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American Mathematical SocietyJohn R. Faulkner

Graduate Studies

in Mathematics

Volume 159

The Role of

Nonassociative Algebra

in Projective Geometry The Role of Nonassociative Algebra in Projective Geometry The Role of Nonassociative Algebra in Projective Geometry

John R. Faulkner

American Mathematical Society

Providence, Rhode Island

Graduate Studies

in Mathematics

Volume 159https://doi.org/10.1090//gsm/159

EDITORIAL COMMITTEE

Dan Abramovich

Daniel S. Freed

Rafe Mazzeo (Chair)

Gigliola Staffilani

2010Mathematics Subject Classification. Primary 51A05, 51A20, 51A25, 51A35, 51C05,

17D05, 17C50.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-159 Library of Congress Cataloging-in-Publication Data

Faulkner, John R., 1943- author.

The role of nonassociative algebra in projective geometry / John R. Faulkner. pages cm. - (Graduate studies in mathematics ; volume 159)

Includes bibliographical references and index.

ISBN 978-1-4704-1849-6 (alk. paper)

1. Geometry, Projective. 2. Nonassociative algebras. I. Title.

QA471.F297 2014

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2014021979

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Contents

Introduction xi

Chapter 1. Affine and Projective Planes 1

§1.1. Preview 1

§1.2. Incidence geometry 2

§1.3. Affine planes 3

§1.4. Projective planes 4

§1.5. Duality 8

§1.6. Exercises 10

Chapter 2. Central Automorphisms of Projective Planes 13

§2.1. Preview 13

§2.2. Projections and automorphisms 14

§2.3. Transvections and dilatations 14

§2.4. Transitivity properties 16

§2.5. Exercises 22

Chapter 3. Coordinates for Projective Planes 23

§3.1. Preview 23

§3.2. Ternary systems 24

§3.3. Two coordinatizations related toG(C)31

§3.4. Transvections and algebraic properties 33

§3.5. Exercises 40

vii viiiContents

Chapter 4. Alternative Rings 43

§4.1. Preview 43

§4.2. Left Moufang rings 44

§4.3. Artin"s Theorem 47

§4.4. Inverses in alternative rings 49

§4.5. The Cayley-Dickson process 50

§4.6. Composition algebras 54

§4.7. Split and division composition algebras 57

§4.8. Exercises 62

Chapter 5. Configuration Conditions 65

§5.1. Preview 65

§5.2. Desargues condition 66

§5.3. Quadrangle sections 71

§5.4. Pappus condition 75

§5.5. Configurations and central automorphisms 78

§5.6. Exercises 84

Chapter 6. Dimension Theory 87

§6.1. Preview 87

§6.2. Dimensionable sets 88

§6.3. Independence and bases 90

§6.4. Strongly dimensionable sets 94

§6.5. Exercises 96

Chapter 7. Projective Geometries 99

§7.1. Preview 99

§7.2. Projective and nearly projective geometries 100 §7.3. Relation to strongly dimensionable sets 102 §7.4. Classification of projective geometries 104

