Hardcover ISBN:  9781470418496 
Product Code:  GSM/159 
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AMS Member Price:  $57.60 
Electronic ISBN:  9781470419332 
Product Code:  GSM/159.E 
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AMS Member Price:  $53.60 

Book DetailsGraduate Studies in MathematicsVolume: 159; 2014; 229 ppMSC: 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 selfcontained, indepth 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 higherdimensional 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.ReadershipGraduate students and research mathematicians interested in nonassociative algebra and projective geometry; physicists interested in division algebras and string theory.

Table of Contents

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


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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 selfcontained, indepth 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 higherdimensional 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