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The Role of Nonassociative Algebra in Projective

John R. Faulkner

Graduate Studies in Volume 159

American Mathematical Society The Role of Nonassociative Algebra in

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. geometry 2 §1.3. Affine planes 3 §1.4. Projective planes 4 §1.5. 8 §1.6. Exercises 10 Chapter 2. Central 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 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. 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 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 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 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 , while the coordinates of a 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 , 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 . 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). 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. Configuration conditions ensure that two geometric constructions give the same point (or line). Thus, configuration 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 field 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 better”. If the “pro- jective dimension” 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 and the transcendency degree of a field extension. This development is based on having the proper collection of “subobjects”, e.g., the subspaces of a vector space or field extensions L/F in K/F with L algebraically closed in K. 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 matrices H(Cn) over a composition algebra C with diagonal entries from the field. There is a determinant-like norm function on H(Cn) if and only if n ≤ 3orC is associative (Theorem 11.7). We also study the group generated by elementary matrices acting on H(Cn) and the rank of an element of H(Cn).

The octonion plane can be constructed using rank 1 elements in H(C3) and the automorphism group can be described in terms of norm semisimi- larities. Moreover, this construction is valid even if the are not a division ring, although the will not be a projective plane. xiv Introduction

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. John R. Faulkner Charlottesville, May 2014 Bibliography

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[43] J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Mathe- matics, 386, Springer-Verlag, New York, Heidelberg, Berlin, 1974. [44] J. Tits, Non-existence de certains, polygones g´en´eralis´es, I, Inventiones Math. 36 (1976), 275–284; II, Inventiones Math. 51 (1979), 267–269. [45] J. Tits, Endliche Spiegelungsgruppen, die als Weylgruppen auftreten, Inventiones Math. 43 (1977), 283–295. [46] J. Tits and R. Weiss, Moufang Polygons, Springer-Verlag, Berlin, Heidelberg, New York, 2002. [47] O. Veblen and J. Young, Projective Geometry, Vol. I, Ginn, Boston, 1910. [48] F. Veldkamp, Projective planes over rings of stable rank 2, Geom. Dedicata 11 (1981), 285–308. [49] E. B. Vinberg, Construction of the exceptional simple Lie algebras, Trudy Sem. Vekt. Tenz. Anal. 13 (1966), 7–9; Amer. Math. Soc. Transl. Ser. 2, 213, Lie groups and invariant theory, 241–242, Amer. Math. Soc., Providence, RI, 2005. [50] R. Weiss, The nonexistence of certain Moufang polygons, Inventiones Math. 51 (1979), 261–266. [51] C. Weibel, Survey of non-Desarguesian planes, Notices Amer. Math. Soc. 54 (2007), 1294–1303. [52] K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov, and A. I. Shirshov, translated by H. F. Smith, Rings That Are Nearly Associative, Academic Press, New York, 1982.

Index

Δ(G), 207 G(C), 24 Δ(M), 208 G(V,f), 139 Γ(V,f), 140 Gh(O), 170 τ(m, x, b), 24 G(A3), 185 τm,x,24 GP , 200 A G (Δ), 10 (G; GX1 ,...,GXk ), 202 [a, b, c], 45 G[S], 204 [a, b], 45 G(G, H,S), 211 Af(R2), 201 H(A, B; C, D), 84 B(S), 208 Hλ, 118 Cent(C, a), 15 H(Cn), 144 Cent(x), 15 HEn(C), 145 op C ,31 Ix,2 Cin,32 l m,4 CD(R,ζ), 50 N (R), 47 C(X), 88 N(x), 147 codimX (Y ), 92 O(V,Q), 126 Cx(S), 94 P | l,2 CentG (C, A), 105 PQ,4 D(G,l), 5 prdim(S), 104 dim(X), 92 PH, 116

D[σ](x), 147 Pf(A), 165 Dv,λ(a), 118 P ∗ Q, 193 der(A), 215 PS , 203 E(A), 4 QS(A, B; C, D; E,F), 71 Eλ(v), 118 Rl, 192 E(V ), 119 sp(Y ), 88 Eu,v(y), 127 Sx(y), 126 fL, 134 S(V,f), 139 F(G), 206 stu3(A, −), 217 G(P, L,I), 2 Trans(a), 19 Gop,3 Tflag(G), 20 Gdual,8 Tgeom(G), 20

