (Hurwitz). The only normed or euclidean algebras over R are

R, C and L, H and M2(R), 0 and O. Proof: Suppose B is a normed algebra. Let Al =- Re B = R. If Al = B, we are finished. If not, choose El E Ai a unit vector, i.e., -Ei = I IEl I I = +1 Let A2 = Al + Al 1. By Lemma 6.15, A2 is a normed subalgebra of B isomorphic to either C = R(+) or L = R(-). If A2 = B, we are finished. Suppose not. Choose E2 E AZ a unit vector, and let A3 =- A2 + A2E2.

Case R(+): Suppose A2 = C = R(+). Then by Lemma 6.15, A3 is a normed subalgebra isomorphic to H = C(+) or M2(R) = C(-).

Case R(-): Suppose A2 = L = R(-). Then E2 = 1, E2 = ±1, and E3 . 1e2 satisfies E3 = f1 since 6162 + E2E1 =

In particular,

Normed Algebras

109

either e2 = -1 or 2 = -1. Say e3 = -1. Then interchange el and e3 (note this does not change A3) in the above constructions so we are back in Case

R(+). Thus, either A3 = C(+) or A3 = C(-). If A3 = B, we are finished. Suppose not. Choose e3 E A3 a unit vector, and let A4 =- A3 + A3e3.

Case C(+): If A3 = C(+) = H, then by Lemma 6.15 A4 is a normed algebra isomorphic to either 0 = H(+) or 0 = H(-). Case C(-): If A3 = C(-) = M2(R), then interchanging either e3 or e4 - e2e3 with 62 reduces us to Case C(+). If A4 = B, we are finished. If A4 # B, then by repeating the above process one more time we obtain

either 0(+) or 0(-) as a normed subalgebra A5 of B. Since 0(+) and O(-) are not normed algebras, this is impossible. J Corollary 6.38. (a) As normed algebras M2(R) = R(-, -) = R(+, -) = R(-, +). (b) The seven normed algebras R(f, f, f), excluding R(+, +, +) = 0, are all the same, namely O.

Remark. Because of the Hurwitz Theorem, all normed algebras are subalgebras of either 0 or O. Consequently, it is frequently clearer to state results that are valid for all normed algebras as results for 0 and O. The next theorem is an example, as are "cross products" in the next section. The weak form of associativity for 0 and O, corresponding to the fact that the associator vanishes if any two of the three variables are set equal, can be strengthened.

Theorem 6.39 (Artin). A subalgebra with unit, generated by any two elements of either 0 or 0, is associative.

Proof: Suppose A is generated by 1, x, y. Let S =_ span{Imx, Im y}.

If dimS = 0, then A = R and the proof is complete. If dimS = 1, choose a nonzero vector e1 E S and note that since 62 = -IIc1Il is real, A = {a + bet a, b E R}. Now it is easy to verify directly that A is associative. If dim S = 2, choose an orthogonal basis 61, 62 for S and note that by repeated use of Corollary 6.13 as in the proof of Lemma 6.15, A :

is spanned by (as a vector space) 1, e1 i e2, and El 62. Now using the facts that [e1, e2, e162], [el, El, e21, etc. vanish, we see that A is associative. Actually we can prove much more.

Proposition 6.40. If S = span {Imx, Im y) is nondegenerate, then the subalgebra A is determined as follows: (6.41)

If dimS = 1, and S is positive, then A = C.

Cross Products

110

(6.42) (6.43) (6.44)

If dim S = 1, and S is negative, then A = L. If dim S = 2, and S is positive, then A = H. If dim S = 2, and S is negative or Lorentz, then A = M2(R).

Further, if dimS = 1 and S is null, then

A= {

(6.45)

l

l

0

6) :a,bER}.

Proof: If dim S = 1 and S is nondegenerate, choose E1 E S to be a unit vector. Then by Lemma 6.15, A = R+Rrl is either C or L depending on II-'111.

Suppose S is degenerate of dimension one. Choose a nonzero null vector u in S and a pure imaginary unit vector e with (u, E) 1 0. Then span{,-, u} is a Lorentz plane = L containing S as a null line. Therefore, we may choose an orthonormal basis El, E2 for L with El + E2 E S. The algebra generated by El, E2 is isomorphic to M2(R) with E2 =

E1

Therefore,

S2{(0 ):bER} and

A{ (a

b) :a,bER}.

Now assume dim S = 2. If S is nondegenerate, then S has an orthonormal basis 61,E2, and by the above arguments and Lemma 6.15, ei-

ther A = L + Lee or A = C + Cr2. It follows that either A = M2(R) or

A=H.

CROSS PRODUCTS The results of this section are formulated for the octonians O. However, all of the results are valid with 0 replaced by the split octonians O. First, we consider the cross product of two octonians.

Normed Algebras

111

Definition 6.46. Given x, y E 0, the cross product of x and y is defined by

xxy=2(yx-xy)=Imyr.

(6.47)

The next lemma justifies the term "cross product".

Lemma 6.48. (a) x x y is alternating on 0.

(b) Ilxxyll=lJxAyllforallx,yE0. Proof: (a) The cross product is alternating because x x x = 0 for all x E 0. (b) Since both sides are alternating, it suffices to prove (b) when x 1 y.

Then 0 = (x, y) = Re iy = (xy + yx) so that x x y = yx. Therefore, X

yII = Ilyxll = IIyII llxll = IIx A yll because of the Cauchy-Schwarz

IIx

equality.

Remark. The proof that Ilx x yll = IIx A yll explains why we needed the conjugates in the definition of x x y; namely, we wanted the vanishing of (x, y) to imply that x x y could be expressed as one term yx. Note that

Re x x y= O for all x, y E 0.

(6.49)

Frequently, the cross product is restricted to Im 0, in which case x x y can be expressed in a variety of interesting ways.

Lemma 6.50. If x, y E Im O, then

xxy =

(6.51)

2[x,y] = xy+(y,x),

where [x, y] = xy - yx is the commutator of x and y.

Proof: (x, y) = - (xy + yx) for X, y E Im O. i 2

Remark 6.52. If x, y E R3 =_ Im H C Im O, then the cross product x x y is just the usual vector cross product on R3 based on the "right hand rule." This explains the choice of sign in the definition of x x y.

Cross Products

112

Lemma 6.53. If x, y E Im O, then (a) x x y E Im O is orthogonal to span{x, y}. (b) x x (x x y) = -IIxIIy + (x, y)x.

Part (b) says that (for IIxII = 1) the square of left cross product by x equals minus orthogonal projection from Im 0 to (span x)-L. See Problem 13, where this propety is used to reconstruct octonian multiplication from the cross product on ImO.

Proof: (x, xy) - (x, yx) = 2IIxII(1, y) - IIxII(1, y) = 0. (b) Let Cs y = x2x y. If y =a x, then CC = 0. If Y I ax, then xy = -yx, so that Cxy = -IIxIIy (a) (x, x x y) =

The natural extension of the cross product to three octonians x, y, z is the alternation of x(yz). (Note that this is not equal to x x (y x z)!) However, by repeated use of Corollary 6.13, the six-term expression obtained by alternating x(17z) can be simplified to two terms. Therefore, we adopt as our definition of x x y x z the following two-term expression.

Definition 6.54. Given x, y, z E 0, the triple cross product of x, y, and z is defined by (6.55)

xxyxz =

2 1--07z) - z(-9x)] .

Lemma 6.56. (a) x x y x z is alternating on O.

(b) IlxxyXZII=IxnyAz1Iforallx,y,zEO. Proof: (a) Since the subalgebra generated by any two elements is associative,

x x y x z= 2 [x(x'z) - zIIxII] = 0, and

z x x x x= 1 [zIIxII-x(xz)] =0.

Obviously, xxzxx=0. (b) Since both sides are alternating, we may assume that x, y, and z are pairwise orthogonal. Repeated use of the key identities in Corollary 6.13 yield -z(yx) = x(yz). That is, (6.57)

x x y x z = x(yz) if x, y, z are orthogonal.

113

Normed Algebras Therefore, (6.58)

jjx x y x zjj = llx(yz)ll = lixil IlyII ilzi _ lix A y A zil.

_1

Now consider the real-valued trilinear form
ci(x, y, z) = (x, yz)

for all x, y, z E Im O.

This form is called the associative 3-form for O. The terminology is explained in the next chapter on calibrations. Since 0 vanishes when any two of the variables x, y, z E Im 0 are set equal, 0 must be alternating. That is,

4 E A3(Im O)*.

(6.60)

Perhaps Gureirch [8] was the first to consider this 3-form (in coordinates as in (6.74)). It is natural to consider the triple cross product restricted to Im O.

Lemma 6.61. If x, y, z E Im O, then (6.62)

Rexxyxz=qS(xAyAz),

(6.63)

Imxxyxz=

1[x,y,z].

Proof: It suffices to prove the lemma when x, y, z are orthogonal. In this case, x x y x z = x(Vz) = -x(yz) by (6.57). Thus Rex x y x z = -(1, x(yz)) = (x,yz) = 4'(xAyAz). Also, x x y x z = -z x y x x = z(yx) by (6.57). Thus F x -y x z = (xy)z. Consequently, Imx x y x z = 2[-x(yz) + (xy)z] = 2 [x, y, z].

THE EXCEPTIONAL LIE GROUP G2 Some of the automorphism groups of the normed algebras are exceptional. Let

(6.64)

Aut(A) _ {g E GL(A) : g(xy) = g(x)g(y) for all x, y E A}

denote the automorphism group of a finite dimensional algebra A. The group G2 is most naturally defined as the automorphism group of the octonians (6.65)

G2

Aut (0),

(6.66)

GZ

Aut(O)

(the split case).

Before examining G2 and G2, we discuss the automorphism groups of

the other normed algebras besides 0 and O. These other automorphism groups will be seen to be orthogonal groups or special orthogonal groups.

The Exceptional Lie Group G2

114

Lemma 6.67. If A is a normed algebra, then Aut(A) C O(ImA) the othogonal group of Im A.

Proof: Let g E Aut(A). First note that g(1) = 1 since g(x) = g(1)g(x) for all x E A. Second, since x2 = (Re x)2 + (IM X)2 + 2(Re x)(Im x) and (IM X)2 = -IJIm xJJ E ReA, it follows that x2 E Re A if and only if x is either real or pure imaginary. Thus, if x E ImA, then g(x)2 = g(x2) = x2g(1) E Re A, so that g(x) is either real or pure imaginary. Thus, Aut(A) C GL(ImA) and g(x) = g(x). Therefore, 11g(x)11 = g(x)g(x) = g(x)g(x) = g(x9) = g(IIxIj) = 11x11g(1) _ 114-

This proves g E O(ImA).

Note that this proof also shows that Aut(O(+)) C O(O(+)).

Corollary 6.68. Aut(C) --- Z2 and Aut(L) = Z2 each Z2 consist of the identity and conjugation.

Lemma 6.69. If g E Aut(A), then g fixes the associative 3-form 0 for A. Proof: Here O(x, y, z) - (x, yz) for all x, y, z E ImA. Suppose g E Aut(A). Then (g(x),g(y)g(z)) = (g(x),g(yz)) = (x,yz)

since g is orthogonal.

If A - H, then the associative form 0 is the unit volume form on ImH corresponding to the orientation determined by {i, j, k}. Thus, if g E Aut(H) C (Im H), then, by Lemma 6.69, det g = 1. Let

C = ( 10 -0) (conjugation),

J= `

1

I

(complex structure),

and

R == (°

1) (reflection or Lorentz structure)

denote the standard orthonormal basis for ImM2(R). Then 0 (the associative form) is the unique unit volume form determined by the orientation

{C, J, R}, since cb(C A J A R) = (C,JR) = -(C,C) = C2 = 1. Thus, if g E Aut(M2(R)) C O(ImM2(R)), then detg = 1, because of Lemma 6.69.

Normed Algebras

115

Proposition 6.70. Aut(H) = SO(ImR) = SO(3, 0). Aut(M2(R)) = SO(ImM2(R)) = SO(1,2).

Proof: See Problem 6.7. J This completes our discussion of the automorphism groups of the normed algebras except for G2 and G2. Sometimes, in order to avoid repetition, results will only be stated for G2. However, these results will be stated in such a way that no modifications will be necessary for G2 other

than replacing 0 by O. The 4-form 0 on Im 0 defined by (6.71)

i,b(x, y, z, w) =

1

2

(x, y(zw) - w(zy))

for all x, y, z, w E Im 0

is called the coassociative 4-form for O. If any two of the four variables x, y, z, w E Im O are set equal, then O(x, y, z, w) = 0. (See Problem 6.8). Thus (6.72)

0 E A4(Im O)*

is a skew tensor of degree 4.

Lemma 6.73. If g E Aut(O), then g fixes the coassociative 4-form 0 E

A4(Im 0)*.

Proof: Apply the facts that g E O(Im O), g E Aut(O), and g(x) = g(x) to the definition of b.

One can choose an orthonormal basis e1,..

. , e7

for Im O, with dual

basis W1, ... , W7 for (Im O)*, so that direct computation yields

(6.74)

O = W123 - W156 - W426 - W453 - W147 - W257 - W367

and (6.75) O = W4$67 - W4237 - W1537 - W1267 - W2536 - W1436 - W1425,

where wijk = wi Awj A Wk, etc.. Therefore,

The Exceptional Lie Group G2

116

Lemma 6.76. 0 A 0 = 7C'1234567

Corollary 6.77. (a) G2 C SO(Im O) = SO(7), (b) G2 C SO(Im O) = SO(3,4).

Proof: g E G2 = Aut(O) has det g = 1 since g fixes 0 and 0 and 0 A 0 is a nonzero volume form. The proof for G2 is analogous. J Remark 6.78. The unit volume element ?4,A1' provides an orientation for Im O. Using this orientation the Hodge star operator * : A Im O -} A Im O is well-defined. One can easily check that

0_4

(6.79)

Similarly, 0 = *0 in the split case O.

As noted above, if g E G2, then g*4, = 0. The converse is more difficult.

Theorem 6.80 (Bryant). (6.81)

G2 = {gEGL(ImO):g'O=0}.

Proof: Assume for the moment that (6.82)

if g E GL(Im0) satisfies g'

then g E O(Im0).

Then, for all x, y, z E Im O, (g(x), g(y x z)) = (x, y x z) = 4,(x A y A z) _ (9*4,)(x A y A z) = 4,(9(x) A g(y) A 9(z)) = (g(x),9(y) x g(z)).

This proves that g preserves the cross product: (6.83)

g(y x z) = g(y) x g(z)

for all y, z E Im O.

Therefore, by Problem 11, g E Aut(O) - G2. It remains to prove (6.82). The identity (Problem 14 (c)) (6.84)

(x _j 0) A (x _j 0) A ¢ = 6IIxIIa,

where A is the standard volume element for Im O, can be polarized to yield (6.85)

(x j 4,)A (y j 4,)A 4, = 6(x, y)A for all x, y E Im O.

117

Normed Algebras

Now suppose g E GL(Im 0) satisfies g*4,

Applying g* to (6.85)

yields

(g-'(X) i 0) A (9-1(y) .j 0) A 0 = 6(x, y) (det g).\, while replacing x by g-1(x) and y by g-1(y) in (6.85) yields (9-1(x) _j 0) A (g-1(y) i 0) A 0= (9-1(x), 9-1(y))A. Therefore,

(g(x), g(y)) = (det

(6.86)

g)-1 (x, y)

for all x, y E Im O.

Therefore, g is either conformal (det g > 0) or anticonformal (det g < 0). Since the signature of 0 is not split, g cannot be anticonformal (see Problem 2.3). Thus, g is conformal with conformal factor A = (det g)-1 > 0. Recall (Problem 2.9(d)) that if g is a conformal transformation with conformal factor A, then (detg)2 = a",

(6.87)

where n is the dimension. Substituting .\ = (det g)-1 and n = 7, we obtain (detg)9 = 1

or

det g = 1.

Thus, we have shown that if g E GL(Im O) satisfies g*qs = ¢, then g E SO(Im O),

(6.88)

completing the proof of Theorem 6.80. This theorem describes G2 as a level set of the function from End(Im 0)

to A3(ImO)* that maps A E End(Im O) to A*4, E A3(Im O)*. The linearization of this map at the identity I E End(Im O) is the map sending A E End(Im O) to D(A) = dt (I + tA)*Ol t=o Now (6.89)

D(A)(x A y A z) = dt 4,((x + tAx) A (y + tAy) A (z + tAz))

LO

_ qS((Ax) A y A z) + q5(x A (Ay) A z) + 4,(x A y A (Az)).

Thus, D(A) is just the action of A on 0 E A3(Im O)*, considering A as a derivation.

The Exceptional Lie Group G2

118

Lemma 6.90. The linear map (6.89), D : End(Im O) --+ A3(Im O)*,

is surjective with 14-dimensional kernel.

Because of the implicit function theorem, we have several intereting consequences.

Corollary 6.91. G2 is a closed 14-dimensional submanifold of End(Im 0)

implicitly defined by G2 = {A E End(ImO) : A*q = 0). The tangent space g2 to G2 at the identity is ker D.

Corollary 6.92. The orbit of 0 under GL(ImO) is an open subset of A3(Im O)*.

Proof of Lemma 6.90: Since dime. End(Im 0) = 49 and dim A3(Im O)* = 35, the map D must have a kernel of at least 14 dimensions; and the kernel is exactly 14-dimensional if and only if D is surjective. Therefore, it suffices to prove that dim ker D < 14.

(6.93)

First, we derive another formula for D(A) in the special case where A = u ®a E (Im O) ®(Im O)* - End(Im O) is simple, i.e., Ax = a(x)u, where a E (Im 0)*, u E Im O.

D(a(9 u)(xAyAz)=a(x).(uAyAz) +a(y)O(x A U Az) +a(z)O(x AyAz) = (u -j ')(a(x)y A z - a(y)x A z + a(z)x A y)

= (a A (u j qS))(x A y A z).

That is, (6.94)

D(u ®a) = a A (u

q).

Consider the linear map B : A3(Im O)* -+ S2 (Im O)*

defined by (6.95)

*B(,3) (x, y) _ (x -1 0) A (y -j 0) A,3

for x, y E Im 0 .

119

Normed Algebras

Assume, for the moment, that (6.96)

BD(A) = A + A* + trace A

has been verified. Then, if A E ker D, (6.97)

A + A* + trace A = 0.

Taking the trace, 2 trace A + 7 trace A = 0, so that (6.98)

A+A* = 0 if A E kerb.

That is, (6.99)

Let L

ker D C Skew(Im O).

{Au : u E ImO), where

forallxEImO. Then L is a 7-dimensional subspace of the 21-dimensional vector space Skew(ImO). It is straightforward to verify that (6.100)

A Y A z)

2 (u, [x, y, z])

(see Problem 6.15). The associator [x, y, z] takes on all values in Im O. Therefore, D(A,,) = 0 implies u = 0. That is, (ker D) n L = {0}. To complete the proof of the Lemma 6.90, the identity (6.97) must be verified. It suffices to assume A = u ® a is simple. Then

1 BD(u 0 a) = 2 B(a A (u]qf),

applied to x x E Im 0 is equal to (6.101)

2 (x j 0) A (x J 0) A a A (u J ).

However, A + A* + trace A applied to x x E Im 0 is equal to (6.102)

a(u)jjxjj + 2a(x)(u,x).

Problems

120

Finally, (6.101) and (6.102) are equal; see Problem 6.14(b). J

PROBLEMS 1. Suppose A is an algebra with an inner product. Assume 1 is not null, and use this fact to define conjugation. Show that if A has the two properties (a) IIxIJ = x x, xy = y

(b) any subalgebra generated by two elements is associative, then A is normed. 2. Verify the Moufang identities:

(xyx)z = x(y(xz)) z(xyx) _ ((zx)y)x (xy)(zx) = x(yz)x

(Lxyx = LZLYLr)

(Rxyx = R.RyR.)

for all x, y, z in 0 or O. Hint: First show that the difference of the two sides vanishes if any two of the variables are equal. 3. Suppose A is a normed algebra. (a) Show that each nonnull element of A has a unique left and right inverse. (b) Given x, y E A with IIxII 0, show that the equations xw = y and wx = y can be uniquely solved for w with w = Yy/JJxJJ and w = y-x/llxll respectively.

Note that (a) does not automatically imply (b), since (a) is true for O(+) but not (b). 4. A complex algebra with unit, equipped with a nondegenerate complex symmetric bilinear form (, ) satisfying JJxyJJ = IlxII Ilyll, where JJxJJ _ (x, x), is called a complex nonmed algebra.

(a) Suppose A, (,) is a normed algebra. Show that A ®R C is a complex normed algebra, where (,) is extended from A to A OR C to be complex bilinear. (b) Consider the following four complex normed algebras: C,

C ®C,

M2(C),

O OF, C.

Show that each complex normed algebra is isomorphic and isometric to one of these four.

Normed Algebras

121

5. Assuming that each prime number p can be written as the sum of four squares, p = ni + n2 + n3 + n4 (n I, n2, n3, n4 E N), show that each natural number n E N can be written as the sum of four squares. 6. Show that, for all x, y, z E 0,

(a) x x y = 1[x, y] - (xiy' - yix'), (b) x x y x z = (x', y'z') + [x, y, z] + xl[z, y] + yi [x, z] +

zl [y, x]

2 2 2 z 7. Use the Cayley-Dickson process to complete the proof of Proposition

6.70.

8. Show that the coassociative form , defined by (6.71), is skew. 9. (a) Let Ss denote the unit sphere in Im O. Show that G2 acts transitively on S6 with isotropy subgroup SU3: G2/SU3 = S6 C IM O. (b) Let V7,2 denote the Stiefel manifold of ordered pairs of orthonormal vectors in Im O. Show that G2 acts transitively on V7,2 with isotropy subgroup SU2: G2/SU2 = V7,2.

(c) Let V7,3(¢ = 0) denote the collection of ordered orthonormal triples el, e2, e3 E Im 0 with O(el A e2 A e3) = 0 (i.e., e3 1 ele2). Show that G2 acts transitively on V7,3(o = 0) with no isotropy: G2 = V7,3(0 = 0) -

10. Consider the algebra homomorphism

0:H®R.H--*M4(R) induced on H OR H by defining ¢, on simple tensors to be qi(p ®q)(x) = pxq for all x E R4 = H.

Let H OR H have the inner product defined by (a (& b, c ® d) _ (a, c) (b, c), and let M4(R) have the inner product defined by (A, B) 4 trace ABt. Prove that ¢ preserves inner products and hence is an algebra isomorphism (cf. Remark 1.33). Hint: Show that trace O(a ® b) = 0 if b equals i, j, or k. 11. Show that

G2 = {g E GL(Im O) : g(x x y) = g(x) x g(y)

for all x, y E Im O}.

Problems

122

Hint: x' x y' = x'y' + (x', y') for all x', y' E Im O. 12. Suppose u E Im O and lull = 1. Show that, for all x, y, z E 0, (a) (xu) x (yu) = u(x x y)u, (b) (xu) x (yu) x (zu) = (x x y x z)u. 13. A cross product on an inner product space V, (,) is usually defined to be a bilinear map x x y from V x V to V that satisfies not only (a) llx x yII _ lIx A ylI but also (b) x x (x x y) = -11xIly+ (x, y)x.

Adopt this (stronger) standard definition. Given a cross product on an inner product space V, (, ) define a product on R ® V by

(xo +x)(yo+y) = xoyo+xoy+yox+ (x,y)+x x y for all xoyo ERandx,yE V. Define (xo + x, yo + y) = xoyo + (x, y). Show that R ® V, (,) is a normed algebra. 14. Let A denote the standard volume element on Im O. (a) Show that (x -j ¢) A (x .1 0) A (x 4,) = 6xI I (x i A). Hint: Set x .= el = i and note that w x .i 0 is the Kahler form on [i]1 under the complex structure right multiplication by i, while x i A is the unit volume element on [z]t _ C3. (b) Show that (u -1 ¢) A (x -1 0) A (x i ¢) = 2 (11zliu + 2(x, u)x) i A. (c) Show that (x 0) A (x -1 0) A 0 = 6lIxllA. Hint: Use (a) and x A (x j A) = jjxjjA. 15. (a) Given a 3-form 0 E A3(R7)* in 7-variables and a volume element A on R7, let (x

0) A (x j 0) A 4, = (x, x)ma

define a real bilinear form on R7. If 0 is nondegenerate (i.e., (, ) 0 is nondegenerate), exhibit either an octonian structure with R7 = Im 0, or a split octonian structure with R7 = Im O, so that 4,(x A y A z) _ (x, yz)O is the associative 3-form. (b) (Gureirch [8]) Show that GL(7, R) acting on A3(R7)* has two open

orbits with isotropy G2 and G2(at the associative three form ¢ and the split associative three form 0, respectively). 16. Suppose N is one of the normed algebras C, H, or O. Let Herm(2, N) {A E M2(N) :IT' =A}. Define P'(N) _ {A E Herm(2, N) : A2 =

123

Normed Algebras

A and trace A = 1}. Given a unit vector a = (al, a2) E N2, define

[a]=a`a=

1a1I2

a2a1

Ia212 a1a2

(a) Show that [a] E PI(N), and that [b] = [a] if and only if b = as for some A E N.

(b) Let S(NP) denote the unit sphere in NP and verify that

S(N) -+ S(N2) -' P1(N) is a fibering of spheres by spheres. That is,

Sl -* S 3 - P1(C) = S2

(Hopf fibration),

S3 -> S 7 -f P1(H) = 54,

S7---*S15- P1(O)S8. (c) Show that the Hopf fibration

S1 -;53-} can also be defined by sending

xES3CH to xixES2CImH. Note: Setting x - z + j w with z, w E C yields a formula for an isomorphism from

P1(C)=C2-{O}/-

to S2CImH=R3

in terms of quaternion multiplication. Namely, [(z, w)]

is mapped to

(z -f jw)i(z - jw) Iz12 + IwI2

E H C ®jC.

17. Consider the representations of G2 induced on Ak(ImO)* by the standard action of G2 on Im 0. Show that these representations decompose as

(a) A2(Im O)* = A7 ®A14, (b) A3(Im O)* = Ai ® A7 ®A27,

Problems

124

where

A7={w., uEImH}, wu = u -

or equivalently wu(x, y) _ (u, x x y),

A2 ,.. 4 = 92,

A7 = {u j 0: u E ImH}, A3 = span 4, and

A3 ®A27 = {D(A) : A E Sym(Im O)},

with D the linear map defined in (6.89) or (6.94).

7. Calibrations

Throughout this chapter, we restrict attention to R" with the standard positive definite inner product (, ). The concept of a "calibration" has been alluded to in the previous discussion of Proposition 4.15. This result states

that a straight line segment in R" minimizes length. The "Fundamental Theorem" of the theory of "calibrations" is a straightforward generalization of this proposition replacing curves by higher dimensional submanifolds in

R". In this chapter, it is assumed that the reader is familiar with topics from advanced calculus, such as submanifolds of R" and Stokes' Theorem.

THE FUNDAMENTAL THEOREM Again, we identify an oriented p-dimensional linear subspace of R" with the element __ el A ... A eP E APR" where el,..., e1, is an oriented orthonormal basis for the p-dimension subspace. Thus, G(p, R") _ { E APR" : = e1 A . A eP for some el, ..., eP orthonormal in R"} denotes the grassmannian of oriented p-dimensional subspaces of R". The dual space of APR" is AP(R")*, the space of p-forms (with constant coefficients). If ei,... , e, is an orthonormal basis for R" and all ... , an is the dual basis for (R")* (for example el =_ -9/ax', ... , e" __ a/ax" the standard basis and ai dxl, ..., a" = dx" the standard dual

basis), then each p-form 0 on an open subset of R" can be expressed as 0 = E111=P 01a', where I = (ii, ... , i,) is a multi-index of length 125

The Fundamental Theorem

126

A a'a, and E' denotes summation over strictly increasing multi-indices, i.e., it < - < ip. The coefficients Oj are functions on the open subset of R', given in terms of ¢ by 0, = O(ej). Definition 7.1. A p-form ¢ on an open subset U of R" is said to be a

III = p, aI = a'1 A

calibration if

(a) d¢ = 0 on U, and, (b) for each fixed point x E U, the form 0x E AP(Rn) satisfies 4)x() < 1 for all E G(p, R"), with the contact set nonempty.

Condition (b) says that the maximum of 0 on the compact set G(p, R°) is one. Recall that a p-form 0 can be integrated over an oriented p-dimensional submanifold M of R". By using a partition of unity (so that one can assume

0 is supported in a coordinate chart), this integral fm 0 can be defined by pulling back 0 (with a parameterization map) to an open subset of RP and then performing ordinary RP integration. Using the inner product (, ) on R" tfiis integral can be expressed in terms of volume measure on M. For each point x E M, let AM E.P(R")*

denote the unit volume form for M, and let M denote the unit volume element for M; that is, M = el A A ep and Am - al A A as', where e1, ... , e, is an oriented orthonormal basis for TxM and al , ... , a' is an or Tz M. Then oriented orthonormal basis for (7.2)

i.e., fm ¢ equals the integral of the scalar valued function O(M) over M with respect to volume measure over M.

Definition 7.3. A closed oriented submanifold M of R" is said to be (area) volume minimizing if, for each relatively compact open subset U of M with smooth boundary 8U, (7.4)

vol(U) < vol(V)

for all other compact oriented p-dimensional submanifolds V with the same

boundary as U, i.e., 6U = 8V. A calibration 0 can be used to distinguish a class of oriented submanifolds M; namely, those with ME G(4)) for each point x E M. Equivalently, 0 restricted to M equals AM, the unit oriented volume form on M.

127

Calibrations

The Fundamental Theorem 7.5. Suppose 0 is a calibration of degree p on R. Each closed oriented p-dimensional submanifold M distinguished by 0, i.e., for all points x E M O (M) is volume minimizing.

Proof: vol(U) = JU 0 =

where

ff()Av 0V 0=

f -f 0=Ju-v c= j

do

c

f

(U_V)IG=0,

because of Stokes' Theorem and the fact that: (7.6)

if do = 0, then 0 = do for some p - 1 form b on R.

THE KAHLER CASE Consider C" equipped with the standard positive definite C-hermitian in-

ner product h. Then h = (,) - iw, where (,) is the standard positive definite real inner product on R2,(R5 CI) and (7.7)

w = 2 d? A dz' .... + 2 dz' Adz'

r

is the standard symplectic inner product on R2' (_s2 C"), which is also called the standard Kii.hler form on C". Note that w may be viewed as the sum of the complex axis lines (7.8)

Ai = 2 dzi Adzi =dxi Ady', j = 1,...,n.

Since forms of degree 2 commute under wedge product, (7.9)

p!

wP = L/,

aI

111=P

is the sum of the complex axis p-planes AI. The unitary group U(n) fixes h, and hence w. Therefore, U(n) also fixes 1,wP. Also note that this form is d-closed since it has constant coefficients.

The Kdhler Case

128

Theorem 7.10. The 2p-form

= 10 on R2n(- C") is a calibration.

The contact set equals the complex grassmannian. That is,

G(q) = Gc(p, C") C GR(2p, C").

(7.11)

Since do = 0, the theorem is equivalent to

Theorem 7.12 (Wirtinger's Inequality).

P

1

for all f E GR(2p, C"),

with equality if and only ifl; E Gc(p, C"). Corollary 7.13. Each complex submanifold of C" is volume minimizing.

First Proof (Federer): The case p = 1 is particularly easy. Let u, v denote an oriented orthonormal basis for e, i.e., 4 = u A v. Then w(u, v) = (Ju, V) < IJUI IvI = 1

(7.14)

by the C.-S. inequality. Moreover, equality occurs in (7.14) if and only if v = Ju. That is, precisely when = u A Ju is a complex line (with the orientation induced from the complex structure on the line). The general case reduces to the p = 1 case. Suppose E GR(2p, C")

and let P = span l; denote the real 2p-subspace corresponding to . Restriction of w to P yields a 2-form wp E A2P* on P. This 2-form can be put in canonical form (see Problem 4.2) with respect to the real inner product on P inherited from R" 5--- Cn: (7.15)

WP

= alai A a2 + A2a3 A a4 + ... + A,,a2r-1 A e2-r

where A1 > A2 > ... > A,. > 0 and a1, ... , aP is an orthonormal basis for

P*. Let e1i ..., eP denote the dual basis for P. Then, by (7.14), for each

j = 1,...,r, (7.16)

Aj =w(e2i_l) e2j) < 1,

with equality if and only if e2j_1 Ae2j represents a complex line. Now

lwP restricted to P is the same as p,wp, which equals Al ...APa'A Aar'. ''herefore, (7.17)

Al ...Ap.

Combined with (7.16), this completes the proof. A second proof of Wirtinger's Inequality utilizes a canonical form for a real 2p-plane in C" under the action of the unitary group U(n) .

129

Calibrations

Lemma 7.18 (Harvey-Lawson). Each oriented 2p-dimensional real subspace 1; E GR(2p, Cn) of R2n = Cn can be expressed in terms of a unitary basis a 1, ... , e as (a) for 2p < n, =e1 A (cos01 Jel +sin01 e2) A... A e2p_1 A (cos 0p Je2p_1 + sin Op e2p),

(7.19)

where the angles satisfy

0 < 91 <

< 9p _ 1

2 and

9p _ 1 < 9p < 7r;

(b) for 2p > n,

1;-ejAJe1A...Ae,.AJe,. (7.19')

A er+1 A (cos 01 Jer+l + sin 01 e,.+2) A(cosOq

Oq en

where the angles satisfy

0 < 91 <

< 9q_1 < 2

and 9q_1 < eq < 7r.

Here r=2p-nand q=n-p. Second Proof of Wirtinger: The Kahler form can be expressed as

anAJan

(7.20)

for any orthonormal basis a1, Jal, .,a', Ja" dual to the orthonormal basis e1, Je1, ..., en) Jen (i.e., e', ..., en is a unitary basis for Cn). Then, by (7.19) and (7.20), (7.21)

11

p

WP(l;) = cos 91 . . . cos Bp,

2p:5 n,

which is less than or equal to one and equal to one if and only if each 9i = 0, j = 1, ..., p. That is, if and only if = el A Jet A A ep A Jep E Gc(p, Cn).

Proof of Lemma 7.18: The proof is by induction on p. First, maximize (Ju, v) over all orthonormal pairs of vectors u, v E span 1; and set cos 91 (Ju, v) equal to the maximum value with 0 < 91 < 7r/2. Since the function f (O) _ (Ju, (cos O)v + (sin O)w) has maximum value at TP = 0 for each w E

The Special Lagrangian Calibration

130

span

with w L span{u, v}, the derivative f'(0) = 0 vanishes. However,

f'(0) _ (Ju, w), so that w 1 Ju. Similarly, w L Jv. Now define el = u. Then v = cos 01 Jel + sin 01 z, z a unit vector orthogonal to both el and Jet. Define e2 - z. As noted above, if w E spank and w L {u,v}, then w L {Ju, Jv}. Therefore w L {e1, Je1, e2, Je2}. Consequently, = elA(cos&jJe1+sin 9ie2}Ar) with rl E GR.(2p-2, V) where V = lei, Jel, e2, Je2}l. In the special case 01 = 0, the proof must be modified: = eiAJeiArl with r) E GR.(2p-2, V), where V = lei, Jell-L. Also note that in the last step of the induction, v cannot be replaced by -v so that one must allow the possibility of cos O, negative, or Op_1 < Op < a.

THE SPECIAL LAGRANGIAN CALIBRATION Consider C" with the standard positive definite C-hermitian inner product h = (, ) - icy, exactly as in the last section. However, also assume that C" is equipped with a complex volume form (7.22)

dz = dz1 A . A dz"

of norm IdzI2 = In this section, we shall examine the real n-form (7.23)

Redz

(dz + dz).

Note that SU(n) fixes this form ¢. Of course, do = 0, since ¢ has constant n coefficients on C" 25 R2". In the coordinates zi = xi + iyi, for C", (7.24)

= Re (dxl + idyl) A

- A (dx" + idy")

can be expanded out and is the sum of the 2"-1 real axis n-planes of the form ddxl Adyj with I U J = {1, ..., n} and I J I even.

Definition 7.25. A real oriented n-plane t E GR,(n, C") is said to be special Lagrangian if l; A(el A ... A e") for some A E SU(n), where = el A . A e" e1, ... , e" is the standard unitary basis for C", i.e., represents the standard oriented subspace R" C C" = R" ® iR". Let SLAG C GR,(n, C") denote the subset of special Lagrangian n-planes. An oriented n-dimensional submanifold M of C" is said to be special Lagrangian if ME SLAG for each point x E M.

Theorem 7.26. The n-form 0 _- Re dz is a calibration on C", and the contact set G(4) = SLAG consists of all the special Lagrangian n-planes.

Corollary 7.27. Each special Lagrangian submanifold M of C" is area minimizing.

Calibrations

131

Remark 7.28. For each fixed angle 9, consider the n-form Oa = Ree-iedz.

(7.29)

For example, if 9 = it/2, then 0 = Im dz. Each such 09 is also a calibration referred to as the special Lagrangian calibration of phase 0. The proof of Theorem 7.26 is via several lemmas that clarify the notion of special Lagrangian subspace. First, the weaker notion of Lagrangian is

investigated. A subspace L of dimension n in C' = R2n is said to be Lagrangian if L is totally null with respect to the Kahler (symplectic) form w. However, there are many equivalent ways of expressing this fact. Let 1 denote orthogonality with respect to (, ) = Re h.

Lemma 7.30. Suppose L is an n-dimensional real subspace of Cn, h. The following are equivalent.

(a) w (u, v) = 0 for all u,vE L. (b) The 1-form E". =1 yjdxj restricted to L is d-closed. (c) If u E L, then Ju L. I

(d) L={Ax:xERnCCn=R"ED iRn} for some A E U(n) unitary. (e) jdz(ul A ... A un)I = 1 if ul, . , u,, is an orthonormal basis for L. Further, if L = graph f can be graphed over Rn C Rn ® iRn, f : Rn -> Rn, then the following can be added to the equivalence.

(f) L = graph V is the graph of a gradient y = (0¢)(x) of a scalar valued function O(x).

Proof: (cf. Problem 5.8) Condition (a) has been taken as the definition of L Lagrangian, i.e.,

w=EdxjAdyj restricted to L should vanish. Since da = -w with n

a=

yjdxj, j=1

and restriction to L commutes with exterior differentiation, (b) is equivalent to (a).

IfL=graph f=_{(x,f(x)):xERn},then, n

alL = E.fj(x)dxj. j=1

The Special Lagrangian Calibration

132

Condition (b) says that this 1-form is d-closed, i.e.,

afj _

Ofi

axi

axi .

This symmetry is true if and only if fi (x)

axi

n,

(x),

for some scalar-valued function 0 on Rn. Thus, (b) and (f) are equivalent. To prove that (a) and (c) are equivalent use the formula (Ju, v) = w(u, v).

Since unitary transformations preserve (, ), J, and w, and since (c) is obviously valid for L = R' C Cn, the condition (d) implies condition (c). Conversely, if L satisfies (c), then an orthonormal basis u1i... , u for L can be used to define a unitary map A E U(n) sending the standard basis

el,...,en for Rf1 into u1i...,u, so that (d) follows. Let uj - Aej denote the image of ej under any complex linear transformation A. Then (dz)(ul A ...A un) _ (dz)(A(el A ...A en)) = detc A.

(7.31)

If (d) is valid, then !; = A(e1 A...Aen) with A unitary, so that I detc Al = 1, which verifies (e). To prove that (e) implies (c) and thus complete the proof of Lemma 7.30, we shall prove the following lemma.

Lemma 7.32. I(dz)(t;)I < 1 for all 1; E GR,(n, Cn), with equality if and only if L = span is Lagrangian. Proof: Assume Al; = ul A .. A u with A E R and u1, ... , un any oriented basis for L = span (not necessarily orthogonal). Define A E GL(n, C) to be the complex linear map sending the basis e 1 , . .. , en for R'i to ul, , 'fin Then, exactly as in (7.31), A

(7.33)

detc A.

However, 1

(7.34)

=IA(elA...AenAJelA...AJen)I=A2I AJI

Calibrations

133

Therefore, (7.35)

I(dz)(e) I2 = 1 A Jt; I.

The classical Hadamard inequality (see Corollary 9.33 in Part II for a proof) states that awl A ... A wkl2 < Iw1I2 . Iwkl2 with equality if and only if w1, ... , wk are orthogonal. In particular, (7.36)

If A JAI <_ lull ...

IJuil ...IJu"I

with equality if and only if

u1i...,u",Jul,...,Ju" are orthogonal (i.e., if and only if u1i ..., u is a unitary basis for C"). Now assume that uii..., u" is an oriented orthonormal basis for l;. Combining (7.35) and (7.36) yields the proof of Lemma 7.32, since L is Lagrangian if and only if u1, ... , u,,, Jut,... , Ju is orthonormal (by (c) above).

Corollary 7.37. Suppose span 1; is Lagrangian, (7.38)

E GR(n, C"). Then

(dz)(e) = detc Al detc Al',

where A is any complex linear map sending the standard basis e1,.. . , e" for R" into an oriented basis u1i ... , u" for e.

Proof. By (7.33), a(dz)(.) = detc A with A > 0. Since I(dz)(C)l = 1, the corollary is immediate.

Proof of Theorem 7.26. Consider E GR,(n, C"). By Lemma 7.32, 0(!;) = Re(dz)(t') < 1(dz)(£)j < 1 so that 0 is a calibration. If equality 0(C) = 1 holds, then I(dz)(t:)l = 1, so that . must be Lagrangian. Therefore, by condition (d) of Lemma 7.30, l; = A(e1 A ... A e,,) for some unitary map A. Corollary 7.37 implies that detc A = 1. Therefore, C E SLAG is a special Lagrangian. Conversely, if 1; E SLAG (by definition 1; = A(e1 A. Ae") some A E SU(n)), then Corollary 7.37 implies

that (dz)(£) = 1 so that 0(f) = 1. Corollary 7.37 provides several useful criterion for a manifold to be special Lagrangian.

The Special Lagrangian Differential Equation

134

Corollary 7.39. Suppose { E GR,(n, Cr) and span is Lagrangian. (a) f£ is special Lagrangian if and only if Im (dz)(e) = 0. (b) t is special Lagrangian if and only if for any (some) complex linear map A sending the standard basis e1, ..., e for R" into an oriented basis for t;:

deter A > 0

(7.40)

(real and positive).

THE SPECIAL LAGRANGIAN DIFFERENTIAL EQUATION In pursuing the analogues between the Kahler (form) calibration and the special Lagrangian calibration, one question naturally comes to mind. What is the special Lagrangian analogue of the Cauchy-Riemann equations? More precisely, suppose (7.41)

M= graph f={x+if (x):xEU0' U'P'" C R'}

is the graph of a smooth function f : U --> W. Then the question becomes: What is the differential equation(s) imposed on f by requiring that M be a special Lagrangian submanifold? The n vectors

u1=_ej +ia

(7.42)

n,

form a basis for the tangent space to M at each point. Thus, one answer to this question is the equation

-Lf ) A ((e1+i!r)A...A(en+iL)) (7.43)

-Lf (el+i 8x1)

A

(en+i

of !

