Geometry Of The Octonions, The
Tevian Dray, Corinne A Manogue
 English
 ePUB (mobile friendly)
 Available on iOS & Android
Geometry Of The Octonions, The
Tevian Dray, Corinne A Manogue
About This Book
There are precisely two further generalizations of the real and complex numbers, namely, the quaternions and the octonions. The quaternions naturally describe rotations in three dimensions. In fact, all (continuous) symmetry groups are based on one of these four number systems. This book provides an elementary introduction to the properties of the octonions, with emphasis on their geometric structure. Elementary applications covered include the rotation groups and their spacetime generalization, the Lorentz group, as well as the eigenvalue problem for Hermitian matrices. In addition, more sophisticated applications include the exceptional Lie groups, octonionic projective spaces, and applications to particle physics including the remarkable fact that classical supersymmetry only exists in particular spacetime dimensions.
Contents:

 Introduction
 Number Systems:
 The Geometry of the Complex Numbers
 The Geometry of the Quaternions
 The Geometry of the Octonions
 Other Number Systems
 Symmetry Groups:
 Some Orthogonal Groups
 Some Unitary Groups
 Some Symplectic Groups
 Symmetry Groups over Other Division Algebras
 Lie Groups and Lie Algebras
 The Exceptional Groups
 Applications:
 Division Algebras in Mathematics
 Octonionic Eigenvalue Problems
 The Physics of the Octonions
 Magic Squares
Readership: Advanced undergraduate and graduate students and faculty in mathematics and physics; nonexperts with moderately sophisticated mathematics background.
Key Features:
 This book is easily digestible by a large audience wanting to know the elementary introduction to octanions
 Suitable for any reader with a grasp of the complex numbers, although familiarity with nonoctonionic versions of some of the other topics would be helpful
 Many open problems are very accessible
 Advanced topics covered are quite sophisticated, leading up to a clear discussion of (one representation of) the exceptional Lie algebras and their associated root diagrams, and of the octonionic projective spaces on which they act