225 226 Index

Trans(H), 109 central involution, 50 T (x, y), 157 chamber, 209 Ug(x), 145 closed subset in a closure family, 88 U[i,j], 210 closed subset of a projective remoteness VL, 134 plane, 192 W ⊥, 125 closure family of subsets, 88 X0,88 codimension, 92 X/Y ,88 collinear points, 2 x#, 158 , 3 x × y, 160 commutator, 45 x ↔ y, 182 composition algebra, 54 Xy, 204 concurrent lines, 2 Z(R), 48 cone of x over S,94 connected, 79 abstract simplicial complex, 207 connected components, 184 additive identity, 29 coordinates, 24 adjacent chambers, 209 coordinatization, 24 adjoint, 158 Coxeter complex, 208 affine geometry, 112 Coxeter diagram, 207 affine plane, 3 Coxeter group, 208 affine transformation, 201 Coxeter matrix, 207 algebra, 54 cubic norm structure, 212 all relation, 120 alternating function, 37 deleted plane of a projective plane, 5 alternative ring, 37 dependence relation, 96 anisotropic cubic norm structure, 212 dependent mset, 90 apartments, 208 dependent set, 90 associated product of a ternary system, derived group, 121 29 Desargues condition, 66 associated sum of a ternary system, 29 diagonal lines of a quadrilateral, 9 associator, 38, 45 diagonal points of a quadrangle, 7 axial automorphism, 14 Dieudonn´e lemma, 121 axis, 175 dilatation, 15, 105 axis of an automorphism, 14, 105 dilatation plane, 17 dilation, 15 Barbilian plane, 195 dimension, 92 base, 90 dimensionable, 89 Bruck-Kleinfeld Theorem, 58 direct span, 89 discriminant, 55 (C, a)-Desargues condition, 67 division ring, 36 (C, a)-transitive, 17, 192 dual basis, 125, 185 C-transitive, 16 dual of a projective plane, 8 canonical remoteness geometry, 182 dual of a statement or definition, 9 Cartan-Dieudonn´e Theorem, 129 dual space, 118 Cartesian group, 31 duality map, 9 Cayley-Dickson process, 51 center, 175 Eichler transformation, 127 center of an automorphism, 14, 105 elation, 15 central automorphism, 14, 105 elementary group, 145 central automorphism geometry, 105 elementary matrix, 145 central duality, 40 elliptic polarity, 214 central duality plane, 40 equality relation, 120 Index 227

Erlangen program, 200 isotropic subspace, 124 extended affine geometry, 113 isotropic vector, 124 extended plane of an affine plane, 4 Jordan polynomial, 44 face of a simplex, 207 finitely closed, 89 K-involution, 54 flag, 2, 202 k-quadrangle section condition, 73 form, 139 k-space, 104 Fundamental Theorem of Projective Geometry, 116 lattice isomorphism, 93 left alternative condition, 37 G-invariant, 120 left cancellation property, 79 , 116 left distributive, 33 generalized n-gon, 209 left inverse, 32 generating subset of a projective left Moufang condition, 37 remoteness plane, 192 left Moufang ring, 44 left nucleus, 47 H-homotopy, 79 left solvable, 31 H-simply connected, 80 line, 2 half n-gon, 210 linear Jordan algebra, 213 harmonic condition, 85 linear subset, 102 harmonic position, 84 linearization, 45 hermitian elementary group, 145 lines meeting at a point, 2 hermitian matrix, 144 little Desargues condition, 67 hermitian matrix plane, 170 little k-quadrangle section condition, 73 homogeneous of degree (n1,...,nr), 135 little quadrangle section condition, 73 homogeneous of degree n, 132 loop, 79 homogeneous space, 200 , 15 matrix of a quadratic form, 54 homomorphism of incidence geometries, Micheev identity, 46 3 middle Moufang condition, 38 homotopy set, 79 middle nucleus, 47 hyperbolic pair, 124 monomial of degree d,48 hyperbolic space, 124 monomorphism of incidence geometries, hyperplane, 104 3 Moufang n-gon, 210 ideal point, 5 Moufang element, 190 idempotent, 57 Moufang ring, 37 incidence geometry, 2, 202 Moufang set, 213 incident, 2 mset, 90 independent mset, 90 multiplier, 139 independent set, 90 inner ideal, 205 n-basis, 184 inverse, 49 nearly projective geometry, 101 invertible, 49, 190 noncommutative determinant, 166 involution, 50 nondegenerate, 124 irreducible spherical , 208 nondegenerate quadratic extension, 55 irreducible subset, 89 nondegenerate quadratic form, 54 isometry, 139 norm of an involution, 50 isomorphism of homogeneous maps, 139 nucleus, 47 isomorphism of incidence geometries, 3 nullity, 29 isomorphism of remoteness geometries, 182 object, 2 228 Index

