**Graduate Studies in Mathematics**

Volume: 159;
2014;
229 pp;
Hardcover

MSC: Primary 51; 17;

Print ISBN: 978-1-4704-1849-6

Product Code: GSM/159

List Price: $67.00

Individual Member Price: $53.60

**Electronic ISBN: 978-1-4704-1933-2
Product Code: GSM/159.E**

List Price: $67.00

Individual Member Price: $53.60

#### You may also like

#### Supplemental Materials

# The Role of Nonassociative Algebra in Projective Geometry

Share this page
*John R. Faulkner*

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 to configuration conditions. It also classifies higher-dimensional geometries and determines 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 the classification of alternative division rings. As tools for the study of the geometries, 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.

#### Table of Contents

# Table of Contents

## The Role of Nonassociative Algebra in Projective Geometry

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Introduction xi12 free
- Chapter 1. Affine and projective planes 116 free
- Chapter 2. Central automorphisms of projective planes 1328
- Chapter 3. Coordinates for projective planes 2338
- Chapter 4. Alternative rings 4358
- Chapter 5. Configuration conditions 6580
- Chapter 6. Dimension theory 87102
- Chapter 7. Projective geometries 99114
- Chapter 8. Automorphisms of 𝒢(𝒱) 115130
- Chapter 9. Quadratic forms and orthogonal groups 123138
- Chapter 10. Homogeneous maps 131146
- Chapter 11. Norms and hermitian matrices 143158
- Chapter 12. Octonion planes 169184
- Chapter 13. Projective remoteness planes 181196
- Chapter 14. Other geometries 199214
- Bibliography 221236
- Index 225240 free
- Back Cover Back Cover1247

#### Readership

Graduate students and research mathematicians interested in nonassociative algebra and projective geometry; physicists interested in division algebras and string theory.