Synopses & Reviews
This book introduces a new class of non-associative algebras related to certain exceptional algebraic groups and their associated buildings. Richard Weiss develops a theory of these "quadrangular algebras" that opens the first purely algebraic approach to the exceptional Moufang quadrangles. These quadrangles include both those that arise as the spherical buildings associated to groups of type E_{6}, E_{7}, and E_{8} as well as the exotic quadrangles "of type F4" discovered earlier by Weiss. Based on their relationship to exceptional algebraic groups, quadrangular algebras belong in a series together with alternative and Jordan division algebras. Formally, the notion of a quadrangular algebra is derived from the notion of a pseudo-quadratic space (introduced by Jacques Tits in the study of classical groups) over a quaternion division ring. This book contains the complete classification of quadrangular algebras starting from first principles. It also shows how this classification can be made to yield the classification of exceptional Moufang quadrangles as a consequence. The book closes with a chapter on isotopes and the structure group of a quadrangular algebra.
Quadrangular Algebras is intended for graduate students of mathematics as well as specialists in buildings, exceptional algebraic groups, and related algebraic structures including Jordan algebras and the algebraic theory of quadratic forms.
Review
"[T]here have been many results on finite generalized quadrangles that fill a well-needed gap in the mathematical literature, but this monograph is much deeper and will enable progress to be made in a difficult technical area where some exotic forms of algebraic groups have hitherto been little understood."--Robert L. Devaney, Bulletin of the AMS
Review
[T]here have been many results on finite generalized quadrangles that fill a well-needed gap in the mathematical literature, but this monograph is much deeper and will enable progress to be made in a difficult technical area where some exotic forms of algebraic groups have hitherto been little understood. Robert L. Devaney
Synopsis
This book introduces a new class of non-associative algebras related to certain exceptional algebraic groups and their associated buildings. Richard Weiss develops a theory of these "quadrangular algebras" that opens the first purely algebraic approach to the exceptional Moufang quadrangles. These quadrangles include both those that arise as the spherical buildings associated to groups of type E
_{6}, E
_{7}, and E
_{8} as well as the exotic quadrangles "of type F4" discovered earlier by Weiss. Based on their relationship to exceptional algebraic groups, quadrangular algebras belong in a series together with alternative and Jordan division algebras. Formally, the notion of a quadrangular algebra is derived from the notion of a pseudo-quadratic space (introduced by Jacques Tits in the study of classical groups) over a quaternion division ring. This book contains the complete classification of quadrangular algebras starting from first principles. It also shows how this classification can be made to yield the classification of exceptional Moufang quadrangles as a consequence. The book closes with a chapter on isotopes and the structure group of a quadrangular algebra.
Quadrangular Algebras is intended for graduate students of mathematics as well as specialists in buildings, exceptional algebraic groups, and related algebraic structures including Jordan algebras and the algebraic theory of quadratic forms.
Synopsis
This book introduces a new class of non-associative algebras related to certain exceptional algebraic groups and their associated buildings. Richard Weiss develops a theory of these "quadrangular algebras" that opens the first purely algebraic approach to the exceptional Moufang quadrangles. These quadrangles include both those that arise as the spherical buildings associated to groups of type E
_{6}, E
_{7}, and E
_{8} as well as the exotic quadrangles "of type F4" discovered earlier by Weiss. Based on their relationship to exceptional algebraic groups, quadrangular algebras belong in a series together with alternative and Jordan division algebras. Formally, the notion of a quadrangular algebra is derived from the notion of a pseudo-quadratic space (introduced by Jacques Tits in the study of classical groups) over a quaternion division ring. This book contains the complete classification of quadrangular algebras starting from first principles. It also shows how this classification can be made to yield the classification of exceptional Moufang quadrangles as a consequence. The book closes with a chapter on isotopes and the structure group of a quadrangular algebra.
Quadrangular Algebras is intended for graduate students of mathematics as well as specialists in buildings, exceptional algebraic groups, and related algebraic structures including Jordan algebras and the algebraic theory of quadratic forms.
Table of Contents
Preface vii
Chapter 1. Basic Definitions 1
Chapter 2. Quadratic Forms 11
Chapter 3. Quadrangular Algebras 21
Chapter 4. Proper Quadrangular Algebras 29
Chapter 5. Special Quadrangular Algebras 37
Chapter 6. Regular Quadrangular Algebras 45
Chapter 7. Defective Quadrangular Algebras 59
Chapter 8. Isotopes 77
Ch apter 9. Improp er Quadrangu lar Algebras 83
Chapter 10. Existence 95
Chapter 11. Moufang Quadrangles 109
Chapter 12. The Structure Group 125
Bibliography 133
Index 134