octonion algebra, 55 rank of a building, 209 opposite geometry, 3 rank of an element of H(Cn), 153 ordered partition, 132 reflection, 126 ordered quadrangle, 71 regular, 183 ordinary line, 5 regular triangle, 178 ordinary point, 5 remote, 182 orthogonal direct sum, 126 remoteness, 170 orthogonal group, 126 remoteness geometry, 182 remoteness subgeometry, 182 Pappus condition, 75 reversed quadrangle section condition, parabolic subgroup, 203 75 , 113 right alternative condition, 37 parallel lines, 4 right distributive, 33 parameterized group of Steinberg type, right inverse, 32 187 right Moufang condition, 37 path in a Dynkin diagram, 204 right nucleus, 47 path of length n,78 right operator invertible, 35 lines, 66 right solvable, 31 perspective from the axis a,66 ring, 36 perspective from the center C,66 root, 210 Pfaffian, 165 rotation property, 79 plane, 104, 112 rotational symmetric space, 218 point, 2 polarization, 133 σ-semilinear map, 140 power associative, 45 σ-semisimilarity, 140 pre-Desargues configuration, 70 section points, 66 primitive group, 120 semilinear group, 116 principle of duality, 9 semilinear map, 116 projection of a point to a line, 100 separate in a Dynkin diagram, 204 projection of one line to another, 7 set with multiplicity, 90 projective dimension, 104 sides of a quadrangle, 7 projective general linear group, 116 sides of a triangle, 6 projective geometry, 100 similarity, 139 projective hermitian elementary group, simplex, 207 172 singular radical, 197 projective pair, 196 Skornyakov Theorem, 58 projective plane, 4 slope, 25 projective remoteness plane, 183 span, 88 projective semilinear group, 116 special Jordan ring, 36, 44 proof by duality, 9 special orthogonal group, 130 proper k-loop, 79 spherical building, 208 split composition algebra, 57 quadrangle, 7, 183 stabilizer subgroup, 121 quadrangle section, 71 standard coordinates, 26 quadrangle section condition, 73 standard coordinatization, 26 quadratic extension, 53 standard extension of the union of quadratic form, 54, 124 nearly projective geometries, 101 quadratic Jordan algebra, 144 Steinberg type, 186 quadrilateral, 9 Steinitz exchange axiom, 96 quaternion algebra, 55 strictly alternative ring, 53 strongly dimensionable set, 95 R-space geometry, 204 structurable algebra, 213 Index 229

sub-mset, 90 subgeometry, 3 symmetric space, 217

1 1 2 -transitive, 121 Teichm¨uller identity, 46 ternary ring, 26 ternary system, 26 thick building, 209 thick generalized n-gon, 209 trace bilinear form T (x, y), 157 trace of an involution, 50 translation, 15 transvection, 15, 105, 175, 192, 210 transvection flag, 20 transvection line, 20 transvection plane, 17, 175 transvection point, 20 triangle, 183 triangle in a projective plane, 6 two-sided inverse ring, 185 unit, 29 ring, 36

Veblen-Wedderburn system, 36 vertex of a simplex, 207 vertices of a quadrangle, 7 vertices of a triangle, 6

