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The Role of Nonassociative Algebra in Projective Geometry

John R. Faulkner University of Virginia, Charlottesville, VA
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Hardcover ISBN: 978-1-4704-1849-6
Product Code: GSM/159
List Price: $72.00 MAA Member Price:$64.80
AMS Member Price: $57.60 Electronic ISBN: 978-1-4704-1933-2 Product Code: GSM/159.E List Price:$67.00
MAA Member Price: $60.30 AMS Member Price:$53.60
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AMS Member Price: $86.40 Click above image for expanded view The Role of Nonassociative Algebra in Projective Geometry John R. Faulkner University of Virginia, Charlottesville, VA Available Formats:  Hardcover ISBN: 978-1-4704-1849-6 Product Code: GSM/159  List Price:$72.00 MAA Member Price: $64.80 AMS Member Price:$57.60
 Electronic ISBN: 978-1-4704-1933-2 Product Code: GSM/159.E
 List Price: $67.00 MAA Member Price:$60.30 AMS Member Price: $53.60 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$108.00 MAA Member Price: $97.20 AMS Member Price:$86.40
• Book Details

Volume: 1592014; 229 pp
MSC: Primary 51; 17;

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.

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

• Chapters
• Chapter 1. Affine and projective planes
• Chapter 2. Central automorphisms of projective planes
• Chapter 3. Coordinates for projective planes
• Chapter 4. Alternative rings
• Chapter 5. Configuration conditions
• Chapter 6. Dimension theory
• Chapter 7. Projective geometries
• Chapter 8. Automorphisms of $\mathcal {G}(V)$
• Chapter 9. Quadratic forms and orthogonal groups
• Chapter 10. Homogeneous maps
• Chapter 11. Norms and hermitian matrices
• Chapter 12. Octonion planes
• Chapter 13. Projective remoteness planes
• Chapter 14. Other geometries

• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1592014; 229 pp
MSC: Primary 51; 17;

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.

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

• Chapters
• Chapter 1. Affine and projective planes
• Chapter 2. Central automorphisms of projective planes
• Chapter 3. Coordinates for projective planes
• Chapter 4. Alternative rings
• Chapter 5. Configuration conditions
• Chapter 6. Dimension theory
• Chapter 7. Projective geometries
• Chapter 8. Automorphisms of $\mathcal {G}(V)$
• Chapter 9. Quadratic forms and orthogonal groups
• Chapter 10. Homogeneous maps
• Chapter 11. Norms and hermitian matrices
• Chapter 12. Octonion planes
• Chapter 13. Projective remoteness planes
• Chapter 14. Other geometries
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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