This answer is not very useful. In fact, in the Kahler case C2, w a similar analysis yields the equation /e1 (7.44)

-i e2 + axl Je2i Jel + 5 e2 + y Je2/ [1+

au av Du + v (8x1 8y1 2

2

av 8u

8x1 8y')

2

135

Calibrations

which is not the elegant Cauchy-Riemann equations: (7.45)

au (9xi

= av ayi '

au = av ayl

ax'

with f = u + iv.

After this unproductive analysis, we can make a fresh start by using Corollary 7.39(a). Since

AMei+i ax

)A...Ae"+i_xP

with A E R, this corollary states: Either M or -M (M with orientation reversed) is special Lagrangian if and only if

M is Lagrangian

(7.46) and

(7.47)

(Im dz) I f ei + i

ax )

A

A (en

i

x

1

1 = 0.

The first condition, M Lagrangian, can be combined/ /with the (first order) partial differential equation (7.47) by using a potential (or generating) function 0.

Lemma 7.48. Suppose M = graph f = {x + if(x) : x E Uopen C R"} and U is simply connected. Then M is Lagrangian if and only if

f=vO

(7.49)

is the gradient of a scalar function 0 on U.

The proof of this lemma is exactly the same as the proof that parts (a) and (f) of Lemma 7.30 are equivalent. Now, if f = V O, then of/ax' is the ith column of the Hessian matrix (7.50)

Hess(O)

(9..).

That is, of/ax' = Hess(q)(ei). Therefore, (with a real) (7.51)

a M = ( e i + i ax) A ... A (en + i = (1 + i Hess(c))(ei A ... A e"),

ax"

The Special Lagrangian Differential Equation

136

so that (7.52)

(dz)(A M) = detc(I + i Hess (q )).

Theorem 7.53. Suppose M - graph f = {x+i f (x) : x E U C R' C Cn } is the graph of a smooth function over a simply connected domain U in R". Then M (with one of the two possible orientations) is special Lagrangian if and only if (7.54)

f = VO for some potential 0 on U,

and either of the following equivalent conditions hold: (7.55)

Im detc(I -I- i Hess(¢)) = 0

on U,

or

[(n-1)/2] (7.56)

E (_1)ka2k}1Hess(O) = 0 on U. k=0

Remark 7.57. The equation (7.56) will be referred to as the special Lagrangian differential equation. Here aj (A) denotes the jth elementary symmetric function of a symmetric matrix A (such as A = Hess(c)). In terms of the eigenvalues A1i ... , An of A, recall that (7.58)

aj (A) = E

III=j

Ail

Aij

If n = 2, then the special Lagrangian differential equation is not a new equation but is just i = trace Hess(O) = 0. The first nonclassical interesting case is n = 3. In this case, the equation is (7.59)

AO = MA(C),

where AO = trace Hess(O) and MA(C) - det Hess(O) is the MongeAmpere operator on ¢.

Proof of Theorem 7.53: The theorem is derived from the characterization of M as special Lagrangian given by (7.46) and (7.47). By Lemma 7.48, M is Lagrangian if and only if there exists a potential function 0 as in (7.54). The equation (7.55) has already been verified; see (7.52). Thus, it remains to show that (7.55) and (7.56) are the same equation. Let A denote the symmetric matrix Hess(O) at a fixed point x E U. The matrix

137

Calibrations

A can be put in canonical form with respect to the orthogonal group O(n). This standard result is often referred to as the Principle Axis Theorem. Its proof is very similar to the proof of Problem 4.2 (a canonical form for O(n) acting on Skew(n, R) = AR), and hence is omitted. The result says: Given

A E Sym(n, R) C Mn(R) a symmetric matrix, there exists g E O(n) such that

gAgt = D

(7.60)

is a diagonal matrix with nonzero entries the eigenvalues Al, ..., An of A.

Now detc(I + i Hess(O)) = detc(g(I + i Hess(q))gt) = detc(I + iD), and hence (7.55) is equivalent to n

Im 11(1+i)3)=0,

(7.61)

J=1

which is the same as (7.56).

Theorem 7.62. Suppose M is a connected oriented n-dimensional real submanifold, of an open subset U of C", which is defined implicitly by smooth functions on U:

M E {z E U : f1(z) = cii ..., fn(z) = cn}.

(7.63)

Then M (with one of the two possible orientations) is special Lagrangian if and only if n

8fafg + afp afq \11 _ 0

(7.64)

ky-

C

OZk OZk

and

(7.65)

for all points zEM.

'

8zk 8xk

Im I detc (i

I

I=0

//

Proof: The normal n-plane to M is Lagrangian if and only if the tangent n-plane to M is Lagrangian (c.f., condition (c) of Lemma 7.30). This normal n-plane is spanned by the vectors (7.66)

up = 2 I O p+ i ayp

\

a

138

Examples of Special Lagrangian Submanifolds

Note that (7.67)

Jug = 2 (i

4) = 2 aafq

a

ay

g

z. .

Therefore, M is Lagrangian if and only if (7.68)

(up, Jug) = 0 for each 1 < p, q < n.

Equivalently, (7.68')

Re n k=1

afp afg = 0, 194 axk

which is Equation (7.62). Now assume M is Lagrangian. Then

p=1,...,n, Jup=i afp ais a basis for the tangent space. The complex matrix A=

(iaafp

\

\/ )

xg

maps the standard basis e1,. .. , e for Rn to the basis Jut, ... , Jun for N H H M, i.e., A M= Jul A ... A Jun with A > 0. Therefore, Im (dz) (A M) AIm detc A by (7.31). Thus, A satisfies (7.65) if and only if d M is special Lagrangian by Corollary 7.39.

EXAMPLES OF SPECIAL LAGRANGIAN SUBMANIFOLDS Rather than discuss general classes of examples, specific examples are dis-

cused in this Section. The first example is invariant under the action of the n - 1 torus = {diag(e191, ..., eie^) : 01 + 02 + + On = 0} contained in SU(n).

Theorem 7.69. Let MM denote the locus of the equations: (7.70)

lzj l' - Ix1l2 = cj,

= 2, ..., n,

and (7.71)

Re z1 ... zn = c1

if n is even,

Calibrations

139

or (7.71')

Im z1

.

if n is odd.

zn = c1

Then M, (with the correct orientation) is a special Lagrangian submanifold.

This provides perhaps the simplest example of an area minimizing object (with a singularity at the origin) that is not a real-analytic variety, that is, not the zero set of a family of real analytic functions. The key property distinguishing a real analytic zero set is that if it contains a half-ray then it must contain the full line. This is because a real-analytic function of one variable t that vanishes for t > 0 must vanish identically.

Corollary 7.72. Suppose n is odd and let

M+={(re'B',... re'°'):01+...+0,=0and r>0} M- _ { (re`B' ... , re'e-) : 01 + .....{- On = it and r > 0} Then the cones M+ and M- are both special Lagrangian, and hence both volume minimizing. Note that M+ and M- are disjoint cones with one the image of the other under the map - Id, so that neither M+ nor M- is a real-analytic variety.

Proof: Set cl =

= cn = 0 in Theorem 7.69 and note that Mo = M+ U M- on Cn - {0}. Proof of Theorem 7.69: Set fl(z) = Re inz1 I z1 1 2 for q = 2,

zn and fq(z) =

Izg12

... , n. Then (see Problem 2a) n

(7.73)

detc t Z afp/

(-1)n j_1

Now verify (see Problem 2a) the conditions (7.64) and (7.65) of Theorem 7.62.

The next example provides a useful tool for the study of pairs of minimizing planes in euclidean space; see the following section entitled Angle Theorem. Given a1, ..., an > 0, let (7.74)

P(a, y) = y2 [(1 + aly2) ... (1 + apy2) - 1] .

Examples of Special Lagrangian Submanifolds

140

Now for each k = 1, ..., n, and each y E R, define (7.75)

rk(y)e:ek(y)

Zk(y) =

with (7.76)

rk(y) =

ak 1 + y2

and y

(7.77)

Bk(y) = ak

o

dy

(1 + aky2)

1'(a, y)

It is convenient to use the calibration 4

Imdzl A

A dz" rather

than Re dz.

Theorem 7.78. The manifold Ma

{wEC" :uwk=tkzk(y) withyER, IER" and Etk=1 ll

is

k=1

= Im dz) special Lagrangian (if correctly oriented).

This example (see Lawlor [11]), introduced by Lawlor and completed by Harvey, is used in the proof of the Angle Theorem 7.134.

Remark 7.79. Let r denote the curve r = {8(y) : y E R},

(7.80)

defined by (7.77). Let (7.81)

Pe = eie R" = J (tie1B1 ,

t E R" }

denote the n-plane in C" obtained by rotating R" by e'9. The key property of the special Lagrangian manifold M. is that each nontrivial intersection (i.e., 0 E r) of Ma with PB is the boundary of a bounded open subset of P9. In fact, for 0 E r, (7.82)

Ma n Pg .

is an ellipsoid.

(tle181

... t"et9") E

,2(y)

1

Calibrations

141

Proof of Theorem 7.78: In addition to giving a proof of Theorem 7.78, we shall sketch the proof that the submanifolds Ma are the only possible special Lagrangian submanifolds with each nontrivial intersection Ma fl Pe a compact hypersurface of P8, thus removing some of the mystery of this example.

If M is to consist of the union of hypersurfaces in a family of the nplanes P9, and M is required to be Lagrangian, then M must be of the form (set wj = Rj eaei )

M(r, h) = w E C" :

(7.83)

R?

j=1

d8j

_A

ds

ds

for some curve r in the 9-space and some function h(s) defined on r (here 9(s) parametrizes r). This can be reformulated as a general fact about Lagrangian submanifolds with degenerate (one-dimensional) projection onto one of the Lagrangian axis planes, using the alternate symplectic coordinates pj = 2R and qj = Bj, j = 1, , n. (Note that dpj Adgj = dxjndyj. See Harvey-Lawson [9] for the proof of (7.83) (cf. Problem 3). r1,

Note that for 9 E r, EB = M(r, h) fl Po is an ellipsoid with radii , r defined by 1

(7.84)

r?

_

dOj

A

if

dO

A

>0 ,

and that otherwise the hypersurface M(r, h) fl Pa is not compact. The vectors (7.85)

with tk

Vk= (-tkz1,0,...,0,t1zk,0,...,0),

Rk/rk and zk = rkeiek, provide tangent vectors to the ellipsoid

EBwEC':wk=tkzk, since N (7.86)

k =2,...,n,

7tk=1}

(tIz1i...,tnz,l) is normal to E9. In addition,

V1 = (tizi, ...,

is also tangent to M. Let L denote the complex linear map sending ej to V j, n, when e1, ... , e is the standrad basis for R" C C'. Since M is Lagrangian, M is 0 = Im dz special Lagrangian if and only if (7,87)

detc L = is

with A > 0,

Examples of Special Lagrangian Submanifolds

142 because of Corollary 7.37.

One can compute that (Problem 2(b)) (7.88)

detc L = ti-1

I1I+ ...

2

zlzri I2 1 (zl ... Z") .

Thus, M is special Lagrangian if and only if each zkzk is of the form (7.89)

zkzk = fki z1

z,,, with fk real and > 0.

In polar coordinates zk = rkeiek, with

ik__10k, Equation (7.89)

becomes (7.90)

rkrk = fkrl . . r sin

and

(7.91)

rkOk = fkr1

r,i cos'.

Since rk(d9k/dh) = 1, Equation (7.91) implies that all the fk must be equal. Reparametrizing, we may assume fk = 1, yielding the system (with

0=E9 ) (7.92)

and

(7.93)

a rk ek = r1 ...rn cos

Let cj = zz (0) > 0, i.e., rj(0) = cj > 0 and 0j(0) = 0, j = 1,. .. , n denote the initial conditions at time t = 0. It remains to solve this system (7.92), (7.93). First, note that if

r(t), 0(t) is a solution, then r(t) = r(-t), 0(t) _ -0(-t) is also a solution. Hence, we need only consider t > 0. Since rkrk is independent of k. (7.94)

rk=ck+u and u(0)=0

for some function u. Also (7.92) and (7.93) imply that r1 derivative zero so that (7.95)

r1rcos0=c1c,a.

r cos 0 has

Calibrations

143

Remark. Alternatively, note that the solution curves to (7.92), (7.93) provide the flow lines of the Hamiltonian vector field associated with the Hamiltonian function H - rl r cos = Re z1 - - z (it is interesting to compare with Theorem 7.69). Since B(0) = cl . c,,/c,2E > 0, the functions Ok(t) and ii(t) must be strictly increasing for t small. Since u = 2rkrk = 2r1 r sin 0, it is also strictly increasing for t > 0 small. Now, in order forr to vanish, sin V = 0 or cos 7P = 1, i.e., r1 .. r = c1 - - c by (7.95). This proves that Tk never vanishes, and hence it is strictly increasing for all t > 0. Assume t > 0 and define y by it - y2. Since ii = 2y' = 2rkik = 2r1 - - - r sin 0, we have -

-

yy = r1

(7.96)

Substituting cos system (7.97)

(7.97')

r sin 0. r into (7.96) and (7.93) yields the

cl

(Cl+y2)...(C2i+y2)-C2...cn y2

1

k = 1, ...

Bk = c1 ... c, / (c2r + y2)

n.

Eliminating t and replacing ck by 1/ak yields (7.98)

ak

flak

P(a, V)

(1 + aky2)

dy

as desired.

Remark 7.99. Equation (7.97) implies that = fy (7.100)

dy

t

M'

where (7.101)

[ Q(y) = y-2 (C1 + y2) ... (Cn + y2) - C2 ... C21

.

If n = 2, the integral diverges as y oo, and the system is solved for all time; while if n > 3, the integral converges and the system is solved for t E [-T,T], where the terminal time T is given by (7.102)

T = cO JO

dy

Q(c,y)

Associative Geometry

144

Note that as cl approaches +oo, the terminal time T approaches zero.

ASSOCIATIVE GEOMETRY In this section, consider the 3-form 0 on Im 0 = R7 defined by (7.103)

Ox, y, z) =- (x, yz) for all x, y, z E Im 0

that was examined in Chapter 6. Recall that 0 E A3(IMO)*, since vanishes when any two of the variables x, y, z are set equal. The analogue of the Wirtinger inequality for 0 is most elegantly described by an equality.

Theorem 7.104 (Associator Equality). For all x, y, z E Im O, (7.105)

O(xAyAz)2+ 1I[x,y,z]12= IxAyAz12.

Proof: Recall that the triple cross product satisfies (7.106)

Ix x y x zj _ Ix A y A zj,

(7.107)

Rex x y x x = O(x,y,z),

(7.108)

Imxxyxz= I[x,y,z].

The associator equality (7.105) then follows.

Definition 7.109. If 1; E GR.(3, Im 0) is the canonically oriented imaginary part of any quaternion subalgebra of 0, then l; is said to be an associative 3-plane. Let ASSOC denote the set of all associative 3-planes. A real oriented 3-manifold M C ImO with ME ASSOC at each point x E M is said to be an associative submanifold of Im O. Given l; = ul A U2 A u3 with u1, u2, u3 an oriented orthonormal basis

fort E GR(3, Im O), Artin's Theorem 6.39 implies that the associator [ul, u2i u3] vanishes if and only if span{1, ul,'u2i u3} is a subalgebra of 0, algebra isomorphic to H. Consequently, given t;' E GR(3, Im O), (7.110)

fit; E ASSOC if and only if [u1i u2, u3] = 0

for some (or equivalently all) basis u1, u2, u3 for span £. Therefore, as a consequence of the associator equality we have that ¢ is a calibration and

that (7.111)

ft; E ASSOC if and only if q5(t;) = fl.

This proves the next result.

145

Calibrations

Theorem 7.112. The associative form 0 E A3(ImO)* is a calibration with the contact set G(O) = ASSOC.

Corollary 7.113. Each associative submanifold of Im 0 = R7 is volume minimizing.

Definition 7.114. An oriented 4-plane 71 E G-(4, Im 0) is said to be coas*0 E A4(Im O)* sociataive if the 3-plane 771 is associative. The 4-form is called the coassociative calibration.

Theorem 7.115. The coassociative 4-form 0 E A4(ImO)* is a calibration, and any real 4-dimensionalsubmanifold of Im 0 whose normal 3-plane (at each point) belongs to ASSOC is volume minimizing.

This theorem can be deduced from Theorem 7.112 (see Problem 4). Interesting systems of partial differential equations arise in associative geometry (see Problem 5) and coassociative geometry, which in some ways are analogous to the Cauchy-Riemann equations. However, we turn now to examples. A particularly interesting example is the Lawson-Osserman coassociative cone.

Theorem 7.116. The cone (7.117)

Vr5xix

M

2

xl + xi:xEH

on the graph of the Hopf map

rl(x) = 2

x%x

xl

where i maps the unit 3-sphere S3 C H to the 2-sphere S2 of radius v'5/2 in Im H, is coassociative, and hence volume minimizing.

Remark 7.118. The function rl : H -+ ImH defined by rl(x) =

2

jxj

is Lipschitz continuous but not of class Cl (continuously differentiable). This makes the fact that its graph is volume minimizing particularly interesting, since a classical result of C. B. Morry says that if the graph of a Cl-function is volume minimizing, then the function is real analytic.

The Angle Theorem

146

A 3-dimensional cone M in R7 = Im O with vertex at the origin is a real 3-dimensional submanifold of R7 - {0} of the form

M-{tx:tER+andsEL}, where L is a 2-dimensional submanifold of the unit 6-sphere S6 C IMO. The real surface L C Ss is called the link of the cone M. Associative cones M have an elegant description in terms of their links L given by Corollary 7.121 below.

Lemma 7.119. At each point x E Ss C Im 0, left multiplication by x, denoted L, is a real linear map from the tangent space T,S6 into itself and Lx = -1 so that Lx is an almost complex structure on T, S"6.

Proof: If u E TxS6, then ux + xu = 2(u, x) = 0. Therefore, ux E Im O. Since (xu, x) _ (u,1) = 0, this proves xu E TiS6.

Lemma 7.120. A two plane i E TxS6 is almost complex if and only if the three plane x A 27 is associative.

Proof: p= u A Lxu implies x A 77 = x A u A xu is associative. Conversely, if x A u A v is associative then v = xu (if x, u, v are orthonormal).

Corollary 7.121. A 3-dimensional cone in Im 0 is associative if and only if the link L is an almost-complex curve in S6 C Im O.

R. Bryant proved that each compact Riemann surface occurs as the link on some associative cone. The proof, using the Newlander-Nirenberg Theorem, is beyond the scope of this book.

Theorem 7.122 (Bryant). Given a compact Riemann surface S there exists a almost-complex embedding of S in the 6-sphere S6 E Im O.

THE ANGLE THEOREM This beautiful result uses a general class of calibrations as well as special Lagrangian geometry in its solution. But before presenting the result, the motivation will be discussed in a heuristic manner. A standard method for understanding a singularity (of, for example, a real analytic variety M) is to study the associated "tangent cone." At a point of M, a "tangent cone" is obtained by considering the intersection of M with successively smaller balls (centered at the point in question) and then enlarging each ball with a rescaling from the small ball to the

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147

unit ball. In the limit, one obtains a cone in the unit ball, which can be considered a cone in Rn, "the tangent cone to M at the point." Thus, it is important to have a good understanding of tangent cones. In associative geometry, Corollary 7.121 reduces the study of (tangent) cones to the study of complex curves in S6 C ImO. In the study of general area minimizing cones, one of the simplest questions one can ask is:

When is the union of a pair of oriented p-dimensional (7.123)

planes = , i E GR(p, R") minimizing?

This question has a remarkably simple answer, as conjectured by Frank Morgan. This answer involves certain characterizing angles 01,. . ., Bp which distinguish one pair of planes l;, q from another such pair 4', rl' (see (7.133') below).

Lemma 7.124. For each pair of oriented planes 4, 71 E GR(p, R2P), there exists an orthonormal basis e 1, ... , e2p for Rep and angles (7.125)

0<01

8p,

Bp <7r-Op_1i

so that 1; = el A

(7.126)

(7.127)

A ep,

77 _ (cos 01 e1 + sin 01 ep+1) A .. A (cos Bp ep -}- sin Op e2p) .

Remark. The statement and proof are easily adapted to the more general case where span 1 = Rm x {0} and r) e GR(p, R' x R). For example, if m < p < n, then rl can be expressed as 27 =(cos01 el+sin01

em+sin Om fm)Afm+iA...Afp

Proof of Lemma 7.124: Choose e1 E P - span 4 and u1 E Q - span rl, both unit vectors, so that (e1, u1) _- cos 01 is maximized. (Note that 0 < (el, u1) < 1 so we may choose 0 < 01 < it/2.) Since (el, ul) is maximized, the orthogonal projection of u1 onto P must be equal to cos01 e1. In particular, (7.128)

u1 = cos 01 e1 + sin 01 ep+l

148

The Angle Theorem

defines a unit vector ep+l E Pl. The equation (7.128) says that (7.129)

if eEP and e ..L e1, then elul.

By symmetry, (7.130)

if u E Q and u 1 ul, then u 1 e1.

Next choose e2 E P, e2 1 e1 and u2 E Q, u2 1 u1 so that (e2, u2) cos 02 is maximized, with 0 < 01 < 02 < 7r/2. Because of (7.130), the projection of u2 onto P only has an e2 component. Similarly, the projection

of u2 onto P1 must be orthogonal to ep+1. Thus, (7.131)

U2 = cos 92e2 + sin Beep+2

defines ep+2 E P1, ep+2 1 ep+1. Continue in this manner. At the last step, choose ep E P and up E Q so that e1, ..., ep is an oriented orthonormal basis for , and ul, ... , up is an orthonormal basis for 17. Since the angle Bp defined by cos 9p = (ep, up) must satisfy I cos Bp < cos 9p-1, the angle 9p

can be chosen so that 9p-1 < 0p < it - 9p-1. Intuitively, the union of the pair!;, 77 is volume minimizing if and only

if a and -r/ are not too close together. (Picture two large disks very close together, almost parallel, and with opposite orientation. Then a narrow strip will have the same boundary and smaller area.) If 91i ... , Bp are the characterizing angles for the pair £, 71, then the characterizing angles for the pair !;, -rl are given by (7.132)

01=01i...,/ip-1Bp-1,

and

?'p=7r-9p.

To prove this fact, replace up by -up in the proof of Lemma 7.124. Thus, -up = - cos 9pep - sin 9pe2p = cos ,ipep - sin ipe2p. Finally, replace e2p by -e2p and note that Op < 7r - Op_1. The next theorem states that the union off and 71 is not volume minimizing if and only if i;, -rl are "close together" in the precise sense

that (7.133)

'Y1+.....FVjp <9r,

or, in terms of the characterizing angles 01i ..., 9p for (7.133')

<9p.

This result was conjectured by Frank Morgan.

and al,

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149

Angle Theorem 7.134 (Lawlor-Nance). The union of a pair 1, 77 E GR(p, Rep) of oriented p- dimensional su bspaces of RP is volume minimizing

if and only if the characterizing angles satisfy the angle criterion: (7.135)

(or equivalently Y'1 +

Op < 01 + - - + Op_1

- +'p > ir).

Proof: First, assume that , rt do not satisfy the angle criterion (7.135), or equivalently, assume that the characterizing angles 451, ..., ipp for the pair 4, -r7 satisfy (7.133) (i.e, 4 and -9 are "close"). We must show that 4 U rj is not volume minimizing. It is convenient to make 4 and -77 symmetrical about a fixed p-plane.

Let 9j denote VJj /2, j = 1, - - -, p, and pick a new orthonormal basis el, ., e2p so that (7.136)

4 = 4(8)

and

- 77

where 4(e) is defined to be (7.137)

4(8) _ (cos B1 e1 + sin B1 ep+1) A .. A (cos 0 ep + sin Op e2p )

The hypothesis (7.133) now takes the form p

E 8j < 9

(7.138)

j=1

Identify RIP with Cp by identifying the ej axis in Rep with the xj axis in Cp (j = 1, ... , p), and identifying the ep+j axis in Rep with the yj axis in CP (j = 1,. .. , p). Assume, for the moment, that for some choice of the constants a =- (al, ..., a,,), the special Lagrangian manifold M. constructed in Theorem 7.78 has nontrivial intersection with PB span 4(B). Then one can show that the portion M of M cut off by Pe span 4 and P_B _ span r7 satisfies vo1(M) = (7.139)

J Im dz = J Imdz - J ve

v-e

Imdz < vol(UB) + vol(U_a),

where Ue denotes the solid ellipsoid in Pe = span 4(0), with the boundary the ellipsoid MaflPe. Thus UB -U_B, and hence 4Ur7 is not area minimizing.

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150

Note that eae, ... ese'.

(7.140)

and -(Imdz)(l (-B)) equal sin> 0,,, which Therefore, both is strictly less than one by the hypothesis (7.138). This proves the strict inequality in (7.139). It remains to prove that the constants al, ... , an > 0 can be chosen so that Ma fl PT is nonempty. Fix a value of y > 0 and consider the function B(y) defined by (7.98) as a function 8(a) of the parameters a, i.e., set v

9(a) = ak

(7.141)

dy

Jo (1 + aky2)

P(a, y)

with (7.142)

P(a, y) = y-2 [(1 + aly2) ... (1 + any2)

It remains to prove that the map a F--* 0(a) is surjective onto the set {0 : E O j < it/2 with 0 < 0j, j = 1, ... , n}. The proof is by induction on

n. If n = 1, then 0(a) = arctan (/y) surjects onto 0 < 0 < 7r/2. Note that a i- 0(a) extends continuously to the closed positive quadrant since setting any one of the variables aj equal to zero in the formula for 9 simply results in the formula for 0(a) in fewer variables. Because of (7.93), the inverse image of the hyperplane (in the positive quadrant) (7.143)

-

H,p = {8 : E 0j = zb and 0 j > 0,

1, ... , n}

under the map 9(a) is contained in the surface

S,, =

{a:(1+aiy2)...(1+any2)=cos-2

It suffices to show that (7.144)

9 : S,p -> H,p is surjective for 0 < 0 < it/2.

The proof of (7.144) is obtained by standard topological methods using the notion of degree of a map. By the induction hypothesis, the degree of the boundary map (0 restricted to the boundary), 0 : 8S,p -+ BH,p, can be shown to be one. Now a standard topological result says that the map 0 : S,1, --} H,p must have the same degree one and hence is surjective.

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151

To prove the remaining half of the theorem, assume that the characterizing angles 01 i ... , Bp for , q satisfy the angle criterion (7.135). Assume that the orthonormal basis e1 ,-- , e2p of Lemma 7.124 is the standard basis for Rep. Let z _ (x, y) with x, y E RP denote coordinates. Following D. Nance [13], consider the p-form (7.145)

c'u _ Re (dx1 + uidyl) A ... A (dxp + updyp) E AP (R2p)*

for each p-tuple ul, ..., up E S2 C ImH of unit imaginary quaternions. Note that if each Uk = i for k = 1, ...,p, then 0 is the special Lagrangian calibration on Cp = Rep.

Proposition 7.146. Each ¢ is a calibration. Moreover, (7.147)

1; (B) _ (cos 01 e1 + sin 01 ep+1) A

A (cos Bp ep + sin Op e2p)

belongs to if and only if the quaterion product v1 ...vp equals 1, where vj = cos O2 + sin Oj uj E H, j = 1, , p.

Proof: Note that (7.148)

q

(

( 9 ) ) = Rev1 ...vp :5 Ivl ...vpj = Ivii ...Ivpl = 1,

with equality if and only if (7.149)

It remains to show that the maximum of ¢,, over the full grassmannian cannot be larger than the maximum of 0. over the special p-planes 1;(B). This is a consequence of Frank Morgan's Torus Lemma, which is discussed below.

First we use this proposition to calibrate both f and £(9) with 0,,. We must find unit imaginary quaternions U]. .., up so that v1 vp = 1, 1, ... , p. If we choose vl,... , vp to be of where vp = cos Bj + sin Bj uj, the form (7.150)

V1 = W1 'w2, V2 = W2 W3, ... , vp = Wpw1,

with w1 i ... , wp unit quaternions then v1

vp = 1 is automatic. The only

other requirement on v1, ..., vp is that (7.151)

Revj = cos9j,

j = 1,...,p.

The Angle Theorem

152

The u1's are defined by Im vj u --JImvj I

(7.152)

p-

Note that with vi given by (7.150) the final condition (7.151) is that (7.153)

(w1iw2) =cos01, ..., (wp,wl) =cosop.

For simplicity, suppose w1i ... , wp E S2 are unit imaginary quaternions. Then the problem is reduced to the following standard problem in spherical

< op, op < r - op_1, trigonometry: Given real numbers 0 < 01 < satisfying the condition op < 01 + . + Op-1, construct p disjoint points Wi, ... , wp on the 2-sphere so that the spherical distance between the points equals the 0's. That is, (7.154)

d(wl, w2) = oi, d(w2, wa) = 03, ..., d(wp, wl) = op.

A picture of a p-sided spherical polygon is helpful (the sides are arcs of great circles).

The useful torus lemma can be described as follows. Let T denote the subset of GR,(p, Rep) consisting of all p-planes of the form 4(0) = (cos 01 el {- sin 01 ep+1) A . . A (cos op ep + sin op e2p).

Given a calibration 0, let GT(c) = {l;(o) E T : 0(1;(0)) = 1} denote the torus contact set as opposed to the (full) contact set G(O) = {l; E GR(p, RZp) : q5(4) = 1}

of Definition 7.1. In coordinates, ej = 8/8xj and dxj = et j = 1, , p, while ep+j = 8/8yi and dy' = ep+,i j = 1, -,p. A form 0 E AP(RZP)* in the span of the forms dx' A dy1, where I and J are disjoint and I U J = {1, ...,p} is called a torus form.

Torus Lemma 7.155 (Morgan). A torus form 0 E Ap(R2p)* is a calibration if and only if 0(t;(O)) < 1

for all t;(o) E T.

Proof: The proof is by induction on p. Renormalize 0 so that M = 1 is the maximum value of 0 on G(p, R2p). Choose t; E G(p, R2p) to be a

153

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maximum point, i.e., 4)(1;) = 1. We must show that the torus contact set GT(4') is nonempty. Put 1; in canonical form (see the remark following Lemma 7.124) with respect to Rep = R2 x R2p-2, with R2 x {0} the span of el and ep+l . That is,

(cos01 u1+sin01 fl)A(cos02 u2+sin02 where 0 < 01 < 02 < a/2; u1i u2 is an orthonormal basis for R2 x {0} and fl, , f2p-2 is an orthonormal basis for {0} x R2p'2. Since 0 is a torus form, ¢ = ei A4D+ep+i A with (D and 0 torus forms on Rep-2. Expressing el and ep+l as linear combinations of ui and u2 yields

0= uina+u2A/i, with a and /j torus forms on R2p-2. Now 4)(1;) = acos01sin02+bsin01cosB2i

where a = a(f2A...Afp) _ 0(uinf2A .Afp) and b = -p(fl A f3A...A fp) _ 4)(flAu2Af3A...Afp). Therefore, 1 = !5(1;) <

a2 cost 61 + b2 sin2 01 < max{lal, IbI} < 1,

so that we must have equality. Therefore 02 = a/2-01 i so that with 0 = 01,

e= (cos 0 ul + sin 0 fi)A(sin

Afp

In particular, a = 1 and/or b = 1 so that by the induction hypothesis either the torus form a or the torus form Q has attained its maximum value 1 on the p - 1 torus, say a(q) = 1. Then 4)(ui A'i) = a(t7) = 1, proving that 0 attains its maximum value 1 on the p-torus T. The Torus Lemma has several interesting reinterpretations and consequences. Note that the space of torus forms is the dual space of span T, so the Torus Lemma can be stated as (7.156a)

for all 0 E (span T)*.

ET

max,,) 001

The Angle Theorem

154

Given a set A, let ch A denote the convex hull of A. Let P denote orthogonal projection onto the linear subspace span T of the vector space APR2P. The Torus Lemma implies (7.156b)

P(ch(G(p, RIP))) = chT

and (7.156c)

ch T = ch(G(p, R2P)) fl span T.

In fact, both (7.156b) and (7.156c) are equivalent to the statement (7.156a) of the Torus Lemma. Assuming (7.156a) we prove (7.156b). Of course,

ch T C P(ch (G(p, R2P))). Now suppose E G(p, R2P). If PC V ch T, then by the Hahn-Banach Theorem there exists 0 E (span T)* with ¢(r7) < 1 for all n E ch T but with O(1;) > 1. This is impossible by (7.156a). Next note that (7.156b)

implies (7.156c) and that (7.156c) implies (7.156a). For a torus calibration the torus contact set GT(4) actually determines the full contact set.

Corollary 7.157. Given a torus calibration 0 and 1; E G(p, R2P), E G(0) if and only if Pi; E ch(GT(q5)).

Proof: If P. E ch(GT(0)), then Pt; = EAj = 1. Therefore, 4(1;) = ¢(Pl;) = E G(O), O(PC) =

Ail j with each j E GT(4) and Aj

1. Conversely, given

1.

By (7.156b), P E ch T. It remains to show that

{'iEchT:¢(7)=1}Cch(GT(¢)). Since the extreme points of a convex set generate the convex set via convex

combinations, it suffices to show that the extreme points of {q E ch T : 0(77) = 1} are contained in ch(GT(O)). However, if rl is such an extreme point, then one can easily verify that 77 is also extreme in ch T. Since the extreme points of ch T are just the elements of T this proves that 17 E GT(O)-

An alternate method for determining G(O) from GT(4), due to Frank Morgan, is described next.

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155

Suppose w E A2 R4 can be written as w = w l A w2 + w3 A w4, where w1, w2, w3, w4 is an orthonormal basis. Let u1i u2, u3, u4 denote the dual

basis. Then w has contact set G(w) = {f E GR,(2, R4) : = u A Ju E Gc(1, C2)} = CP1 with respect to the complex structure on R4 defined by Jet = e2, Je3 = e4 More generally if 0 E AP (R')* is of the form ' = wAw5 A AwP-2, with w = w1 A w2 + w3 A w4 as before, then G(O) = G(w) A u5 A

A up-2

will also be referred to as a CPI-contact set.

A subset B C G(p, R') with the following property will be called CP1-closed.

If K is any CP1-contact set containing two points of B, then the whole (CP1-contact) setK is a subset of B.

Proposition 7.158. The contact set G(c/) of a calibration ¢ E AP(Rn)* is CP1-closed.

This is a consequence of the "First Cousin Principle"; see Problem 8. Note that an arbitrary intersection of CP' closed sets is again CP1-closed.

Proposition 7.159. Suppose 0 E AP(R2P)* is a torus calibration. The contact set G(O) is the smallest CPI-closed set containing the torus contact set GT(O).

Proof: Given any set B with the two properties; B is larger than GT(cf) and B is CP'-closed, we must show that G(q5) is a subset of B. Again, the proof is by induction on p. Suppose C E G(q5), i.e., O(£) = 1. We must show that 1; E B. We proceed exactly as in the proof of the Torus Lemma to show that

l:= (cos9u1-I-sin0 f1)A(sin0u2-1-cos0 f2)Af3A.. A fp.

Case 1:If9=0,then= u1Af2A...Afpand0(.)=a(f2A...Afp)=1, so that AfpEG(a). Let B' _= {rl E G(p- 1, R2P'2) : u1 Al E B}. Then B' contains GT(a) and B' is CPI-closed. Therefore, by the induction hypothesis B' contains G(a). Since Oo E G(a) C B', this proves that 1; = u1 A rlo E B.

Case 2: If 0 < B < a/2, then a = b = 1 and both u1 A f2 A

.

A fp

A fp belong to G(O). As in the previous case, by induction both belong to B. Both also belong to the CP1-contact set K = G((u1 A f2 + f1 A U2) A f3 A A fp). Therefore, the hypothesis on B implies that B contains all of K. Since 1; E K the proof is complete. and f1 A u2 A f3 A

156

Generalized Nance Calibrations and Complex Structures

GENERALIZED NANCE CALIBRATIONS AND COMPLEX STRUCTURES The Nance calibrations 0,,, defined in (7.145), provide a large class of "torus calibrations" using the quaternions, or more precisely, a collection of unit imaginary quaternions ul, ... , up. This construction can be generalized as follows.

Consider the vector space M2n(R) of matrices with positive definite inner product given by

(A, B) = 2n trace AY. Let P : M2,,(R) -+ span 1 denote the orthogonal projection onto the span of the identity matrix 1 E M2,,(R). The orthogonal complex structures on R2n are given by

Cpx(R2n) - {J E M2n(R) : Jt = J-1 and J2 = -1}

.

Definition 7.160. The p-form 0. E AP(R2p)* defined by (7.161)

ou = P [ (dx1 + u'dy1) A ... A (dxP + updyp)] ,

where u1i ..., up E Cpx(R2n), is called a generalized Nance form.

Proposition 7.162. Each generalized Nance form ¢u is a calibration. Moreover, (7.163)

1;(B) E G(giu) if and only if Al . . Ap = 1,

where Aj = cos Bj + sin 6j uj E M2, (R), j = 1, ... , p.

Proof: If u E Cpx(R2s) C M2n(R) and B E R, we set

A=cos0+sin6u. Then At = cos 6 - sin B u, and

AAt = cost B + sine B = 1,

so that A E O(2n) is also an orthgonal transformation. If u1, ..., up E Cpx(R2') and 01 i ... , Bp E R are given, define Aj = cos Bj + sin Bj uj, j = 1,.. . , p. Then each Aj E O(2n) and hence the product Al . Ap E O(2n).

Calibrations

Therefore, Al . note that

157

AP is a unit vector in M2n(R). To complete the proof, P[(cos Bl + sin B1 u1) .. (cos Bp + sinO up)]

= (A, ... Ap, 1) < 1, with equality if and only if Al

Ap = 1; and then apply the Torus Lemma.

In order to understand why this construction-using complex struc-

tures Cpx(R2n) rather than unit imaginary quaternions S2 C ImHgeneralizes the Nance construction, we examine the set Cpx(R2n) of orthogonal complex structures in more detail. Since each complex structure determines a canonical orientation on R2n, the set Cpx(n) naturally divides into two disjoint sets of complex structures

Cpx(n) = Cpx+(n) U Cpx (n),

(7.164)

with Cpx+(n) the subset of Cpx(n) consisting of complex structures that induce a given orientation on R2n, and Cpx (n) the complex structures that induce the opposite orientation on The group O(2n) acts on Cpx(n) by R2n.

(7.165)

J E Cpx(n) maps to gJg'1 E Cpx(n) for each g E O(2n).

The isotropy subgroup of O(2n) at a particular J E Cpx(n) is just (7.166)

U(n) = GL(n, C) fl O(2n),

where GL(n, C) = {A E GL(2n, R) : JA = AJ}. Given a complex structure J E O(2n) on R2n, there exists a (complex) unitary basis

for C' = R2n. Thus,

(7.167)

el,... , en, Jel,... , Jen

is an (real) orthonormal basis for R2n. With respect to this orthonormal basis the matrix for the complex structure J is (7.168)

J = ( _01 01)

where Rn = V ® JV with V = spanf{el,..., en} induces the 2 x 2 blocking of J. This proves that 0(2n) acts transitively on Cpx(n). Since O(2n) has two connected components and U(n) is connected, the quotient O(2n)/U(n) has two connected components. In summary we have the following proposition.

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158

Proposition 7.169. The set Cpx(n) of orthogonal complex structures on R21 has two connected components Cpx+(n) and Cpx (n), with

Cpx(n) = O(2n)/U(n), Cpxt(n) = SO(2n)/U(n).

(7.170)

In low dimensions, the space of complex structures Cpx(n) can be represented in terms of the quaternions and octonians. For example, Ru, right multiplication by a unit imaginary quaternion, is an orthogonal complex

structure on H - R4. Proposition 7.171.

Cpx+(2)={Lu:uES2CImH}=S2 and

Cpx-(2)={Ru:uES2CImH}= S2, where H has the standard orientation.

Proof: First note that, for each u E S2 C ImH, Lu E Cpx+(2). This is because the complex structure Li induces the standard orientation on H. Hence, each L. also induces the standard orientation on H, since S2 is connected.

Now suppose J E Cpx+(2) is given and identify R4 = H.

Define

J 1 and choose a unit vector e2 1 1, e1. Then, because of the CayleyDickson process, 1, e1, e2 and e3 = e1e2 is an oriented orthonormal basis for H. Now Jet must equal ±e3. However, since 1, J1, e2, Jet must induce e1

the standard orientation on H, Jet = e3. This proves J = Let. Similarly, Cpx (2)

{Ru : u E S2 C ImH} - S2. J

Alternatively, Proposition 7.171 can be proved using the fact (7.170) that each J E Cpx+(2) is of the form J = gLig-1 with g E SO(4). Because of (1.34'), g(x) = pxq' for some unit vectors p, q E H. Thus J = Lpip-. Replacing the quaternions H by the octonians 0, each Ru with u E

S6 C Im 0 determines a (6-sphere of) complex structures on Rs = 0 (compatible with the standard orientation on 0). However, Cpx+(O) SO(8)/U(4) has dimension 12, so that Ss is too small.

Definition 7.172. Given l; = u A v E G(2, 0), let J£ = RuR-). z First, note that Jg is well-defined (independent of the choice of u, v with = u A v) since the right hand side vanishes when u = v. Second, the basic octonian formula R,X + R,, R,-. = 2(u,v)

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159

implies that, for u, v orthonormal, J£ = R Rr = -R R,-,, so that

J

(7.173)

R,,RV = -1.

This proves that JJ E Cpx(4).

Proposition 7.174. (n = 3)

Cpx+(3) =

(n = 4)

Cps+(4) = {Je : 1; E G(2, O)} = G(2, 0).

u E S6 C Im O} - P6(R)

The proof will be given in Chapter 14 Low Dimensions. Remark 7.175. Consider the case n = 4. Given 1; = u A v E G(2, 0) with u, v orthonormal.

Jfu = v and Jfv = -u.

(7.176)

Thus, in the plane span{u, v}, Jf is just counterclockwise rotation by 7/2. Suppose x E span{u, v}1. Then v(ux) = -v(xu) = x(vu), so that

Jfx =

(7.177)

if x E span{u, v}1,

where u x v = (vu - Vv). In summary, the complex structure A is

i

(1) counterclockwise rotation by 7r/2 on the 2-plane span !;', and (2) right multiplication by u x v on (span l;)1-, where u, v is any oriented orthonormal basis for C.

PROBLEMS 1. Use the canonical form given by Lemma 7.18 to give a second proof of Theorem 7.26: (Re

1

for all f E GR,(n, C"),

with equality if and only if 1;' E SLAG. 2. (a) Complete the proof of Theorem 7.69, computing

dete \\liq J, P/

and then verifying conditions (7.64) and (7.65) so that Theorem 7.62 is applicable.

