**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: $72.00

AMS Member Price: $57.60

MAA Member Price: $64.80

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

List Price: $67.00

AMS Member Price: $53.60

MAA Member Price: $60.30

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#### Supplemental Materials

# The Role of Nonassociative Algebra in Projective Geometry

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*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.

#### Readership

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

#### 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