§7.5. Exercises 112

Chapter 8. Automorphisms ofG(V) 115

§8.1. Preview 115

§8.2. The Fundamental Theorem 116

§8.3. Subgroups of Aut(G(V)) 118

§8.4. Simple groups 120

§8.5. Exercises 121

Contentsix

Chapter 9. Quadratic Forms and Orthogonal Groups 123

§9.1. Preview 123

§9.2. Quadratic forms 124

§9.3. Orthogonal groups 126

§9.4. Exercises 129

Chapter 10. Homogeneous Maps 131

§10.1. Preview 131

§10.2. Polarization of homogeneous maps 132

§10.3. Exercises 140

Chapter 11. Norms and Hermitian Matrices 143

§11.1. Preview 143

§11.2. Hermitian matrices and HE

n (C) 144

§11.3. Norms onH(C

n ) 146

§11.4. Transitivity of HE

n (C) 153

§11.5. Trace and adjoint 157

§11.6.H(C

3 ) 160

§11.7. Exercises 165

Chapter 12. Octonion Planes 169

§12.1. Preview 169

§12.2. The construction of octonion planes 170

§12.3. Simplicity of PHE

3 (O) 174

§12.4. Automorphisms of octonion planes 177

§12.5. Exercises 178

Chapter 13. Projective Remoteness Planes 181

§13.1. Preview 181

§13.2. Definition and examples 182

§13.3. Groups of Steinberg type 186

§13.4. Transvections 192

§13.5. Exercises 195

Chapter 14. Other Geometries 199

§14.1. Preview 199

§14.2. Erlangen program 200

§14.3. The geometry ofR-spaces 201

xContents

§14.4. Buildings 206

§14.5. Generalizedn-gons 209

§14.6. Moufang sets and structurable algebras 213

§14.7. Freudenthal-Tits magic square 214

§14.8. Exercises 218

Bibliography 221

Index 225

Introduction

Discovering a connection between two apparently disjoint areas of mathe- matics has always held a fascination for me. Just as a mental twist provides the punch line of a joke, a theorem giving an unsuspected link between two areas of mathematics is both enlightening and satisfying. The main goal of this book is to provide a largely self-contained, in-depth account of the linkage between nonassociative algebra and projective planes with particular emphasis on octonion planes. There are several new results and many, if not most, of the proofs are new. A knowledge of linear algebra, basic ring theory, and basic group theory is required, as well as the ability to follow a detailed proof, but otherwise, except in Chapter 14, the development will be from first principles. Thus, a course based on this book would be accessible to most graduate students and would give them introductions to two areas which are often referenced but not often taught. Some of these students might continue in nonassociative algebra or use the geometry as a step towards research areas such as buildings or algebraic groups as indicated in Chapter 14. The link between algebra and geometry goes back to the introduction of real coordinates in the Euclidean plane by Descartes. We also will introduce coordinates in a class of axiomatically defined geometries. The axiomatic approach to the Euclidean plane is seldom used after a high school course because a truly rigorous development is very demanding while the Cartesian product of the reals provides an easy-to-use model. However, we shall find it advantageous to start with a simple axiomatization of our geometries to set the scope of our investigation and then determine which algebraic structures can serve as coordinates. These coordinates are not limited to xi xiiIntroduction algebras over the reals or fields of characteristic 0, or even to nonassociative rings. Our original axiomatization will be restricted to planar geometries for the same reason that high school students study the Euclidean plane. It is easier. However, we will find later that although the classification of higher- dimensional geometries requires more machinery, their structure is actually simpler. Indeed, the coordinates of a higher-dimensional projective geom- etry form an associative division ring, while the coordinates of a projective plane can be more exotic. Unlike the projective case, strictly nonassociative coordinates occur in some of the nonplanar geometries in Chapter 14. (Note that by convention "nonassociative" means "not necessarily associative", so "strictly nonassociative" is used to rule out associative rings.) Although exceptional Lie (or algebraic) groups or Lie algebras are not mentioned explicitly except in Chapter 14, the simple group associated with the octonion plane in Chapter 12 is, in fact, of typeE 6 (see [37,Proposition

11.20 with Proposition 12.3, Corollary 12.4, and Theorem 12.7]). Also,

there are connections to physics through Lie groups and the use of projective geometries as quantum logic (see [5, p. 833]), although these topics will not be discussed here. I strongly recommend that the reader have a scratch pad handy to sketch parts of the geometric proofs and to keep track of some of the nonassociative identities. The exercises present a lot of additional material not found in the main development, often with directions to the reader for supplying the proof. Each chapter has an informal preview section that introduces the reader to the coming material. We give below an overview of the contents. We begin with affine planes which have the incidence properties of Eu- clidean planes, but we quickly pass to the equivalent notion of projective planes. Projective planes have the advantage that the projection of one line to another from a point is a bijection, which is not true, in general, in affine planes. Looking at automorphisms of projective planes which extend projections leads to the notion of a central automorphism. Coordinates can be introduced into any projective plane, but, in general, the algebraic structure of the coordinates is rather weak. However, the existence of increasing sets of central automorphisms results in an increasing structure on the coordinates, ranging through Cartesian groups, Veblen- Wedderburn systems, nonassociative division rings, left Moufang division rings, alternative division rings, and associative division rings. In particular, a projective plane in which every projection extends to an automorphism has an alternative division ring as coordinates (Theorems 2.8 and 3.17). We employ a trick using special Jordan rings to get identities in left Mo- ufang (and hence alternative) rings. In particular, Micheev"s identity shows

Introductionxiii

that a left Moufang division ring is alternative (Theorem 4.2 and Corollary

4.3). This algebraic result eliminates a potential class of projective planes.