Problems

160

(b) Compute the complex determinant detc L of the linear map L used in the proof of Theorem 7.78. 3. (HL [9]-assumes knowledge of harmonic functions on surfaces in R3, minimal surfaces, and normal bundles.) Given a surface S in R3 and a smooth function F on a neighborhood of S, let

A(S,f)-{(x,tnx+VF(x))ER6:xES, tER}. Here n,x is the unit normal to S at x E S and f ___ Fps. (a) Show that A(S, f) depends only on f - FIs and not on the ambient extension F of f. (b) Show that if S is a minimal surface and f is an harmonic function on S, then A(S, f) is a special Lagrangian 3-manifold in Rs = C3. 4. Deduce the Coassociative Theorem 7.115 from the Associative Theorem 7.112.

5. Show that M=Ix +f(x)e:xEUCImH}, the graph off:U -;H over an open domain U in ImH, correctly oriented, is associative if and only if

+ axl e, 9 Lz

ax2 e, k + a s eJ = 0 on U.

6. (a) When n = 2, explicitly solve (using elementary functions) the system of differential equations (7.92), (7.93) for zl(t),z2(t).

(b) Find a real orthogonal coordinate change in R4 - C2 so that the special Lagrangian manifold M,, described in Theorem 7.78 equals

{wEC2:alwi-a2wz=1}. 7. (a) At the point

/ 2 i+el, rER,

rI

compute a basis for the normal 3-plane to the cone M defined by (7.117) (the cone on the graph of the Hopf map). Verify that this normal 3-plane is associative.

(b) Show that M is fixed by HU(1) acting on 0 = H ® He, with p E HU(1) sending a + be to pap + (bp)e. (c) Prove Theorem 7.116.

Calibrations

161

8. (a) (The First Cousin Principle) Suppose 0 E AP (R")* is a calibration and q(el A . A ep) = 1, where el,..., e is an orthonormal basis. Prove that 0(t;) = 0 for all first cousins: =eaA(ei](eiA...Aep)),

1 =1,...,pandi=p+1,.

ofelA...Aep. (b) Prove Proposition 7.158. 9. Consider a 7-torus M = Im O/A where A is a rank 7 lattice in R7

Im O equipped with the associative calibration 0 E A3(M). Prove that the lattice may be chosen so that M has a compact associative submanifold.

8. Matrix Algebras

This chapter collects information about matrix algebras that will be needed in our discussions of spinors and pinors. More knowledgeable read-

ers may wish to treat this as a reference chapter rather than reading it in detail. The matrix algebras

MN(R), MN(C), MN(H) are examined in this chapter. The algebras MN(R) and MN(H) will always be considered real algebras, while the algebra MN(C) may be considered a real algebra or a complex algebra.

REPRESENTATIONS At first, consider all three of the algebras MN(R), MN(C), MN(H) to be real algebras.

Definition 8.1. Given a real algebra A and a vector space V over F R, C, or H), a real linear map (8.2)

p : A -+ Endp(V)

sending 1 to the identity on V that satisfies p(ab) = p(a)p(b) 163

for all a, b E A,

Representations

164

is called an F-representation of A. Two F-representations, pl

: A --->

EndF(Vl) and P2 : A --> Endp(V2) of the real algebra A are, equivalent if there exists an invertible F-linear map f : Vi -+ V2

(8.4)

satisfying (8.5)

P2(a) = f o pi(a) o f-'

for all a E A.

Such an operator f is called an intertwining operator.

The kernel of a representation p is a two-sided ideal in A (which is proper since 1

ker p).

Lemma 8.6. Each two-sided ideal in one of the matrix algebras MN(R), MN(C), or MN(H) is either {O} or the entire matrix algebra.

The proof is a straightforward calculation, left as an exercise. An algebra with this property is said to be simple.

Corollary 8.7. Each representation of MN(R), MN(C), or MN(H) is injective.

Examples (The Standard Representations) (8.8)

p(a)x = ax for a E MN(F) and x E FN

defines an F-representation of MN(F) called the standard representation of MN(F). Here F is either R, C, or H.

Given a matrix a E MN(C), let a denote the conjugate matrix. The C-representation of the real algebra MN(C), defined by (8.9)

p(a)z = az for a E MN(C) and z E CN,

is called the conjugate representation of MN(C). Note that each H-representation, or each C-representation of A, is automatically an R-representation. As R-representations, the standard representation of MN(C) and the conjugate representation of MN(C) are equivalent; the intertwining operator C : CN --> CN is conjugation on C extended to CN by acting on each component. However, since p(i) = i

and p(i) = -i, the intertwining operator cannot be chosen to be complex linear. Therefore, these two representations are not equivalent as C-representations of the real algebra MN (C).

Matrix Algebras

165

Definition 8.10. Suppose pl : A -+ EndF(Vl) and P2 : A --r EndF(V2) are two F-representations of A. Then p = PI ® p2 : A --+ EndF(Vl ® V2),

defined by p(a) = pl(a) ® p2(a), is also an F-representation of A. An F-representation of this form p = p1 ® p2 is said to be F-reducible. If an F-representation p of A is not reducible, then it is called F-irreducible.

Theorem 8.11. Suppose F is it, C, or H. The standard representation of MN(F) on FN is the only irreducible R-representation of MN(F), up to equivalence.

Proof: Let I denote the subspace of MN(F) consisting of those matrices with nonzero entries confined to the first column, and identify I with FN. Note that I is a left ideal in MN(F). Also note that (8.12)

if b E I (nonzero) is given, then each a E I can be written as a = cb for some c E MN(F).

In particular, if K is a left ideal contained in I, then either K = {0} or K = I. (Thus, I is called a minimal left ideal.) Let E denote the matrix with all entries zero except for the entry 1 in the position first row-first column. Since p is injective (Corollary 8.7),

p(E) # 0. Suppose v1 E V satisfies p(E)vl # 0. Define f : I ? FN -+ V by

(8.13)

f (b) = p(b)vl

for all b E I.

First, we show that f is injective. Let

J - {a E MN(F) : p(a)ve = 0}. Note that J, and hence ker f = Ifl J, is a left ideal in MN (F). Since E E I, but E V J, the left ideal In! is proper in I. However, since I is a minimal left ideal, this proves that I n J = {0}, i.e., f is injective. Now define V1 = image f = {p(b)vl : b E I}. Note that p(a)Vl C V1 for each matrix a, since p(a)p(b)vl = p(ab)vl. Let pi(a) = p(a)jv,. Then (8.14)

pi(a) = f o a of-'.

This is because for each u = p(b)vl E V1, p(a)u = p(a)p(b)vi = p(ab)vl = f(ab) = f (a f -1(u)). This proves that pl is equivalent to the standard R-representation of MN(F) on FN -_ I. If V1 = V the proof is complete. Next, we prove that (8.15)

if p(E)V C V1, then V1 = V.

Representations

166

If p(E)V C V1, then for any matrices a and b,

p(aEb) V = p(aE)p(b)V C p(aE) V = p(a)p(E)V C p(a)V)i C V1.

Since the two-sided ideal in MN(F) generated by E is all of MN(F), this proves that p(1)V C V1 or V = V1. V1. Exactly as above, define V2 Now choose v2 so that p(E)v2 {p(a)v2 : a c I}, note that p(a)V2 C V2, and prove that p2(a) - p(a)lv, defines an R- representation P2 of MN(F) equivalent to the standard representation. Next, since p(E)v2 ¢ Vi it follows that Vi n V2 = {O}. If Vi n V2 # {O},

say p(ai)vl = p(a2)v2 with ai, a2 E I-{O}, then by (8.12) E = cat for some matrix c. Therefore, p(E)v2 = p(ca2)v2 = p(c)p(a2)v2 = p(c)p(ai)vi = p(cai)vi E V1 since cal E I.

Continuing in this manner, we obtain V1 ® ... ® Vk = V, so that Pi ®' - - ®Pk = p is reducible unless V = V1. L

Corollary 8.16. The real algebra MN(C) has exactly two irreducible C-representations-the standard C-representation and the conjugate Crepresentation.

Proof: Suppose p : MN(C) -+ Endc(V) is any C-representation of MN(C). The operator p(i) E Endc(V) has square -1. Therefore, the eigenvalues are ±1 where I is the complex structure on V. Let

Wf={xEV:p(i)x=±Ix} denote the eigenspaces, and note that if p(i)x = ±Ix, then p(i)p(a)x = p(a)p(i)x = ±Ip(a)x, so that p(a)WW C Wt for all a E MN(C). Let p (a) = P(a) l wf . Then

p = p+ ®p_

as C-representations.

Thus we may assume that either p(i) = I or p(i) = -I if p is an irreducible C-representation. If P = P1 ® P2 is reducible as an R-representation, with V = Vi ® V2i then each subspace V is complex with respect to I, since I = ±p(i) and p(i) maps Vj to V j. That is, p = P1 ® P2 is also reducible as a C-representation.

Therefore, we may assume that p is R-irreducible, and hence Requivalent to the standard representation of MN(C) on CN via an intertwining operator f : CN -+ V, which is real linear. Now f o i o f -1 = p(i) = ±1, or f o i = ±1 o f. That is, f is C-linear if p(i) = I and C-antilinear if

P(i) = -I.

Matrix Algebras

167

Corollary 8.17. The standard representation of MN(H) on HN is the only irreducible H-representation, up to equivalence.

Proof: Suppose p : MN(H) --r EndH(V) is an H-representation of MN(H) on a right H-space V. Recall that (8.18)

MN(H) ®a. H - EndR(HN) - M4N(R),

with (a (9 A)(x) - axX for a E MN(H), X E HN, and A E H. Now p : M4N(R) - EndH(HN) --} EndR,(V), defined by (8.19)

p(a

))(v) = p(a)J, for all v E V,

is an R-representation of M4N(R) on V. Thus by Theorem 8.11, P=P1Ej...EPk,

V

=Vi®...®Vk,

where each representation pj : M4N(R) - Enda(HN) -t Enda(V') is Requivalent to the standard representation of M4N(R) via an intertwining operator fj : R4N - HN -; Vj. That is, (8.20)

pj(A)=fjoAof11.

If A E H, then right multiplication by A on R4N = HN is given by

rA,=1®a, and p(ra) = p(1 0 a) is RA, right multiplication by A on V (see (8.19)). Therefore, by (8.20), fj o ra = RA o fj. That is, each fj : HN --+ V is equivalent, as an H-representation, to the standard H-representation on MN(H). LJ

UNIQUENESS OF INTERTWINING OPERATORS The center of an algebra A is, by definition, (8.21)

cenA={aEA:ab=bafor all bEA}.

Let R C MN(R) and C C MN(C) denote all scalar multiples of the identity, and let R C MN(H) denote all real multiples of the identity. The centers of the matrix algebras are listed in the next lemma.

Uniqueness of Intertwining Operators

168

Lemma 8.22. Center

Algebra MN (R) MN(C) MN(H)

R C

R

Proof: The proof for N = 2 is Problem 1 (note that H has center R). The general case follows immediately, because, for a - (a;2) in the center, each 2 x 2 submatrix of the form

a ash } a1i

ail

must belong to the center of M2(F).

Corollary 8.23. (a) The intertwining operator for two irreducible R-representations of MN(R) is unique up to a nonzero scale A E R*. (b) The intertwining operator for two equivalent irreducible C-representations of MN (C) is unique up to a nonzero scale A E C*.

(c) The interwining operator for two irreducible H-representations of MN(H) is unique up to a nonzero real scale A E R*. Proof: (a) Suppose p'(a) = f1 o p(a) o fi 1 = f2 o p(a) o f2 1, where f1 and f2 are both intertwining operators. Then f2 1 0 f1 E EndF(V) commutes with all p(a) E EndF(V). Since p : MN(F) -* EndF(V) is an isomorphism, the corollary follows. Suppose A is a subalgebra of B. The centralizer of A in B is defined to be (8.24)

{bEB:ab=baforallaEA}.

Lemma 8.25. (a) The centralizer of Endc(CT') in EndR(CN) is C. (b) The centralizer of EndH(HN) in EndR,(HN) is H-acting on the right as scalar multiplication. (c) The centralizer of Enda(RN) in EndFt(CN) is M2 (R).

Proof: (See Problem 2.)

Matrix Algebras

169

Corollary 8.26. (a) The intertwining operator for two irreducible R-representations of MN(C) is unique up to a nonzero complex scale A E C. (b) The intertwining operator for two irreducible R-representations of MN(H) is unique up to a nonzero right multiplication RA, A E H*.

Proof: Analogous to the proof of Corollary 8.23.

Ll

AUTOMORPHISMS The automorphisms of the algebras MN(R), MN(C), and MN(H) can be computed as a special case of Theorem 8.11 and its two corollaries. If a is a real automorphism of one of these algebras, then (8.27)

p(a) = a(a)

defines an F-representation of the algebra on FN. If V is a complex vector space, let Endc(V) denote the space of complex antilinear maps from V to V.

Corollary 8.28 (Automorphisms). (a) (Existence) Each automorphism a of the algebra MN(R) is inner. That is, there exists h E MN(R) with (8.29)

a(a) = hah-1 for all a E MN(R).

(Uniqueness) The intertwining operator h in (8.29) is unique up to a

real scalar multiple. (b) Suppose a is an automorphism of the algebra MN(C) considered as a real algebra. Then a is either complex linear or complex antilinear. (Existence) If a is complex linear, then a is inner. That is, there exists h E MN(C) with (8.30)

a(a) = hah'1

for all a E MN(C).

If a is complex antilinear, then there exists h E Endc(C") satisfying (8.30).

(Uniqueness) In both cases the intertwining operator h is unique up to complex scalar multiples. (c) (Existence) Each automorphism a of the real algebra MN(H) is inner. That is, there exists h E MN(H) with

Inner Products

170

(8.31)

a(a) = hah-' for all a E MN(H).

(Uniqueness) The intertwining operator h is unique up to a real scalar multiple.

INNER PRODUCTS An inner product e (of one of the seven types presented in Chapter 3) on a vector space V over F determines an antiautomrophism of Endp(V) called the adjoint (and denoted by a* or Ad(a)): (8.32)

e(ax, y) = e(x, Ad(a)y)

for all x, y E V.

The adjoint in each of the seven cases is explicitly given in Problem 2.7. Up to a scalar multiple no other inner product can determine the same adjoint.

Theorem 8.33. Suppose V is an F-vector space with F = R, C, or H. Given two inner products el and e2 on V that determine the same an tiautomorphism of the algebra EndF(V) by taking adjoints, el and e2 must be of the same type and differ by a constant multiple c }e 0: (8.34)

E2 (X, y) = c el (x, y),

where c E R* if the type is R-symmetric, R-skew, C-hermitian symmetric,

H-hermitian symmetric, or H-hermitian skew, and c E C* if the type is C-symmetric or C-skew.

Proof: Let a, = Reei and a2 = Ree2. Note that if a is an inner product on a complex vector space V, then (8.35)

(e(x, y), i) = - Re ,-(x, y)i = - Re e(x, iy);

while if a is an inner product on a right H-vector space, then (8.36)

(e(x, y), u) = Re e(x, y)u = Re e(x, yu)

for all scalars u E H. In particular, al and a2 are real inner products on V. Let bi : V --+ V* be defined by (bi (x)) (Y) = ai (x, y), j = 1, 2.

Matrix Algebras

171

Given a E EndF(V), let a* E EndF(V*) denote the dual map from V* to V*. Let Adi(a) denote the adjoint of a with respect to aj. That is, aj (ax, y) = aj (x, Adi (a)y). Let b : V - (V*)* --.> V* denote the map dual to bj. Note that aj(ar,y) = (bj(ax))(y) = (ax)(b*(y)) = x(a*bj* (y)), while aj (x, Adj (a)y) = (bj (x))(Adj (a)y) = x(bp Adj(a)(y)). Therefore, (8.37)

Adj (a) =

By hypothesis, Ad,(a) = Ad2(a) for all a E EndF(V) C EndF(V). Thus, by taking the dual of (8.37), for all a E EndF(V).

blabi 1 = b262 1

Therefore, b2b1 E EndF(V) is in the centralizer of EndF(V), and the theorem follows from Lemma 8.25 as described below.

If F - R, then bl = cb2 with c E R*, and hence El = cE2. If F - C, then bl = b2c with c E C*. Therefore,

aI(x, y) = al(cx, y),

and it follows easily that el and 62 differ by a constant using (8.35). Further, if El, 62 are C-hermitian, then the constant must be real. If

F - H, then bzbl = R, right multiplication by a scalar c E H*, because of part (b) of Lemma 8.25. Therefore, al(x, y) = a2(xc, y), and hence El(x, y) = E2(xc, y) = c e2(x, y) by Corollary 8.26. Since both el and

62 satisfy E(xa, y) = Ae(x, y), the constant c must be in the center of H, i.e., c E R*. An F-inner product E on a vector space V with scalar field F induces an R-symmetric inner product on EndF(V).

Definition 8.38. Suppose V, E is an inner product space with scalar field F and real dimension N. Define (8.39)

(a, b) = N traceR a*b for all a, b E Endp V,

where a* denotes the E-adjoint of a.

Lemma 8.40. (, ) is an R-symmetric inner product on Endp(V), and (8.41)

(ab, c) = (b, a*c)

for all a, 6, c E EndF(V).

Proof: Note that (,) is symmetric if traceR. a = traceR a*. If V, e is a real inner product space, then traceR a = traces a* by Problem 2.7. In general,

Inner Products

172

Ree is an R-inner product on V and (Re e)(ax, y) = (Re e)(x, a* y), since e(ax, y) = E(x, a*y). Therefore, the general case reduces to the real case. To show that (, ) is nondegenerate it suffices to prove that if traceR, ab = 0 for all a E MN(F), then 6 = 0, which follows by considering matrices a with all but one entry zero. Finally, (8.40) is immediate since (ab)* = b*a*.

J

Definition 8.42. Suppose V, E is an F-inner product space. Given x, y E V, define the product x O Y E EndF(V) by (8.43)

(x 0 y)(z) = xE(y, z) for all z E V.

Note that xOy=0ifandonlyifx=y=0. Lemma 8.44. For all a E EndF(V), (a) (ax) O y = a(x (D y),

x O (ay) _ (x O y)a*,

(b) (x O y)(z (D w) = (xe(y,z)) 0 w,

(c) (x O y)* = y O x if e is symmetric, (d) (x O y)* = -y O x if a is skew.

Proof: ((ax) O y)(z) = axe(y,z) = a((x O y)(z)),

(a)

and

(x 0 (ay))(z) = xe(ay, z) = xe(y, a*z) = (x (D y)(a* z). (b)

(x (D y)(z (D w)(u) _ (x (D y)(Xe(w, u)) = xe(y, ze(w, u)) = xe(y, z)e(w, u) _ ((xe(y, z)) (D W) (U).

In all cases, but e C-symmetric or C-skew, (c) and (d) are proven by noting 4U, (x (D y)*v) = e((x 0 y)(u),v) = e(xe(y,u),v) = e(y,'u)E(x,v)

= e(x,v)e(y,u) = e(ye(x,v),u) = e((y (D x)(v),u)

_ E&(u, (y (D

x)(v))

The cases a C-symmetric or e C-skew are proven by deleting the conjuga-

tions in the previous proof. J Theorem 8.45. Suppose V, e is an F-inner product space. Then (1, x O y) = (dimF V)-1 Ree(y, x)

for each x, y E V.

Matrix Algebras

173

Proof: We give the proof in the most difficult case F = H. Let (, ) denote a positive definite H-hermitian inner product on V. Choose an H-orthonormal basis with respect to (, ), say U1, ... , u)n E V, and let v1, ..., v4n denote the corresponding real orthonormal basis. That is vl = U1, V2 = u1i, v3 = U1, j, v4 = ul k, v5 = u2, v6 = u2i, etc. Then the real dual basis vi, ... v*,, is given by v,*(x) = Re (vj, x), j = 1, ... , 4n. Therefore, 4n

4n

traceR, x O y =

Re (vj, (xe(y, vj ))

vj* [xe(y, vj )] j=1

j=1

4n

4n

_

Re .j(vj,x)e(y,VA = j=1 1

j=1

Re [e(y, vj)(vi, x)]

1

4n

=Ree (YVi(VIZ)) j=1

However, since ui(ui, x) = u(u, x), etc., n

4n

Evj(vj,x) = 4Euj(uj,x) = 4x, j=1

j=1

completing the proof, since dimR, v = 4 dimH V.

Corollary 8.46. Suppose V, e is an F-inner product space of F-dimension n. (8.47)

(x (D y, a) = 1 Re e(ay, x) for all a E EndF(V). n

Proof: (x O y, a) _ (a*(x 0 y), 1) = ((a*x) 0 y, 1)

=

1

Ree(y,a*x) = n Ree(ay,x).

Corollary 8.48. Suppose V, c is an F-inner product space of F-dimension n. If c is F-symmetric or F-skew, then (8.49)

(x0y,zcw) = n

Re [e(x,z)e(y,w)]

Problems

174

If E is F-hermitian, then

(xOy,z(D w) = 1 Re [e(x,z)e(y,w), n

(8.50)

Proof: We give the proof of (8.50). The proof of (8.49) is exactly the same except for the last equality. By Corollary 8.46, (x O y, z O w) =

1

Re E((z (D w) y, x) =

1

n

Re E(zE(w, y), x)

= 1 Re [e-(X-,Z)'-(Y, w), . -J n

PROBLEMS 1. Show that M2(R) has center R, M2(C) has center C, and M2(H) has center R. 2. Prove Lemma 8.25. 3. Show that MN (F) x MN (F), F = R or H, has exactly two irreducible F-representations, up to equivalence. Namely, given (a, b) E MN(F) x MN (F), p+(a, b)(x) = ax

for all x E FN,

p_ (a, b)(x) = bx

for all x E FN.

and

4. Let Vl denote HN with the usual right scalar multiplication RA, A E H. Let V2 denote HN with right multiplication by A E H given by

Rµaµ-1, where µ E H is a fixed unit quaternion. Let p, (a = 1,2) denote the R-representation of MN(H) on Va defined by p,,,(a)v = an for all v E Va. Find (explicitly) the intertwining operator f for these two H-representations.

PART II. SPINORS

9. The Clifford Algebras

We start with V, (, ), 1111, a euclidean vector space of dimension n and signature p, q, and with the orthogonal group O(p, q) acting on V. The Clifford algebra Cl(V) is constructed in this chapter. It has considerably more structure than just that of an algebra. In fact, if algC1(V) denotes the underlying associative algebra with unit, then each algebra alg Cl(V) is nothing more than one, or occasionally two, copies of a matrix algebra, i.e., MN(R),MN(C), or MN(H). Some patience is required, as this important fact is not proven until Chapter 11! The reader is assumed to be familiar with the exterior algebra n

AV

AV P=O

and the tensor algebra

®VEOPV P=O

over V. Since the extent of this familiarity may vary, we include a brief reminder of these constructions.

For example, the tensor algebra is defined by OPV = FIR. Here F x V (p times) is the vector space of all real-valued functions on V x that vanish except at a finite number of points. Let 4(vl, ... , vP) E F denote the function that vanishes everywhere except at the single point 177

The Clifford Algebras

178

(vi, ... , vp), where it has the value one. Thus, F consists of all expressions of the form L..N1 aiq(vi,..., v,), with each ai E R. These expressions are subject to certain rules. This is made precise by quotienting out by all the rules or expressions one wishes to vanish. Thus, R is defined to be the vector subspace of F generated by

(vii...)Vi_l,x+y,Vi+1i...,Vn) - 0 (vi.... ) vi-1, x1 vi+1, ... , vn) -

(vi, ... , Vi-1, Y, Vi+1) ... , Vn)

and

0 (vi, ... , vi-11 cx) Vi+l, ... , vn) - CO (v1, ... , vi-1, x, Vi+l, ..., vn)

with c E R. The equivalence class ¢(v1,. .. , vn)+R is denoted vie .. -®vp E ®PV.

The tensor algebra ®V is graded, associative, with unit, and has an inner product that we also denote by (, ). Let I(V) denote the two-sided ideal in ®V generated by all elements x ® x E ®2V with x E V. The quotient algebra AV - ®V/I(V) is called the exterior algebra over V, and the equivalence class of vi ® -. 0 vp is denoted v1 A ... A vp. In fact, AV = Zp=0 APV is a graded, associative, anticommutative algebra with unit (and with inner product also denoted (, )). The exterior algebra AV may also be identified with a vector subspace

of ®V, the space of skew tensors. For example, x A y = a x 0 y- y ®x], x, y E V V. More generally,

(9.1) x1A

Ax,=Alt(x1®

®xp)=-

Esignoxo(1)0.

.®xo(p).

Elements of APV are called p-vectors and those of the form v1 A.. .Avp

are said to be simple or decomposible. The set of all unit (i.e., lull = ±1) simple p-vectors is called the grassmannian of oriented, nondegenerate, pplanes in V, and denoted by G(p, V). Of course, when p = 1, this is just the unit sphere in V. Given w E V, let Ew : AV -+ AV, left exterior multiplication by w, be defined by

Ewu=to Au foralluEAV. The adjoint I. : AV -> AV, defined by (w A u, v) _ (u, Iw v)

for all u, v E Av,

is called interior multiplication by w and sometimes is denoted by

I(u)=wLU.

Clifford Algebras

179

Observe that

I,,,v=(w,v) ifvEV, and that interior multiplication I. is an antiderivation on AV; that is, P

(9.2) Iw (Vi A ..Avp) = E(-1)k-1(w,vk) V1 A...nvk_lnvk+lA...Avp k=1

on simple p-vectors, and II (u A v) = Iw(u) A V + (-1)deguu A I,, (v)

for all u, v E AV with u of pure degree. In particular,

(9.3) IIwIlu=wA(wLu)+wL(wAu) or IIwlI=EwoI.+I,,,oEE for w E V and u E AV. This is called a chain homolopy for E. if IIwII 0 0.

Definition 9.4. Given a euclidean vector space V, (,) of signature p, q, the Clifford algebra Cl(V) is the quotient ®V/I(V), where I(V) is the two-sided ideal in ®V generated by all elements: (9.5)

x ®x + (x,x)

with x E V.

If V - R(p, q), then Cl(V) will be denoted Cl(p, q). The multiplication on C1(V) will be denoted by a dot. Polarizing x ® x + (x, x) E I(V) yields (9.6)

x ®y + y 0 x + 2(x, y) E I(V). The Fundamental Lemma for Clifford algebras follows.

Lemma 9.7. Suppose ¢ : V -+ A is a linear map from V into A, an associative algebra with unit A. If (9.8)

O(x)O(x) = -IIxII for all x E V,

then 0 has a unique "extension" (also denoted 0) to an algebra homomorphism of Cl(V) into A.

V (9.9)

1

Cl(V)

4'

A

The Clifford Algebras

180

Proof: The fundamental lemma for tensor algebras says that 0 has a

unique extension to ®V as an algebra homomorphism, which we also denote

by 0: ®V -+ A. The hypothesis O(x)O(x) = -JJxJJ is equivalent to requiring that the kernel of 0 : ®V -+ A contains I(V). Thus ¢, descends to a well-defined map on the quotient Cl(V) - ®V/I(V). I Frequently, it is more convenient to use Lemma 9.7 in the following form.

Remark 9.10. Suppose e1,.. . , e, is an orthonormal basis for V and O(ei) E A satisfy

(a) O(ei)2 = -IIei!I, and (b) q(ei)4(ej) + ¢(ej)g(ei) = 0 for all i # j. Then r¢ has a unique linear extension 0 : V --+ A that satisfies q(x)2 = -JJxJ], and hence Lemma 9.7 is applicable.

Proposition 9.11. The subspace AV of ®V is isomorphic (as a vector space, not as an algebra!) to Cl(V) - ®V/I(V) under the quotient map. In particular, the natural map V i-+C1(V)

is an inclusion. Moreover, if x E V and U E AV

Cl(V), then (Clifford product)

(9.12)

That is, Clifford multiplication is an enhancement of exterior minus interior multiplication.

Proof: The quotient map restricted to AV maps onto C1(V); that is, each tensor can be expressed modulo 1(V) as a skew tensor. This is automatic for tensors of degree zero and one. Each 2-tensor x (D y is the sum of a symmetric 2-tensor and a skew 2-tensor. The symmetric part belongs to I(V) modulo a tensor of degree zero. The proof is completed by induction. To prove that AV injects into Cl(V), consider the map ¢ : V -+ End(AV) defined to be exterior minus interior multiplication. Given x E V, ¢(x) _- E,x - Ix E End(AV).

Note that [5(x) o O(x)] (u) _ (E. - Ix) (E. - Ix) (u) _ - (E., o Ix + Ix o E,x) (u) = -IlxIJu

Clifford Algebras

181

by (9.3). That is, O(x) o O(x) _ -11x1j, so by the Fundamental Lemma, 0 extends to an algebra homomorphism 0 : Cl(V) --> End(AV). Now consider the linear map 0 : End(AV) , AV defined by evaluation at 1 E AV. The map 0 o 0 : Cl(V) , AV is a right inverse to the natural map from AV to Cl(V). To prove this, choose u E AV of the form (9.13)

u = ui A

.

. A up,

with u 1 , . . .,up orthonormal.

Let [u] denote the corresponding element of Cl(V). Since

u=

sign o uo(1) ®... ® uo(p), sign Qo (uo(i)) o ... o 0 (uo(p))

0 ([u]) =

sign v (Eu,(,) - I-,(,)) o ... o (Eu,(P) Since ul, ... up is orthonormal, evaluating at 1 E AV yields 1

-

sign vuo(1) A ... A ua(p) = ul A ... A up. 0

Because V has a basis with each basis vector of the form (9.13), this proves that the composite

AV - Cl(V) 4 End(AV) - AV is the identity map on AV. Finally, we prove (9.12). Note that cS(x u) = ¢(x) o 0(u) and hence, (0 o 0) (x u) = 0(x)(0(u)(1)) _ 0(x)(u) _ (Ex -.1,,,) (u), since 0(u)(1) = (b o 0)(u) = u.

THE CLIFFORD AUTOMORPHISMS An important special case of the Fundamental Lemma of Clifford algebras is where the target algebra A is also a Clifford algebra.

The Clifford Involutions

182

Proposition 9.14. Suppose V1, (, )1 and V2, (, )2 are two inner product spaces. Each linear map f : V1 -+ V2 that preserves the inner products has a unique extension (also denoted f) to an algebra homomorphism (9.15)

f : Cl(V1) -+ CI(V2)-

Proof. f(x) - f(x) _ -(f(x), f(x))2 = -(x,x)1 = -IIxII1 for all x E V1, so that f satisfies (9.8). -1 Remark 9.16. Each linear map f : V1 --> V2 also has a unique extension to an algebra homomorphism (9.17)

f : AV1 --> AV2.

The extensions (9.15) and (9.17) are the same under the canonical identifications Cl(V1) - AV1 and CI(V2) - AV2. Remark 9.18. Suppose that V3i (, )3 is another inner product space and g : V2 -+ V3 is a linear map preserving the inner products then the composite h = g o f : V1 -+ V3 has a unique extension H to an algebra homomorphism of C1(V1) to Cl(V3). By the uniqueness part of Proposition 9.14, h must be the composite of the extension of f with the extension of g. In order to emphasize the fact that Cl(V) is much more than an algebra, let alg CI(V) denote Cl(V) considered just as an associative algebra with unit. In particular, Aut(alg Cl(V)) denotes the algebra automorphisms of C1(V).

Definition 9.19. A Clifford automorphism of C1(V), f E Aut(Cl(V)), is an algebra automorphism ofalg Cl(V) that also maps V to V.

Theorem 9.20. Aut(Cl(V)) = O(V).

Proof: Suppose f E Aut(Cl(V)). Then f E O(V), since (f(x),f(x)) _ -f(x)f(x) = -f(x2) = f((x,x)) = (x,x)f(1) = (x,x), for all x E V. Con-

versely, each f E O(V) has a unique extension to an algebra automorphism of Cl(V) by Proposition 9.14 and Remark 9.18 applied to g = f'1.

THE CLIFFORD INVOLUTIONS The Clifford algebra C1(V) comes equipped with a particularly distinguished automorphism and two anti-automorphisms.

Clifford Algebras

183

Definition 9.21. The isometry x r+ -x on V extends to an automorphism of the Clifford algebra Cl(V), because of Theorem 9.20. This automorphism

of C1(V) will be denoted by x i-+ x and is referred to as the canonical automorphism of CI(V).

Note that x= x because of Remark 9.18. Define the even part of Cl(V) by (9.22)

{x E C1(V) : i = x

Cleven (V)

and the odd part of C1(V) by

Clodd(V) = {x E C1(V) : x = -x).

(9.23)

Under the canonical vector space isomorphism AV - C1(V), Cl even (V) = Aeven V

(9.24) and

Clodd(V)

(9.25)

- AoddV

Defintion 9.26. The anti-automorphism of ®V defined by reversing the order in a simple product, i.e., sending vi ® ®vp to vp ® ®vl, maps I(V) to I(V), and hence determines an anti-automorphism of the Clifford algebra C1(V) _ ®V/I(V). This anti-automorphism obviously squares to and referred the identity (i.e., is an involution). It will be denoted by z to as the check involution.

Note that the composition V o - o V is an automorphism of Cl(V) that equals minus the identity of V. Thus, by the uniqueness in Lemma

9.7, V o - o V =-. This proves that V and -. commute. The second anti-automorphism is defined to be the composite A = V o is referred to as the hat involution x x. We say that u E C1(V) is of degree p if u E APV C AV

_

o V and

Cl(V).

Proposition 9.27. Ife E Cl(V) is of degree p, then ii = fu, u = fu, and u = ±u with the plus or minus depending only on p mod 4 as indicated in the following table.

p mod 4 (9.28)

0

1

2

+

-

+

A

+

V

+

3

-

+ +

The Clifford Inner Product

184

Proof: It suffices to verify the last row of the table for check V. Also, we may assume that u is an axis p-plane. Since ei ej = -ej - ei for i # j, and + 1 = p(p - 1), the check of the Ith axis p-plane is p - 1 + p - 2 -I(ei,

...

2 eip) v = eip

. .. . lei, = (-1)2P(P-1)ei1

.. eip.

THE CLIFFORD INNER PRODUCT The canonical vector space isomorphism AV = Cl(V) provides the Clifford algebra Cl(V) with a natural inner product (, ), namely, the natural inner product on AV. This will be referred to as the Clifford inner product. Thus,

if el, ... , e is an orthonormal basis for V, then lei : I increasing}, with ei ei, ... eip, is an orthonormal basis for Cl(V). Note ei1 ... ei, = A eip are the same. The square norm on Cl(V) determined by the inner product will be denoted 11 11, as usual for AV. The inner product in Cl(V) can be computed using the hat involution:

el A

Proposition 9.29.

forallx,yEC1(V).

(x,y)

Proof: Let (x, y)' _ (1, i y). The bilinear forms (,) and (, )' on Cl(V) are equal because lei : I increasing} is an orthonormal basis for (, )' as well as (, ). (el, ei)' = (1, eip

... ii,

ei,

...

eip)

= I1ei1II ... Ileipll = Ilei, A ... A eipll = (ei, ei),

since it u = -u2 = IIulI for all u E V. Similarly, (ei, ej)' can be seen to

vanish if I

J, completing the proof. J

The adjoint of Clifford multiplication by a is just Clifford multiplication by a.

Proposition 9.30. Given a E CI(V), (a - u, v) = (u, a v)

and (u - a, v) = (u, v a)

for all u, v E Cl(V).

Proof: (a u, v) = (1, (a u)^ v) _ (l, u a v) = (u, a

v).

We shall say u E Cl(V) is a simple product if u = ul ... u1,...,up E V.

at,, with

Clifford Algebras

185

Proposition 9.31. If u E C1(V) is a simple product u = ul ... up then (9.32)

Hull

Proof: up llxll for all x E V. In particular, it u is a scalar multiple of 1, so that Proposition 9.29. J Given it E Cl(V), let (u)k E AkV denote the orthogonal projection of u on AkV.

Corollary 9.33 (Hadamard). F o r all u1, ... , up E V, P-1

Iluill ... Ilupll = Ilui A ... A upll + E II(ul

...

up)kll-

k=0

Thus, if V, (,) is positive definite, (9.34)

Iul A ... A upl <- lull ... lupl

with equality if and only if u1i..., up are orthogonal.

Proof: Note that u1i ... , up are orthogonal if and only if u1 ... up = u1 A...Aup. The Clifford inner product can also be expressed in terms of a trace (see Theorem 9.65).

THE MAIN SYMMETRY Suppose e1,. .. , e is an orthonormal basis for R(r, s) (n = r + s), and E, E1,. .., E is an orthonormal basis for either R(r + 1, s) (with E2 = -IJEJl = -1) or R(s, r+1) (with E2 = -IJEll = 1). Define q5(ei) =-EE1, j =

1,...,n. Then ¢(ei)5(ei) = EE;EEj = -E2EiEE =

if i # j.

Also, 4(ei)O(ei) = -E2EE = E2IIEill. We wish to apply Lemma 9.7 (in the form discussed in Remark 9.10) and extend 0 to an algebra homomorphism. The remaining condition to verify is (9.35)

O(ei)c(ei) = -Ileidl

for i= 1,...,n.

There are two cases to consider. First, if E2 = -1 and IlE.ill = lleill for all j, then (9.35) is valid and ¢ extends to an algebra homomorphism (9.36)

0 : Cl(r, s) -* Cl(r + 1, s).

The Main Symmetry

186

Second, if E2 = 1 and IIE,JI _ - II ejII for all j, then (9.35) is valid and 0 extends to an algebra homomorphism

¢ : Cl(r, s) - Cl(s, r + 1).

(9.37)

Theorem 9.38. The maps defined above provide algebra isomorphisms: C1(r, S) - Cl(r + 1, s)even

(9.39)

and Cl(r, s) = Cl(s, r + 1)even,

(9.40)

which preserve the hat involutions.

Proof: Certainly the image of ¢ is contained in the even part of the target Clifford algebra. Each product EEti belongs to the image, and hence

E;Ej = ±EE;EEE also belongs to the image. Thus, 0 surjects onto the even part. By counting dimensions, 0 must be an isomorphism. Note that (9.41)

¢ preserves even degrees.

(9.42).

0 adds one to odd degrees

Therefore, the hat involutions are preserved. Combining these two isomorphisms yields the main symmetry for both Cl' eveand Cl.

Cl(r, Seven - CI(S' r) even

Cllr -1, s) = Cl(s - 1, r). Moreover, these isomorphisms preserve the hat involutions.

Remark 9.46. In fact, the isomorphism Cl(r, seven = Cl(s, r) even preserves degree, while the isomorphism Cllr - 1, s) = Cl(s - 1, r) preserves A2mRn ®

A2.n-'R".

187

Clifford Algebras

THE CLIFFORD CENTER The center of the Clifford algebra Cl(V) is

cen Cl(V) - {a E C1(V) : a x = x a for all x E Cl(V)}.

(9.47)

Since Cl(V) has a basis of simple products of vectors in V,

cen Cl(V) =

(9.47')

V}.

The twisted center of the Clifford algebra Cl(V) is twcen Cl(V)

(9.48)

{a E Cl(V) : a x = x a for all x E Cl(V)}.

Note that (9.48')

Of course, the real scalars R = A° V always belong to the center.

Lemma 9.49. If n = dim V is even, then cen Cl(V) = A°V

and twcen Cl(V) = A"V.

If n = dim V is odd, then

cen Cl(V) = A°V ® A'V and twcen CI(V) _ {0). Remark 9.50. Perhaps a more convenient way to remember Lemma 9.49 is in terms of a unit volume element A E A" V. First, note that the center and the twisted center always belong to A° V ® An V. Second, note that (9.51)

(9.52)

(n odd) A commutes with all of the Clifford algebra Cl(V). (n even)

A commutes with Cleven(V) and anticommutes with Clodd(V)

All the information in Lemma 9.49 is contained in this remark. Proof. First note that for any non-null vector e E V, we have the following (9.53) (9.54) (9.55) (9.56)

If a is even and a e = e a, then a does not involve e. If a is even and a e = -e a, then a does involve e.

If a is odd and a e = -e a, then a does not involve e. If a is odd and a e = e a, then a does involve e.

Self Duality

188

For example, to prove (9.53), express a as a = x + e y, where x, y E Cl(V) do not involve e. Then a e = x e + e y - e = x e - e2 - y, since y is odd; and e - a = e x + e2 y = x e + e2 y, since x is even. Therefore, y = 0, so that a = x does not involve e. for V. An element a Now choose an orthonormal basis belongs to the center if and only if a commutes with each vector e if and only if both the even and the odd part of a commute with each vector ej. Thus, it suffices to consider the even and odd parts of a separately. Now if a is even and in the center, repeated application of (9.53) implies

that a E A°V = R does not involve any of the e1, ..., e,,. The other cases are similarly handled.

SELF DUALITY A choice of orientation for R(r, s) is equivalent to a choice A of one of the two unit volume elements for R(r, s). However, the conditions A2 = 1 and A2 = -1 are independent of the choice, A or -A.

Proposition 9.57. Suppose A is a unit volume element for R(r, s). A2 = 1 if r - s = 0, 3 mod 4,

(9.58) (9.59)

A2=-lifr-s=1,2mod4.

Proof: First note that as = (-1)r for A - el

er+,. Second, recall from

Proposition 9.27, that

A=Aif n=r+s= 0,1 mod 4,

-A if n-r+s=2,3mod 4, Thus, if n = 2p is even, then A = (-1)PA, so that A2 =

(-1)r-p

=

(-1)(r-°)/2; while if n = 2p + 1 is odd, then A = (-1)PA, so that A2 =

(-l)r-P = (-1)(r-3+1)/2. 0

Definition 9.60. Suppose r - s = 0, 3 mod 4, or equivalently A2 = 1. An element a E Cl(r, s) is said to be self-dual if as = a and anti-self-dual if

as = -a. Let (9.61)

Cl(r, s)f = {a E Cl(r, s) : as = ±a}

Clifford Algebras

189

denote the self-dual and anti-self dual parts of the Clifford algebra Cl(r, s).

Proposition 9.62. Suppose r - s = 3 mod 4. Then Cl(r, s) = Cl(r, s)+ ® Cl(r, s)-,

(9.63)

and both Cl(r, s)+ and C1(r, s)' are two-sided ideals in C1(r, s).

In fact, the only case in which Cl(r, s) has a nontrivial proper twosided ideal is this case r - s = 3 mod 4 (see Problem 11.4 and Lemma 8.6).

Remark 9.64. Cl(r, s)±, with the multiplicative identity defined to be I (I ± a), is an algebra (assuming r - s = 3 mod 4).

TRACE The Clifford inner product can be expressed in terms of the trace given a representation of the Clifford algebra.

Theorem 9.65. Suppose p : Cl(r, s) --+ EndR(V) is an R-representation of the Clifford algebras Cl(r, s) on a vector space V of real dimension N. Then (9.66)

trace, p(ab) for all a, b E Cl(r, s),

(a, b) =

N

unless r - s = 3 mod 4 and trace, p(a) # 0. Proof: Since (a, b) = (1, ab), it suffices to show that (9.67)

traceR p(a) for all a E Cl(r, s).

(1, a) =

N

Because traceR p(a) is a linear functional on C1(r, s), there exists an element c E Cl(r, s) such that (9.68)

N traceR, p(a) = (c, a) for all a E Cl(r, s).