Witt index, 126 y-intercept, 25

Selected Published Titles in This Series

159 John R. Faulkner, The Role of Nonassociative Algebra in Projective Geometry, 2014 152 G´abor Sz´ekelyhidi, An Introduction to Extremal K¨ahler Metrics, 2014 151 Jennifer Schultens, Introduction to 3-, 2014 150 Joe Diestel and Angela Spalsbury, The Joys of Haar Measure, 2013 149 Daniel W. Stroock, Mathematics of Probability, 2013 148 Luis Barreira and Yakov Pesin, Introduction to Smooth Ergodic Theory, 2013 147 Xingzhi Zhan, Matrix Theory, 2013 146 Aaron N. Siegel, Combinatorial Game Theory, 2013 145 Charles A. Weibel, The K-book, 2013 144 Shun-Jen Cheng and Weiqiang Wang, Dualities and Representations of Lie Superalgebras, 2012 143 Alberto Bressan, Lecture Notes on Functional Analysis, 2013 142 Terence Tao, Higher Order Fourier Analysis, 2012 141 John B. Conway, A Course in Abstract Analysis, 2012 140 Gerald Teschl, Ordinary Differential Equations and Dynamical Systems, 2012 139 John B. Walsh, Knowing the Odds, 2012 138 Maciej Zworski, Semiclassical Analysis, 2012 137 Luis Barreira and Claudia Valls, Ordinary Differential Equations, 2012 136 Arshak Petrosyan, Henrik Shahgholian, and Nina Uraltseva, Regularity of Free Boundaries in Obstacle-Type Problems, 2012 135 Pascal Cherrier and Albert Milani, Linear and Quasi-linear Evolution Equations in Hilbert Spaces, 2012 134 Jean-Marie De Koninck and Florian Luca, Analytic Number Theory, 2012 133 Jeffrey Rauch, Hyperbolic Partial Differential Equations and Geometric Optics, 2012 132 Terence Tao, Topics in Random Matrix Theory, 2012 131 Ian M. Musson, Lie Superalgebras and Enveloping Algebras, 2012 130 Viviana Ene and J¨urgen Herzog, Gr¨obner Bases in Commutative Algebra, 2011 129 Stuart P. Hastings and J. Bryce McLeod, Classical Methods in Ordinary Differential Equations, 2012 128 J. M. Landsberg, Tensors: Geometry and Applications, 2012 127 Jeffrey Strom, Modern Classical Homotopy Theory, 2011 126 Terence Tao, An Introduction to Measure Theory, 2011 125 Dror Varolin, Riemann Surfaces by Way of Complex Analytic Geometry, 2011 124 David A. Cox, John B. Little, and Henry K. Schenck, Toric Varieties, 2011 123 Gregory Eskin, Lectures on Linear Partial Differential Equations, 2011 122 Teresa Crespo and Zbigniew Hajto, Algebraic Groups and Differential Galois Theory, 2011 121 Tobias Holck Colding and William P. Minicozzi II, A Course in Minimal Surfaces, 2011 120 Qing Han, A Basic Course in Partial Differential Equations, 2011 119 Alexander Korostelev and Olga Korosteleva, Mathematical Statistics, 2011 118 Hal L. Smith and Horst R. Thieme, Dynamical Systems and Population Persistence, 2011 117 Terence Tao, An Epsilon of Room, I: Real Analysis, 2010 116 Joan Cerd`a, Linear Functional Analysis, 2010

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/gsmseries/. There is a particular fascination when two apparently disjoint areas of mathematics turn out to have a meaningful connection to each other. 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. The development should be accessible to most graduate students and should give them introductions to two areas which are often referenced but not often taught. On the geometric side, the book introduces coordinates in projective planes and relates coordinate properties to transitivity properties of certain automorphisms and XSGSR½KYVEXMSRGSRHMXMSRW-XEPWSGPEWWM½IWLMKLIVHMQIRWMSREPKISQIXVMIWERHHIXIV- mines their automorphisms. The exceptional octonion plane is studied in detail in a geometric context that allows nondivision coordinates. An axiomatic version of that context is also provided. Finally, some connections of nonassociative algebra to other geometries, including buildings, are outlined. On the algebraic side, basic properties of alternative algebras are derived, including XLIGPEWWM½GEXMSRSJEPXIVREXMZIHMZMWMSRVMRKW%WXSSPWJSVXLIWXYH]SJXLIKISQIXVMIW an axiomatic development of dimension, the basics of quadratic forms, a treatment of homogeneous maps and their polarizations, and a study of norm forms on hermitian matrices over composition algebras are included.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-159

GSM/159 AMS on the Web www.ams.orgwww.ams.org