The Cayley-Dickson process is a doubling construction which after several iterations results in an octonion ring,an 8-dimensional alternative algebra over its center. A major result, due independently to Skornyakov and to Bruck and Kleinfeld, is that an alternative division ring is either associative or an octonion ring. Con"guration conditions ensure that two geometric constructions give the same point (or line). Thus, con"guration conditions play the same role for projective planes that identities do for nonassociative algebras. In fact, the Pappus condition is equivalent to the plane having a "eld for coordinates, the Desargues condition (or the quadrangle section condition) is equivalent to the plane having an associative division ring for coordinates, and the little Desargues condition (or the little quadrangle section condition) is equivalent to having an alternative division ring for coordinates. Projective geometry is an example of bigger is betterŽ. If the pro- jective dimensionŽ is 3 or more, the coordinates are associative and the automorphism group is easily described. In order to even talk about di- mension, we present an axiomatic development of dimension modeled on the dimension of a vector space and the transcendency degree of a "eld extension. This development is based on having the proper collection of subobjectsŽ, e.g., the subspaces of a vector space or "eld extensionsL/F inK/FwithLalgebraically closed inK. We shall see that the existence of a strong version of dimension is essentially equivalent to being a union of projective geometries (Theorems 7.2 and 7.3). Certain algebraic machinery is needed to study octonion planes. We develop the basic properties of quadratic forms and orthogonal groups, in- cluding the Cartan-Dieudonn´e Theorem (Corollary 9.10). We also present an approach to homogeneous maps and their polarizations based on mul- tilinear maps, rather than the standard use of polynomials. This allows a basis-free development which works equally well in infinite dimensions.

Finally, we look at hermitian matricesH(C

n ) over a composition algebra Cwith diagonal entries from the field. There is a determinant-like norm function onH(C n We also study the group generated by elementary matrices acting onH(C n and the rank of an element ofH(C n The octonion plane can be constructed using rank 1 elements inH(C 3 and the automorphism group can be described in terms of norm semisimi- larities. Moreover, this construction is valid even if the octonions are not a division ring, although the incidence geometry will not be a projective plane. xivIntroduction In this more general setting, the subgroup generated by transvections is sim- ple (Lemma 12.8 and Theorem 12.10). Octonion planes and similar planes constructed from associative two-sided inverse rings are examples of projec- tive remoteness planes, extending the notion of projective planes. Some of the results about transvections and the group they generate can be obtained in the setting of projective remoteness planes. Finally, in Chapter 14, we assume a more extensive background and give a sketchy introduction to other geometries involving nonassociative algebras, since the complete treatment would require at least another book.

JohnR. Faulkner

Charlottesville, May 2014

Bibliography

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Δ(G), 207

Δ(M), 208

Γ(V,f), 140

τ(m,x,b), 24

m,x ,24

A(Δ), 10

[a,b,c], 45 [a,b], 45 Af(R 2 ), 201

B(S), 208

Cent(C,a), 15

Cent(x), 15

C op ,31 C in ,32

C(X), 88

codim X (Y), 92 C x (S), 94 Cent G (C,A), 105

D(G,l), 5

dim(X), 92 D (x), 147 D v,λ (a), 118 der(A), 215

E(A), 4

E (v), 118

E(V), 119

E u,v (y), 127 f L , 134

F(G), 206

G(P,L,I), 2

G op ,3 G dual ,8G(C), 24

G(V,f), 139

G h (O), 170 G(A 3 ), 185 Gquotesdbs_dbs33.pdfusesText_39

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