The property traceR AB = traceR, BA implies that (c, ab) = (c, ba) for all a, b E Cl(r, s). Therefore, (c&, b) = (ac, b), or c& = ac for all a, so that c E center Cl(r, s). Consequently, if the dimension n = r + s of R(r, s) is even, then c E R, so that (9.69)

(r + s even)

1

traceR, p(a) = c(1, a).

The Complex Clifford Algebras

190

Setting a = 1, yields (9.66). While if the dimension n = r + s is odd, then c = a + /3a with a,,3 E R. Thus, (9.70)

(r -l- s odd)

N trace, p(a) = a(1, a) + 3(A, a).

Setting a = 1 yields a = 1, while setting a = A yields

_ (I) traces p(A) Since r + s is odd, r - s is also odd and hence equals either 1 or 3 mod 4. If r - s = 1 mod 4, then A2 = -1 and hence p(.X)2 = -1. In this case, p(A) is a complex structure on V and therefore traces, p(A) = 0 is automatic. U

Remark 9.71. This proof shows that N)s (9.72)

N traceR, p(a) _ (1 a) + (

(tracea p(A))(A, a)

for all a E Cl(r, s), no matter what the signature r, s or the representation. In particular, for any representation p of a Clifford algebra Cl(r, s), (9.73)

traceR, p(v) = 0

for each u E R(r, s).

In fact, for any u E Cl(r, s) - AR(r, s) which does not have a degree 0 or degree n = r + s part, (9.73) also holds.

THE COMPLEX CLIFFORD ALGEBRAS Suppose V is a complex vector space with nondegeneate symmetric complex bilinear form (, ). Then, proceeding exactly as in the real case, the complex Clifford algebra Clc(V) is defined by (9.74)

Clc(V) - OV/I(V),

where I(V) is the two-sided ideal in ®V generated by (9.75)

x ®x + (x, x)

with x E V.

All of the results of this chapter carry over, with the field R replaced by the field C, except for the following two items. First, the Hadamard Inequality (9.34) for the positive definite case must be deleted. Second, a unit volume element A for V complex is only determined up to a multiple

Clifford Algebras

191

e" A, so that Proposition 9.57 loses any significance. However, the concepts of self-dual and anti-self-dual are now valid in all dimensions and independent of the choice of volume element. Some of the results that remain valid for Clc(V) have been reformulated as exercises.

Since all of the complex inner product spaces C(r, s) of the same dimension n = r + s are isometric, and the isometry has a unique extension to a complex Clifford algebra automorphism, there is (essentially) only one complex Clifford algebra for each dimension, denoted Clc(n).

Proposition 9.76. Let R denote conjugation (or the reality operator) on E (r, s) = R(r, s) OR C, the complexification of R(r, s). Since (Rx, Rx) _ (x, x) for all x E C(r, s), R has a unique extension to a complex antilinear isomorphism of Clc(n), which is the conjugation for (9.77)

Clc (n) = Cl(r, s) OR C.

Proof: The fundamental lemma for complex Clifford algebras does not apply to R since R is not complex linear and does not preserve (, ). However, R has a unique extension to a complex antilinear algebra automorphism of ®V (define R(i) _ -i). This extension R maps x ® x + (x, x)

to R(x ® x + (x, x)) = Rx ®Rx + (x, x) = Rx ®Rx + (Rx, Rx), and hence maps I(C(r, s)) to I(C(r, s)). Therefore, R descends to the quotient yielding a complex antilinear algebra automorphism of Clc(n). Since R fixes R(r, s), R fixes the real subalgebra Cl(r, s) of Clc(n) generated by R(r, s). Since R is complex antilinear, R is -1 on iCl(r, s) C Clc(n). Now, counting dimensions,

Clc(n) = Cl(r, s) ® iCl(r, s), and hence

Clc(n) = Cl(r, s) ®R C. iJ

PROBLEMS 1. Use (9.6) to show directly that the natural map of V to CI(V) is injective.

2. Under the canonical isomorphism of vector spaces AV = Cl(V), show that, for all vectors xl, x2i x3, x4 E V (a) x1 x2 = x1 A x2 - (z1, x2), (b) x1 x2 ' x3 = x1 A X2 A x3 - (x2, 53)x1 + (x1, 53)x2 - (x1, 52)x3,

Problems

192 (c)

X1 X2'x3'X4=x1AX2AX3Ax4 -- (x1, x2)23 A x4 + (x1, 23)x2 A x4 - (x1 i 24)x2 A x3 - (x3, x4)x1 A Z2 + (x2, 24)21 A 23 - (52i 23)x1 A x4 + (x1, 52) (x3, 54) - (Si, 53)(52, x4) + (Si, 54)(52, X3)-

3. (a) Show that CI(1, 0) and C are isomorphic as algebras with inner product and that A (hat) on C1(1, 0) corresponds to bar (conjugation) on C. (b) Show that Cl(0, 1) and L are isomorphic as algebras with inner product and that A (hat) on C(0, 1) corresponds to bar (conjugation) on L.

Note that V (check) is the identity on Cl(1, 0) and Cl(0,1) so that A =- on both Cl(1, 0) and Cl(0,1). 4. (a) Show that as algebras with inner products Cl(2, 0) = H,

and that the isomorphism can be chosen so that Cleven (2, 0) - C C H,

Clodd(2, 0) = jC C H,

and the hat on Cl(2, 0) corresponds to the bar (conjugation) on H. (b) Show that as algebras with inner products

Cl(1,1) = M2(R), and that the isomorphism can be chosen so that Cleven (1,1) '" R® R C M2(R) as diagonal 2 x 2 matrices,

and the A (hat) on Cl(1, 1) corresponds to the bar (conjugation) on the normed aglebra M2(R). Alternatively, show that the isomorphism Cl(1,1) = M2(R) can be chosen so that

\

r/

Cleven(1,1)-Lj `b

l

b a)

:a,bER}

the `space of/L-linear maps,

Clodd(1,1)- {

I

ab

ba

I

:a,bER}

the space of L-antilinear maps,

193

Clifford Algebras

and, as before, the A (hat) on C1(1, 1) corresponds to the bar (conjugation) on the normed algebra M2(R). (c) Show that as algebras with inner product Cl(O, 2) = M2(R)

and that the isomorphism can be chosen so thatl

16

CI even(0, 2)

C

Clodd(1 1)

((6 a)

ab ) : a, b E R }

(C-linear)

: a, b E R y (L-antilinear),

and the A (hat) on Cl(O, 2) corresponds to bar (conjugation) on the normed algebra M2(R). 5. Suppose u E V is nonnull. Given x E APV show that u L x = 0 if and

only if x=tLyforsome yEAP+1V. 6. Give the proof of Remark 9.16. 7. Complete the proof of Lemma 9.49. 8. The centralizer of A in B is defined to be

{bEB:ab=baforallaEA}. (a) Show that a E Cl(p, q) belongs to the centralizer of Cl(p, q)even in Cl(p, q) if and only if a x = x a for all elements x E G(2, R(p, q)) C AZR(p, q).

(b) Show that the centralizer of Cl(p, q)even in Cl(p, q) is A°R(p, q) A"R(p, q) = span {1, Al. (c) Show that the center of Cl(p, q)even is A°R(p, q) = R if n = p + q is odd and A°R(p, q) ® A"R(p, q) if n = p + q is even. 9. Give the proof of Proposition 9.62. 10. Suppose that p : C1(V) --* EndR(W) is an injective R-representation of Cl(V) on W, where W, e is an inner product space; and consider p as the identity so that Cl(V) C EndR(W) is a subalgebra of EndR(W). (a) Assume that, given a E Cl(V), ,-(ax, y) = e(x, ay) for all x, y E W.

If V is not positive definite, show that there exists an anti-isometry of W, and hence that the signature of W is split (if W has a signature).

Problems

194

(b) Assume that, given a E Cl(V), e(ax, y) = s(x, ay) for all x, y E W. If V is not negative definite, show that there exists an anti-isometry of W, and hence that the signature of W is split (if W has a signature). 11. Suppose R(p, q) is oriented and let A denote the (unique) unit volume element. The Hodge star operator* on AR(p, q) is defined by a A (*b) = (a, b) A, for all a, b E AR(p, q).

Show that *b = bA, for all b E AR(p, q) = C1(p, q).

That is, * and (right) multiplication by A agree on elements of degree 0 or 3 mod 4, while * equals (right) multiplication by -A on elements of degree 1 or 2 mod 4.

10. The Groups Spin and Pin In this chapter, we construct several groups contained in the Clifford algebra Cl(V). First, consider the multiplicative group Cl*(V) of invertible elements in the Clifford algebra Cl(V). (Later we shall see that this group is just one or two copies of a general linear group over either It, C, or H depending on the signature.) Since u2 = -11ull

for each vector u E V C C1(V),

a nonnull vector u E V belongs to Cl*(V), with inverse u-1 = -u/11ull, while a null vector u E V has no inverse in Cl(V). Definition 10.1. The Pin group is the subgroup of Cl*(V) generated by unit vectors in V. That is, (10.2) Pin

eachuj EVandFujJ=1}.

Note that each a E Pin is either even or odd, since it is a simple product of vectors.

Defintion 10.3. The spin group is defined by Spin = Pin fl Cleven(V) That is, (10.4)

Spin ={aECl*(V):a=ul and Juj1 = 1}. 195

u2, with eachuj E V

The Groups Spin and Pin

196

The alternate notation Pin (V), Spin (V) is slightly more precise. If V - R(p, q) is the standard model, then the notation used will be Pin (p, q) and Spin (p, q). Clifford multiplication by a vector on the left is just exterior minus interior multiplication, therefore; (10.5) while

xy = -yx if x, y E V are orthogonal,

(10.6)

xy = yx

if x, y E V are colinear.

This proves the next lemma.

Lemma 10.7. If u E V is nonnull, then R,,, reflection along u, is given in terms of Clifford multiplication by (10.8)

Rux = -uxu-1 for all x E V.

_

Motivated by this lemma consider the twisted adjoint representation Ad of the group Cl*(V) on C1(V): (10.9)

Adax =_ axa-1

for all x E Cl(V).

Note that (Ad,, Adb)(x) = abx6'la-1 = Adab(x), so that Cl*(V) GL(Cl(V)) is a group representation. The adjoint representation Ad of Cl*(V) on C1(V) is defined by (10.10)

Adax =- axa-1

for all x E Cl(V).

Theorem 10.11. (a) The sequence (10.12)

O(V) -.. I.

1 -+ Z2 -+ Pin

is exact. (b) The sequence

(10.13)

is exact.

1 --> Z2 --+ Spin

naf

SO(V) -+ I.

aa}

The Groups Spin and Pin

197

Proof: If u E V is nonnull, then Adu = R.,, E O(V). Therefore, if a E Pin, then Ada E 0(V). The weak form of the Cartan-Dieudonne Theorem (Remark 4.25) states that each orthogonal transformation can be expressed as the product of reflections along nondegenerate lines, so that id is surjec-

_

tive.

Now suppose a E Pin and Ad,, = Id. That is, assume ax = xa for all x E V. First, suppose a is odd. Then -ax = xa for all x E V. Thus ai = xa for all x E Cl(V), so that a belongs to the twisted center. But (Lemma 9.49) the twisted center of Cl(V) cannot have an odd part. Second, suppose a is even. Then ax = xa for all x E V and hence for all x E C1(V);

that is, a belongs to the center of C1(V). Therefore, by (Lemma 9.49),

aEA°V=R.

Finally, since a = ul

u, is a simple product of unit vectors, by

Proposition 9.29, (10.14)

IIall _ IIu1Il .. Ilurll = ±1

for all a E Pin.

{1, -1} C Pin, completing the proof of part (a). Part (b) follows easily from part (a) and the fact that det Ru = -1 for each reflection R . Cl This proves a E ZZ

Remark 10.15. Note that the full strength of the Cartan-Dieudonne Theorem 4.23 implies that (10.16)

Pin = {a E Cl* (V) :a = ul

ur, r < n}.

THE GRASSMANNIANS AND REFLECTIONS Recall that G(r, V), the grassmannian of all unit, oriented, nondegenerate r-planes through the origin in V, consists of all simple vectors in A' V that are of unit length. That is,

uEG(r,V)if u = with u1i...,ur E V and Jul = 1 (Ilull = f1). Of course, we may choose u1,. .. , ur orthonormal without changing u. The

span of u, denoted span u, is just the span of {u1,. .., ur}. Given u E G(r, V), reflection along u, denoted R,,, is defined by (10.17) and (10.18)

RuxE -x ifxEspan u, if xE(span u)1.

Lemma 10.7 can be generalized.

The Grassmannians and Reflections

198

Lemma 10.19. If u E G(r, V) is a unit nondegenerate r-plane in V, then

Ruv=Adu x forallxEV. Remark 10.20. Therefore, each reflection Ru E O(V) along a subspace span u of V is replaced in the double cover Pin(V) of O(V) by either of the two elements Eu E G(r, V) C A''V C Cl(V) in the Clifford algebra.

Proof of Lemma 10.19. Each u E G(r, V) can be expressed as u = ul A ... A u,., where ul, -, u,- is chosen to be an oriented orthonormal basis for the nondegenerate subspace span u C V. Since ul, ..., U. are othogonal, R = R,,, o ... o Rug and u = ul A - - A u,. = ul u,., so that Adu = Adu, o. oAdur.. Finally note that&, = Adu, for each j by Lemma

10.7. J

By definition, the group Pin is generated by the element G(1, V) C Pin. Although the definition of Spin appears to be "generative," it suffers

from the defect that the generators u C G(1, V) are not in Spin. This defect can be corrected in several ways. First, if e is any unit vector and S"-1 denotes the unit sphere in V, then e S"-1 generates Spin, since

w=±eue ES"-1. In addition, we have

Proposition 10.21. (n = dim V > 3). The group Spin is the subgroup of Cl*(V) generated by G(2, V). That is, (10.22)

Spin =

with each

u; E G(2, V) C AV - Cl(V)}.

Moreover, r can be chosen even and < n.

This proposition is the "double cover" of a theorem in Chapter 4. However, the proof is so short that it is repeated here. Proof: First, suppose n = 3. Each a E Spin can be expressed as a = u1u2 with u1i u2 E V unit vectors. Let A denote a unit volume form for V. Then

vi - u1A E G(2, V) and V2 - Au2 E G(2, V). Thus, a = ulu2 = fvlv2i where A2 = fl. Second, suppose n > 4. Then, given a product u1u2 of unit vectors 111, u2 E V, there exists a unit vector v orthogonal to both ul and u2 (see Problem 4.10(b)). Hence u1u2 = ±ulvvu2 = (±ul A v) (v A u2). Li If n is odd, then Spin is generated by another sphere as well as by the grassmannian G(2, V) and e Sn-1.

199

The Groups Spin and Pin

Proposition 10.23. (n > 3) Suppose n - dim V is odd. The group Spin is generated by the sphere G(n - 1, V) in A"-1V See Chapter 14 for applications (for example, Spin(7) C Cleven(8) = M8(R) is generated by {Ru : u E S6 C ImO}). Proof: Since n is odd, the unit volume element A is in the center of Cl(V). Therefore, for u, v an orthonormal pair and A2 = f 1, u A v = uv = (±Au)(Av),

with

± Au, Av E G(n - 1, V).

ADDITIONAL GROUPS FOR NONDEFINITE SIGNATURE If the signature p, q is not definite (i.e., p > 1 and q > 1), then, in addition to the groups Spin and Pin, there are three other closely related groups (cf. Definition 4.48).

Note that the square norm (or Clifford norm) IIaII - (a, a) = (aa,1) _ (aa, 1) is multiplicative on Pin because of (10.14), i.e., IIabII = IIaII IIbil

for all a, b E Pin.

That is, the Clifford norm (10.24)

II

-

II

: Pin -+ Z2

is a group homomorphism. The twisted Clifford norm, defined by

(a, b)' - (a, b)

for a, b E Cl(p, q),

provides another group homomorphism from Pin to Z2. The kernels of these group homomorphisms are subgroups of Pin.

Definition 10.25. (10.26)

Pin^

{a E Pin as = 11

(10.27)

Pin'

{a E Pin : as = 1}.

Note that the intersection of any two of the three subgroups of Pin, (10.28)

Spin,

Pin^,

and

Pin",

is the same as the intersection of all three subgroups. This common intersection is denoted

Additional Groups for Nondefinite Signature

200

(10.29)

Spino = {a E Spin : a& = 1}

and called the reduced spin group.

Remark 10.30. If V is positive definite, then (10.31)

Pin' = Pin,

Pin" = Spin,

and

Spin° = Spin;

while if V is negative definite, then (10.31')

Pin" = Pin, Pin" = Spin, and Spin° = Spin,

so that the only groups are Spin and Pin. II on Pin descends to the

Since loll = II -all, the Clifford norm II

homomorphism (10.32)

II

IIs : O(V) --* Z2,

called the spinorial norm (of an orthogonal transformation-see Definition 4.55). To prove this fact, assume that

o...oR1,,, EO(V) is expressed as the product of reflections. By definition, the spinorial norm of A is equal to the parity of number of reflections that are along timelike (11u11 = -1) directions, which of course is also equal to the Clifford norm of u l u,.. Similarly, the twisted Clifford norm on Pin descends to the homomorphism (det A)IIAIIs on O(V). The next result is immediate from Theorem 10.11 (cf. Lemma 4.54).

Proposition 10.33. (p, q > 1) The following sequences are exact.

-*1

(10 . 34)

1-

(10.35)

1-*Z2-*Pin "-b01--*1

(10 36) .

Z2

2

-*S pi n °

SOO

-_.Pin vAd ).O+

- *1

.

There are certain distinguished classical groups containing Spino, Spin, Pin", or Pin, with equality occurring in low dimensions. We shall concentrate on the case Spino, leaving the other cases to the reader.

201

The Groups Spin and Pin

Definition 10.37. The classical companion group, denoted Cp(V), is defined by (10.38)

Cp(V) - {a E Cl.... (V) : as = 1}.

The connected component of the identity element in Cp(V) is a subgroup that is called the reduced classical companion group and is denoted by Cp°(V).

Note: Spin°(V) C Cp°(V). By Problem 2(d), Spin°(r, s) is connected (unless p < 1 and q < 1). Therefore Spin°(V) C Cp°(V). The identification of the groups Cp°(r, s) as classical groups is a major part of Chapter 13.

THE CONFORMAL PIN (OR CLIFFORD) GROUP Definition 10.39. The conformal pin group (or Clifford group) consists of all simple products of nonnull vectors. That is, (10.40)

with each uj E V nonnull}.

The proof of Theorem 10.11 applies to the conformal Pin group yielding

Proposition 10.41. The sequence (10.42)

1 -+ R* --+ Cpin

na

CO(V) -+ Id

is exact.

The conformal Pin group has a very useful alternative description, as the subgroup of Cl*(V) that maps V to V under the twisted adjoint representation.

Theorem 10.43. (10.44)

Cpin = {a E Cl*(V) : Ada(V) C V}.

The proof of this result depends on the next lemma.

The Conformal Pin (or Clifford) Group

202

Lemma 10.45. The following are equivalent: (a) as = IIaII, (b) as = Ilall,

(c) as E R, (d) Qa E R. Proof: For each a E C1(V), hall = (1, ad) = (1, aa), so the lemma follows easily.

_J

Corollary 10.46. If a E Cl(V) satisfies the equivalent conditions of Lemma 10.45, then IlabIl = IIaII IIbII

for all b E Cl(V).

Proof: Ilabll = (1, abab) = (1, baab) = I1all(1, bb) = IIail IIbII u

Proof of Theorem 10.43: Let

I'- {aECI*(V):axa-1 EV ifxEV} denote the group occurring on the right hand side of (10.44). We wish to show that the subgroup Cpin of F is all of I'. Because of the exact sequence (10.42),

1--+R*-*CpinTd CO(V)-+Id, it suffices to prove that

1 -* R* -+ F - CO(V)

(10.47)

is exact. Assume a E F and Ada

Id. That is,

ax=xa forallxEV.

(10.48)

Let a+ denote the even part of a, and a_ denote the odd part of a. Then (10.48) is equivalent to (10.48')

a+x = xa+ and a_x = -xa_

for all x E V.

The Groups Spin and Pin

203

Thus, the even part of a is in the center of Cl(V), and the odd part of a is in the twisted center of Cl(V). However, referring back to Lemma 9.49, the twisted center of C1(V) cannot have an odd part, and the even part of the center is always R*. Thus,

1-.R*-}F

GL(V)

is exact. Next, we show that

aaER forallaEF

(10.49)

Because of (10.47), as E R* if as E ker Ad. The hypothesis a E r says that axa-1 E V if x E V. Therefore, the check anti-automorphism fixes axa-1 if x E V. However, the check of axa-1 is a-1x& so that axa-1 = a-1xa or (aa)x = xaa, proving that as E ker Ad = R*. Now by Lemma 10.45, if a E F, then a-1 = a/11all. Thus, 11axa-'11

= IIaII-2IIaxaII

IIxII,

because of Corollary 10.46 and the fact that-and -are isometries. This proves that Ad : r --r CO(V), thus completing the proof of Theorem 10.43. U

Sometimes it is useful to adopt the following as alternate (equivalent) definitions of the groups Pin, Spin, and Spin°.

Corollary 10.50. Pin = {a E Cl*(V) : Ada(V) C V and 1lall = ±1} Spin = {a E Cl*(V)even Ada(V) C V and h ai = ±11 Spin° = {a E Cl*(V)even : Ada(V) C V and hall = 11.

DETERMINANTS In several different situations it is useful to compute the determinant of an element of Spin or Pin acting on a vector space W. The idea of the proof is the same in the two cases considered below.

Suppose p : G -+ EndR,(W) is a real representation of a group G on a vector space W. Further, suppose G is generated by elements u in a subset S with (10.51)

u2=±l

for each u E S.

Problems

204

Note that this is true for G - Pin with S - G(1, V); and true for G - Spin with S - G(2, V) by Proposition 10.21 (if n > 2). Then, p(a)2 = ±1 for all a E S.

(10.52)

Consequently,

detRp(a) = ±1 for all a E S,

(10.53)

and hence for all a E G. To prove (10.53) consider two cases. Case 1: (10.54)

If p(a)2 = -1, then detR p(a) = 1, because p(a) is a complex structure on W.

Case 2: (10.55)

If p(a)2 = 1, then p(a) is reflection along W_ through W+, where Wf - {x E W : p(a)z = ±x} are the eigenspaces of p(a). Therefore, (10.56)

detR p(a) = (_l)dim W_

PROBLEMS 1. Verify Remark 10.30.

2. Suppose V, (,) has signature p, q, and el and e2 are orthonormal in V.

(a) Show that cos 0 + sin 0el e2 = (- cos 0/2 el + sin 0/2 e2)(cos 0/2 el - sin 0/2 e2),

ifllelll=lle2ll=1, and that cos 0 + sin 0e1e2 = (cos 0/2 el - sin 0/2 e2)(cos 0/2 el + sin 0/2 e2), if llel ll = Ile2ll = -1-

(b) If either p > 2 or q > 2, find a subalgebra of Cl(V) that is isomorphic to C with Spin(V) fl c Sl the unit circle in C C Cl(V).

The Groups Spin and Pin

205

(c) If either p > 2 or q > 2, show that

Spin°(V) -+ SOT(V) is a nontrivial double cover, i.e., show that -1 can be connected to +1 in Spin°(V). (d) If either p > 2 or q > 2, show that Spin°(p, q) is connected. 3. Give the proof of Proposition 10.41. 4. Compute, explicitly as (classical) subgroups of Cl*(V), (a) Pin and Spin if p = 2, q = 0, or p = 0, q = 2.

(b) Pin, Pin^, Pin", Spin, and Spin° if p = 1, q = 1. 5. (a) Show that 1 --+ Z2 --+ Pin

Ad

O(V) -> 1

is exact if n is even. (b) Show that for n odd,

O(V)-}1 is exact where K = {±1, ±A} with A a unit volume element. Note that K = Z2 X Z2 if p - q = 3, 7 mod 8, while

K=Z4 ifp-q=1,5mod8. 6. (a) Show that the natural inclusion of R(p, q) into R(p + 1, q) induces an inclusion of algebras C1(p, q) C Cl(p + 1, q). (b) Prove that Pin(p, q) is included in Pin(p + 1, q) by Pin(p, q) where

{a E Pin(p + 1, q) : Ada(ep+1) = ep+i

e1, ... , ep, ep+i; ep+2, ... ,

is the

standard basis for

R(p + 1, q). Thus, Pin(p, q)

C

Pin(p + 1, q)

1Ad O(p,q)

C

O(p+1,q).

1Ad

7. Suppose p : Spin(r, s) --* Endc(W) is a group representation with dimension n = r + s > 3. Show that deter p(a) E {±1, ±i} for all a E Spin(r, s).

Problems

206

8. Define the complex Pin group by

u, with Pin(n, C) = {a E C1 (n) : a = ul uj E C" and Ijj11 = 1, j = 1,...,r} and the complex Spin group by

Spin(n, C) - Pin(n, C) n

Cleven(n)

Show that (see Problem 4.17)

(a) 1 -+ Z2 -+ Pin(n, C) - O(n, C) -+ 1 is exact, (b) 1 --+ Z2 - Spin(n, C) Ad SO(n, C) --+ 1 is exact, (c) Pin(n, C) _ {a E C1c(n) : Ada(Cn) C C" and IIall = 1}. 9. Suppose A E O(V) can be written as the product of reflections A = R,,, o ...o R,,, with ul, . . ., u, linearly independent. Let a = ul u,. E Pin(V). Let (a)k denote the degree k component of a in AkV C AV Cl(V). Note (a)k = 0 fork > r and (a),. = ul A . A u,.. o R, is any other representation of A as the Suppose A = R , o product of reflections. (a) Show that k < r implies ul,..., u,. are linearly dependent, so that k > r (i.e., r is the Cartan-Dieudonne rank of A). (b) Show that if k = r, then ul A . Au, = ±vl A A v,. (Thus, there is a well-defined vector space span{ul, ... , u,} = span{vl,... , v,.} associated with A.) 10. Show that, for n > 3, Spin°(p, q) is generated by G+(2, V) U G- (2, V) (the definite 2-planes). 11. Show that the centralizer of Spin°(p, q) in Cl(p, q) is A°R(p, q) A"R(p, q), and hence the center of Spin°(p, q) is equal to (a) {±1} = Z2 if n is odd or q is odd. (b) {f1, ±A} = Z2 ®Z2 if p - q = 0 mod 4 and n, q are even. (c) {±1, ±A} = Z4 if p - q = 2 mod 4 and n, q are even.

11. The Clifford Algebras Cl(r, s) as Algebras

Recall the algebra isomorphisms (11.1)

(11.2)

Cl(1,0)= C, Cl(2, 0) - H,

C1(0,1)=L=R®R

Cl(1, 1) = M2(R),

Cl(0, 2) - M2(R)

established in Problems 9.3 and 9.4. Each of the Clifford algebras Cl(r, s) is isomorphic, as an associative algebra with unit, to a matrix algebra. Some

of the important extra structure of Cl(r, s), that is hidden in the corresponding matrix algebra will be exposed in terms of "pinors" in Chapter 13.

Theorem 11.3. The Clifford algebras Cl(r, s) of signature r, s are isomorphic, as associative algebras with unit, to the matrix algebras listed below.

r - s mod 8: 0,6 2,4 1,5 3 7

Cl(r, s) MN (R) Cl(r, s) = MN (H) Cl(r, s) = MN (C)

C1(r,s) = MN(H) ® MN(H) Cl(r,s) MN(R) ® MN (R) 207

Clifford Algebras Cl(r, s) as Algebras

208

Remark 11.4. The integer N is, of course, trivial to compute in terms of the dimension n = r + s, since the Clifford algebra Cl(r, s) has dimension 211 as a real vector space. Before giving the proof of Theorem 11.3, we examine some reformulations and corollaries.

Some readers may find the following tabular form of Theorem 11.3 convenient for reference.

Table 11.5. Matrix Algebras Isomorphic to Cl(r, s) M16(H)

e

M16(R) M16(C) M16(H)

M128(R)

M64(R)

M5(H)

a

M16(H) M32(C) M64(R)

e

M12,3(R) M12s(C)

M32(R)

Ms(H) M16(C) M32(R)

M4(H)

a

M64(R) M64(C) M64(H)

M32(R)

M2(H)

e

e

M64(R)

Ms(H)

M4(H) M4(H)

256(R)

M12,3(R)

M16(H)

M8(C) M8(H)

a

M32(H) M64(C) M128(R)

M16(R)

M4(H) M8(C) M16(R)

M2(H)

e

M32(H)

M32(R) M32(C) M32(H)

M16(R)

M8(R)

M2(H) M4(C) M8(R)

M

e(R)

e

M32(H)

M16(H)

M16(R) M16(C) M16(H)

M32(H)

Ml (H)

M

M2(C) M4(R)

ED

M8(R)

Ms(C) Ms(H) M ED M4(H)

M2(R) M2(R)

M

M16(H) M32(C)

M8(H)

M4(R)

(R) M4(R) M4(C)

M4(H)

M H)

M8(H) M16(C) M32(R)

M2(H)

ReR M2(R) M2(C) M2(H) R

C

r

H

H)

M2(H)

M16(R)

M4(H) M8(C) M16(R)

H E) H M2(H) M4(C) Ms(R)

e (R)

Ms (R)

M

(R)

M16(R)

Clifford Algebras Cl(r, s) as Algebras

209

The even part of the Clifford algebra, Cl(r, seven may be read of from the entry to the left of Cl(r, s) in this table since (see Theorem 9.38) (11.6)

Cl(r, )even - Cl(r - 1, s),

r > 1.

Cl(s -1, 0),

s > 1.

For an entry Cl(0, seven

(11.7)

Note that (11.7) is equivalent to extending the table one column to the left, in the obvious manner, so that (11.6) can be applied in the case

r=0.

Corollary 11.8. The even Clifford algebras Cl(r, seven are isomorphic, as associative algebras with unit, to the matrix algebras described below.

r - 2 mod 8: 0

1,7 3,5 2,6 4

Cl(r, s)even - MN (R) ® MN(R) Cl(r, S)even - MN(R)

Cllr, seven - MN(H) Cl(r, )even L-- MN (C)

Cl(r, ,)even - MN (H) ® MN (H)

Given a choice of unit volume element A for R(r, s), recall that an element a E Cl(r, s) is said to be self-dual if Aa = a and anti-self-dual if as = -a. Also recall that Cl+(r, s) denotes the space of self-dual Clifford elements and Cl- (r, s) denotes the space of anti-self-dual Clifford elements. If Cl(r, s) or Cl(r, seven is isomorphic to the sum MN(F) ® MN(F) of two simple matrix algebras, the summands can be easily identified in Clifford terms as the self-dual and anti-self-dual parts, as follows.

Lemma 11.9. (a) In the following cases,

r-smod8: 4

MN(H) ® MN(H) Cl(r, s) = MN (R) 0 MN(R) Cllr, Steven = MN(H) ® MN(H)

0

Cl(r, S)even S-:: MN

3

7

Cl(r, S)

® MN (R)

a unit volume element A for R(r, s) must correspond to either (1, -1) or (-1, 1).

The Pinor Representations

210

Selecting A - (1, -1) yields

r-smod8: 3

C1(r, s)+

- MN(H) ®{0}

{0} ® MN(H) Cl(r, s)+ = MN (R) ® {0} C1(r,s)- = {0} ® MN(R) C1(r, s)even = MN(H) ® {0} Cl(r, s)even {0} ® MN (H) C1(r, s)-

7 4 0

Cllr, s)+ en - MN(R) ®{0} Cl(r, s)even - {0} ® MN (R)

.

(b) In the cases where Cl(r, s) = MN(C)(i.e., r - s = 1, 5 mod 8) or the cases where Cl(r, seven - MN(C)(i.e., r - s = 2, 6 mod 8), a unit volume element A for R(r, s) must correspond to ±i E MN(C). In these cases, the standard choice is A - i. Proof: (a) Suppose A = M is one of the isomorphisms listed in this lemma. Of course, the identity 1 E A must correspond to the identity 1 E M, and -1 E A must correspond to -1 E M. The remaining elements in the center

of M that square to 1 are (1,-1) and (-1, 1). The remaining elements in the center of A that square to 1 are ±A since A M and M has center RE) R. (b) The elements of the center of MN(C) that square to -1 are ±i. Hence ±i must, correspond to ±A E A.

THE PINOR REPRESENTATIONS The irreducible representations of the Clifford algebra Cl(r, s) are called the pinor representations, while the irreducible representations of the even parts Cl(r, seven are called the spinor representations.

Definition 11.10. The pinor representation of Cl(r, s) and the space of pinors P(r, s) are defined as follows. (If the dimension n = r + s is odd, assume a choice of unit volume element for R(r, s)-an orientation-has been made.)

For r - s = 0, 6 mod 8, an irreducible representation Cl(r, s) = EndR,(P), is called the pinor representation, and the real vector space P is called the space of pinors.

Clifford Algebras C1(r, s) as Algebras

211

For r - s = 2,4 mod 8, an irreducible H-representation C1(r, s) = EndH(P),

is called the pinor representation, and the right H-space P is called the space of pinors.

For r - s = 1, 5 mod 8, an irreducible C-representation

C1(r, s) - Endc(P), with the unit volume element in C1(r,s) corresponding to the complex structure on P, is called the pinor representation, and the complex vector space P is called the space of pinors. For r - s = 3 mod 8, an irreducible H-representation C1(r, s)+ = EndH(P+) is called the positive pinor representation, while an irreducible H-represen-

tation C1(r, s)- - EndH(P_) is called the negative pinor representation. The right H-space P = P+ $ P_ is called the space of pinors. Note that C1(r, s) - EndH(P+) ® EndH(P_), and that the unit volume element is 1

0

0 -1 E EndR(P+) ®EndH(P-).

For r - s = 7 mod 8, an irreducible representation C1(r, s)+ = EndR(P+) is called the positive pinor representation, while an irreducible representation

C1(r, s)- - EndR,(P_) is called the negative pinor representation. The real vector space P P+ ® P_ is called the space of pinors. Note that

C1(r, s) - End(P+) ® End(P_) and that the unit volume element is 1

0

0

-1 E EndR(P+) ®EndR(P_).

212

The Pinor Representations

Now Theorem 11.3 may be rewritten:

Theorem 11.3 (Pinor Version). The Clifford algebras Cl(r, s) are isomorphic, as an associative algebras with unit, to the endomorphism algebras listed below,

r - s mod 8 0, 6

2, 4

C1(r, s) = EndH(P) C1(r, s) = EndH(P)

3

Cl(r, s) = Endc(P) Cl(r, s) = EndH(P+) ® EndH(P_)

7

C1(r, s) = EndR,(P+) ® EndR,(P_ )

1, 5

Remark 11.11. If r-s = 1, 5 mod 8, the isomorphism Cl(r, s) Endc(P) is required to map the unit volume element in Cl(r, s) to i E Endc(P). That is, this is the standard irreducible C-representation of Cl(r, S). The conjugate C-representation of Cl(r, s) defines the conjugate pinor space P: (11.12)

Cl(r, s) = EndH(P),

where the unit volume element in Cl(r, s) maps to -i E EndH(P). In all odd dimensions, Cl(r, s) has exactly two irreducible representations (with the appropriate notion of equivalence of representations) that define two spaces of pinors, P, P or P+, P_ as the case may be. The question of how unique or canonical the space P of pinors is may be answered by the uniqueness results of Chapter 8 for intertwining operators, which are summarized in the next remark. Remark 11.13. Suppose that A is one of the real algebras MN(F) with

F =_ R or F = H, and that pl

:A

EndF(P1), P2 : A --* EndF(P2) are

two irreducible representations of A. Then there exists an F-linear map f : P1 --* P2 (the intertwining operator) with

p2(a)=fop1(a)of-' for allaEA. Moreover, f is unique up to a real scalar multiple c E R*. If it were not for the flexibility of this scale c E R*, the concept of pinor would be canonical.

Next, consider the algebra A = MN(C). First recall that, for any C-representation, p : A -> Endc(W)

sends i E A to either i E Endc(W) or -i E Endc(W), and that (up to C-equivalence) there are exactly two irreducible C-representations:

p : A - Endc(W) where p(i) = i (standard), and p : A- Endc(W) where p(i) = -i (conjugate).

213

Clifford Algebras Cl(r, s) as Algebras

Endc(P2) are two equivalent C-representations of A. Then by definition there exists a C-linear map f : P1 -> P2 (the intertwining operator) with Suppose pl : A - Endc(P1) and P2 : A

p2(a) = f o pl(a) o f-1,

for all a E A.

Moreover, f is unique up to a complex scalar multiple c E C*. Again, if it were not for the flexibility of this scale c E C*, the map f would be unique, making the concept of the pinor space P canonical. In summary, the pinor space P is "projectively canonical."

THE SPINOR REPRESENTATIONS The definition of the space of spinors is analogous to Definition 11.10 for pinors, except that now a choice of volume element is required for even dimensions.

Definition 11.14. The spinor representation ofCl(r, s)eve" and the space of spinors S(r, s) is defined as follows. (If the dimension n = r + s is even, assume that a choice of unit volume element for R(r, s)-an orientationhas been made.) For r - s = 1, 7 mod 8, an irreducible representation p : Cl(r, seven = Endrt(S) is called the spinor representation, and the real vector space S is called the space of spinors.

For r - s = 3, 5 mod 8, an irreducible H-representation p : Cl(r, seven = Ends(S)

is called the spinor representation, and the right H-vector space S is called the space of spinors.

For r - s = 2, 6 mod 8, an irreducible C-representation p : Cl(r, seven - Endc(S), with the unit volume element A equal to the complex structure i on S, is called the spinor representation, and the complex vector space S is called the space of spinors. An irreducible C-representation

p : Cl(r,s)even = Endc(S),

The Spinor Representations

214

with the unit volumeelement A equal to -i where i is the complex structure on S, is called the conjugate spinor representation, and S is called the space of conjugate spinors.

For r - s = 0 mod 8, an irreducible representation p+ : CI(r, s)+ en = EndH(S+)

is called the positive spinor representation, while an irreducible representation p_ : C1(r, seven - End1 .(S_)

is called the negative spinor representation. The real vector space S S+ a S_ is called the space of spinors. For r - s = 4 mod 8, an irreducible H-representation p+ : C1(r, s)+en

EndH(S+)

is called the positive spin or representation, while an irreducible H-representation p_ : C1(r,s)even = EndH(S_)

is called the negative spinor representation. The right H-vector space S S+ a S_ is called the space of spinors. The spinor representation(s) of C1(r, seven restrict to representations of the group Spin(r, s) C Cl(r, Seven It is convenient to use the same symbol for these representations of Spin(r, s) (i.e., p or p,;5 or p+, p_, as the case may be). Now Corollary 11.8 may be rewritten:

Corollary 11.15. (The spin representations).

r-smod8: 0

4

p+ e p_ : Spin(r, s) C Cllr, Seven EndH(S+) a EndR,(S_) pf C1(r, s)f en = EndH(S±) p+ e p_ : Spin(r, s) C Cl(r, S) even

EndH(S+) a EndH(S-) pf Cl(r,.)even = EndH(S±) 2,6

p

: Spin(r, s) C Cllr, seven Spin(r, s) C Cl(r, s)even

1,7

p

: Spin(r, s) C C1(r, s)even = EndR,(S)

3, 5

p

: Spin(r, s) C Cl(r, s)even = EndH(S)

p

Endc(S) Endc(S)

Clifford Algebras Cl(r, s) as Algebras

215

THE FIRST PROOF We shall give two proofs of the important Theorem 11.3. The first proof is by induction on the dimension n - r + s and is based on the following three lemmas.

Lemma 11.16. Cl(1, 0) - C, C1(0,1) = R ® R, Cl(2, 0) = H, Cl(1, 1) M2(R), Cl(0, 2) = M2(R). Proof: A set of "'y matrices" for each one of these five Clifford algebras Cl(r, s) is listed below: For Cl(1, 0), For Cl(0, 1),

e1iEC.

For Cl(2, 0),

e1=rEL. e1 j, e2=kEH.

For Cl(1, 1),

e2

For Cl(0, 2),

1

e1

(01

e2 = (10

0

E M2(R).

-10) E M2(R).

U

Lemma 11.17. (11.18) Cl(r + 1, s + 1) = Cl(r, s) ® Cl(1,1) = Cl(r, s) OR M2(R). (11.19)

Cl(s, r + 2) = C1(r, s) ® Cl(0, 2) = Cl(r, s) OR M2(R).

(11.20)

Cl(s + 2, r) = Cl(r, s) ® CI(2, 0) = Cl(r, s) OR H.

Proof: Suppose

is an orthonormal basis for R(r,s) C C1(r, s). Let u1i u2 denote an orthonormal basis for V C Cl(V), where V is a twodimensional inner product space, and let '1= U1u2 denote the unit volume element.

Define W C Cl(r, s) OR C1(V) to be the inner product space with orthonormal basis (11.21)

e1 ®72, ..., e ®7J, 1 (2) U1, 10 U2,

and signature given by (11.22)

Ile2 07711 = i2IIeiII,

II1®u.II=

IlujIl,

j =1,...,n, j=1,2.

Note that any two of the basis vectors in (11.21) anticommute and that (e.i®,)2=e2®772=-77211e111

(

11 2 3)

_-lief ®7711,

.

(1®v1)2=1 ®uj=- ilu.ilj=-ll1 ®uiil, j=1,2.

The First Proof

216

Therefore, by the Fundamental Lemma of Clifford algebras the natural inclusion 0 : W -; Cl(r, s) ® Cl(V) extends to an algebra homomorphism 4 : Cl(W) -} Cl(r, s) 0 C1(V).

This map 0 must be surjective, since the basis for W generates the algebra Cl(r, s) ® CI(V). Therefore, by a dimension count, 0 must be injective and hence provides the desired isomorphism Cl(W) = Cl(r, s) ® C1(V).

The signature of W follows from (11.22), reversing that of R(r, s), when V - R(2, 0) or V - R(0, 2), since in both these cases ,772 = -1. J

Lemma 11.24. (11.25)

Mk(F) OR Mm(R) = Mkm(F) for F = R, C, or H.

(11.26)

H OR H

M4(R).

(11.27)

C OR H

M2(C).

(11.28)

COR CC®C.

Proof: The isomorphism (11.25) is a consequence of these two special cases:

(11.25a)

Mk(R) 0 M,, (R) = Mk,,,(R)

and (11.25b)

F®Mm(R)-M,(F) forF - CorH.

The algebra homomorphism ¢ : H OR H -> M4(R), defined by (11.29)

q(p®q)(x)-pxq forallxER4-H,

is an isomorphism since 0 is an isometry. (See Problem 1.) Let 0(i®1) (left multiplication by i) determine the complex structure on H - R4 - C2. Then the subalgebra C OR H of H OR H corresponds to the subalgebra M2(C) of M4(R) under the map 0. This proves (11.27).

