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The Role of Nonassociative Algebra in Projective Geometry The Role of Nonassociative Algebra in Projective Geometry John R. Faulkner Graduate Studies in Mathematics Volume 159 American Mathematical Society The Role of Nonassociative Algebra in Projective Geometry https://doi.org/10.1090//gsm/159 The Role of Nonassociative Algebra in Projective Geometry John R. Faulkner Graduate Studies in Mathematics Volume 159 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani 2010 Mathematics 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 516.5—dc23 2014021979 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 191817161514 To Nancy 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 to G(C)31 §3.4. Transvections and algebraic properties 33 §3.5. Exercises 40 vii viii Contents 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 of G(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 Contents ix 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 HEn(C) 144 §11.3. Norms on H(Cn) 146 §11.4. Transitivity of HEn(C) 153 §11.5. Trace and adjoint 157 §11.6. H(C3) 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 PHE3(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 of R-spaces 201 x Contents §14.4. Buildings 206 §14.5. Generalized n-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 xii Introduction 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 type E6 (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 Introduction xiii that a left Moufang division ring is alternative (Theorem 4.2 and Corollary 4.3).
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