Since C2 = C ® C j = H, via the complex structure 0(i (9 1) (left multiplication by i), the matrix 0(l (9 i) (right multiplication by i) equals

Clifford Algebras Cl(r, s) as Algebras

the 2 x 2 complex matrix (i

i

217

). (Note (z + wj)i = iz - i(wj). Thus,

C 0 C = C ® C, contained in M2(C) as the diagonal 2 x 2 matrices. U

The reader is invited to give the proof of Theorem 11.3, based on these three lemmas, in Problem 2. A second proof of Theorem 11.3 will be given in the course of developing additional information about the Clifford algebras in the next chapter.

THE SPINOR STRUCTURE MAP ON P(r, s) Definition 11.30. An invertible real linear map s on P(r,s) is called a spinor structure map if s2 = ±1 and (11.31)

a = sas` z for all a E Cl(r, s) C EndR(P).

Thus, a spinor structure map on the space P(r, s) of pinors is sufficient extra structure to determine Cl(r, s)even as a subalgebra of EndF(P(r, s)). Recall that, if the dimension n = r + s is even, then (11.32)

a = AaA-'

for all a E Cl(r, s),

while if the dimension n =- r + s is odd, then (11.33)

Aa = as for all a E Cllr, s);

where A is a unit volume element.

EVEN DIMENSIONS If the dimension n =- r + s is even, then the spinor structure map s is defined to be a choice A of the two unit volume elements. The next theorem

describes how s = A determines the spinor space S(r, s) from the pinor space P(r, s). This theorem is stated so that each of the four cases r - s = 0, 2, 4, 6 mod 8 can be referred to independently of the other three cases. Consequently, there is a certain amount of repetition.

Theorem 11.34 (The Spinor and Pinor Representations in Even Dimensions). Define the pinor structure map s to be a choice of unit volume element A.

For r - s = 0 mod 8: Consider the pinor representation (11.35)

Cl(r, s) = EndR,(P).

Even Dimensions

218

Define S+ and S_ to be the eigenspaces of A:

St = {x E P : Ax = -

(11.36) ±X}-The decomposition

P = S+ ®S_

(11.37)

determines a 2 x 2 blocking (11.38)

A= ( ac d)

for each A E EndR,(P),

and isomorphisms: (11.39)

C1(r, seven

(O b) : a E EndR,(S+), b E EndR,(S_) } ,

(11.40)

a C1(r, s)""' = {(0 0)

: a E EndR,(S+) } = Enda(S+),

(11.41)

C1(r, seven - 1 ( 0

:bE

b)

Enda(S_)} = Enda(S_)

that are the spinor representations. Note that (11.42)

A=

1

0

0

-1)

and that (11.43)

dimR, S+ = dimR, S_.

For r - s = 4 mod 8: Consider the pinor representation (11.44)

C1(r, is) c--- EndH (P).

Define S+ and S_ to be the eigenspaces of A: (11.45)

Sf={xEP:Ax =±x}.

Clifford Algebras C1(r, s) as Algebras

219

The decomposition

P = S+ ®S_

(11.46)

determines a 2 x 2 blocking

A=

(11.47)

(2

for each A E EndH(P),

d)

and isomorphisms

Cl(r, s)even = f

(a b

(11.48)

) : a E EndH(S+), b E EndR(S_) }

EndH(S+) ® EndH(S_),

(11.49)

C1(r, s)+en

(11.50)

Cllr s)even

(O

0)

JJJ

: a E EndH (S+) } - EndH(S+),

f(O b) : b E EndH(S_) } = EndH(S_)

that are the spinor representations. Note that

A-

C1

0

-1 1

and that dimH S+ = dimH S_.

(11.52)

For r - s = 2 mod 8: Consider the pinor representation C1(r, s) = EndH(P),

(11.53)

with the right H-structure on P given by I, J, K. Utilizing the complex structure I on P, define S and S to be the eigenspaces of A:

S={xEP:Ax=xI}, S={xEP:)x=-xI}.

(11.54)

Then C1(r,s)even C EndH(P) maps S to S and S to S, and the induced isomorphisms (11.55)

p : C1(r, seven = Endc(S),

C1(r, s)even = Endc(

Odd Dimensions

220

are the spinor representations. As R-representations, p and p are equivalent via the intertwining operator J:S

(11.56)

For r - s = 6 mod 8: Consider the pinor representation Cl(r, s) = EndR(P).

(11.57)

Define S and '9 to be the eigenspaces of A:

(11.58) S={sEP®RC:ax=ix}, Then Cl(r, s)even C EndR,(P) C Endc(P ®R C) maps S to S and '9 to S, and the induced isomorophisms (11.59)

p : Cllr, seven = Endc(S),

p : Cl(r, s)e°eII = Endc(S)

are the spinor representations. As R-representations, p and p are equivalent via the intertwining operator

C:S

(11.60)

the natural conjugation on P OR C.

Proof: The cases r - s = 0, 4, 6 mod 8 are exercises (see Problem 6). Suppose r - s = 2 mod 8. Since A E Cl(r, seven commutes with A, A preserves the eigenspaces S and S of A. Let a = p(A) denote A E Cl(r, Seven restricted to S, and a = p(A) denote A E Cl(r, seven restricted to S. Note that J : S --+ S is an anticomplex linear isomorphism. Since A commutes with J, (11.61)

a(xJ) _ (ax)J for all x E S.

The theorem now follows easily. Note that p(A) = I and p(A) = -I.

ODD DIMENSIONS Now we turn to the odd dimensional cases. Several questions arise. Does a spinor structure map exist? How (non-) unique is it? Finally, how does s determine the spinor space S(r, s)? These questions are most easily answered by turning to the inverse question first: How to construct the pinor representation from the spinor representation? The unit volume element A provides the key to the answer to this question, since A commutes with Cl(r, S)even

Clifford Algebras Cl(r, s) as Algebras

221

Lemma 11.61. If r - s = 1, 5 mod 8, then Cl(r, s) - C1(r, seven OR C,

(11.63)

with A = i.

If r - s :=3,7 mod 8, then C1(r, s) = Cl(r, )even OR L,

(11.64)

with A = r,

or equivalently, (11.64')

Cl(r, s) = Cllr, seven ® (R ®R),

with A = (1, -1).

These algebra isomorphisms are the identity on Cl(r, S)even.

Proof: Since Cl(r, s) = Cl(r, s)even ® A Cllr, seven and A commutes with Cllr, Seven

Cl(r, s) = Cl(r, s)even ®R A,

where A is the two-dimensional algebra spanR{1, A}.

If r - s = 1, 5 mod 8, then A2 = -1 and A - C with A = i; while if

r - s= 3,7 mod 8, then A2=1 and A=L=RED Rwith A=r=(1,-1). This lemma provides part of the second proof of Theorem 11.3 that is given in the next chapter.

Corollary 11.65. The even dimensional cases of Theorem 11.3 imply the odd dimensional cases of Theorem 11.3.

Proof: Of course, MN(F)®R(RED R) - MN(F)®MN(F) and MN(R)®R C - MN(C). Exactly as in Lemma 11.24, MN(H) OR C - M2N(C).

Theorem 11.66 (The Spinor and Pinor Representations in Odd Dimensions). For r - s = 1 mod 8: Given the spinor representation (11.67)

Cl(r, s)even

EndR,(S),

define (11.68)

P = S OR C.

Since Cl(r, s) = Cl(r, s)even OR C, with J1 = i, there is an induced isomorphism

Cl(r, s) - Endc(P)

Odd Dimensions

222

that is the pinor representation. The natural conjugations on P = S®R,C is a spinor structure map. Note

SE{x-P:sx=x}. All other spinor structure maps are of the form ete s, 0 E R.

For r - s = 3 mod 8: Given the spinor representation (11.69)

C1(r, s)even - EndH(S),

define P± - S so that P - S ® S. Since C1(r, s) ^' C1(r, seven ® (R ®R) with A = (1, -1), there is an induced isomorphism C1(r, s) = EndH(S) ® EndH(S)

that is the pinor representation. The natural reflection

(0 11 0)

1

that maps (x, y) E P to (y, x) E P is a spinor structure map. All other spinor structure maps are of the form 0

Peers=±

e_er C

0

eer

onP=_S®S. For r - s = 5 mod 8: Given the spinor representation (11.70)

C1(r, s)even - EndH(S),

define P to be S with the complex structure I. Since

C1(r, s) = C1(r, s)even ® C,

with A = I, there is an induced isomorphism (11.71)

C1(r,s) - Endc(P)

that is the pinor representation. Given the isomorphism (11.70), the isomorphism (11.71) induced by (11.70) is given explicitaly by (11.72)

(a 9 z)(x) = axz for all x E P,

Clifford Algebras C1(r, s) as Algebras

223

where a E Cl(r, seven - EndR(S) and z E C = spanR{1, I}. The map J (right multiplication) is a spinor structure map and all other spinor structure maps are of the form eIBJ.

For r - s = 7 mod 8: Given the spinor representation Cl(r, s)even = EndR(S),

(11.73)

define P± - S so that P = S ® S. Since Cl(r, s) - Cl(r, s)even (g (R ® R) with A = (1, -1), there is an induced isomorphism

Cl(r, s) - EndR(S) ® Endri(S) that is the pinor representation. The natural reflection

s= that maps (x, y) E P to (y, x) E P is a spinor structure map. All other spinor structure maps are of the form 0

.eSTs = ±

(e-8T

eeT 0

on P=S®S. Proof: Except for the uniqueness results for a spinor structure map s, all other parts of the theorem are immediate consequences of Problem 1(c) and Problem 7.

Note that (n odd) a spinor structure map s for Cl(r,s) C EndR(P) must lie outside of cen Cl(r, s) = spanR{1, A} but still lie inside of the centralizer of Cl(r, s)even in EndR(P). This leaves very little choice for s, and the precise range of possibilities for a spinor structure map s can be deduced from the next lemma (Problem 8). iJ Lemma 11.74.

r-s

cen Cl(r, s) - spanR It, Al

centralizer of Cllr, seven in EndR(P(r, s))

C with A

M2(R) M2(R)

mod 8 1

i

5

R® R with A r Cwith A

I

7

R ®R with A - r

(1, -1)

3

(1, -1)

H M2(R).

Problems

224

Proof: In the case r - s = 1 mod 8, the lemma is just the statement that (11.75)

the centralizer of EndR(S) in EndR,(S OR C) is M2(R).

In fact, 1, i, s, and is form a vector basis for this centralizer, where s denotes

the natural conjugation on S OR C. In the case r - s = 5 mod 8, the lemma follows from the fact that (11.76)

the centralizer of EndH(S) in EndR(S) is H, the field of right scalar multiplications,

which is part (b) of Lemma 8.25.

The cases r - s = 3 mod 8 and r - s = 7 mod 8 are similar. The centralizer of

(° ) : a E EndF(S)

EndF(S)

}

a

in EndR(S ® S) is M2(R) for both F = R and F = H since H has center R.

PROBLEMS 1. Let H OR H have the inner product defined by (a 0 b, c ® d) (a, c) (b, d). Let M4(R) have the natural positive definite inner product defined by

(A, B) = 4 trace ABt.

(a) Prove that the map

H ®R H --> M4(R) defined by (11.29) preserves the inner products. (b) Prove that 0 is an isomorphism. (c) Suppose VH is a right H-space with I, J, K a standard basis for the scalars. Let VC denote VH with the complex structure I. Show that (aOz)(x) = axz for a E EndH(VH), z E C - span {1, I}, x E V, defines an isomorphism EndH(VH) OR C

EndC(VC).

2. Using Lemma 11.16, Lemma 11.17, and Lemma 11.24, give the proof of Theorem 11.3.

Clifford Algebras Cl(r, s) as Algebras

225

3. Prove that Table 11.5 is symmetric about the lines q = p + k with k = 3 mod 4, and that CI(p, q) = Cl(p - 4, q + 4) as algebras. 4. Equip C" with the standard C-symmetric inner product

(a) Define the associated complex Clifford algebra, denoted Clc(n).

(b) Prove that Clc(1) - C ® C and Clc(2) = M2(C) as complex associative algebras with unit. (c) Prove that, as complex associative algebras with unit,

Clc(2p) - MN(C) with N = 2P, and

Clc(2p+ 1) = MN(C) ® MN(C)

with N = 2P,

by showing

Clc(n + 2)

Clc(n) ® Clc(2).

(d) Prove that Cl(p, q) OR C - Clc(n) for all signatures in a given dimension n = p + q. 5. Prove that Table 11.5 has the following periodicity isomorphisms:

Cl(p+1,q+1) - C(p,q)®M2(R), C1(p + 8, q)

C(p, q) 0 Mis(R),

Cl(p, q + 8) 25 C(p, q) 0 M16(R).

6. Give the proof of Theorem 11.34 for the case r - s = 0 mod 8. Moreover, verify that the isomorphism Cl(r, s) = EndR(P) determines the following isomorphisms: Cil(r, Seven 25

1( a

Cl(r, S)odd - r ( 0 b

Cl(r, s)+ -

Cl(r, s)_ -

a

1( b 0

1(0

a E Enda(S+), b E EndR,(S_) I , a E IIomR,(S_, S+), b E HomR,(S+, S_)

a E EndR(S+), b E HomR,(S+, S_) } . a E HomR,(S+, S_), b E EndR,(S_)

Cl(r, S)+en

(a

a E EndR,(S+)

Cl(r, Seven =

f(0

bEEndR,(S_)}.

=

l

0

Problems

226

7. (a) Given an algebra isomorphism A °-' EndR(S), show that there is an induced isomorphism A OR C '- Endc(S OR C). If s denotes the natural conjugation on S OR C, then show that for a E A, z E C, s(a ®z)s = a ®z.

(b) Given an algebra isomorphism A = EndR(S), show that

A®(R(D R)=A®A-{ (0 b) :a,bEA}CEndH(S®S), 111 \

JJJ

and that

8. Using Lemma 11.74, verify that the only possible spinor structure maps (for odd dimensions) are those described in Theorem 11.66. 9. (a) Suppose V is one of the 2-dimensional spaces R(2, 0), R(1, 1), or R(0, 2), and that W is the vector space defined in the proof of Lemma 11.17. Recall from Lemma 11.17 that

Cl(W) = C1(r, s) 0 Cl(V)

(as algebras).

Prove that the extra structure in C1(W) of the three involutions and the inner product is given by (1)

(a ®b) = a ®b,

(2)

(a ®b)^ = a ®

(3)

(a ®b)v = a ®

(4)

(a (9 b, c ®d) = (a, c) (b, d).

(b) Show that, as associative algebras with unit,

Cl(p - 1, q) = CI(q - 1, p) (symmetry about the line q = p + 1). Moreover, show that this isomorphism can be chosen to preserve the hat involutions and the inner products.

12. The Split Case C1(p, p)

The case Cl(r, s) with r = s is particularly easy to analyze, independent of the inductive method used in the last chapter. Furthermore, the split case can be used to determine information about the complex Clifford algebra Clc(2p) = C1(p, p) ®R. C and the other real Clifford algebras

Cl(r, s) (r + s = 2p) as subalgebras of Cl(p, p) ®R, C. This chapter is independent of Chapter 11 and may be read first.

A MODEL FOR Cl(p,p)

Let R(p, p) = R(p, 0) x R(p, 0) with z = (u, v) E R(p, p) and IIzI I Iu!I

- IIvII. Let Eux =- uAx denote left exterior multiplication by u E R(p, 0)

for x E AR(p, 0). Let 1,,x = u L x denote left interior multiplication by u E R(p, 0) for x E AR(p, 0). Then Lu Eu - Iu denotes left Clifford multiplication by u in Cl(p, 0) L ARP, and Lu = Eu + Iu denotes left Clifford multiplication by u in Cl(0, p) = ARP.

Lemma 12.1. Lu and Ly anticommute for all u, v E BY = A'RR.

Proof 1: Lu Lv + LU Lu = (E,, - Iu)(E, + Iv) + (E, + II)(Eu - Iu) = Eu IU + I Eu - (I Eu + E 1u)

=(u,v)-(v,u)-0. 227

A Model for Cl(p, p)

228

Here we use the fact that

E. I,, + I, E. = (u, v), obtained from (9.3) by polarization (i.e., replace w by u + v in (9.3)). J

Proof 2: By Problem 1, Lv x = R+1. Therefore

(LV Lv +LvLu)x=uxv+uxv=0 for all x.

Lemma 12.1. Let e1,.. . , ep denote the standard orthonormal basis for lep is all of R(p, 0). The algebra generated by E,,,.. ., Eej, Endn.(ARR).

Proof: The proof is by induction on p. If p = 1, then computing the four 2 x 2 matrices for the operators IEEe, Iej Ee, Eele with respect to the basis 1, e for AR yields the result. Now assume the lemma is true for p and let Ali E Enda(ARP) denote the linear map that sends e j to e j and eK to zero for K # J (i.e., the matrix with a single 1 in the I, Jth position). Let e =- ep+l and observe that

IeAijEe maps e; to el, Aji1. maps aAej to el, EeAli maps e; to e A ej, EeAijIe maps eAej to eAej,

(12.3)

while mapping all other eK or e A eK to zero.

Theorem 12.4. There exists an algebra isomorphism Cl(p, p) = Enda(AR'),

(12.5)

extending the map

: R(p, p) --* EndR,(ARR) defined by

(12.6)

Proof: Because of Lemma 12.1,

(Lu +

(Lu +

(Lu )2+ (LU )2+Lu Lv +L Lu = - (Ilull - Ilvll)

The Split Case Cl(p, p)

229

for all u, v E RP. The Fundamental Lemma of Clifford Algebras implies that 0 extends to an algebra homomorphism in the following diagram:

R(p, p) - Enda, (ARP) n

(12.7)

Cl(p, P)

Because of Lemma 12.1, the image of the map 4 : Cl(p, p) -+ Enda(ARP) must be all of Endf,(ARP). Therefore, by a dimension count, 0 must be an isomorphism. _1 Consider the map 0 : C1(p, p) -> EndR,(ARP) as an identification.

Corollary 12.8. Let el, ..., eP denote the standard orthonorrnal basis for R(p, 0).

L...... Lep, Lei ... I Len

(12.9)

provides the standard basis (of y matrices) for R(p, p) C EndR,(ARP) Cl(p,p)-

Definition 12.10 (The Split Case). The vector space P(p,p) of pinors is defined to be ARP, and the isomorphism

Cl(p, p) = Enda(P)

(12.11)

is called the pinor representation of Cl(p, p).

Since the dimension n = 2p is even, the canonical automorphism of Cl(p, p) is given by (12.12)

a=Aaa-1 for all aECl(p,p),

where A is a unit volume element for R(p, p). Recall that there are two choices for A and that making a choice of A is equivalent to choosing an orientation for R(p, p).

Definition 12.13 (The Split Case). Assume that an orientation for R(p, p) is given, and let A denote the positive unit volume for R(p, p). (12.14)

S+ E {xEP:Ax=x}

is the space of positive spinors, and

Pinor Inner Products for C1(p, p) - Enda(P(p, P))

230

S_= {xEP:.1x=-x}

(12.15)

is the space of negative spinors.

The map x = .1x for x E P is sometimes called the canonical pinor involution. This notation x -+ x =- Ax is consistent with the notation a

a for a E Cl(p, p) because

ax = Aax = )aA-lax = ax.

(12.16)

The decomposition

PS+®S_

(12.17) determines a 2 x 2 blocking

b)\

(12.18)

d

for each A E EndR,(P).

Note that (12.19)

Lemma 12.20. (12.21)

Cl even(p, p) = 1(0 6) : a E Endit(S+), b E EndR,(S_) } J 11

(12.22) Clodd(p, p) C

a E Enda(S_, S+), b E EndR,(S+, S_)

0

° and Proof: Consult (12.12)

111

(12.19). J

Corollary 12.23. dimS+ = dimS_. Proof: If s± = dim St, then s+ + s? = dim Cl even(p, p) = dim Cl odd (p p) _ 2s+s_.

J

PINOR INNER PRODUCTS FOR Cl(p, p) - EndR,(P(p, P)) The pinor space has inner products i and E that determine the hat and check involutions as adjoints with respect to these inner products.

Let (, ) denote the inner product on P =- AR(p, 0) induced by the standard inner product on R(p, 0). Let o = el A A ep denote the unit volume element for R(p, 0). Utilizing the Clifford multiplication on P AR(p, 0) - Cl(p, 0), given x E P, let cx E P denote the Clifford product of a and x in CI(p, 0).

The Split Case Cl(p, p)

231

Definition 12.24. The hat bilinear form e on P = AR(p, 0) is defined by (12.25)

(p odd) t(x, y) _ (x, o y)

for all x, y E P,

(12.26)

(p even)

e(x, y) _ (.i, oy)

for all x, y E P.

The check bilinear form e on P = AR(p, 0) is defined by E(x,y)

e(x y)

(12.27)

Hat/Check Theorem 12.28. Both e and a are nondegenerate bilinear forms on the space P of pinors. The Clifford hat anti-automorphism, A A, is equal to the adjoint on EndR(P) with respect to the bilinear form e. That is, given A E Cl(p, P) End(P), A is determined by (12.29)

e(Ax, y) = e(x, Ay)

for all x, y E P.

e(Ax, y) = e(x, Ay)

for all x, y E P.

Similarly, (12.29')

The bilinear forms 9 and e on P are either symmetric (with split signature) or skew, as indicated in the following table. Table 12.30

pmod4

9

e

0

symmetric

1

skew skew

symmetric symmetric

2 3

symmetric

skew skew

Moreover,

(12.31) S+ and S_ are nondegenerate and orthogonal if p is even, and (12.32)

S+ and S_ are totally null if p is odd.

Proof: Since (,) is nondegenerate on P =- AR(p, 0), and both x + x and x E--* ax are invertible, the bilinear forms e and e are nondegenerate. Suppose (12.29) is true. Then (12.29') follows directly:

e(Ax, y) = e(Ax, y) = e(Ax, y) = e(x, Ay) = e(x, Ay).

Pinor Inner Products for Cl(p,p) = Endp,(P(p, p))

232

Let A - At denote the anti-automorphism of End(P) determined by t. Suppose we can show that

if A E R(p,p) C Cl(p,p) = End(P), then At = -A.

(12.33)

Then (At)" is an automorphism of Cl(p,p) that extends the identity on R(p, p), so that by the uniqueness part of the fundamental lemma of Clifford algebras (At)^ = A for all A E Cl(p,p). Therefore, At = A^ for all A E Cl(p, p).

The next lemma implies, as a corollary that At = -A for A E R(p,p), completing the proof of (12.33).

Lemma 12.34. Suppose u E R(p, 0) and x E AR(p, 0). (12.35) (p odd )

u A (ox) = -o(u L x)

and

u L (ax) = -o(u A x).

(p even)

u A (cx) = o(u L x)

and

u L (cx) = o(u A x).

(12.36)

Note that Lemma 12.34 implies (12.37)

(12.38)

(p odd

(Eu-lu)o=o-(Eu-Iu),

(Eu + Iu)o = -o(Eu + Iu), (p even)

(Eu - Iu)o = -o(Eu - Iu), (Eu + Iu)o = a(Eu + Iu)

Corollary 12.39. (E.-II)t=-(Eu-Iu)

and

(Eu+Iu)t =-(Eu+Iu) for all u E R(p, 0).

Proof: Suppose p is odd. Then

e((Eu - I3)x, y) = ((E. - II)x, oy) _ -(x, (Eu - Iu)cy) _ -(x, o(EE - .t )y) = -E(x, (Eu - Iu)y), and E((Eu + I1)x, y)

((Eu + II)x, oy) = (x, (Eu + Iu)oy) _ -(x, o(Eu + lu)y) = -E(x, (Eu + Iu)y)

233

The Split Case Cl(p,p)

The proof for p even is Problem 2(a). J Proof of Lemma 12.34: Suppose uAx = 0. Then x is of the form x = uAy

and hence u L ax = 0. Therefore, u A (ax) = uax and a(u L x) = -oux. Suppose U L x = 0. Then u A ax = 0. Therefore, U L (ax) = -uax and a(u Ax) = aux. If p is odd, then ua = au and (12.35) follows. If p is even, then ua = -au and (12.36) follows. The proof of the Hat/Check Theorem 12.28 is completed as follows: If either f or E has a signature (i.e., is symmetric), then by Problem 9.10 the signature must be split. Table 12.30 is deduced as follows. First, suppose p is even so that E(x, y) _ (x, ay). Then (12.40)

E(y, x)

(y, ax) = (ax, y) = (ax, y) _ (x, &y).

Here & is the CI(p, 0) hat of the unit volume element a for R(p, 0). Since

& = a if p = 0 mod 4, and & = -a if p = 2 mod 4, this verifies that e is symmetric if p = 0 mod 4 and that E is skew if p = 2 mod 4. The analysis of t for p odd is Problem 2 (b). Using the facts that (12.41)

A2 . 1,

(12.42)

A = A = A, for p even,

(12.43)

A = A = -A, for p odd,

both (12.31) and (12.32) follow. For example, if p is even and x E S+, Y E S_, then e(x, y) = -E(Ax, Ay) = -E(x, AAy) = -E(x, y),

proving that S+ and S_ are orthogonal.

The bilinear form e on P can also be understood in terms of (,) and a. Alternatively, t can be analyzed in terms of e using the relationship (12.44)

E(x, y) = E(Ax, y), with A =

(1 -0) 0

.

1

The details are left as Problem 3. [J

THE COMPLEX CLIFFORD ALGEBRAS (CONTINUED FROM CHAPTER 9) The complex Clifford algebra Clc(2p) is obtained from the complex vector space Cep equipped with a square norm (see (9.74)).

The Complex Clifford Algebras

234

Theorem 12.45. As complex associative algebras with unit,

Clc(2p) = Mv(C)

(12.46)

with N = 2p.

Also, given an isomorphism

Clc(2p) - Endc(Pc), there exist complex inner products ec and ec on PC, so that ec(ax, y) = 9C (X, ay)

(12.47)

and (12.48)

ec(ax, y) = 1c(x, dy)

for all a E Clc(2p) and x, y E Pc. Both ec and ec are C-symmetric or C-skew exactly when their counterparts, a and t in the split case, are R-symmetric or R-skew as indicated in Table 12.30. Proof: It is convenient to use the square norm on C2p = C(p, p) paralleling the real split case. Let

z - (u, v) E C(p, 0) x C(p, 0) = Cep have square norm

-vp.

llzll= Let (12.49)

Lu E - I E Endc(AC")

(12.50)

Lu = E, + I E Endc (ACp)

with interior multiplication I,, determined by the square norm llull = u2 +

+u2 on Cp.

Exactly as in Lemma 12.1,

Lu Lu + Lv Lu = 0 since L,+, and LV acting on ACp are just the complexifications of the real operators Lu and LV acting on AR". The map 0: C(p,p) --> Endc(ACP) defined by O(z) _ O(u, v) = Lu + Lv has a unique extension

0 : Clc(2p) --+ Endc(AC")

The Split Case Cl(p, p)

235

that is a complex algebra isomorphism. This proves that Clc(2p) MN(C). Because of the results of Chapter 8, any other isomorphism Clc(2p)

Endc(Pc) of complex algebras is equivalent, via a complex linear intertwining operator, to the isomorphism Clc(2p) = Endc(ACP).

(12.51)

Thus, we need only construct ec and ec on ACE. Now simply take ec (and tc) to be the complexifi cations of the real inner products e and e on ARE. H Corollary 12.52. The complex Clifford algebras Clc(2p), in even dimensions, contain no proper nontrivial two-sided ideals.

Proof: Because of (12.46), it suffices to show that MN(C) has no proper nontrivial two-sided ideals. This was done in Chapter 8.

Corollary 12.53. The real Clifford algebras Cl(r, s), in even dimensions r + s = 2p, contain no proper nontrivial two-sided ideals.

Proof: Since Clc(2p)

Cl(r, s) ®R C is the complexification of the real algebra Cl(r, s), Corollary 12.53 is an immediate consequence of Corollary 12.52.

Cl(r, s)(r+s = 2p) AS A SUBALGEBRA OF Clc(2p) = Cl(p,p)®RC Each Clifford algebra Cl(r, s) of even dimension n = r + s = 2p is of the form Cl(p + k, p - k) with r - s = 2k, k E Z. These Clifford algebras are real subalgebras of Clc(2p). By Proposition 9.76, (12.54)

Cl(p + k, p - k) OR C = Clc (2p).

The conjugation or reality operator fixing Cl(p + k, p - k) in Clc (2p) will be denoted R, when the signature r, s = p+k, p-k has been prescribed. No matter the signature, the model taken for Clc(2p) = Cl(p+ k, p k) OR C will be the complexification of the split case Cl(p,p). Thus, (12.55)

Clc(2p) - Cl(p, P) OR C

is realized as the complexification of the matrix algebra (12.56)

Cl(p, p) L EndR(ARE).

236

Cl(r, s)(r + s = 2p) as a Subalgebra of Clc(2p) - Cl(p, p) OR C

Namely,

Clc(2p) - Endc(ACP),

(12.57)

where the pinor space for Clc(2p), ACP = ARP OR C,

(12.58)

is the complexification of the pinor space ARP for Cl(p, p). The conjugation, or reality operator for Cl(p, p), contained in Clc(2p) Cl(p, P) OR C will be denoted C. The complexification ACP t--- ARP (DR C also comes equipped with a conjugation, the natural extension of the conjugation on CP - RP ® C. (12.59) This conjugation on ACP that fixes ARP will be denoted C (or x = Cx for x E ACP). In turn, the conjugation C on ACP induces a conjugation on Endc(ACP): (12.60)

a - CaC for all a E Endc(ACP),

which fixes EndR(ARP) in Endc(ACP). Here EndR(ARP) is naturally considered a subspace of Endc(ACP) by extending each real linear map of ARP to a complex linear map of ACP - ARP ® C. Thus, (12.61)

Ca ="d for all a E Clc(2p) - Endc(ACP).

Remark 12.62. Note that the map a -+ Ca = a is a real algebra automorphism of Clc(2p) = Endc(ACP), that is complex antilinear. Also note that each element A E EndR(ACP) has a unique decomposition (12.63) A = a + bC into a complex linear map a E Endc(ACP) and an anticomplex linear map bC, where 6 E Endc(ACP) is complex linear. Let a+, ... , e+, e 1- , . . . , ep denote the standard basis for R(p,p) considered as a subspace of EndR(ARP) - C1(p, p). That is, et = L , left Clifford multiplication by ej (12.64)

on Cl(p,0) = ARP,

and

(12.65)

e- __ Lei,

left Clifford multiplication by ej on Cl(0, p) = ARP.

The signature of R(p, p), considered as a subspace of C(p,p) C Endc(ACP) = Clc(2p) can easily be modified.

237

The Split Case Cl(p, p)

Definition 12.66. Let (12.69)

(12.70)

(k positive)

ei , ... , ep , ie1 , ... iek

(k negative) e+ , ... , eP+k ,

ieP+k.+1

;

ek}1, ..., ep

, ... , ie4 , el , ... , ep

define a standard basis for R(p + k, p - k) as a subspace of C(p,p) C Endc(ACP) = Clc(2p).

Proposition 12.69. The real subalgebra of Endc(ACP) generated by R(p + k, p - k) is isomorphic to Cl(p + k, p - k).

Proof: The inner product (, ) on C(p, p), restricted to R(p + k, p - k) is real-valued with signature p + k, p - k. Thus the Fundamental Lemma of Clifford Algebras applies, yielding an algebra homomorphism

0 : Cl(p+k,p- k) --* Endc(ACP)

(12.70)

that extends the inclusion

R(p + k, p - k) C Endc(ACP) given by Definition 12.66. Since the dimension n - 2p is even, Cl(p + k, p - k) contains no proper nontrivial two-sided ideals, by Corollary 12.53.

Therefore, the kernel of 0 is trivial. J

THE PINOR REALITY MAP In order to identify Cl(p + k, p - k) as a matrix algebra, it is useful to compute the centralizer of Cl(p + k, p - k) in EndR.(ACP), i.e., all real linear maps of ACP that commute with Cl(p + k,p - k). Of course, the centralizer always contains C, all complex multiples of the identity. Note that EndR(ACP) is larger yet than ClC(2p) ; Endc(ACP).

Theorem 12.71. The centralizer of Cl(p+k, p- k) in EndFt(ACP), which canonically contains C, is the algebra (12.72) (12.73)

M2 (R)

H

if k = 0,3 mod 4 (r - s = 0,6 mod 8), if k = 1, 2 mod 4 (r - s = 2, 4 mod 8).

Remark 12.74. Recall from Chapter 6 that both M2(R) and H are normed algebras and that the square norm is uniquely determined by the algebraic structure. Theorem 12.71 will be proved after some lemmas are established.

The Pinor Reality Map

238

Lemma 12.75. A E Enda(ACP) commutes with Cl(p + k, p - k) if and only if A is of the form A = a+bpC, with a, b E C and i defined as follows: ek (k even positive)y = ei (12.76) (k odd positive) y = ei ...e+ ek+1 .. eP , (12.77)

(k even negative) µ = ep k+i...ep (k odd negative) p = ei ep kei

(12.78) (12.79)

ep .

Proof: Given A E EndR,(ACP ), it has a unique decomposition (see (12.63))

A = a + 6C, with a, b E Endc(ACP). Suppose A commutes with all u E Cl(p + k, p - k), or equivalently, with all u E R(p + k, p - k). Since Cu = iC, this is equivalent to au = ua and bu = ub for all u E R(p + k, p - k). (12.80) Since a commutes with i and Endc(ACP) = Clc(2p) - Cl(p + k, p - k) ® i Cl(p -f k, p - k), a belongs to the center of Clc(2p). Thus, a E C is a complex scalar (because of either the complex version of Lemma 9.49 or the fact that Clc(2p) L--- MN(C)). Referring back to the choice (Definition 12.66) of basis for R(p + k, p - k) C Cl(p + k, p - k) C Endc(ACP) - Clc(2p), the condition bu = ub becomes (k positive) b commutes with ei , ... , ep , ek+l, ... , eP (12.82) and anticommutes with e1 , ..., ek . (12.81)

(k (12 . 83)

negative) b commutes with ei , ..., ep k, ei , ..., ep

and anticommutes with ep k+1 .. , eP Since these conditions are valid for b if and only if they are valid for both the even part of b and the odd part of b, it suffices to consider the cases of b even and b odd separately. Suppose k is positive and b is even. Then, by (9.53), b does not involve e+ l ..., ep , ek+l, ., ep , while by (9.54), b does involve ei , ..., ek. Thus, b is a scalar multiple of ei ... ek , and hence k must be even.

Suppose k is positive and b is odd. Then by (9.56), 6 does involve

ei , ... , ep , ek+1, ... , eP , while by (9.55), b does not involve el , ... , ek . Thus, b is a scalar multiple of ei . . . . eP ek+l

ep , and hence 2p - k and k must be odd. The proofs for k negative are similar and so are omitted.

239

The Split Case Cl(p, p)

Definition 12.84. Suppose p is defined as in Lemma 12.75. Then (12.85)

R= e'apC E EndR(ACP),

for 9 E R

is called a choice of pinor reality map for Cl(p + k, p - k).

Lemma 12.86. Suppose R is a pinor reality map for Cl(p+k, p-k). Then (12.87)

(12.88)

Ri = -iR, i.e., R is complex antilinear.

Ra = RaR-1 for all a E Clc(2p) = Cl(p + k, p - k) OR C,

R2 = 1 for k = 0, 3 mod 4 (r - s = 0, 6 mod 8), (12.89)

R2=-1 fork =1,2mod4(r-s=2,4mod 8).

Proof: Since e'Bp E Endc(ACP)

Clc(2p), R = e'BpC is complex

= a if a E Cl(p + k, p - k), and since antilinear. By Lemma 12.75, = -b if b E i Cl(p+k, p- k). Therefore (12.88) is valid. Ri = -iR, Finally, to prove (12.89), note that R2 = e'BpCe1 ItC = p2, and that p is a unit volume element for RaR-1

RbR-1

(k even positive) (k odd positive) (k even negative) (k odd negative)

R(0, k) C C1(0, k) c--- AR(O, k),

R(p, p - k) C Cl(p, p - k) = AR(p, p - k), R(k, 0) C Cl(-k, 0) AR(-k, 0), R(p + k, p) C Cl(p + k, p) = AR(p + k, p).

Now (12.89) follows immediately from Proposition 9.57.

Proof of Theorem 12.71.

The centralizer of Cl(p + k, p - k) in

EndR(ACP) is C ® RC, where R anticommutes with i and R2 = ±1 depending on k mod 4, so the theorem follows. O It is convenient to choose the reality map to have an additional property. Recall the hat bilinear form e on the pinor space ARP for Cl(p, p) (the split case), as well as the complex extension, denoted Ec, which is a complex inner product on the pinor space ACP = ARP 0 C for Clc(2p).

Pure Spinors

240

Lemma 12.90. The pinor reality map R for Cl(p+ k, p - k) can be chosen so that

Ec(Rx,Ry)=ec(x,y) forallx,yEACP. Proof: Let R = e'BpC denote a reality map. Then F(Rx, Ry) = E (Ce-'Opx,Ce-te,jy) _ e (e-aepx e-4epy) = e2tee(izx,Ay) = e2ieII/'lIe(x, y).

Since p is the product of unit simple vectors, IIpII = ±1. If IIµII = 1, take

R = ±IX, while if IIFiHI = -1 take R = ±ipC. J

A SECOND PROOF OF THE CLASSIFICATION THEOREM A second proof of the important Theorem 11.3 can be given using the reality map R on the pinor space ACP for Clc(2p) = Cl(r, s) ®n, C.

Theorem 12.91. If r - s = 0, 6 mod 8, then Cl(r, s) = Ends (ARCP) . If r - s = 2,4 mod 8, then Cl(r, s) EndH (ACP). Remark 12.92. Combined with Corollary 11.65 this theorem provides a second proof of Theorem 11.3.

Proof: Case 1 (r - s = 0, 6 mod 8): Choose a reality map R for CI(r, s). Then R2 = 1. Therefore, R is a conjugation. In general, conjugations on a complex vector space V are characterized by two properties: (12.93)

Ri = -iR and

R2 = 1.

This is because the negative eigenspace for R is just i times the positive eigenspace for R, which implies that these two eigenspaces are of the same dimension. Let (12.94)

ARC

{x E AC :Rx=x}

denote the set fixed by the reality map R. Since R is a conjugation and Ra = RaR-1 (for all a E Endc(ACP)) fixes Cl(r, s), (12.95)

Cl(r, s) = Ends(ARCP).

The Split Case Cl(p, p)

241

Case 2 (r - s = 2, 4 mod 8): The centralizer of Cl(r, s) in EndR,(ACP ) C Endc(ACP) - Clc(2p) is H C ® RC. Therefore, H (the centralizer) defines a right quaternion structure on ACP, and since Cl(r, s) is exactly the subset of Endc(ACP) that commutes with R,

Cl(r, s) - EndH(ACP).

(12.96)

PURE SPINORS In general, the orbit structure of Pin acting on the pinor space P is extremely complicated. However, there is one particular orbit that warrants special attention, the so-called pure spinors. We shall examine the split signature case and the complex case. See Lawson-Micheleson [12] for a slightly different treatment of the complex case and some nice applications to geometry. Also consult Proposition 13.78 and Problem 13.8 in the next chapter for a further elaboration of the material in this section in terms of squares of spinors. Let N denote the space of all p-dimensional totally null vector sub-

spaces of R(p,p). Recall, by Problem 2.4, that the maximal dimension possible for a totally null subspace of R(p, p) is p. Consider a pinor representation: (12.97)

Cl(p, p) - EndR,(P).

Given a pinor x E P, let (12.98)

N. = {z E R(p, p) C Cl(p, p) : z(x) = 01.

Note that for x # 0, Nx is always totally null, since 0 = z2(x) = -jjzjjx for all z N.

Definition 12.99. A pinor x E P(p, p) is said to be pure if dimN, = p, i.e., NN E N. Let PURE(p) or PURE denote the set of all pure pinors.

Consider Pin(p) _- Pin(O,p) as a subgroup of Pin(p,p), where O(p) is the subgroup of 0(p, p), which acts as the identity on the first factor R(p, 0) C R(p, p). That is, A E O(p) sends (u, v) to (u, Av), and Pin(p) is generated by the unit sphere in R(0, p) C R(p, p). See Proposition 13.78 for a further elaboration of the next theorem in terms of the square of pure spinors.

Pure Spinors

242

Theorem 12.100. Pin(p) acts transitively on PURE/R* with trivial isotropic subgroup. O(p) acts transitively on N with trivial isotropic subgroup. The map sending x E PURE to Nx E N induces an equivarient isomorphism X : PURE/R* -* N: (12.101) Xa(NN) = Nax =

aNxa-1

for all a E Pin(p)and all x E PURE.

The orbit PURE consists of two connected components of dimension a p(p - 1); one, denoted PURE+, is a subset of S+, the space of positive spinors, while the other, denoted PURE-, is a subset of S_, the space of negative spinors.

Proof: First we show that O(p) acts transitively on N with trivial isotropy subgroup. Suppose N E N is totally null of dimension p. Then N fl ({0} x RP) = {0} implies that N = {(u, Au) : u E RP} is the graph over RP x {0} of a linear map A E End(RP). A graph N = graph A is totally null if and only if ((u, Au), (v, Av)) = (u, v) - (Au, Av) = 0

for all u, v E R(p, 0), i.e., if and only if A E O(p). Now it is obvious that O(p) acts transitively on N with trivial isotropy subgroup. In particular, since O(p) has exactly two components, this proves that N consists of two connected components, each of dimension 2p(p - 1). The p-planes in one component are called a -planes, while the p-planes in the other component are called Q -planes. Suppose a E Pin(p, p) and x E PURE are given. Let z, w E R(p, p) be

related by z = a-1wa or w = aza-1. Then the following are equivalent: w E Nax, wax = 0, azx = 0, zx = 0, z E N, w E aNxa-1. This proves that Pin(p) C Pin(p, p) maps the set of pure pinors PURE into itself and that the map from PURE/R* to N is equivariant, i.e., Na., = Next we shall show that there exists a positive spinor s+, which is aNxa-1

pure, and that if Nx = N,+ for any other pure pinor, then x = A x+ with A E R*, is a scalar multiple of s+. Assuming this fact, the proof of the theorem is completed as follows. Each totally null plane N E N is of the form N = aN,+a-1 = Na,+ for some a E Pin(p), since O(p) acts transitively on N. Therefore, the map from PURE to N is surjective. The equivariance, plus the fact that Nx = N,+ if and only if x = As+ with A E R*, implies that PURE/R* 25 N. Since Spin(p) C Spin(p, p) has two connected components, one Spin°(p) = Spin(p) n Cleven(p p) consisting of even Clifford elements and the other Spin(p)flClodd(p p) consisting of odd Clifford elements, the orbit

of Spin(p) through s+ E S+ must consist of a component PURE+ in S+ and a component PURE- in S_.

The Split Case Cl(p, p)

243

To prove the existence of a pure positive spinor s+, we use the model

Cl(p, p) = EndR(P) of this chapter with P =_ AR(p, 0). Consider the positive spinor s+ - 1 E AevenR(p, 0). A vector z - (u, v) E R(p, p) C Cl(p,p) = End(P) acts on s+ by

z(s+) _

(Eu+0 , - Iu-v) (s+) = u + v.

Therefore, N,+ _ {(u, -u) : u E RP}, which is p-dimensional (i.e., s+ E

If x E PURE and Ny = N,+, then for all z = (u, -u) E N,+, z(x) = -I2u(x) = -2u -i x vanishes, i.e., u i x = 0 for all u c PURE).

P must belong to A°R(p, 0), i.e., x = )s+ for some A E R*, completing the proof of the R(p, 0) This proves that the element x E AR(p, 0) .

theorem.

Now we consider the complex case. Suppose Vc is a complex inner product space and assume that the dimension n - dimc VC = 2p is even. Let NC denote the space of all totally null p-planes through the origin in V. (Note that Nc is a complex submanifold of Gc(p, VC) and hence

is projective algebraic.) Assume that Clc(Vc) 25 Endc(Pc) is a pinor representation. A pinor x E Pc is said to be pure if NN __ {z E VC C Cl(Vc) : z(x) = 0}

(12.102)

is of complex dimension p, i.e., Nx E N. Let PUREC denote the set of pure spinors.

The complex inner product space Vc may be expressed as Vc V(p, p) OR C with V(p, p) = R(p, p) split. The model Cl(p, p) = EndR.(AR(p, 0))

can be complexified, and some parts of the proof of Theorem 12.100 carry over to the complex case. For example, (a) (12.103)

(b)

there exists a pure positive spinor s+ E S+, and Nx = N,+ if and only if x = As+ for some ,1 E C*

are proved exactly as in the split case. However, just because N E Nc is totally null N fl ({0} x CP) need not be equal to {0}. That is, it is no longer true that each N E Nc can be graphed over CP x {0} C CP x CP, as in the split case. Consequently, the model (12.104)

Vc

V(2p) OR C,

with V(2p) = R(2p) positive definite,

244

Pure Spin ors

is more convenient. Now each N E NC can be graphed over V = V(2p) C

V ® iV = Vc, since iV cannot contain any null vectors other than 0. Suppose N = graph A = {u + iAu : u E V} is the graph of A E Endlt(V). Then N is totally null if and only if (u+iAu, v+iAv) = (u, v) - (Au, Av)+ i((Au, v) + (u, Av)) = 0. That is, A E O(2p) and At = -A, or equivalently, A E O(2p) and A2 = -Id. Thus, A E Cpx(2p) is an orthogonal complex structure. This proves that

Nc = Cpx(2p).

Recall that Cpx(2p) - 0(2p)/U(p) with g E O(2p) sending J E Cpx(2p) to gJg-1. Therefore, NC O(2p)/U(p) with g E O(2p) sending N E NC to gNg-1. The space (12.105)

NC = Cpx(2p) = O(2p)/U(p)

is called the twistor space (at a point on a manifold), i.e., the twistor fiber.

Remark 12.106. Suppose J E Cpx(V). Note that N = graph J = {u + iJu : u E V} is just V1"0(J), the +i eigenspace of J (extended to VC). As in the split case, it follows easily that Na. = aNxa-1 for z E PUREC and a E Pin(2p, C). The group Pin(2p) is naturally a subgroup of Pin(2p, C), since V(2p) C Vc. This completes the proof of the following result.

Theorem 12.107. Pin(2p) C Pin(2p, C) acts transitively on PUREC/C*. The map x sending x E PUREC to NN E NC - Cpx(2p) is an equivariant isomorphism x: (12.108)

PUREC/C* - NC - Cpx(2p) - O(2p)/U(p).

Also, PUREC = PURE+ U PURE- with PURE a connected subset of SC.

Remark 12.109. In particular, given s+ E PUREC (with associated complex structure J) and an element a E Pin(2p): (12.110)

Xa(s+) E U(p) if and only if as+ = cs+ with c E C*.

Proposition 13.79 has a complex analogue which says that (12.111)

if 8 E PUREC then s 6 s= cdz1 A ... A dzp,

with c E C* and z1,. .. , zp a choice of complex linear coordinates for Vc(2p). The square of s is defined by (s o s) (x) = se(s, x). It then follows easily that (12.112)

{a E Pin(2p) : as = s} = SU(p),

The Split Case Cl(p, p)

245

with the isomorphism given by the vector representation X. That is, the isotropy subgroup of Pin(2p) at a pure spinor is SU(p) (cf. Problem 8).

PROBLEMS 1. Given u E V and x E Cl(V), show that

-Fua x. This formula has the following interesting interpretation: Right Clifford multiplication on x by a vector u is the same as first replacing x by i and then left multiplying by u using Clifford multiplcation based

on the inner product -(, ) of signature s, r rather than r, s. 2. (a) Give the proof of Corollary 12.39 if p is even. (b) Complete the proof that to is skew if p = 1 mod 4, t is symmetric if p = 3 mod 4. 3. Complete the proof of the e portion of Table 12.30.

4. (a) Prove that, for r - s = 1 or 5 mod 8, Cl(r, s) contains no proper nontrivial two-sided ideals.

(b) Prove that, for r - s = 3 or 7 mod 8, the only proper nontrivial two-sided ideals of Cl(r, s) are Cl}(r, s). (c) Suppose R(r, s) A (nontrivial)

n Cl(r, s)

/

is given exactly as in the Fundamental Lemma of Clifford Algebras. Show that 0 is injective if r - s # 3, 7 mod 8, while the kernel of 0 is either {0}, Cl(r, s)+, or Cl(r, s) - if r - s = 3, 7 mod 8. 5. In the model P (p, p) = AR(p, 0), Cl(p, p) EndR,(P) of this chapter, show that the set of pure positive spinors PURE+ is equal to the subset R* Spin(0, p) of AevenRP = S+ 6. Prove that the span of PURE+ is S+(p, p). 7. Show that a positive spinor x E S+(4, 4) is pure if and only if x is null, i.e., r(x, x) = 0. 8. Suppose J E Cpx(2p) is an orthogonal complex structure on the positive euclidean space R(2p). Suppose el, Jet, ... , eP, Jep, is an orthonormal basis for R(2p) and let al, 01, ... , aP, PP denote the dual basis. Then w = al A ,61 + ... + aP A /3P is the standard Kii.hler

Problems

246

form. Let X : Spin(2p) --+ SO(2p) denote the vector representation. Identify AR(2p) and AR(2p)* using the inner product. Let A - a1 A Q1 A ... A ap A /3p denote the unit volume.

(a) Show that

0 = 2-p/2(1 + a1 A )1)

. (1 + Cep

A/3P) E Spin(2p) C Cl(2p)

is independent of the orthonormal basis. (b) Show that xm = J, 02 = A , and that 0-1 = OA = ¢. (c) Define U(p) {a E Pin(2p) : X. E U(p)}. Show that

U(p) _ {a E Spin(2p) : a4 = Oa} and

U(p) = {a E Spin(2p) : aw = wa}.

(d) Using (12.111), show that the isotropy subgroup of Pin(2p) at a pure spinor is SU(p). (e) Show that x-'(SU(p)) has two connected components.

9. (a) Suppose N E N is a p-dimensional totally null plane in R(p,p) and a E GLR,(N). Show that there exists A E 0(p, p) with AIN = a; i.e., show that a extends to an isometry of R(p, p). Hint: Use the model N = RP x {0} C RP x RP with I I(u, v)II = uv.

(b) Show that any linear isomorphism a : N, -+ N2 of totally null p-planes N1, N2 E N can be extended to an isometry A of R(p,p), i.e.,

AEO(p,p). (c) Prove Witt's Theorem: Suppose V and W are isometric euclidean vector spaces (i.e., same dimension and signature). Then each isometry a : V --> W, from a subspace V of V onto a subspace W of W extends

to an isometry A from 7 to W.

13. Inner Products on the Spaces of Spinors and Pinors

The hat and the check anti-automorphism of Cl(r, s) can be described in terms of extra structure on the space P(r, s) of pinors. These two antiautomorphisms agree on Cl(r, s)even and will be referred to as the canonical involution of Cl(r, This canonical involution on Cllr, seven can be described in terms of extra structure on the space S(r, s) of spinors. seven.

THE SPINOR INNER PRODUCT The spinor inner product, denoted c, is an inner product on the space S(r, s) of spinors with the property that the adjoint with respect to a is the canonical involution of Cl(r, s)eVen. The next result establishes the existence of spinor inner products and classifies the types. It is interesting to note that all of the different types of inner products introduced in Chapter 2 can occur as spinor inner products.

Theorem 13.1. There exists an inner product e on the space of spinors S(r, s) called the spinor inner product with the property that, given a E Cl(r, Seven (13.2)

(ax, y) = r(x, ay) for all x, y E S. 247

The Spinor Inner Product

248

Table 13.3. Spinor Inner Products

r - s = 0 mod 8:

p mod 4 0

e =- c+ E) e-, both

1

e reflective, S_ = S+

2

e - e+ ® e_, both et R-skew

3

e reflective, S_ = S+

of R-symmetric

S-S+®S_ real Cllr, S)even =

EndR(S+) (D EndR(S_)

r - s = 1,7 mod 8:

0

C1(r, S)even = EndR(S)

1

2 3

r- s= 2,6 mod 8:

1

2

3

r-s=3,5 mod 8:

1

2 3

r - s = 4 mod 8:

C-symmetric C-hermitian (symmetric) C-skew C-hermitian (skew) e

H-hermitian skew H-hermitian symmetric H-hermitian symmetric H-hermitian skew

p mod 4 0

S=-S+eS_ 1

C1(r, s)

e

p mod 4 0

Cl(r, S)even - EndH(S)

R-symmetric R-skew R-skew R-symmetric

p mod 4 0

C1(r, Seven = Endc(S)

e

p mod 4

2

EndH(S+) e EndH(S_) 3

e

e - e+

e_, both

,-:k H-hermitian skew e reflective, S_ S+ E =- -+ E) e-, both E± H-hermitian symmetric

e reflective, S_ = S+

Inner Products on Spaces of Spinors and Pinors

249

The type of the spinor inner product a is described in the Table 13.3. Here n - r + s = 2p defines p if the dimension n is even, and n - r + s - 2p + 1 defines p if the dimension n is odd. If the signature of R(r, s) is definite (i.e., positive, in case s = 0, or negative in case r = 0), then a is definite.

(In particular, a must be of the type that has a signature.) In all other cases, where e has a signature, the signature ofe is split.

Remark. Since the spinor inner product e is only determined up to a change of scale (see Theorem 8.33), it is not the signature r', s' of a that is an invariant but the absolute value Ir'-s'I. However, we continue to use the terminology "e has a signature." In the cases e - e+ ®E-, another abuse of terminology occurs in Theorem 13.1. Here either e+ or e_ may be changed by a nonzero real constant, so that, strictly speaking, e - e+ ® e_ does not have a signature even in the absolute value sense, Ir' - s'I, discussed above. However, if e± has a signature r±, s±, then I r± - st I is a true invariant. The proof of Theorem 13.1 (constructing the spinor inner products e) will be given as a corollary of Theorem 13.17 (constructing the pinor inner products e and e).

The next remark applies to the algebra A = Cl(r, s)even for either r - s = 0 mod 8 (with F-= R), or r - s = 4 mod 8 (with F - H). It gives the definition of the term e-reflective used in Table 13.3.

Remark 13.4. Suppose that the algebra A is of the form A = MN(F) x MN(F) --- EndF(V+) ® EndF(V_). Suppose e : V+ x V_ -* F is a nondegenerate hermitian F-bilinear form (or equivalently, b(u)(v) - e(u, v) for all u E V+, v E V_ defines an F-linear isomorphism b : V+ --+ V-*). Then e is said to be reflective, and the antiautomorphism of the algebra A sending (a, 6) E A to (b*, a*) E A, defined by

e(au, v) = e(u, a*v) (13.5)

and

e(u,bv)=e(b*u,v) foralluEV+,vEV_,

is called the (reflective) a involution of A. See Problem 1.7 for the definition of W*, when W is a right H-space. Suppose a reflective e : V+ x V_ --> F is given. Then there exists an (unique up to a change of scale) F-hermitian symmetric inner product

on V - V+ e V_ (also denoted e) with the property that the reflective e involution is the adjoint with respect to this inner product on V. Namely, (13.6)

c(z1, x2) = e(x1, y2) + 6(x2, yi) for all z1 = (x j, yr), x2 = (x2, y2) E V - V+ ®V_,

250

Spin Representation and (Reduced) Classical Companion Group

defines the inner product on V. Note that we could have defined this inner product on V by E(zl, Z2) = E(xl, Y2) - E(x2, yl)

so that E would be F-hermitian skew. Thus, the property of the inner product on V being symmetric or skew is of no significance, and we simply say E is reflective on V - V+ ® V_.

THE SPIN REPRESENTATION AND THE (REDUCED) CLASSICAL COMPANION GROUP Cp°(r, s) As noted after (10.38), the reduced spin group Spin°(r, s) is a subgroup of the classical companion group (13.7)

Cp(r, s) - {a E

Cleven(r, s)

: a& = 1}.

Since a is the adjoint of a with respect to the spinor inner product E, this group Cp(r, s) is indeed a classical group, namely, the group that fixes the inner product e. Thus Theorem 13.1 provides a classical description of this companion group Cp(r, s) (see Problem 1). The subgroup of the classical companion group, equal to the connected component of the identity element in Cp(r, s), is called the reduced classical companion to Spin°(r, s) and denoted by Cp°(r, s). Recall from (10.53) and Problem 10.7 that for the spin representations

p in dimension n > 3, the determinant (real or complex) of p(a), with a E Spin, is in the finite set {f1, ±i}. Therefore, for a E Spin°(r, s), each such determinant must be one, because Spin° (r, s) is connected. Each of these reduced classical companion groups (is a classical group that) can be read off from the next theorem. For example, if r-s = 0 mod 8 and n = rd-s = 2p with p, = 0 mod 4 then Cp°(r, s) = SO1(S+) X SOt(S_) is the reduced classical companion of Spin°(r, s) (cf. Problem 5). The main result of the section Cartan's Isomorphisms in Chapter 14 is obtained by reading off the (reduced) classical companion group Cp°(r, s) in low dimensions.

Inner Products on Spaces of Spinors and Pin ors

251

Theorem 13.8 (The Spin Representations). r - s = 0 mod 8: p = 0 mod 4

p = 2 mod 4

1 -+ Z 2 1 -+ Z2

{1 ,

Al -+ S p in°(r

,

S)

P

+ SO1(S +),

{1, -Al -+ Spin°(r, s) -°_-+ SOT (S_),

Z2

{1, A) -+ Spin°(r, s) -°+ Sp(S+, R),

1 -* Z2

{1, -A} -+ Spin°(r, s) -°__+ Sp(S_,R),

1

except for Spin° (2,2)

p = 1, 3 mod 4

1 - Z2

{1, A} -* Spin°(r, s) °--> SL(S+, R),

{ 1, -A) _+ Spin°(r, s) -°_+ SL(S+, R), except for Spin(1,1) = GL(1,R).

1 --* Z2

r - s = 4 mod 8:

p = 0 mod 4 p = 2 mod 4

1 -+ Z2

1, A} ---+ Spin°(r, s) P* SK(S+),

1 -+ Z2

{1, -A} -+ Spin°(r, s) -°-+ SK(S_),

1 --+ Z2

{1, Al -+ Spin°(r, s) -°-+ HU(S+),

1 -+ Z 2 {1 , -A} - + S pin°(r , s) -° =+ HU(S_ ), except for Spin(4,O) = Spin(0,4),

p = 1 , 3 mod 4

- Z 2 = {1 1 - Z 2 = {1 1

A} -+ Sp in°(r, s )

H ), , -A} -+ S p in°(r , s ) -° =* SL(S +, H) . ,

°-+* SL (S +,

r - s= 1 7 mod 8: ,

p= 0,3mod4 p = 1,2 mod 4

Spin°(r, s) C SO1(S), Spin°(r, s) C Sp(S, R).

r - s= 2 6 mod 8: ,

p = 0 mod 4

p= 1,3mod4 p = 2 mod 4

1 --* Spin°(r, s) -°- SO(S, C), 1 -+ Sp iri°( r, s ) -° + SO ( S , C ), 1 - Spin°(r, s) -°-+ SU(S), 1 -* Spin°(r, s) -°-+ SU(S), except for Spin(2,O) = Spin(0,2) = U(1), 1 -+ S p in°( r, s) -f + Sp( S , C ),

1 -+ S p i n °( r, s) -° + Sp( S , C ).

r - s = 3,5 mod 8: p= 0,3 mod 4

p= 1,2mod4

Spin°(r, s) C SK(S), Spin°(r, s) C HU(S).

The Pinor Inner Products a and e

252

Proof: The only part of Theorem 13.8 that does not follow from the spinor

inner product theorem and the extra determinant restrictions discussed above is the fact that (13.9)

ker p+ = {1, A},

kernel p_ = {1, -A}

for r - s = 0 mod 4 and the dimension n - r + s > 6. This follows from the next theorem (cf. Problem 10.11).

Theorem 13.10. If the dimension n - r + s > 5 and p : Spin°(r, s) ---; EndR,(W)

is a nontrivial representation of the (reduced) spin group Spin°(r, s), then (13.11)

ker p C cen Spin°(r,s).

This is a standard result proved in most textbooks on Lie algebras (the complex Lie algebra so(n, C) has no nontrivial two-sided ideals for n > 5, i.e., so(n, C) is simple for n > 5).

THE PINOR INNER PRODUCTS a AND The extra structure on P(r, s) required to describe the hat anti-automorphism, mapping a to a, is an inner product on P(r, s), denoted e, with the property that given a E Cl(r, s), (13.12)

e(ax, y) = e(x, ay) for all x, y E P.

The extra structure on P(r, s) required to describe the check antiautomorphism, mapping a to a, is an inner product on P(r, s), denoted e, with the property that given a E Cl(r, s), (13.13)

e(ax, y) = e(x, ay) for all x, y E P.

Recall from Definition 11.30 and Theorems 11.34 and 11.66 that there

exists a spinor structure map s E Enda(P) that determines the canonical automorphism of Cl(r, s) by (13.14)

a = sas-1 for all a E Cl(r, s).

Consequently, either one of E or a can be used to determine (define) the other by the formula: (13.15)

E(x, y) - E(sx, y)

for all x, y E P.

Inner Products on Spaces of Spinors and Pinors

253

For example, if e satisfying (13.12) is given and t is defined by (13.15), then

e(ax,y) - E(sax,y) = E(asx,y) = E(sx,ay) - E(x,ay).

(13.16)

Because of Theorem 8.33, 9 and z are unique up to a change of scale (as usual, the cases r - s = 3,7 mod 8 will be somewhat different). In particular, if both 9 and e are given, then a change of scale may be required in order for (13.15) to be true. The next theorem describes the pinor inner products e and e.

Theorem 13.17. There exist inner products e and ton the space of pinors P(r, s) with the property that, for each a E Cl(r,s) (13.18)

E(ax, y) = E(x, ay)

and

E(ax, y) _ ff(x, ay) for all x, y E P.

The type of the pinor inner products t and e is described in Table 13.19. Here n = r + s = 2p defines p if the dimension n is even, while n - r + s 2p + 1 defines p if the dimension n is odd. If R(r, s) is positive definite (i.e., s = 0), then 9 is (positive) definite. If R(r, s) is negative definite (i.e., r = 0), then is (positive) definite. In all other cases, if t or t has a signature it must be split. The proofs of the spinor Theorem 13.1 and the pinor Theorem 13.17 are intertwined in the following five steps. Step 1. Suppose n is even and that the E portion of the Pinor Theorem is valid. Then the t portion of the Pinor Theorem is valid. Step 2. Suppose n is even. The inner product t is constructed from the inner product Ec of Theorem 12.45 using the pinor reality map R.

Step 3. Suppose n is odd. The Spinor Theorem is deduced from the Pinor Theorem for even dimensions. Step 4. Suppose n is odd. The Pinor Theorem is deduced from the Spinor Theorem. Step 5. Suppose n is even. The Spinor Theorem is deduced from the Pinor Theorem for odd dimensions.

The Pinor Inner Products E and E 254

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Inner Products on Spaces of Spin ors and Pinors

255

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The Pinor Inner Products E and e

256

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Inner Products on Spaces of Spinors and Pinors

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The Pinor Inner Products t and e

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Proof of Theorems 13.1 and 13.17. Step 1. Assume n = r + s = 2p is even. Also, for the moment assume that 9 has been constructed and satisfies Theorem 13.17. Let A denote the choice of unit volume element for R(r, s).

Definition 13.20. (n even). The (check) pinor inner product t on P(r, s) is defined by (13.21)

e(x,y)=e(Ax,y) for allx,yEP.

As noted above in 13.16, if a E Cl (r, s), then

e(ax, y) = e(x, ay) for all x, y E P, because of the corresponding fact for e. Also note that e is F-bilinear if and only if e is F-blinear (13.22) and e is F-hermitian bilinear if and only if t is F-hermitian bilinear. (13.23)

Lemma 13.24. If n = r + s = 2p is even, then (a) (p even) e is symmetric if and only if a is symmetric, and e is skew if and only if a is skew, (b) (p odd) e is symmetric if and only if a is skew, and e is skew if and only if a is symmetric. We give the proof for the nonhermitian cases. The proof for the hermitian cases is essentially the same. Proof: Because of Proposition 9.27, (13.25) A = .\ if p is even and A _ -A if p is odd. Assume that e(y, x) = ±e(x, y) with the + if a is symmetric, and the - if e is skew. Then e(y, x) _ e(Ay, x) = e(y, Ax) = ±e(ax, y),

which, because of (13.25), equals ±&(Ax, y) = fe(x, y) if p is even, and equals ::Fe(ax, y) = ::Fe(x, y) if p is odd.

If r - s = 0, 4 mod 8, then A2 = 1 and the eigenspaces

St= {xEP:Ax=±x} are (by definition) the spaces of positive and negative spinors.

Inner Products on Spaces of Spinors and Pinors

259

Lemma 13.26. If r - s = 0, 4 mod 8 (and n = r + s = 2p), then (a) (p even) S+ and S_ are orthogonal for both e and e, (b) (p odd) S+ and S_ are totally null for both e and e. Proof: Suppose p is even. Then, for x E S+, y E S_,

s(x, y) _ -e(Ax, ay) _ -e(x, aAy) _ -e(x, A2y) = -e(x, y), since A = A. The proof of (a) for a is identical since A = A.

Suppose p is odd. Then, for either x, y E S+ both positive, or for x, y E S_ both negative, e(x, y) = e(Ax, Ay) = e(x, 5 Ay) = -e(x, A2y) _ -e(x, y),

since A = -A. The proof of (b) for

is identical since A = -A.

If r - s = 2,6 mod 8 then A2 = -1. Note that (13.27)

A is an a isometry if and only if A is an e isometry.

(13.28)

A is an a anti-isometry if and only if A is an e anti-isometry.

Both of these statements are true because; if e(Ax, Ay) = ±e(x, y), then

'(ax, Ay) = e(A2x, Ay) = ±e(Ax, y) = ±e(x, y).

Lemma 13.29. If r - s = 2,6 mod 8 (and n = r + s = 2p) then (a) (p even) A is an anti-isometry for a and e, (b) (p odd) A is an isometry fore and 9.

Proof: e(Ax, Ay) = e(x, AAy), which equals e(x, A2y) = -e(x, y) if p is even, and which equals e(x, A2y) = e(x, y) if p is odd, because of (13.25).

Step 2.

In the even dimensional cases, it remains to construct a satisfying the Pinor Theorem 13.17. Recall from Chapter 12 that Cl(r, s) can be realized as a subalgebra of

Endc (ACP) - Clc(n) when n =- r + s = 2p. Also recall that there exists a reality operator R that has the property that

(13.30) Ra

RaR-1, for all a E Endc(ACP) - Cl(n) = Cl(r, s) ®a C,

The Pinor Inner Products E and e

260

where 7Z is the natural conjugation on Cl(r, s) OR C. Both the fact that each of the three involutions sending a to a, a, and a on Clc(n) leave Clc(r, s) C Cl(n) fixed, and the fact that each of these involutions when restricted to Cl(r, s), yield the corresponding involutions on Cl(r, s), will (implicitly) be assumed in the following. The complex inner product ec on ACP constructed in Chapter 12 will provide the basis for the construction of a on R(r, s). The facts that we shall need about ec are listed here for convenience. Given a E Clc(n) = Endc(ACP),

ec(ax, y) = ec(x, ay) for all x, y E ACP.

(13.31)

The reality operator R E EndR,(ACP) for Cl(r, s) may be chosen so that

ec(Rx, Ry) = ec(x, y) for all x, y E ACP.

(13.32) Also, (13.33)

if p = 0, 3 mod 4, then ec is C-symmetric; if p = 1, 2 mod 4, then ec is C-skew.

Case r - s = 0, 6 mod S. Here R2 = 1 and Cl(r, s) = EndR,(P), where the space of pinors is defined to be P - {x E ACP : Rx = x} (see (12.94) and (12.95)). Note that because of condition (13.32), ec restricted to P is real-valued and nondegenerate. Definition 13.34. (r - s = 0, 6 mod 8) The hat inner product on P(r, s) is defined to be

e = ecIp. The condition (13.18) fore follows immediately from the analogous condition (13.31) on ec. Finally, because of (13.33), e is R-symmetric when p = 0, 3 mod 4 and a is R-skew when p = 1, 2 mod 4.

Case r - s = 2, 4 mod S. Here R2 = -1 and Cl(r, s) = EndH(P), where the space of pinors P(r, s) is defined to be ACP equipped with the right H-structure given by I =- i and J = R (see (12.96)).

Definition 13.35. (r - s = 2,4 mod 8). The hat inner product a on P(r, s) is defined to be (13.36)

E(x, y) _ -ec(xJ, y) + jec(x, y) for all x, y E P.

The condition (13.18) fore follows from the analogous condition (13.31) on ec because J commutes with all a E Cl(r, s).

Inner Products on Spaces of Spin ors and Pinors

261

Lemma 13.37. Suppose W is a right H-space and cc is a complex bilinear form on W with respect to the complex structure I (right multiplication) on W, that satisfies (13.38)

ec(xJ, yJ) = ec(x, y).

Let (13.39)

h(x, y) _ -Ec(xJ, y) + je(x, y) for all x, y E W.

Then (13.40)

h is H-hermitian symmetric if and only if cc is C-skew;

(13.41)

h is H-hermitian skew if and only if cc is C-symmetric.

Proof: Since cc is complex bilinear and I, J as well as i, j anticommute, it is easy to show that (13.42)

h(xI, y) = -ih(x, y) and h(x, yI) = h(x, y)i.

Also, (13.43)

h(xJ, y) = ec(x, y) + jec(xJ, y) = -jh(x, y)

is automatic. Finally, to prove that (13.44)

h(z, yJ) = h(x, y)j,

the hypothesis (13.38) must be used:

h(x, yJ) = -Ec(xJ, yJ) + jec(x, yJ) _ -ec(x, y) - jec(xJ, y) = jec(x, y)j - cc(xJ, y).7 = h(x, y)j. Thus, h is H-hermitian. The proofs of the remainder of (13.40) and (13.41) are similar and so are omitted (also cf. Lemmas 2.72 and 2.78).

Because of Lemma 13.37 and (13.33), e is H-hermitian skew when p = 0, 3 mod 4 and is H-hermitian symmetric when p = 1, 2 mod 4.

The Pin or Inner Products t and 9

262

Combining Step 1 and Step 2, this completes the proof of the Pinor Theorem 13.17 when the dimension n = r + s is even.

Step 3. Next we give the proof of the Spinor Theorem 13.1 when n = r + s 2p + 1 is odd. The proof is based on two ingredients. The first ingredient is the pair of isomorphism: CI(r - 1, s) if r > 1,

(13.45)

Cllr, seven

and (13.45')

Cl(r, s)even = C1(s - 1, r) if s > 1,

from Theorem 9.38. (The important fact is that these isomorphisms preserve the hat anti-automorphisms.) The second ingredient is the d portion of the Pinor Theorem 13.17 for even dimensions.

Case r - s = 1, 7 mod 8. Then r - 1 - s = 0,6 mod 8, so that Cl(r, seven = CI(r - 1, s) 25 EndR(P(r - 1, s)), and we may take S(r, s) _P(r - 1, s). The spinor inner product a on S(r, s) is, by definition, the hat inner product 9 on P(r - 1, s). Finally, a is R-symmetric if p = 0, 3 mod 4,

and a is R-skew if p = 1, 2 mod 4 by Theorem 13.17 applied to 9 on P(r - 1, s). If r = 0, consider the isomorphism C1(r, s)even - Cl(s - 1, r) EndR(P(s - 1, r)) instead.

Case r - s = 3,5 mod S. Then r - 1 - s = 2,4 mod 8, so that Cl(r, $)even e Cl(r - 1, s) EndH(P(r - 1, s)), and we may take S(r, s) to be the right H-space P(r-1, s). The spinor inner product e on S(r, s) is, by definition, the hat inner product a on P(r -1, s). Finally, a is H-hermitian symmetric if p = 1, 2 mod 4, and a is H-hermitian skew if p = 0, 3 mod 4 by Theorem 13.17 applied to P(r-1, s). If r = 0, consider the isomorphism Cl(r, s)even

CI(s - 1, r)

EndH(P(s - 1, r)) instead.

Step 4.

Next the Pinor Theorem 13.17 for odd dimensions will be deduced from the Spinor Theorem 13.1 for odd dimensions.

Case r - s = 1 mod 8. Recall from Theorem 11.66 that P(r, s) S(r, s) ®R, C and that Cl(r, s) = Endc(P); and Cl(r, s)even 25 EndR,(S), where A = i is a unit volume element and s, the spinor structure map, is the natural conjugation sending x to Y on P S OR C. Let cc denote the complexification of the spinor inner product a on S. Consulting the r - s = 1 mod 8 entry in Table 13.3 associated with Theorem 13.1, proves

that (13.46)

cc is C-symmetric for p = 0, 3 mod 4, cc is C-skew for p = 1, 2 mod 4.

263

Inner Products on Spaces of Spinors and Pinors

It is natural to require that e and a be related by e(x, y) = e(x, y), for all x, y E P, because of (13.15).

Definition 13.47. (r-s = 1 mod 8) Let n = r + s

2p+1 define p. The pinor inner products a and e on P(r, s) are defined by: First, if p, is even (13.48)

e(x, y) - ec(x, y),

e(x, y) = ec(x, y) for all x, y E P,

and, second if p is odd (13.49)

e(x, y) = ec(x, y), e(x, y) = ec(x, y), for all x, y E P.

Now we prove that

e(ax, y) = E(x, ay) for all x, y E P.

(13.50)

If a E Cl(r, $)even, then (13.49) is automatic. Suppose is E Cl(r, s)odd, then seven. Then a E Cllr, (13.51)

(p even)

e(iax, y) - ec(iax, y) _ -iec(ax, y) _ -iec(x, ay) = -ec(x, iay) = -e(y, iay),

where -ia = is since i = -i (the dimension n - 2p + 1 = 1 mod 4). (13.52)

(p odd) e(iax, y) = ec(iax, y) = ec(x, iay) = e(x, iay),

where i& = is since i = i (the dimension n 2p + 1 = 3 mod 4). The type of a and a listed in the r - s = 1 mod 8 portion of Theorem 13.17 follows from (13.46) and the definitions of e and e. Case r - s = 3 mod 8. Recall from Theorem 11.66 that S(r, s) is

a right H-space and that P(r, s) = P+(r, s) ® P_(r, s) with P±(r, s) S(r, s). The unit volume element is A = (o _°) and the spinor structure map is s = (° o). Also recall that Cl(r, s)'- EndH(P+) a EndH(P_) and Cl(r, s)even

-

EndH(S)

embedded

diagonally

in

EndH(P+)

® EndH(P_). Let e denote the spinor inner product on S. Because of Theorem 13.1, (13.53)

a is H-hermitian skew if p = 0, 3 mod 4, and e is H-hermitian symmetric if p = 1, 2 mod 4.

The Pinor Inner Products t and e

264

Defintion 13.54. The pinor inner products t and a on P - S ® S are defined by (13.55)

(p even)

e(zl, z2) = t(szl z2)

(13.56)

(p odd)

E = E ®e and E(z1i Z2) = E(szli z2).

i

and e = e ®E.

It remains to prove that (13.57)

(p even)

e(Azi, z2) = e(z1, Az2)

(p odd)

E(Azl, z2) = e(z1, Az2)

and (13.58)

for A = (a, b) with a, b E EndH(S) and z1 = (x1, y1), z2 = (x2, y2) E P

S®S. Both cases are automatic if A (a, a) E Cllr, seven. Each B (a, -a) E Cl(r, s)odd is of the form B = AA with A = (a, a) E Cl(r, seven If p is even, the dimension 2p + 1 = 1 mod 4 and A = -A, while if p is odd then 2p+ 1 = 3 mod 4 and ) = A. Now (13.57) and (13.58) follow easily. Case r - s = 5 mod S. Recall from Theorem 11.66 that S(r, s) is a right H-space and that P(r, s) equals S(r, s) equipped with the complex structure I. The spinor structure map on P is s - J and the unit volume element is A - I. Let a denote the spinor inner product on S. Because of Theorem 13.1, (13.59)

e is H-hermitian skew if p = 0, 3 mod 4, and e is H-hermitian symmetric if p = 1, 2 mod 4.

Definition 13.60. The pinor inner products E and t are defined to be complex-valued inner products by (13.61)

(p even)

e(x, y) = E(x, y) + it (x, y),

and (p odd) e(x, y) = e(x, y) - je(x, y)

(or je(x,y)

(13.62)

E(x,y) + je(x,y))

forallx,yeP - S. Note that in both cases, (13.63)

E(x, y) = E(xJ, y)

for all x, y E P.

Inner Products on Spaces of Spinors and Pin ors

265

Now we prove that (13.64)

e(ax, y) = e(x, ay) for all x, y E P.

If a E Cl(x, S)even then (13.64) is automatic. Each 6 E Cl(r, s)odd is of the form b = as with a E Cl(r, $)even. Therefore, E(bx, y) = e(aax, y) = E(.Xx, ay) = E(xI, ay) = -iE(x, ay).

If p is even, this implies that e(bx, y) _ -ie(x, ay) = -e(x, Aay), which

(since the dimension n = 2p + 1 = 1 mod 4 implies a = -a) equals e(x, aay) = e(x, by). If p is odd, this implies that e(aAx, y) = ie(x, ay), which, since the dimension 2p + 1 = 3 mod 4, equals e(x, )ay) = e(x, aay). The type of e and e follows from the type of E listed in (13.59) because of the next lemma.

Lemma 13.65. Suppose h - a + j/3 is an H-hermitian form and a and,8 are complex valued. If h is H-hermitian symmetric, then a is C-hermitian (symmetric) and /3 is C-skew. If h is H-hermitian skew, then a is Chermitian (skew) and /3 is C-symmetric.

Remark 13.66. The inner product a is said to be the first complex part of h while ,3 is said to be the second complex part of h (cf. Chapter 2).

Case r - s = 7 mod 8. This case is so similar to the case r - s = 3 mod 8 that the proof is omitted. This completes the proof of the Pinor Theorem 13.17. It remains to give the proof of the Spinor Theorem 13.1 in the even dimensional cases.

Step 5. Using the hat preserving isomorphisms

Cl(r, seven N Cl(r - 1, s) Cl(r, seven

Cl(S - 1, r),

the r - s odd mod 8 portion of the Spinor Theorem 13.1 can be read off from the even portion of the Pinor Theorem 13.17 (Problem 3). This completes the description of the types for the spinor inner products E and the pinor inner products a and e. The discussion of the signatures of these inner products will be given later in the section entitled Signature, as an applicatian of pinor multiplication.

Pinor Multiplication

266

PINOR MULTIPLICATION The inner products t and a on P(r, s) enable one to multiply pinors, as described in Definition 8.42. An application to calibrations is given in Chapter 14, and an application to pure spinors in the split case is given at the end of this section. The

product x O y induced by e will be denoted x 6 y, and the product x 0 y induced by t will be denoted x 6 y. Therefore, if x, y E P are given, then x 6 y and x 6 y E EndF(P) are defined by (x a y)(z)

(13.67)

xE(y,z)

and (x 6 y)(z) = xe(y,z)

(13.68)

for all z E P. For all signatures r, s, the pinor representation enables us to consider C1(r, s) as a subalgebra of EndF(P(r, s)), with equality C1(r, s) = EndF(P)

unless r-s= 3 mod4. Ifr-s=3 mod 4, then Cl(r, s)

EndF(P+) ® EndF(P_) C EndF(P).

However, in all cases, p(a) (which we also denote by a in the present context) has real trace zero. Therefore Theorem 9.65 implies that

Lemma 13.69. Consider Cl(r, s) C EndF(P(r, s)). The Clifford inner product (,) can be expressed as a real trace: (13.70)

(a, b) = (dim,, P)-1 trace,, ab for all a, b E Cl(r, s).

Remark 13.71. Note that this implies that the twisted Clifford inner product is also given by a trace: (13.72)

(a, b) = (dim,,

P)-1

trace,, ab.

This lemma states that the natural R-symmetric inner product on EndF(P) induced by t (see Defintion 8.38) is exactly the same as the Clifford inner product on Cl(r, s). Therefore, all the results of the section Inner Products in Chapter 8 are applicable to P,E and EndF(P), (, ). These results are summarized in the next two theorems. Let N = dimF P.

Inner Products on Spaces of Spinors and Pinors

Theorem 13.73. (x 6 y, a) =

267

Re e(ay, x), for all a E Cl(r, s) and N

x,y E P. Note that e(ay, x) has the order of x and y reversed. Theorem 13.74. For signatures where t is F-hermitian on P(r, s), (x 6 Y' Z 6 w) = N Re

(13.75)

z)E(y, w))

For signatrues where a is F-symmetric or F-skew on P(r, s), (x 6 Y' Z 6 w) =

(13.76)

Re

((x, z)E(y, w) I

.

N Here x, y, z, w E P(r, s).

Remark 13.77. Because of (13.72), if the Clifford inner product (a, b) is replaced by the twisted Clifford inner product (a, b), the hat product x 6 y

by the check product x 6 y, and t by t, then both Theorem 13.73 and Theorem 13.74 remain valid.

Other properties of pinor multiplication are easily read off Lemma 8.44.

Proposition 13.78. For all a E Cl(r, s) (i) (ax) 6 y = a(x 6 y), and x 6 (ay) = (x 6 y)a, (ii) (x 6 Y) (z o w) = (xe(y, z)) 6 w, (iii) (x o yj = y 6 x if e is symmetric (hermitian or pure), (iv) (x 6 y) = -y 6 x if a is skew (hermitian or pure). Recall the notion of a pure spinor in the split case P(p, p). The square of a pure spinor represents the associated null plane in AR(p, p).

Proposition 13.79. Suppose s E PURE C P(p, p) is a pure spinor. Then the squares o s E Cl(p, P) AR(p, p) is of the form s o s = z1 A Azp with N. = span{zl,..., zp} = {z E R(p, p) : z(s) = 0} totally null. Conversely, each E C1(p, p) = AR(p, p) of the form = zi A . A xp with N = span{zl, , zp} totally null is the square of a pure spinor.

Proof: Since, for all a E Pin(p), a(s 6 s)a-1 = (as) 6 (as), we need only show that for some s+ E PURE+, s+ o s+ = zl A. ..A zp with zl, , zp E N,+ (see Theorem 12.100). As in the proof of Theorem 12.100, note that s+ = 1 E AR(p, 0) = P(p, p) is a pure positive spinor. By (13.67) and Definition 12.24, (13.80)

(s+ 6 s+)(x) = (1 6 1)(x) = f(1, x) 1= (1, ox)

1.

Signature

268

Therefore, s+ o s+ = Pa, where P is orthogonal projection onto the line

-1(ei, -ej) so that z1,.. . , zp is a zp = 1e, o o Ier Therefore, zl basis for N,+. Recall that zj = lei . A ep E AR(p, 0) = P to ±1 and the maps the volume element c el A through 1 E AR(p, 0) = P. Let zj

orthogonal complement of v to zero. This proves that s+ 6 s+ = ± .

J

SIGNATURE First note that, by Problem 9.10, (13.81)

If s > 1 (i.e., R(r, s) not positive definite), then 9 must have split signature (if it has a signature).

(13.82)

If r > 1 (i.e., R(r, s) not negative definite), then a must have split signature (if it has a signature).

Consequently, unless r = 0 or s = 0, the spinor inner product a must have split signature (if it has a signature) Case s = 0 (i.e., R(r,O) positive definite). Then the Clifford inner product (, ) on Cl(r, 0) ARr is positive definite. Suppose x is a null vector in P, i.e., &(x, x) = 0. By Theorem 13.74,

(x6y,x&y)=0 for all YEP. Since (,) is positive definite, x 6 y = 0 for all y. Thus x = 0. This proves that e has no nonzero null vectors, so that t must be definite (and by adjusting the scale we may assume that t is positive definite).

Case r = 0 (i.e., R(O,s) negative definite). Let (,) denote the Clifford inner product on Cl(0, s) AR'. The twisted Clifford inner product (a, b) (a, b) is positive definite. Using the product x 6 y and Theorem 13.74 it follows that a cannot have nonzero null vectors. The signature part of Theorem 13.1 (cf. the Remark that follows this Theorem), for the spinor inner product a on S(r, s), follows because of the hat preserving isomorphism Cl(r, 0)even - CI(r - 1, 0).

Table 13.19 can be used to determine exactly when e and e have a signature.

Inner Products on Spaces of Spinors and Pin ors

269

Proposition 13.83. (i) The pinor inner product t has a signature if and only if either s =

0 mod4orr=3mod4. (ii) The pinor inner product t has a signature if and only if either r =

0 mod4ors=1mod4.

PROBLEMS 1. List the classical companions Cp(r, s), defined by (13.7), as classical groups.

2. Deduce the r - s = 0, 4 mod 8 cases of the Spinor Theorem 13.1 from the r - s = 0, 4 mod 8 cases of the Pinor Theorem 13.17. 3. (a) Deduce the r - s = 0 mod 8 case of the Spinor Theorem 13.1 from the r - s = 7 mod 8 cases of the Pinor Theorem 13.17. (b) Deduce the r - s = 2, 6 mod 8 case of the Spinor Theorem 13.1 from the r - s = 1, 5 mod 8 cases of the Pinor Theorem 13.17. (c) Deduce the r - s = 4 mod 8 case of the Spinor Theorem 13.1 from the r - s = 3 mod 8 case of the Pinor Theorem 13.17. 4. Suppose r 36 0 mod 4. Show that the check inner product i on P(r, 0) never has a signature, using Table 13.19. 5. Suppose r- s = 0 mod 8 and r+ s = 2p with p = 1 or 3 mod 4. Show that the (reduced) classical companion (p > 1) is

SL(p, R) a f

\0

(at)-1) : a E SL(p, R) } .

6. Suppose Cl(V) EndF(P) is the pinor representation for a positive definite inner product space V.

(a) Prove that G - {a E Cl(V) : as = 1} is a subgroup of O(Cl(V)), and hence compact. (b) Pick any positive definite inner product on P and define E to be

the average of this inner product over the compact group G. Show that e is the pinor inner product (up to a scale). 7. A real vector space V equipped with a linear map J, with j2 = 1 and dim V.. = dim V_, where Vf = {x E V : Jx = Ex} is called a L-vector space. Assume r - s = 0 mod 8. (a) Show that P(r, s) is an L-vector space.

270

Problems

(b) Combine 9 and a to obtain an "L-valued inner product" on P(r, s). Define L-hermitian symmetric, etc., and describe the "type" of this inner product (depending on p mod 4 with r + s = 2p defining p). 8. State and prove a complex analogue of Proposition 13.79 concerning squares of pure spinors.

14. Low Dimensions

The results of the previous chapters are particularly interesting in the low dimensional cases. These special cases are examined in some detail in this chapter.

CARTAN'S ISOMORPHISMS Recall from Chapter 1 some low dimensional isomorphisms of the groups defined in that chapter (see Proposition 1.40):

SO(2) = U(1) - SK(1) - S' and CSO(2) = GL(1, C) = SO(2, C),

SO(4) - HU(1) HU(1) and CSO(4) = GL(1, H) H', Sp(1, R) = SL(2, R) - SU(1, 1) and SL(2, C)!-:- Sp(1, C), SU(2) - HU(1) = SL(1, H) - S3 and SOT (3, 1) 2_-' SO(3, C).

The final isomorphism SO? (3, 1) - SO(3, C), which was proved in Chapter 3, is also a consequence of two of the spin isomorphisms presented

below-see Spin°(3, 1) and Spin(3, C), and note that both are SL(2, C), which has center Z2 = {±1}. Also SU(2) = HU(2) followsfrom Spin(3, 0) HU(1) and Spin(0, 3) = SU(2) (see Remark 14.113).

271

272

Cartan's Isomorphisms

In low dimensions, the spin groups Spin° are not new but are isomorphic to classical groups. Recall from Problem 10.2(d) that Spin°(r, s) {a E Spin(r, s) : as = 1} is the connected component of the identity in Spin(r, s) (n > 2), except when both r = 1 and s = 1.

Theorem 14.1 (The Spin Isomorphisms).

n=r+s=2: = U(1)

Spin (2)

Spin°(1, 1) . f 0

t

=

Spin (3) = HU(1) (? SU(2)) Spin°(2,1) SL(2, R) Spin (4) = HU(1) x HU(1) Spin°(3, 1) = SL(2, C) Spin°(2, 2) SL(2, R) x SL(2, R) Spin (5) Spin°(4,1) Spin°(3, 2)

Spin (6) Spin°(5,1) Spin°(4, 2)

HU(2) HU(1, 1) Sp(2, R) SU(4)

= SL(2, H) SU(2,2)

pf : Spin°(3, 3)/Z2 - SL(4, R)

n=r+s-8: pf : Spin°(6, 2)/Z2 = SK(4).

Proof: These results can be read off from Theorem 13.8 by comparing dimensions and connectivity. Recall the type of the spinor inner product implies:

Spin°(6) C U(4)

Spin°(5,1) C GL(2, H), Spin°(4, 2) C U(2,2), Spin°(3, 3) /Z2 C GL(4, R).

The extra information required in these cases is that each element of Spin° has determinant equal to one. This follows easily from the fact that

Low Dimensions

273

Spin° is connected and the fact that the determinant takes on a finite number of values for any representation of Pin (see the section Determinants in Chapter 10). In summary, using the spinor inner products (and in the four cases

listed above, information about the determinants), the group Spin° is a subgroup of the particular classical group (the (reduced) classical companion) described in Theorem 14.1. To complete the proof, equality is obtained by counting dimensions and using connectivity of this (reduced) classical companion associated with the group Spin°. The last case, signature 6,2 in dimension 8 is a special case of Theorem 14.3 presented below. Ll

Theorem 14.2 (Dimension Seven). Spin(7) C SO(8) C Ms(R) Spin°(6,1) C SK(4) C M4(H) Spin°(5, 2) C SK(4) C M4(H) Spin°(4, 3) C SOT (4, 4).

Proof: This result follows from Theorem 13.8 exactly parallel to the proof of the previous theorem.

Theorem 14.3 (Dimension Eight). First, Spin°(7,1) C SO(8, C) and Spin°(5, 3) C SO(8, C).

Second, Spin(8) C SO(8) x SO(8). Moreover, the the positive spin representation 1 --+ Z2 = {1, Al -+ Spin(8)

P±,

SO(8) -+ 1,

the negative spin representation 1 --+ Z2 = {1, -A) -+ Spin(8) -_+ SO(8) -+ 1, and the vector representation

1 --+ Z2 = {1, -1} -a Spin(8) --* SO(8) -+ 1 are exact sequences of groups.

Also, Spin°(4,4) C SO(8) x SO(8). Moreover, the positive spin representation 1 --+ Z2 =_ {1, A} -+ Spin°(4, 4) °+ SOT(4, 4) _+ 1,

the negative spin representation 1 -+ Z2 =- {1, -A} -+ Spin°(4, 4) _+ SOT (4,4) 1, and the vector representation SOT(4, 4) -+ 1 1 -+ Z2 = {1, -1} --f Spin°(4, 4)

Cartan's Isomorphisms

274

are exact sequences of groups.

In addition, Spin°(6, 2) C SK(4) x SK(4). Moreover, the positive spin representation 1 -+ Z2 = { 1, A} -+ Spin°(6, 2) °-++ SK(4) -- 1

and the negative spin representation

1 --+ Z2 - {1, -a} -+ Spin°(6, 2) p-+ SK(4) , 1 are exact sequences of groups. For each one of these Spin groups in dimension eight, all of the group representations listed above are inequivalent. Proof: Most of the information in this theorem follows from Theorem 13.8 in a parallel manner to the proof of Theorem 14.1. It remains to prove that the representations listed above are not equivalent. For example, suppose p+ : Spin(8) --+ Ms(R) S--- EndR(S+)

and

p_ : Spin(8) --+ M8(R) - EndR,(S_)

were equivalent, with intertwining operator f : S_ -+ S+. Then p+(g) = f o p_ (g) o f -1 for all g E Spin(8). In particular, this implies that ker p+ _ ker p_ which is false. Theorem 14.1 has a complex analogue.

Theorem 14.4. Spin(2, C) = C* Spin(3, C) SL(2, C) Spin(4, C) = SL(2, C) x SL(2, C) Spin(5, C) - Sp(2, C) Spin(6, C) = SL(4, C).

Hint of Proof: For n = 2p even, complexify the split case, C1(2p) Cl(p,p) ®R, C, and refer to the split cases Spin°(p, p), p = 2 and 3, in Theorem 14.1. For n = 2p + 1 odd, refer to the Spin°(p + 1, p) case in Theorem 14.1. The details are omitted. There is an alternate elementary proof of this theorem. This proof for Spin(6, C) is outlined in Problem 1. In addition to the isomorphisms presented above:

Low Dimensions

275

Theorem 14.5. The following groups are isomorphic. SK(2) = SU(2) x SU(1,1)/Z2 SK(3) = SU(3,1)/Z2. Proof: See problem 2. -J

TRIALITY In dimension eight, for both the positive definite case and the split case, the spin group Spin° has three inequivalent 8-dimensional representations: the positive spin representation p+, the negative spin representation p_, and the vector representation X. There are automorphisms of the group Spin called the triality automorphisms, which interchange these three representations. Of course, they cannot be inner automorphisms since the three representations are distinct (Theorem 14.3). Thus, by definition, the automorphisms are outer. There is an elegant description of these triality automorphisms in terms of the octonians 0 for Spin(8) and the split octonians O for Spin°(4,4). We shall pursue the positive definite case Spin(8) using O. However, except for notational changes, the development will apply to Spin°(4, 4) and O as well.

As an added bonus, we discuss a very useful concrete model for Cl(8, 0) (or alternatively, Cl(4, 4)). As will be showm, all three of the 8-dimensional

euclidean spaces V - R(8), S+, and S_ can be identified with O. First, the space of vectors, V(8) C Cl(8, 0), is identified with 0 as follows.

Definition 14.6. Let V(8) C Enda(O (D 0) denote the positive definite 8-dimensional euclidean space defined by

\ The square norm on V(8) is defined by jjA(u)jj

R,

0

A(u)

-R-U

0

)

Note that (14.7)

A(u)A(v) . I

-

OR"

_Ru

1juJll,

where

Triality

276

so that A(u)A(u) = -Ijull 1 for all u E 0.

Thus, by the Fundamental Lemma of Clifford Algebras, the map A 0 , EndR,(O (D 0) extends to an algebra homomorphism A : CI(O) --} EndR(O ® 0). By Lemma 8.6 and Theorem 11.3, CI(8) = CI(O) has no two-sided nontrivial ideals. Thus the algebra homomorphism is injective. A dimension count shows that this map A is onto, so that A is a isomorphism: CI(O) c--- EndR(O (D 0).

(14.8)

Therefore, the space of pinors may be taken as

P=0®O. The isomorphism (14.8) will frequently be composed with the natural isomorphism CI(O) = A(O) (as vector spaces).

Lemma 14.9. A unit volume element ,\ for V(8) - 0 is given by:

A= ('

-' ) E End,,(O ®O) = CI(O) = A(O).

Proof 1: Consider the standard orthonormal basis eo = 1, el = i, e2 = j, e3 = k, e4 = e, e5 = ie, e6 - je, e7 - ke

for O. Using octonian multiplication compute, A(eo) ... A(e7) = (1

-Ol)

.

Proof 2: Note A belongs to the twisted center of CI(O) by (9.48'). Lemma 9.49 implies that A E A130 and Theorem 9.65 implies that A is of unit length.

Remark. Right multiplictaion by i, denoted R;, induces a complex structure on 0. The orientation induced by this complex structure is the same as the orientation induced by this standard basis.

Corollary 14.10. If 0 is identified with V(8) C EndR,(O ® 0) by (14.11)

V(8) - (I

-& 0

J

:uEO},

then

CI(O) - Enda(S+ ® S_)

277

Low Dimensions

and Cleven(0)

= EndR(S+) ® EndR(S_) =

a

b)

: a, b E

}, EndR(O)1(0

where S+ _-- 0 is the first copy of 0 and S_ = 0 is the second copy of 0

in P=0®0. Proof: Note that A = (o _°) commutes with (d n) if and only if c = d = 0.

Lemma 14.12. The standard inner product on 0,

(x, y) = Reiy for all x, y E 0,

(14.13)

when applied to P = O®O so that the two factors S+ = 0 and S_ = 0 are orthogonal, can be adopted as the pinor inner product t on P = S+ ® S_ . Proof: It suffices to show that, for z = (x, y) and z' =_ (x', y') E 0 ® 0, (14.14)

t (A(u)z, z') = -9(z, A(u)z') for all u E O,

where t is defined by

(z, z') - (x, x') + (y, y').

(14.15)

Since A(u)z = (yu, -xu) and -A(u)z' = (-y'u, x'u), the result (14.14) follows from RU = R-.

Corollary 14.16. Under the isomorphism Cl(O) = EndR(O (D 0) determined by Definition 14.6, the hat anti-automorphismis equal to the adjoint

(transpose) with respect to the standard norm on 0 ® 0. In particular, Theorem 14.3 for Spin(8) yields (14.17)

Spin(8) C SO(O) x SO(O)

SO(S+) x SO(S_).

Note that {g E Cl(O)even : 99 = 1)

0(0) X 0(0).

We shall adopt both the notation

9 = (9+, g-) E Spin(8), as well as 9 = (go,9+,9-) E Spin(8), where by definition g+ = p+(g) and g_ _- p_(g), while go = X9 denotes the vector representation of g. The vector representation X will also be (14.18}

denoted by po = X.

Octonian multiplication can be used to give an important characterization of Spin(8).

Triality

278

The Triality Theorem 14.19. Suppose (go, g+, g_) is a triple of orthogonal linear maps on O. Then (9+, g-) E Spin(8), with go the vector representation of (g+, g_) if and only if g+(xy) = g_(x)go(y)

(14.20)

for all x, y E O.

Remark 14.21. If (go, g+, g_) is a triple of orthogonal transformations on 0 satisfying (14.20), then each of the three must have determinant one, because the theorem implies that g - (go, 9+, g_) E Spin(8) and Spin(8) is connected.

Proof: Given

A = (0

) E EndR.(O) ® Endp,(O) - Cl(O)e°e°, b

then, by Corollary 10.50, A E Spin(8) if and only if

AA = 1

(i.e., a, b E 0(0)), and XA(u) E 0 for all u E 0,

where

XA(u)=A(_p 0)A is the vector representation. Thus, for A (0 6) to belong to Spin(8), we must have that a, b E 0(0) and that: for each u E 0, there exists a v E 0 such that (14.22)

That is, (14.22')

(-R

=A(_R

0)

0

)A.

R=a

Considering this as a map sending u to v, it must be the vector representation go = po(A) of A. Applying (14.22') to 1 E 0, we have v = a(bt(1)u).

Therefore, if g = (g+, g-) E Spin(8), then (14.23)

v = go(u) __ g+(gt (1)u) for all u E 0

279

Low Dimensions

defines the vector represenation go = Xg. Now (14.22') with a = g+ and b = g_ can be rewritten as wgo(y) = g+(g' (w)y)

(14.24)

for all w, y E 0.

Setting x = gt (w) yields the desired equation (14.20). Conversely, suppose (14.20) is satisfied by (go, g_, g+). Set A__

/g+ I

0

0 g-

.

If u E 0 is given, then (with v = go(u)) the equation (14.22) is satisfied, since the equations (14.22), (14.22'), and (14.24) are all equivalent. iJ Since conjugation, c(x) = x, is an orthogonal transformation, the representation po = CO po o c is 0(8) equivalent to po. Similarly, p' = c o p+ o c

is 0(8) equivalent to p+. Note that if h . R and h' _- c o h o c then h' = Lu. The identity g+(xy) = g_(x)go(y), for g E Spin(8), implies that

9+(xy) = g+(xy) = g+(9x) = 9-(9)go(x) = go(x)9 (y)Therefore, (14.25)

a(go, g+, 9-) _ (g'_, g+, 90)

defines an automorphism a : Spin(8) --+ Spin(8). Similarly, (14.26)

Q(go, g+, 9-) _ (go, g-, 9+)

defines an automorphism 3 : Spin(8) -> Spin(8). Note that the product r = a/3 is given by (14.27)

r(go, 9+, g-) = (g+, g''-, 90),

and is of order three. r is called the triality automorphism of Spin(8)-

Since a2 = j32 = r3 is the identity homomorphism of Spin(8) and cep = r, , a = r2, it follows that a and ,3 generate a group of automorphisms of Spin(8) that is isomorphic to S3, the symmetric group on three letters. These automorphisms are all outer since they act nontrivially on the center, {±1, ±A} = {(1,1, 1), (1, -1, -1), (-1, 1, -1), (-1, -1,1)}, of Spin(8). This concrete realization of Spin(8) using the octonians has many applications. We give one example involving complex structures on R$ (see Proposition 7.174). Let Reff denote the orthogonal reflection along the plane (in order to avoid confusion with right octonian multiplication).

Triality

280

Proposition 14.28. Spin(8) is generated by triples (Refg, Jt , J£)

with £ E GR,(2, O),

where the complex structures J and J£ are defined by

J£ and 1

J£

=

2 (Rv Ru - Ru Ro)

if s=uAv. Proof: Proposition 10.21 states that GR,(2, V) C Spin(8) generates Spin(8). Given g = (14.29)

E GR,(2, V) C Spin(8), we must show that and

go=Reff,

Given two unit vectors it, v E V = 0, identify it with 0

Ru

Ru

0

'

and v with 0 R, 0J C-Ra as in the above discussion. Then the Clifford product is given by

(14.30)

uv=

R. Ru 0

0

-R-R

Note g = it v E Spin(8), so that (14.30) can be rewritten as (14.31)

(Ref ,, .Ref,,, -R,,&,

E Spin(8) for all u, v E O.

Finally, given l; E G-(2, 0), choose it, v orthonormal with 1; = u AV = it v and apply (14.31) and (6.14), to complete the proof of (14.29).

Now we can prove Proposition 7.174 (n = 4): (14.32)

Cpx+(4) = {JE :

E GR,(2, O)} = Gx(2, 0).

Low Dimensions

281

Proof of Proposition 7.174 (n = 4): First note that the complex structure Ri = J+i induces the same orientation on 0 as the standard basis eo, ..., e7 defined above. Therefore, R, E Cpx+(4). Since GR(2, 0) is connected, each complex structure J+ induces the same orientation on 0.

It remains to show that the map sending

E GR(2, 0) to J+ E

Cpx+(4) is an isomorphism. This map is just the positive spinor representation p+ restricted to GR(2, 0). Since ker p+ = {1, A) and A GR(2, 0), p+ restricted to GR(2, 0) is injective. Consider the action of Spin(8) on GR(2, V) given by sending to

geg-' for each g = (go,g+,g_) E Spin(8). Let K denote the isotropy subgroup at a point o. Then considering the vector representation and the positive spinor representation yields an equivarient isomorphism between the quotient

GR(2, 0) = SO(8)/(SO(2) x SO(6))

(14.33)

and the quotient Cpx+(4) = SO(8)/U(4).

(14.34)

In the following Remark this isotropy subgroup K of Spin(8) at .o is computed explicitly for fo = 1 A i (see (14.41)).

Remark 14.35. The octonian description of Spin(8) presented in the Triality Theorem 14.19 naturally leads to octonian descriptions of the two isomorphisms (14.36)

Spin(6) = SU(4) and

Spin(5) = HU(2)

as follows.

Recall (Problem 10.6) that the subgroup of Spin(n) that fixes a vector in the vector representation is Spin(n - 1). Thus

(14.37) Spin(7) = {g E Spin(8) : go(1) = 1} (14.38) Spin(6) _ {g E Spin(8) go(1) = 1 and go(i) = i} (14.39) Spin(5) = {g E Spin(8) go(1) = 1, go(i) = i, and go(j) Also, let Spin(2) denote the following subgroup of Spin(8): (14.40)

Spin(2) = {g E Spin(8) : go(x) = x if x 1 span{1, i}}.

Then it follows that (14.41)

K = Spin(2) x Spin(6).

Triality

282

The triality identity (14.20) can be used to compute p+ (G) and p- (G) where G is one of the above subgroups of Spin(8). First, we obtain

Spin(7) = {g E Spin(8) : g+ = g_}

(14.42)

from (14.20) by setting y = 1. Next, note that each element g of the group Spin(2) defined by (14.40) must have go a rotation Rote through an angle 0 in the 1, i plane. Thus, go is the product of two reflections along lines in span{1,i}. For example, Consequently, either g or -g is equal to the Clifford go = Ref 1 o Ref, product of 1 E V(8) with e'`B/2 E V(8), i.e., 0

1

O

Re-;e/a

( -1 0) (-Reie/2

_

Re:e/2

0

0

-Re_ie/2

0

Therefore, with V) = z + a, (14.43)

Spin(2) = {(Rote,,, Reio, Re_i*)

E R.) .

Next we show that the positive spinor representation p+ yields an isomorphism (14.44)

(Spin(2) x Spin(6))/Z2 = U(4),

when S+ = 0 is given the complex structure Ri, and Z2 = {1,a). If g E Spin(6) C Spin(7), then the triality identity (14.20) with y - i implies that g+ = g_ commutes with the complex structure Ri, so that g+ = g_ E U(4). Now a dimension count combined with the connectivity of U(4) completes the proof, showing p+ from Spin(2) x Spin(6) to U(4) is surjective. Using Problem 10.7 and the fact that ker p+ = {1, A} proves that Spin(6) = {(ho, h, h) : h E SU(4), and ho is given by (14.45)

ho(x) - h(1)h(x), for all x E V(8) - 0).

In particular, Spin(6) S--- SU(4).

Finally, the triality identity implies that if g E Spin(5), then g+ = g_ commutes with both Ri and R. Thus, g+ = g_ E HU(2), where 0 = H2 is provided with the (right) quaternionic structure inducted by I - Ri, J

R and K (14.46)

RiRj. Again connectivity and a dimension count show that

p+ = p_ : Spin(5) - HU(2).

Low Dimensions

283

TRANSITIVE ACTIONS ON SPHERES In low dimensions, n < 6, the spin groups agree with their (larger) classical companions-see Theorem 14.1. For slightly larger dimensions, vestiges of these isomorphisms remain. Consider, for example, Spin(7) C SO(8). In this case, SO(8) is the larger classical companion (determined by the spinor

inner product e on S = Its). The group Spin(7) is "almost" as big as its classical companion SO(8). One way of making this precise is to show that Spin(7) also acts transitively on spheres in R.

This action as well as the other transitive actions to be discussed in this section are listed below for convenient reference. (14.47)

Spin(7)/G2

(14.48)

Spin(8)/G2 = S7 X S7.

(14.49)

Spin(8)/Spin(7) = S7.

(14.50)

Spin(9)/Spin(7) =

S7.

S15.

Also, recall

G2/SU(3) - S6

(14.51)

from Problem 6.9(a). First, a model for C1(7) is constructed, using the octonians.

A Model for Cl(7) Identify V(7) = ImO with the following subspace of EndR(O) EndR(O):

Let

A(u)

(14.52)

` _ -I )

for u E ImO.

This map A : V(7) = Im 0 --+ EndR(O) ® EndR(O) extends to an algebra isomorphism (14.53)

Cl(7) = EndR(O) ® EndR(O) = EndR(P+) ® EndR(P_)

by the Fundamental Lemma of Clifford Algebras because A(u)A(u) _ -I1uII 1.

Transitive Actions on Spheres

284

The natural inner product on P = 0 ® 0 can be adopted as the pinor inner product E = e ®c, since R;, = -R,, for u E Im 0 (one must verify that E(A(u)z, z') = -t(z, A(u)z')). The unit volume element .A for V(7, 0) = Im O can be chosen to be (o °1) = A(el) A(e7), consistent with the orientation on Im O determined by the basis e1, ... , e7 defined above (see the proof of Lemma 14.9). Obviously, CI(7)even is contained in Enda(O), embedded diagonally in EndR(O) ® EndR(O) = C1(7). Counting dimensions yields

Cl(7)even = Enda(O).

(14.54)

Because of Corollary 10.50, g E Cl(7)even - EndR(O) belongs to Spin(7) if and only if (14.55a)

gg = 1

(i.e., g E 0(0))

and (14.55b)

given u E Im O - V(7), there exists w E Im O = V(7)

such that (14.56)

Rw.

The only possibility for w (apply both sides of (14.56) to 1 E 0) is (14.57)

w = g(g-1(1)u).

Therefore, if g E Spin(7) C 0(0), then (14.58)

Xg(u) = g(g-1(1)u)

for all u E ImO

is equal to the vector representation (14;59)

X = Ad : Spin(7) -+ SO(7).

Moreover, (apply (14.56) to g(v)) (14.60)

g(vu) = g(v)X9(u) for all u E ImO and v E O.

(Note that (14.58) can be used to extend xg to 0 with xg(l) = 1, and then (14.60) is valid for all u E 0.) Conversely, if (14.60) is satisfied by g E 0(0), with xg defined by (14.58), then (14.56) is valid with w = X5(u) (take v = g-1(z) and apply both sides of (14.56) to z). This proves the following lemma.

Low Dimensions

285

Lemma 14.61. An element g E 0(0) belongs to Spin(7) if and only if g(uv) = g(u)X9(v)

(14.62)

for all u, v E 0,

where

X9(v)=g(g-1(l)v) forallvEO

(14.63)

defines the vector representation of Spin(7) on V(7) = Im 0. Now because of the Triality Theorem 14.19

Corollary 14.64. (14.65)

Spin(7) _ {g = (g+, g-) E Spin(8) : g+ = g-} _ {g = (go,g+,g-) E Spin(8) :go(1) = 1}.

To complete our list of characterizations of Spin(7), note that if

(R, 0

?) E G(1, Im O),

-Ru

then

)EGIm0).

(0

Lemma 14.66. Spin(7) is generated by {R. : u E Ss C Im 0}. Proof: By Proposition 10.23, Spin(7) is generated by AG(1, Im O) = G(6, Im O) C Cl(7)even 5 EndR(O).

Theorem 14.67. Consider the spin representation of Spin(7) on S Then Spin(7) acts transitively on the 7-sphere

R8.

S7=Ix ER8:IIxII=1}, and the isotropy subgroup at a point is G2: (14.68)

Spin(7)/G2 = S7.

Proof: To compute the isotropy subgroup of Spin(7), the octonian model for Cl(7) is useful. Let K denote this isotropy subgroup of Spin(7) at 1 E S 25 0. Suppose g E K. Then, by (14.63), X.9(v) = g(g-1(1)v) = g(v). Therefore, by (14.62), g(uv) = g(u)g(v), so that g E G2. Conversely, if

Transitive Actions on Spheres

286

g E G2, then, by Lemma 6.67, g E 0(0). Since g(1) = 1 it follows that X9 = g, and (14.62) is valid. Therefore, conditions (14.55a and b) are satisfied and g E Spin(7). This proves that K = G2. We give two proofs that Spin(7) acts transitively on S7. First, recall that Spin(6) = SU(4). The spin representation of Spin(7) on Rs, when restricted to SU(4) _-- Spin(6) C Spin(7), is the standard representation of SU(4) on C4, where the volume element A for R(6, 0) provides the complex structure on Rs = C4. Note that Cl(6)e°e° = Endc(C4) and CI(6)even C Cl(6) = CI(7)even = EndR,(Rs). Since SU(4) acts transitively on S7 C C4, this proves that the larger group Spin(7) also acts transitively on S7. Second, the orbit of Spin(7) through 1 E S7 C 0 is Spin(7)/G2 which has dimension 7 = 21 - 14. Since S7 is connected and Spin(7) is compact,

the orbit must be all of S7. J Theorem 14.69. Consider the spin representation p = p+ ®p_ of Spin(7) on S+ ®S_ = Rs ®Rs. The orbit through a point (a, b) E S+ ® S_ is {(x, y) E S+ ® S_ : IxI = Jai and IyI = IbI}.

(a) If a, b # 0, the isotropy subgroup is G2 and

(14.70)

Spin(8)/G2 = S7 X S7.

(b) If exactly one of a, b vanishes, then the isotropy subgroup is a copy of Spin(7) and (14.71) Spin(8)/Spin(7) = S7.

Proof of (a): Spin(7) is diagonally embedded in Spin(8) by Corollary 14.64. Thus, an element g E Spin(8) can be chosen that sends a to jai because Spin(7) acts transitively on the 7-sphere. Consequently, we may assume that a E 0 - S+ is nonzero and real. Consider the subgroup (14.72)

H = {g = (go, g+, g_) E Spin(8) : g+(1) = 1}

of Spin(8). Applying the triality automorphism r(go, g+, g-) = (g+, g'_' go) shows that rH = {(go, 9+, g-) E Spin(8) : go(1) = 1}.

Low Dimensions

287

Now go(1) = 1 if and only if g+ = g_ (see Corollary 14.64). Therefore TH = Spin(7).

(14.73)

Thus, H = r2(Spin(7)) is an embedding of Spin(7) into Spin(8) send-

ing h E Spin(7) to the triple (h, Xh, h') E Spin(8). In particular, h' E Spin(7) can be chosen to send b to jjbjj since Spin(7) acts transitively on spheres. This proves that Spin(8) acts transitively on

S7xS7={(x,y)ES+ED S_:lxj =jalandjyI=bbl}. Let K denote the isotropy subgroup of Spin(8) at (a, b) - (r cos 9, rsin 9) E S7 X S7 C O ®O,

with r > 0 and 0 < 0 < ir/2. Suppose g = (g+, g_) E K. Recall that g+(uv) = g_(u)go(v) for all u, v E 0. Since g_(1) = 1, g+(v) = go(v). Therefore, go(l) = g+(1) = 1 also. Consequently, g+(u) = g_(u). This proves go = g+ = g_ E G2. Conversely, if go = 9+ = g_ E G2, then g - (g+, g_) fixes (a, b) since a, b are real. Therefore, K = G2. J Proof of (b): Since p+ : Spin(8) -+ SO(8) is surjective, Spin(8) certainly acts transitively on S7. Suppose (a, 0) E S. x {0}, with a real and nonzero. The isotropy H of Spin(8) at (a, 0) E S+ ® S_ is given by (14.59) so that H = r2(Spin(7)) as desired.

The proof for (0, a) E {0} x S_, with a real and nonzero, is similar

and so omitted. J

A Model for Cl(9)

Let S=-0®0 and PP=S®R,\ C. Choose (14.74)

V(9) = j Z

1

Ru

I: r E R, u E 01 C Endc(P).

If

A(r,u)= i

r

(Ru

R"l

,

r

then A(r, u)A(r, u) = -(r2 + 1u12)Id. Therefore, by the Fundamental Lemma of Clifford Algebras (14.75)

Cl(9) = Endc(P).

Transitive Actions on Spheres

288

Also note that (14.76)

Cl(8)even = EndR(S),

and the unit volume element for V(9) may be chosen to be i E Endc(P), the complex structure on P = S OR C. This model can be developed further, yielding a characterization of Spin(9) via Corollary 10.50 (see Problem 5). However, what we shall need below is a characterization of Spin(9) using Proposition 10.23. This Proposition says Spin(9) is generated by .G(1, V(9)) = G(8, V(9)), where A - i is the unit volume element. That is,

Lemma 14.77. The group Spin(9) is generated by the 8-sphere (14.78)

) :rER,uE0, andr2+Iu12=1}

f(

in EndR(O ® 0) - CI(9)even This lemma can be used to prove that Spin(9) acts transitively on S's

Theorem 14.79. Consider the spin representation (14.80)

p : Spin(9) --> SO(16),

of Spin(9) on S - R's. Then Spin(9) acts transitively on the 15-sphere

S15_Ix ER16:IIxii=1}, and the isotropy subgroup at a point is Spin(7): Spin(9)/Spin(7)

(14.81)

Sis

Proof: Because Spin(8) C Spin(9), Theorem 14.69 says that a unit vector in 0 ® 0 can be mapped to the arc (cos 9, sin 9), with 0 < 0 < a/2 by an element of Spin(8). Lemma 14.77 says, in particular, that con B sin O

sin 0) - cos B J

E Spin(9)

for each 0.

This implies that each point on the arc (cos 9, sin 0), 0 < 0 < a/2, can be mapped to the point (1, 0) by an element of Spin(9). This proves Spin(9) acts transitively on the unit sphere S's on R's

Low Dimensions

289

Let K denote the isotropy subgroup of Spin(9) at (1, 0) E S = 0 ® O. Since Spin(8) C Spin(9), part (b) of Theorem 14.69 says that K fl Spin(8) = r2(Spin(7)).

(14.82)

This proves that r2(Spin(7)) C K.

(14.83)

Therefore,

: Spin(9)/r2(Spin(7)) -+ Sls is a covering map, since dim Spin(9) - dim Spin(7) = dim S15. Since S15 is simply connected, it must be one to one. it

THE CAYLEY PLANE AND THE EXCEPTIONAL GROUP F4 Consider the relation - on F" - {0} defined by a - 6 if as = b for some scalar A E F.

(14.84)

If F = R, C, or H then - is an equivalence relation and Pr-1(F) is the projective space of all F-lines through the origin in F". However, since 0 is not associative the "Cayley projective spaces P"-1(O)" are not well-defined. Alternate (equivalent) definitions of P"-1(F), for F =- R, C, or H, are available (cf. Problem 4.15). Instead of considering a line L - [a] in F", consider the orthogonal projection A from F" onto L. (Equip F" with the standard F-hermitian symmetric inner product.) Then A satisfies At = A,

(14.85)

A2 = A,

and

traceF A = 1.

Recall the notation Herm(n, F) _ {A E M"(F) : At = A). Now projective space can be described as a subset of the real vector space

Herm(n, F). (14.86)

P"-1(F) = {A E Herm(n, F) : A2 = A and tracer A = Q.

To complete the proof of (14.86), note that if A2 = A with A E Herm(n, F),

then A has eigenvalues 0 and 1, so that A is orthogonal projection from F" onto L = {x E F" : Ax = x}. In particular, tracer A = dimp L. If A E M,, (F) represents orthogonal projection onto L and dimF L = 1, then, choosing any unit vector a E L, (14.87)

A=aa

Ax = a(a, x) for all x E F". Here a E F" represents a column vector. Now we turn to the Cayley plane. or

Cayley Plane and Exceptional Group F4

290

Definition 14.88. The Cayley plane is defined by (14.89)

P2(O) _ {A E Herm(3, 0) : A2 = A and traceo A = 1}.

Lemma 14.90.

(14.91)

P2(O) _ {a at : at = (a1i a2, a3) E 03 with all = 1 and [al, a2, aa] = 0},

which is a 16-dimensional compact submanifold of Herm(3, 0).

Proof: Suppose rl A

x3 x2

23 r2

22

21

r3

x1

E Herm(3, 0).

Then A 2

=

r1+112211+11x311

(rl + r2)x3 +'E211

(rl + r3)72 + 23x1

(r1 + r2)23 + 21x2

r2 + lIx1II + lIx311

(r2 + r3)x1 + x3x2

(r1 + r3)x2 + 21x3

(r2 + r3)71 + 22x3

r3 + Ilx1II + IIx21i

Now suppose A2 = A and traceo A = 1. Then r1x1 = x322i r2x2 = 71x3, and r3x3 = 21x2. Since not all of r1i r2, r3 can vanish, Artin's Theorem implies that for A E P2(0) the entries x1i x2, x3 belong to a quaternion subalgebra H C 0. Thus, A E Herm(3, H) C Herm(3, 0). Because of this fact, (14.87) is applicable. Choose at E H3 with A = a Qt and hail = 1. This proves that P2(O) is contained in the right hand side of (14.91). Finally, if at = (a1 i a2, a3), jjajj = 1, and [a1, a2, a3] = 0, then A =_ a -d'

satisfies traceA = 1 and A2 = A so that A E P2(0). To prove that P2(O) is a 16-dimensional manifold, note that it is covered by the three charts U1 _

.

at

_ (1, a2, a3) E 03 = 02,

at

(al, 1, a3) E 031 = 02,

-aa-I

_ U3= U2

Doll

jll

II

a

LI

Low Dimensions

291

Definition 14.92. The group F4 is defined to be the automorphism group of the (Jordan) algebra Herm(3, 0), equipped with the symmetric (or Jordan) product A o B = (AB + BA). a

A linear map g E GL(Herm(3, 0)) is an automorphism if (14.93)

g(A o B) = g(A) o g(B)

for all A, B E Herm(3, 0).

Note that A o A = AA, so that the notation A2 is unambiguous. In particular, (14.93) implies that (14.93')

g(A2) = g(A)2.

This simpler condition of "preserving squares" suffices to guarantee that g E F4, because polarization of (14.93) yields (14.93). Given

A=

rl

73

12

x3

r2

xl rs

(X2 xl

E Herm(3, 0),

define

traceA (14.94)

rl + r2 + r3,

IIAII = rl + r2 + r3 + 211x112 + 21x212 + 21x312, o2(A) = (rlr2 - (x312) + (tits - (x212) + (r2r3 - 1x112),

det A = rlr2r3 + 2x1, x2x3) - rl Ix1I2 - r2Ix212 - r3Ix312, in analogy with Herm(3, R). Note that (i.e., calculate directly) (14.95)

IIAII = trace (A2),

r2(A) = 1((trace A)2 - trace A2)

and

(14.95')

det A = 6 (trace A)3 - 2 (trace A2)(trace A) + 3 trace A3,

for all A E Herm(3, 0). Here and in the following A3 = A o A o A denotes

the Jordan cube, so that A3 = "(A2 A + A A2) in terms of ordinary matrix multiplication.

Cayley Plane and Exceptional Group F4

292

Lemma 14.96. If g E F4, then for all A E Herm(3, O).

trace g(A) = trace A, Consequently, F4 fixes 11

-

11,

0-2, and det.

Proof: It suffices to show that F4 fixes trace because of (14.95) and (14.95'). Since each g E F4 maps 1 to 1, it suffices to show that if trace A = 0, then trace g(A) = 0. One can show by direct calculation that each A E Herm(3, 0) satisfies A3 - (trace A)A2 + 0-2 (A)A - det A = 0.

(14.97)

Let

C - {A E Herm(3, O) : A3 = aA + b for some a, b E R}, and

Q - {A E Herm(3, 0): A2 = aA + b for some a, b E R}. Note both C and Q are invariant under F4. Now assume that trace A = 0. Because of (14.97) this implies A E C. Therefore B - g(A) E C so that (14.98)

B3 = aB + b for some a, b E R.

Subtracting (14.97) (with A replaced by B) from (14.98) yields

(trace B)B2 - (o2(B) + a)B + (det B - b) = 0. Therefore, either trace B = 0 or B E Q. If B E Q, then A = g-1(B) E Q. This proves that g maps {A : trace A = 0} N Q into {A : trace A = 0}. Finally, note that {A trace A = 0} . Q is a nonempty open subset of {A : trace A = 0} (choose A diagonal with trace A = 0 and distinct :

eigenvalues).

Theorem 14.99. F4 acts transitively on the Cayley plane P2(O) with isotropy subgroup equal to (an isomorphic copy of) Spin(9) at the point El =-

(14.100)

1

0

0

0 0

0 0

0 0

E P2(O),

F4/Spin(9) = P2(O).

Low Dimensions

293

Proof: Spin(9) is generated by

Ru) :rER,, uEO, andr2+JU12=1}

r

l

r)

Ru

J

because of Lemma 14.77.

Let C denote conjugation on both factors of

0 ® O. It is more convenient to consider the copy C C. Spin(9) C of Spin(9). Since

r

C'

u-r

Ru

(R).c=(;

the group C Spin(9) C is generated by

r Lu

Lul :rER., uEO,

rJ

JJ

Suppose

9=

r

(Lu-r )

is a generator for C Spin(9) C. Then g(x) =

rxl + ux2l I

uxl - rx2JJJ

for

(x2)

E O®O,

determines the spinor representation of C Spin(9) C. Also, (14.101) p Xg

Lv

k Lv -p

-

(r2 + lul2)p + 2r(u, v) L2rpu-rev-uvu

L2rpu-rev}uvu - (1u12 + r2) p - 2r(u, v)

determines the vector representation of C Spin(9) C on p Lu-

L" ) : P E R, V E 0 -P

Extend Xg to act on

\

r2 L (Lu r3) :r2 r3ER,vE0 } by defining

Xg(0 0)-(0 1 0)

ll

Cayley Plane and Exceptional Group F4

294

The action of C C. Spin(9) . C on Herm(3, 0) is defined as follows. Given

A-

(14.102)

rl

zl 72)

x1

r2

v

X2

'U

r3

E Herm(3, O)

and

a generator for C Spin(9) C,

(14.103) T

define t 9(r

g(A) =

\

1x)

Xg

where a =

a) ())

L"

r2

'

(Lu-

I

r3 /

and x = (X2) x1

Now we can prove that

Lemma 14.104. C C. Spin(9) C C F4. Note that this implies that Spin(9) C IsotE, (F4).

Proof: Given A E Herm(3, 0) and g a generator for C Spin(9) C as in (14.102) and (14.103), let

G-

(14.105)

1

0

0

0

r

u

0

u -r

Then, using (14.101) for Xg(a), a direct calculation shows that

GAG=

(14.106)

r1

(9(x)

g(x)t

g(A).

Xg(a)

Suppose U - (u;j) E M,,(O), with span{uij, 11 contained in a subalgebra isomorphic to C C O. (For example, U - G defined by (14.105).) Then for any element z E 0, span{uij,1, z} is contained in an associative subalgebra of O. In particular, ulzu2u3zu4 is well-defined for all U1i U2, U3, U4 E span{u;j, 1), so that

(14.107)

(ulzu2)(u3zu4) = ul[z(u2u3)z]u4

Polarization yields (14.107')

(u1 Zu2)(U3wU4) + (ulwU2)(u3ZU4)

= u1[z(u2u3)w]u4 + ul[w(u2u3)Z]u4

Low Dimensions

295

This implies that (14.108)

(UAU)(UBU) + (UBU)(UAU) = U(AU2B)U + U(BU2A)U,

for all A, B E Ilerm(n, 0). Applying this identity, with U - G defined by (14.104), yields

g(A)g(A) = (GAG)(GAG) = GA2G = g(A2),

since G2 = 1. Thus g E F4. This proves that C Spin(9) C is a subgroup

of F4. J Since F4 preserves the trace, both the conditions, A2 = A and trace A = 1, defining P2(O) are preserved by F4. Next we prove that F4 acts transitively on P2(O). Suppose A E P2(O): A

(x

a)'

x

and a=

(x2)

(v2

V

r3

Choose g E C Spin(9) C C F4 so that x9 (a) = (o ° ). This is possible because x : Spin(9) --> SO(9) acts transitively on the 8-sphere in each 9plane {a E Herm(2, 0) : trace a = constant). The isotropy of C Spin(9) C at (o °) is the same as the isotropy at (o °) , which contains C End(O ®O) has orbits S7 x {0}, S7 X S7, and {0) x S7. In all cases, there exists h E Spin(8) so that h(xi, x2) = (yi, y2) with yl, y2 E R. This proves that the orbit of F4 (in fact, C Spin(9) C C F4) through any point A E P2(O) contains a point B E P2(O) with each entry bij E R, i.e., B E Herm(3, R), Given g E SO(3) C M3(R), define (14.109)

p9(A) = gAg`

for all A E Herm(3, 0).

This defines SO(3) as a subgroup of F4. If A is real, then we may choose g so that B =_ p9 (A) is diagonal. The conditions B2 = B and trace B = 1 imply that B has eigenvalue 1 with multiplicity one and eigenvalue 0 with multiplicity 2. Finally, by applying a permutation zr E SO(3) to B, we obtain El, proving that F4 acts transitively on P2(O).

It remains to show that each g E F4, which fixes El, belongs to C Spin(9) C. Given A (r'x3Y) a E Herm(3, 0), note that (14.110)

2AoEl -A=

(r'0

0

)

a/

Cayley Plane and Exceptional Group F4

296

Thus, if g E F4 fixes El, then g maps the subspace of Herm(3, 0) defined by a = 0 into itself. Therefore, (14.111)

r

f(x)t

f(x)

h(a)

g(A) = (

where h E SO(9), acting on Herm(2, 0) while leaving the identity fixed. Now by utilizing an element of Spin(9), we may assume that h = 1. It remains to show that f is the identity. Suppose A is of the special form 0

71

72

xi

0

0

X2

0

0

Since h - 1, g fixes the basic diagonal matricies E2 and E3. Note that

2AoE2=

0

71

xi

0

0 0

0

0

0

Thus, 2A o E3 = A if and only if x2 = 0. Consequently if x2 = 0, then y2 = 0, where y = f (x). Similarly, if xl = 0, then yl = 0. This proves that yi = fl(xi) and Y2 = f2(x2). Since Ix12

A2 =

0

0 0

'1x2

0

x271

0

0

the condition g(A2) = g(A)g(A) becomes

21x2 = fl(xl)f2(x2) for all x1i x2 E O.

If u = fi(1), then u-1 = f2(1), and it follows that fl(xl) = xiu and f2(x2) = x2u-1. The above condition on f becomes ('lu)(u-172) = 21x2

for all 21,x2 E 0, which implies that u = 1. Therefore f(x) = x. This completes the proof of Theorem 14.99.

The complex Lie group F4 can be defined to be the automorphism group of the complexified Jordan algebra Herm(3, O)®R,C - Herm(3, OC). Here Oc - Oc ®R, 0 denotes the algebra of complexified octonians. Note

that Oc contains the split octonians O as a subalgebra. The noncompact

297

Low Dimensions

or split case of F4, denoted F4 is defined to be the automorphism group of the Jordan algebra Herm(3, O). In addition to F4 and F4, there is a third real form of F4 c, which we denote by F4. Let

Herm'(3, 0) = 1A E M3(Oc) : A =

f

r1

-ix1

ix1

r2 x3

Zx2

-i22 1

x3

r3

and x1i x2, x3 E O, rl, r2, r3 E R1 -

(Here i denotes the i E C in the complexification Oc = 0 (9R C of 0.) Define F4 to be the automorphisms of the Jordan algebra Herm'(3, 0). Note that all three Jordan algebras: Herm(3, 0),

Herm(3, O),

and

Herm'(3, 0)

have the same complexification Herm(3, Oc). The results of this section have analogues for F4 and F4. For example, define

P2(O) -= {A E Herm(3, O) : A2 = A and traceo A = 11. Then the orbit of F4 through 1

0

0

0

0

0

0

0

0

is P2(O) with isotropy subgroup Spin°(5,4): (14.112)

P2(0) a-- F4/Spin°(5,4).

Also,

(14.112)'

F4/Spin(9) L'5 {A E Herm'(3, O) : A2 = A and traceA = 1}

The proofs of (14.112) and (14.112)' can be adapted from the proof of P2(O) L, F4/Spin(9), with suitable modifications and additions.

Clifford Algbebras in Low Dimensions

298

CLIFFORD ALGEBRAS IN LOW DIMENSIONS The models presented in the previous sections for Cl(7), Cl(8), Cl(3, 4), Cl(4, 4), and Cl(9) are quite useful. Similar models for the other low dimensional Clifford algebras are available. Some of these models are collected

in this section (dimensions 2, 3, and 4 and the various signatures). jr, these low dimensions, the algebraic structure of Cl(r, s) along with the canonical involutions uniquely determines the choice of vectors A'R(r, S) and hence uniquely determines the full Clifford structure. This is because the hat involution is minus one on A'R(r,s), and plus one on A3R(r, S). Thus, in dimensions n < 4, A'R(r, s) is uniquely defined as the subspace of Clodd(r, s) fixed by the hat involution, sending a to a.

Dimension 2 and Signature 2, 0 or 0, 2 Spinors. Cl(2, 0)even = Cl(O1 2)even = C,

where a choice A of unit volume element corresponds to i E C. A canonical choice for the spinor space S (a complex 1-dimensional vector space with Cl(2, 0)even = Endc(S)) is possible once an orientation has been selected. Just take S = Cl(2, 0)even with complex structure A, and let Cl(2, O)even act on S by left multiplication. Thus

Spin(2)={aEC:hail=1}=Sl and the canonical involution (hat = check) on Cleven tion on C.

C is just conjuga-

Pinors. The two larger algebras Cl(2, 0) and Cl(0, 2) containing Cl even C are the two Cayley-Dickson doubles of C. First, Cl(2, 0) = Cl(2, 0)even ® Cl(2, 0)odd

can be rewritten as:

H

C

®

Cl

where Cl = span{ j, k}. In particular, the space of vectors R(2, 0) _ A'R(2, 0) C Cl(2, 0) is just C'. One can take P = Cl(2, 0) with right H-structure H - Cl(2, 0). Second, the pinor space P(0, 2) is a real 2-dimensional vector space with Cl(0, 2) - EndR(P). Now Cl(2, 0) - Cl(2, 0)even ® Cl(2 0)odd

can be rewritten as:

Low Dimensions

299

M2(R)

=

C

®

C'

where Cl = span { C, R}, C = (o _°) conjugation, R = (° o) reflection, and i = J = (° -o). In particular, C' = span{C, R} = A1R(0, 2), i.e., the set of complex antilinear maps in M2(R) is the space of vectors. In both cases, the vector representation of Spin(2) S' is equivalent to the representation X : Spin(2) -. Enda(C) defined by X,,(z) = a2z. Remark. Suppose M is a 2-dimensional (positive definite) Riemannian manifold. Then, via the canonical isomorphism Cl(2, 0) = A(R(2, 0), the exterior bundle ATM is naturally a bundle of H-algebras, with the subbundle AeVe"TM = A°TM ® A2TM a bundle of C-algebras_ Since in this dimension AevenTM acts on TM by Clifford multiplication, AevenTM is

naturally a subbundle of End(TM). Note M is oriented if and only if AevenTM = M x C is trivial. (Similarly, ATM is naturally a bundle of M2(R)-algebras.) In this dimension (if M is oriented), AeVe"TM = M x C can be taken as the bundle of spinors M x S with the bundle isomorphism Cleven(2)

Endc(S).

Dimension 2 and Signature 1,1 Spinors. Cl(1, 1)even R ® R - L (double numbers of Lorentz numbers), where a choice of unit volume element corresponds to r E L. A canonical choice for the spinor space S = S+ ® S_ (two real vector spaces S+ and S_ with Cl(1, 1)even = Enda(S+) ® EndF(S_)) is possible once an orientation has been selected. Just take St = Cl(1,1)f en = {a E Cl(1, 1)even :Aa = ±a}, and let Cl(1,1)even act on S by left multiplication. The canonical involution on Cleven - L is conjugation on L. Now

Spin°(1, 1) = {fe''e : 9 E R} = R ® Z2 Spin(1,1) = {±er9 0 E R} U {±rere : 0 E R}

Pinors. The pinor space P is just S and Cl(1, 1) - M2(R) EndR,(P), with Cl(1, 1) - Cl(1,1)even ® cl(1,1)odd

corresponding to

M2(R) = L

®

L1

where r =- (o _°) and Ll = span{R, J} with R = (° o) and J (° o). That is, L C M2(R) consists of the L-linear maps and Ll C M2(R) consists of the L-antilinear maps.

Clifford Algbebras in Low Dimensions

300

Dimension 3 and Signature 3, 0 or 0, 3 Signature 3, 0. Cl(3, 0) - M1(H) ® M1(H),

(14.114)

with r = (o _°) a choice of unit volume element. Also S =_ H with P = H ® H. The hat inner product t on P = H ® H is the standard H-hermitian symmetric inner product. Thus, the hat anti-automorphism applied to (o equals (o °) . Now Cl(3, 0)even 25 M1(H) considered to be °) diagonally embedded in M1 (H) ® M1 (H) - Cl(3, 0). Thus, {a E H : 1Iall = 1} = S3 C H = C1(3, 0)even

Spin(3)

That is, Spin(3) = HU(1). The vectors A1R(3, 0) can be distinguished as a subspace of AR(3, 0) - Cl(3, 0) by noting that A E A1R(3, 0) if and only if a is odd and a is orthogonal to the unit volume element A = (o _°) . Thus, Al R(3, 0)

p _0 I

: u E Im H } C M1(H) ®M1(H)

Cl(3, 0).

Conversely, adopting this as the definition of A'R(3, 0) C M1(H)®M1 (H), it is easy to deduce (from the Fundamental Lemma for Clifford Algebras) the isomorphism (14.114) and all of the rest of the information above about Cl(3, 0). The vector representation Ad : Spin(3) --* SO(3) is given by Adau =

aua, for all u E ImH, where a E Spin(3) - S3 = HU(1) C H. Remark. If M is a 3-dimensional (positive definite) Riemannanian manifold, then the vector bundle E = Aeve.TM = Cleven(TM) is naturally a bundle of H-algebras. Letting E act on itself on the left embeds E as a subbundle of EndR(E). Let C denote the centralizer of E in Enda(E). Then C is a bundle of H-algebras that acts on E on the right, giving E the structure of a right H-vector bundle with coefficient (or scalar) bundle C. Because of dimensions, Cleven(TM) - EndH(E), so that E may be taken as the global bundle of spinors for M.

Signature 0, 3. Cl(0, 3) - M2(C), where A = i is a choice of unit volume element. Thus P = C2. The check inner product t is the standard C-hermitian symmetric positive definite inner product on C2; thus A = A* = At. Therefore, consulting the Table

Low Dimensions

301

in Lemma 9.27, we see that A1R(0, 3) must consist of those A E M2(C) that satisfy

(i.e., traces A = 0)

A is orthogonal to 1 and

A=A*, i.e., A is hermitian symmetric). This proves

A1R(0, 3) = Hermo(C2),

the space of trace free hermitian symmetric 2 x 2 matrices. Note that the Pauli spin matrices

_ (1 (0 U1-

1

0

0.2

'

_

0

i

i

0

_

'

0

1

Q3- (0 -1)

form a basis for A'R(0, 3) C M2 (C) = Cl(0, 3), with A = i = U1620'3 the volume element. Since A1R(0, 3) = Hermo(C2), A2R(0, 3) = i Hermo(C2)

so that CI(0, 3)even =

{(z I.

w

z I :z,wEC}.

Therefore,

Spin(3) =

{(z w

z/

I

: z, w E C with Iz12 + 1w12 = 1 }

SU(2).

JJJ

The vector representation Ad : Spin(3) - SO(3) is given by AdAh = AhA*

for all h E Hermo(C2),

where A E Spin(3) = SU(2). Remark 14.113. The isomorphism Cl(3, 0)even - CI(0, 3)even induces an isomorphism of the two realizations of Spin(3), namely, HU(1) t--- SU(2).

Thus, the Clifford algebra point of view leads naturally to an isomorphism, which was first discussed in Proposition 1.40; the isomorphism between Cleven(0 3) =

((W and

z

: z, w E C} C M2(C) = C1(0, 3),

Clifford Algbebras in Low Dimensions

302

Cleven (3, 0) = Mi(H).

Remark. One can show that (a) Pin(3, 0) = Z2 x S3 C H ® H with Z2 = {(1, 1), (1, -1)}, (b) Pin(0, 3) = {A E U(2) : detc A = f1} C M2(C),, (c) Center Pin(3, 0) = Z2,

(d) Center Pin(0, 3) = {f1,±i} = Z4. Thus by (c) and (d), Pin(3, 0) and Pin(0, 3) are not isomorphic as abstract groups.

Dimension 3 and Signature 2, 1 or 1, 2 Spinors. C1(2, 1)even - Cl(1, 2)even - M2(R).

The space of spinors S = R2, and the spinor inner product e is the standard volume form (R skew) on S = R2. Thus, the canonical involution Cleve. = M2(R) is the cofactor transpose, and on

Spin°(2,1) Spin (2,1)

SL(2, R)

(A E M2(R) : detA = ±1}.

Pinors for Signature 2, 1. Cl(2, 1) - M2(C) = C1(2, 1)even OR C.

The pinor space p is C2 and the unit volume element A equals i E M2(C). The hat inner product t on C2 is the standard complex volume element

e = dzl A dz2 on C2. Thus, the hat involution on Cl(2, 1) = M2(C) is the cofactor transpose. Consequently, A'R(2, 1) C M2(C) Cl(2, 1) is distinguished as

{iA:AEM2(R)andtrace A=0}={

Cix

l \` iz

-iy l :x,y,zER}. ix ) JJ

JJ1

Pinors for Signature 1, 2. Cl(1, 2) - M2(R) ® M2(R) = Cl(1, 2)even ®R L

The pinor space is given by P = P+ ® P_ with P± = R2, and the hat inner product E on P equals e ®e, where e = dxl A dx2 is the standard

Low Dimensions

303

volume element on R2. Thus, the hat involution is the cofactor transpose applied to both 2 x 2 real matrices in C1(1,2) = M2(R) ® M2(R). The space A1 R(1, 2) of vectors can be distinguished in Cl(1, 2) and equals Al R(l, 2) = { I 0

-A )

: A E M2(R) and trace A = 0 } .

Remark. Thus, in both cases, signature 2, 1 and 1, 2, the vector representation Ad : Spin(2, 1) -> SO(2, 1) is given by

AdAB =_ ABA-'

for all B E M2(R) such that trace B = 0,

where A E Spin(2, 1) = SL(2, R).

Dimension 4, Signature 4, 0 and 0, 4 Spinors. Cl(4, 0)even - Cl(0, 4)even - M1 (H) ® M, (H).

The space of spinors S =_ S+ ® S_ with S+ = H, so that Cleve. EndH(S+)®EndB(S_). The spinor inner products of are the standard Hhermitian symmetric (positive definite) inner product on S± = H. Thus, Cleve. the canonical involution on LI M1(H) ® M,(H) is conjugation of both factors. Therefore, Spin(4) = S3 X S3

a

0) :IIail=IIbhl=1}

C M1(H)ED M1(H).

0l The unit volume element is A _ (1o_11

Pinors. As algebras both

C1(4, 0) - M2(H), and Cl(0, 4) = M2(H).

Thus, P(4, 0) and P(0,4) are both H2. The standard H-hermitian symmetric positive definite inner product on H2 corresponds to the hat inner product t on P(4, 0) = H2 while it corresponds to the check inner product

Clifford Algbebras in Low Dimensions

304

t on P(0, 4) - H2. Consequently, V(4, 0) = A1R(4, 0) is distinguished as the subspace (14.115)

(/ -Lu _ l

V(4, 0) = S

l

1

u E H) C Cl(4, 0) = M2(H),

i.e., those odd elements`that are H-hermitian skew; while V(0,4)

)

A1R(0, 4) is distinguished as the subspace (14.116)

V(0, 4) _

f0u 1

: u E H } C Cl(0, 4) = M2(H).

Conversely, adopting (\14.115) as the definition of V(4, 0) C M2(H) and

(14.116) as the definition of V(0, 4) C M2(H), it is easy to deduce (from the Fundamental Lemma of Clifford Algebras) the isomorphisms C1(4, 0)

M2(H) and C1(0, 4) = M2(H) and all the rest of the information above about Cl(4, 0) and Cl(0, 4).

Dimension 4, Signature 1, 3 and 3,1 (Special Relativity) Spinors. C1(3, 1)even - Cl(1, 3)even = M2(C),

and the unit volume element A for R(3, 1) or R(1, 3) can be chosen to be i E M2(C). The space of spinors S is a 2-dimensional complex vector space with isomorphisms Cl(3,1)even - Cl(1, 3)even t--- Endc(S).

The spinor inner product c on S is C-skew. (Hence, we may take S = C2 and s = dzi A dz2, the standard complex volume form on C2.) Therefore, the canonical involution on Cleven = M2(C) is the cofactor transpose sending

A=`a \c b) d

to

A*

Thus, Spin°(3,1) = Spin°(1, 3) = SL(2, C) and

Spin (3,1) = Spin (1, 3) = {A E M2(C) : detc A = ±11. The reduced spin group Spin°(3,1) = SL(2, C) acts on the space S, e of spinors preserving the C-skew inner product e. The conjugate represen-

tation of SL(2, C) is on the space S of conjugate spinors. (Pc = S ®S is the space of complexified pinors.)

Low Dimensions

305

In this important case, the spinor representations yield a classification of all representations of SL(2, C) with important implications in general relativity. A brief description is included. Start with S, J a complex vector space with complex structure J. As usual, consider the complexification Sc = S®R,C = 510®S°,1 decomposed into the +i eigenspace Sl,0 and the -i eigenspace S°,1 for J (extended to be complex linear on S), along with the canonical conjugation on Sc.

Now assume dimc S = 2, let S = S1'°, S = S°'1, and assume that S is equipped with a complex volume element (a C-skew inner product e). Representations of the group SL(S) = SL(2, C) will be denoted by the vector space alone, i.e., the action is assumed to be obvious. For example, SL(2, C) has a natural induced action on SP the pth symmetric tensor product of S, and on S the qth symmetric tensor product of S. It is a standard fact in representation theory that (1) SP ®Sq (the space of p, q spinors) is irreducible, p, q > 0, (2) there are no other irreducible representations of SL(2, C).

For example, a defines an isomorphism b of S with dual space S* that is the intertwining operator for S = S*. Also, e provides an isomorphism between A2S and C, the trivial representation. The vector representation of Spin°(3, 1) SL(2, C) is obtained as follows. Let H = (S ®S)R denote the subspace of S ® S A1,1Sc fixed by conjugation. Let 2 (Im h) A (Im h) _ (h, h)e A E E A2'2SC

define a bilinear form ( , ) on H. One can identify H with the space of hermitian symmetric 2 x 2 matrices with square norm equal to the determinant. In particular, H = R(1, 3) is Minkowski space. Note that replacing e by eae dos not change the metric on H. Using e, the space of p, q spinors (with the SL(2, C) action) can be identified with the space Cp q [z, z] of polynomials which are homogeneous of degree p in z E S and homogeneous of degree q in x E S (with the natural action of SL(2, C) on C[z, z]).

Remark. On a Lorentzian manifold M equipped with a spinor bundle

S, e, note that: (i)T*M®C-S®S, and s®S-S2®A2S-S2®CS2'0 ®S°'°. More generally, one can show that: (ii) S ®SP = SP+1 These facts, (i) and (ii), enable one to define four different (if p > 1 and q > 1) differential operators on p, q spinor fields s based on decomposing the covarient deviative Ds of s as follows: ®Sp-1

Ds E TTM 0 Sp,q = Sp-1,q-1 ® SP+l,q-1

®Sp+l,q+l

®

SP-1,q+1.

Clifford Algbebras in Low Dimensions

306

q

.p

Figure 14.117

If q = 0, then there are just two differential operators. the one taking a p, 0 spinor field to a p - 1, 1 spinor field is called the Dirac operator and denoted by ill. The other, taking a p, 0 spinor field to a p+ 1, 1 spinor field is called the twistor operator. The Dirac equation

P f = 0,

where f is a p, 0 spinor field,

is called the equation for a massless particle of spin s = p/2. The case p = 2 (s = 1) is that of an electron. Pinors for Signature 1, 3: The pinor space P is a real 4-dimensional vector space. A choice of unit volume element A provides a complex structure I A on P. The spinor space S can be taken to be P with the complex structure I. The spinor inner product a is complex skew and

e=E+ie, ore=Ree. It is possible to choose a conjugation c on P so that e(cx, cy) = e(x, y) for all x, y E P.

Note that

AoddR(1, 3) = Endc(P)

(the space of all complex antilinear maps of P) can be expressed as

Endc(P) = {Ac: A E Endc(P)} .

307

Low Dimensions

Now c2 = 1 and c is complex antilinear. Therefore, given A E Endc(P), t(Acx,y) = Ree(Acx,y) = Res(cx,A*y) = E(cx,A*y) = e(x,cA*c2y) _ 9(x, A * cy). This proves that

(Ac)' = A*c for all A E Endc(P). Therefore,

A'R(l, 3) = {Ac : A E Endc(P) and A* = -A}

.

That is V(1, 3) - A1R(1, 3) must be the subspace

(14.118)

V(1,3)=1

l\y

of EndR,(P). The square norm (14.119)

z

-xc:zEC,andx,yER} zl J

on V(1, 3) is given by

IlAcIl = (Ac)(Ac)^ = AcA*c = AA* = detc A,

which equals -jjzjj - xy if

A= ( y -y ) Conversely, adopting (14.118) as the definition of V(1, 3) C EndR,(P)

M4(R), where V(1, 3) is equipped with the square norm given by (14.119), it is easy to deduce the isomorphism Cl(1, 3) - EndR,(P) and all the rest of the information presented above about Cl(1, 3).

Pinors for Signature 3,1: The pinor space P is a 2-dimensional (right) quaternionic vector space; so that CI(3, 1) - Endc(P). One can show that, under an isomorphism Endc(P) - M2(H), the space of vectors A'R(3, 1) is isomorphic to

-w/ :x,yERandwEC} Left multiplication by i, denoted Li, corresponds to a choice of unit volume element.

Squares of Spinors and Calibrations

308

SQUARES OF SPINORS AND CALIBRATIONS The pinor multiplication introduced in Chapter 13 is a useful tool for constructing calibrations. The Clifford element 0 - xoy E C1(r, s) obtained from multiplying two pinors x and y may also be considered as a differential form 0 E AR(r, s)*; under the natural identifications of Cl(r, s) with AR(r, s), and AR(r, s) with AR(r, s)*. In this section, we consider the positive definite case. The special case n = 8p is particularly interesting. Recall

CI(8k) - Endn.(P), with P = S+ ® S_ decomposing into the real vector spaces S±, of positive and negative spinors. The spinor inner product c on S+ and S_ is positive definite and real. Given a differential form 0 E AR(n)*, let cfk E AkR(n)* denote the degree k part of 0.

Theorem 14.120. Consider R(n) with n = 8p. Given a unit positive spinor x E S+, let (14.120)

0 = 161'x OX E Cl(8p) - AR(n)*.

Then

(a) 0 E > A4mR(n)* only has components of degree 4m. (b) Each ck is a calibration, i.e., ¢k(') < 1 for all t: E G(k, R(n)). (c) Equality Ok(1;') = 1 occurs if and only x, (d) The isotropy subgroup of 0(n) that fixes 0 is isomorphic to the subgroup Kx of Spin(n) that fixes the spinor x.

Corollary 14.122. Under the hypothesis of Theorem 14.120, the degree k-form ck is a nontrivial calibration (k = 4m) if and only if there exist an 1; E G(k, R(n)) such that x = x. The contact set

G(4) - { E G(k, R(n)) : q(4) = 1} is equal to {!; E G(k, R(n)) C Spin(n) : t;x = x}, which via the map X is isomorphic to {Refspa=,t : E G(k, R(n)) and RefBpan£(0) = 4'} C SOW-

Proof. By Theorem 13.73 (dims P = 161'), (14.123)

q(t;) = 16P(xox,t;) =s(l;x,x) for all 1; E Cl(n).

This formula is very useful.

Low Dimensions

309

(a) If E Cl(n)°dd then x E S+ implies E S_ so that If has degree = 2 mod 4, then = - and hence

e(x, x) =

x) = 0.

x),

so that again 0({) = 0. (b) By the Cauchy-Schwarz inequality, Equation (14.123) implies that (14.124)

jxj = 1

for all

E G(k,R(n)) C Cl(n)

where y12 = e(y, y) is the spinor norm on S+. (c) Also, (14.125)

1 if and only if ex = x.

Note that Jex! = 1 in (14.124) because £ E G(k, R(n)) C Spin(n) for = 1. k = 0 mod 4, which implies that (d) Consider the vector representation x : Pin(n) --> O(n). Given a E

Pin(n) and v E R(n) then Xa(v) = ava-1. More generally, given a E

Pin(n), Xa E O(n) acts on AR(n)* = AR(n) by Xa(O) = a0a-1 = 16Pa(x 6 x)a. By Lemma 8.44(a), this equals 16(ax) 6 (ax). Therefore, if Xa(0) = 0 then (ax) o (ax) = x o x. By the definition of spinor multiplication, this implies that ax = cs for some scalar c E R. Since a preserves the pinor norm, c = ±1. The element a E Pin(n) must be either even or odd. If a is odd, then ax E S_ and ax = ±x is impossible. Thus ax = ±x and

{a E Pin(n) : X.,(') = ¢} _ {a E Spin(n) : ax = ±x}. Since, under the vector representation X, both a and -a have the same image, this proves that (14.126)

X:Kx={aESpin(n):ax =x}--*{gEO(n):g*c6=¢}

is surjective. Since -1 ¢ Ks,, this is an isomorphism. Using the octonian model for C1(8) = EndR,(O ®O) with S+ = 0, the square of a positive spinor is easily calculated. First note that Spin(8) acts transitively on the unit sphere in S+, so that any two squares are SO(8)equivalent in AR(8)*. Therefore, we may choose the multiplicative unit in S+ = 0, denoted 1+, as a unit positive spinor without loss of generality.

Squares of Spinors and Calibrations

310

Theorem 14.127. (a) 16 1+ o 1+ = 1 + 4D + a, where ID E A4O* is defined by (14.127)

c(u1 A U2 A U3 A u4) _ (u1, u2 X U3 X U4),

and a is the unit volume element determined by the orientation on O. (b) The 4-form is a calibration (called the Cayley calibration) on Rs 0, that is fixed by the group Spin(7) {g E Spin(8) : g+(1) = 1}.

Proof: By (14.123)/,

p 7 16(1+ 6 1+)(u1 A U2 A U3 A U4) = E(RuR1u,R,,3Ru,1+, 1+), which equals (((U4 u3)'U2)'U1, 1) = ((U4U3) u2, 'U1)

= (U1, u2 ('U3u4)) =' (ui A U2 A u3 A U4)

Therefore, the degree 4 component of 16 1+ 6 1+ is C Similarly, one shows that the degree 8 component is

if u1i u2, U3, U4 are orthogonal. A.

See Bryant and Harvey [6] for other examples of pinor multiplication in dimension 8. In the next multiple of 8, dimension 16, all unit positive spinors are not the same under the action of Spin(16). The orbit structure of Spin(16) acting on S+ is very interesting. Each orbit can be used to construct calibrations by squaring a spinor in that orbit. See Dadok and Harvey [7] for the details. The case dimension n = 0 mod 8 can be used to understand the case dimension m = 7 mod 8 (see the next Theorem and Problem 9). Recall that if a unit vector eo E V(n) is chosen so that V(n - 1) = eo , then sending a vector u E V(n-1) to eo - u E Cl(n) induces an isomorphism Furthermore, if Cl(n) = EndR,(P) is a pinor of Cl(n - 1) 25 representation with P - S+®S_, then Cl(n)e°eII = Enda(S+)®EndR,(S_). Therefore, an identification S - S+ - S_ induces an isomorphism Cl(n)even.

Cl(n - 1) = EndR,(S) ® EndR(S),

which is a pinor representation of Cl(n - 1). Now, a unit positive spinor x E S+(n) can also be considered a unit

pinor y-(z,0)EP(n-1)-S®S.

311

Low Dimensions

Theorem 14.129. Let 16" x o x E CI(n),

(14.130)

n= 8p,

and

16" y 6 y E Cl(n - 1).

(14.131)

Let the orthogonal decomposition

0-eona+ define a, Q in AR(n - 1)*. Then +)3.

(14.132)

Proof: By Theorem 13.73 y) = E(ex, z)

(14.131)

If i; has degree 1 or 2 mod 4, then

for all

E Cl(n - 1).

_ - so that

0. We must show

that if degree l; = 0 mod 4,

(14.134)

and (14.135)

a(eo A t;)

if degree

= 3 mod 4.

Let u1, ...,

denote an orthonormal basis for V(n - 1) and eo, el an orthonormal basis for V (n). Then Cl(n-1) embeds in C1(n) by sending ui E Cl(n-1) to eo e3 E Cl(n). If E' _- ui, ui, ui, ui4 E Cl(n - 1) - AR(n - 1), then the corresponding element, also denoted e, in Cl(n) = AR(n) is equal to

ul,..., en- 1 -

eoui, eoui, eoui, eoui, = uil ui, ui, ui,.

Therefore, O(C) = E(cx, x) = /3(e). If C = uil ui, ui, E C1(n - 1) '= AR(n - 1), then the corresponding element of Cl(n) AR(n) is eoui, eoui, eoui, = eoui, ui, ui,. Therefore, b(£) = ¢(eo ) = a(e).

Problems

312

PROBLEMS 1. A complex volume element A on C4 determines a nondegenerate complex inner product (, ) on A2C4 by

Show that the induced action x of SL(4, C) on A2C4 provides a double cover

SL(4, C) x 0(6, C). 2. Verify the isomorphisms in Theorem 14.5. 3. (a) Show that the (triality automorphism) r defined by (14.27) is an automorphism of Spin(8) of order three that permutes the representa-

tions x, p+, and p-. (b) Show that (go, g+, g_) E Spin(8) if and only if one (all) of the following triples belongs to Spin(8): 9o,9+,9-

(9+,9-',9o) (9-190,9+)

,

,

9_,9+,90

,

(go, 9-, 9+)

(9+, 90, 9- )

4. (a) Show that Spin(3, 4) acts transitively on S7 = S+ U S? ,

St -{xE0:IIxII=dd}CO-S, with isotropy subgroup at 1 E O - S equal to split G2. (b) Show that Spin°(3, 4)/G2 25 S. 5. Show that, under the spinor representation p = p+ ® p_ of Spin(4, 4)

on S-S+®S_ 0®0,

{(x, y) E O x o : IIxII = ±1 and IIyII = ±1}. (b) Spin (4, 4) /Spin(3, 4) 95 {x E O : IIxII = ±1}. (a) Spin(4, 4)/G2

6. Show that Spin(5, 4)/Spin(3, 4)

{x E R(8,8) : IIxII = ±1}.

7. Use Corollary 10.50 to characterize Spin(9) in the model Cl(9)even EndR,(O ® 0) defined by (14.74).

313

Low Dimensions

8. (a) Show that, under the action of F4 on Herm(3, 0), each A E Herm(3, 0) may be put in the canonical form 11rl

A=

0 0

0

0

r2

0

0

r3

with rl, r2i r3 E R and rl < r2 < r3-

(b) If two (but not all three) of the eigenvalues rl, r2i r3 are equal, then the orbit through A is F4/Spin(9) = P2(O). (c) If all three of the eigenvalues ri, r2, r3 of A are distinct, then the orbit through A is

F4/Spin(8) - An Ss bundle over P2(0). 9. Prove that 16 1+ 6 1+ considered as an element of C1(7) = AR(7)* is equal to 1 + ¢ + V) + A, where 0 is the associative calibration, 0 is the coassociative calibration, and A is the unit volume form on Im O.

REFERENCES 1. Arnold, V. L, Mathematical Methods of Classical Mechanics, SpringerVerlag, New York, 1978. 2. Atiyah, M. F., R. Bott, and A. Shapiro, Clifford modules, Topology, 3 (1964), 3-38. 3. Besse, A. L., Einstein Manifolds, Springer-Verlag, Berlin, 1987.

4. Bryant, It. and It. Harvey, Submanifolds in hyperkahler geometry, Jour. Amer. Math. Soc., 2 (1989), 1-31. 5. Bryant, R. and It. Harvey, Stabilizers of calibrations, Rice University preprint, 1989.

6. Bryant, R. and R. Harvey, Geometry of G2 and Spin(7) structures, Rice University preprint, 1989.

7. Dadok, J. and It. Harvey, Calibrations and spinors, Rice University preprint, 1989.

8. Gureirch, G. B., Foundations of the Theory of Algebraic Invariants, P. Noordhoff LTD., Groningen, The Netherlands, 1964. 9. Harvey, R. and Lawson, H. B., Jr., Calibrated geometries, Acta Math. 148 (1982), 47-157. 10. Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. 11. Lawlor, G., The Angle Criterion, Invent. Math., 95 (1989), 437-446. 12. Lawson, H. B., Jr. and M. Michelsohn, Spin Geometry, Princeton University Press, Princeton, New Jersey, 1989.

13. Nance, D., Sufficient conditions for a pair of n-planes to be areaminimizing, Math. Ann. 279 (1987), 161-164. 14. O'Niell, B., Semi-Riemannian Geometry, Academic Press, New York, 1983.

15. Penrose, It. and W. Rindler, Spinors and Space-Time (2 vols.), Cambridge University Press, Cambridge, 1986. 16. Salamon, S., Riemannian Geometry and Holonomy Groups, Wiley, New York, 1989.

315

Subject Index

A Adjoint map, 47, 52, 170 Adjoint representation, 54 Algebra center of, 168 centralizer, 168 complex Clifford, 190, 225, 233

exterior, 178 Jordan, 291 Lie, 48, 53 representation of, 163 simple, 164, 235 tensor, 177 Almost-complex manifold, 84 "weak", 84 Almost-quaternionic manifold, 86 coefficient bundle, 86 Angle between vectors Lorentzian, 62 positive definite, 61 Angle theorem, 140, 149 Annihilator of a subspace, 27

Anti-automorphism, 47, 182 Anti-self dual, 97, 188, 209 Artin's theorem, 109 Associative 3-plane, 144 calibration, 113, 144 submanifolds, 144 Associator, 104 Atlas, 82 Automorphism group of complex numbers, 114 of Lorentz numbers, 114 of octonians, 116 of quaternions, 115 of split octonians, 116 B Backwards Cauchy-Schwarz inequality, 62 Backwards triangle inequality, 64 Basis theorem, 29, 40, 46 Bilinear form check, 231

317

Subject Index

318

hat, 231 hermitian, 19 hermitian skew, 20 hermitian symmetric, 20 nondegenerate, 20 pure H-symmetric, 17 pure skew, 20 pure symmetric, 20 Bryant's theorem, 146

center of, 187 centralizer of, 223, 237 check involution, 183, 252 classical companion group, 201 250

Clifford group, 201 conformal pin group, 201 degree of an element, 183 even dimensional, 235 even part, 183, 230 fundamental lemma, 179 C hat involution, 183, 252 Calibration, 126, 308 inner product, 184, 189 associative, 113, 144, 313 irreducible representations, 212 Cayley, 310 isomorphisms with coassociative, 115, 145, 313 contact set, 126 endomorphism algebras, 212 isomorphisms with matrix fundamental theorem, 127 generalized Nance, 156 algebras, 207 Nance, 151, 156 models in low dimensions, 275, special Lagrangian, 130 287, 298-307 Canonical automorphism, 181, 247 odd part, 183, 230, 232 split, 227 Cartan's isomorphisms, 14, 271, 274 twisted center of, 187 volume element, 187 Cartan-Dieudonne theorem, 67 Clifford automorphisms, 181 Cauchy-Schwarz inequality, 61 Lorentzian, 62 Clifford group, 201 positive definite, 61 Clifford multiplication, 180 Clifford norm, 199 Cauchy-Schwarz equality, 59 Cayley calibration, 310 Coassociative Cayley plane, 290, 292 4-plane, 145 Cayley-Dickson process, 104 calibration, 115, 145 Center Coefficient bundle, 86 of an algebra, 167 Commutator (octonian), 111 of Clifford algebra, 187 Complex Clifford algebra, 190, Centralizer of subalgebra, 170 225, 233 Clifford algebra, 223, 237 Complex manifold, 34, 83 with volume, 86 Check involution, 183, 231 Complex normed algebras, 120 Classical companion group, 201, Complex Riemannian manifold, 88 250 Clifford algebra Complex structure, 83 orthogonal, 156, 271 canonical automorphism, 183, Complex symplectic manifold, 88 247

Subject Index

Conformal manifold, 87 Conformal pin group, 201 Conjugate representation, 164 Constancy of light axiom, 64 Cross product, 110, 122, 124 triple cross product, 112 Curve length of, 61 proper time, 63 D Determinant, 9 for spin representation, 203, 205 Dirac operator, 306 Double, or Lorentz, numbers, 69, 107

Dual map, 9

319

Grassmannian, 75, 79, 125, 178, 197

Group associated geometry, 82 classical companion, 201, 250 Clifford, 201 complex general linear GL(n, C), 4 complex orthogonal O(n, C), 7 complex symplectic Sp(n, C), 7 complex unitary U(p, q), 7, 34 conformal, 13 conformal orthogonal CO(p, q), 13

conformal pin, 201 connected components, 16, 46 enhanced H-general linear

GL(n, H) H*, 5 E Electromagnetic field, 65 Equivalence of representations, 41, 164

Euclidean vector space with signature p, q, 57 Cauchy-Schwarz equality, 59 Euclidean positive definite, 57 Lorentzian, 57 Exceptional Lie group F4, 289 G2, 113

Exponential map, 50 Exterior algebra, 178 Exterior multiplication, 178

F Fibrations of spheres, 123 First cousin principle, 155, 161 Free fall, 63 Future timelike vector, 62 G y matrices, 180, 215

enhanced hyper-unitary (or quaternionic unitary) HU(p, q) HU(1), 11 exceptional, F4, 289 exceptional G2, 113 hyper-unitary HU(p,q), 7, 35 Lorentz SO1(3,1), 66 low-dimensional coincidences, 14, 271 orientation preserving general

linear GL+(n, R), 5 orthogonal O(p, q), 6 Pin, 195, 197

Pin, 199 Pin', 199 quaternionic general linear GL(n, H), 4 quaternionic unitary (or enhanced hyper-unitary) HU(p, q) HU(1), 11 real general linear GL(n, R), 4 real symplectic Sp(n, R), 6 reduced classical companion, 201, 250

Subject Index

320

reduced special orthogonal SOt(p, q), 13, 76 reduced spin, Spin°(p, q), 200 skew H-unitary SK(n, H), 8, 38 spacelike reduced orthogonal O+(V), 76 special C-linear SL(n, C), 5, 9 special complex orthogonal SO(n, C), 9 special conformal SCO(p, q), 13 special H-linear SL(n, H), 6 special orthogonal SO(p, q), 9, 78

special R-linear SL(n, R), 5, 9 special unitary SU(p, q), 9 Spin, 195, 198 symplectic Sp(p, q), 8 timelike reduced orthogonal

0-(V), 76 H Hadamard's inequality, 133, 185 Hat involution, 183, 231 Hodge star operator, 66, 96, 116,

exterior algebra, 58, 184 H-hermitian skew, 8, 21, 37 H-hermitian symmetric, 21, 34

on EndF(V), 171 pinor, 253-257 polarization of, 58 R-skew, 6, 21 R-symmetric, 6, 21 spinor, 247 tensor algebra, 58 Instantaneous observer, 65 Interior multiplication, 178 Intertwining operator, 41, 166, 169 Irreducible representation

algebra, 165

group, 56 Isometry, 23 anti-isometry, 38 Isotropy subgroup, 41

J Jordan algebra, 291

194

Hopf fibration, 123, 145 Hurwitz theorem, 107 HyperKahler manifold, 91

I Inner product space, 21 basis theorem, 29 canonical form, 29 standard models, 22 Inner product C-hermitian skew, 21 C-hermitian symmetric, 7, 21, 32

C-skew, 7, 21 C-symmetric, 7, 21 Clifford, 184 degenerate, 39

K Kahler form, 33, 89, 127 Kahler manifold, 34, 89 L

Lagrangian submanifold, 99, 131, 135 Lie algebra, 48, 55 Lie bracket, 48 Light cone, 43, 64 Light rays, 64 Lorentz, or double, numbers, 69, 109 Lorentzian manifold, 57, 87, 304 Lorentzian vector space, 23, 57 time orientation, 62 triangle inequality, 64 Low dimensional coincidences geometries, 96

Subject Inder

321

groups, 14, 272

M

real and imaginary parts, 103 split octonians O, 107 split quaternions M2(R), 107 Null basis, 73

Magnetic field, 65 Null cone, 64 Matrix algebras M,, (R), M, (C), Null subspace, 23 M,, (H), 3 Maximal negative subspace, 24, 28 Maximal positive subspace, 24, 28, 0 38 Observer, 65 Maxwell's equations, 66 Octonians, 107 Metric equivalence associative 3-plane, 144 flat isomorphism b, 27, 47 associative calibration, 113, 144 sharp isomorphism #, 28 associative submanifold, 144 Minkowski space, 23 automorphism group, 116 Monge-Ampere operator, 136 coassociative 4-plane, 145 Morgan's torus lemma, 152 coassociative calibration, 115 cross product, 111 Moser's theorem, 86 triality, 275 Orbit, 41 N Nance calibrations, 151 Orientation Negative subspace, 23 spacelike, 76 timelike, 76 Norm total, 76 Clifford, 199 of vector, 58 Oriented real manifold, 82 spinorial, 200 Orthogonal complex structures, twisted Clifford, 199 156, 281 Normed algebras, 101 Orthogonal projection, 28 alternative, 104 Orthogonal transformation, 6, 67 Artin's theorem, 109 Orthogonal vectors, 24 associator, 104 Orthonormal basis, 30 Cayley-Dickson process, 105 commutator, 111 P complex normed algebra, 120 Pauli spin matrices, 301 complex numbers C, 68, 107 Pin group, 195, 197, 203 Pin group, 199 conjugation, 103 cross products, 110, 122 Pin - group, 199 Pinor fibrations of spheres, 123 canonical involution, 230 Hurwitz theorem, 108 inner product, 254-257 Lorentz numbers L, 69, 107

octonians 0, 107 polarization, 101 quaternions H, 107

reality map, 239 representation, 210

Polarization

Subject Index

322

for normed algebras, 101 of inner product, 58 Positive subspace, 23 Projective space, 80, 123, 289 Cayley plane, 290, 292 quaternionic, 85 Proper time, 63 Q

Quaternionic Kahler form, 93 Quaternionic Kahler manifold, 92 Quaternionic manifold, 85 cofficient bundle, 86 Quaternionic n-space H", 3 complex structures on, 10 Quaternionic skew hermitian manifold, 93

R Real manifold, 82 with volume, 86 Real symplectic manifold, 34, 87 Reality operator, 239, 259 Reducible representation algebra, 165 group, 56 Reflection along a subspace, 67, 74, 197 Representation of a group, 41 Representation of an algebra, 163 conjugate standard, 164 equivalence, 164 irreducible, 165 reducible, 165 Rest space, 65 Riemannian manifold, 34, 87 S

Self-dual, 97, 188, 209 Sherk theorem, 67 Signature, 24, 28 of standard models, 29

positive definite, 23 split, 24 Simple algebra, 164, 235 Simple p-vector, 178 Simple product, 178, 184 Skew tensor, 178 Spacelike orientation, 76 Special Kiihler manifold, 90 Special Lagrangian calibration, 131 differential equation, 136 submanifold, 130, 136, 137, 138

subspaces, SLAG, 130, 134 Special relativity, 62 spinors, 304 twin paradox, 63 Sphere complex (or hyperquadric), 44 H-skew, 45 negative, 42 positive, 42 unit, 42 Spin group, 195 action on spheres, 283 classical companion, 201, 250 coincidences with classical groups, 272-275, 281 generating set, 198 reduced, 200, 203, 273 spin(7), 282 spin(8), 275-281 spin(9), 288 Spin representations, 251 Spinor inner product, 248 signature of, 249, 268 table of, 248 Spinor multiplication, 266 of pure spinors, 267 Spinor conjugate, 213 inner product, 247

Subject Index

323

negative, 214, 230 positive, 214, 229 products, 172, 266 pure, 241, 243 representation, 213, 217, 221 space, 213

structure map, 217 Spinorial norm, 77, 200 Split Clifford algebra, 227 Split signature, 24 Squares of spinors, 172, 266, 308 Standard representation, 164 Symplectic manifold complex, 88 real, 34, 87

Transitive action, 42, 283 Triality, 275, 278, 285 Twin paradox, 63 Twisted center 187 Twisted Clifford norm, 199 Twistor operator, 306 Twistor space, 244

V Vector

norm of, 58 Volume form, 8, 187 Volume minimizing submanifold, 126

W T Tangent cone, 146 Tensor algebra, 177 Timecone, 62 Timelike orientation, 76 Torus lemma, 152 Total oreintation, 76 Totally null subspace, 26, 29r Trace, 51

"Weak" almost complex manifold, 84

Wirtinger's inequality, 128 Witt's theorem, 246 Worldline of a particle, 62 Worldline of a photon, 64

Y Yau's theorem, 90