The geometry of the octonions / Tevian Dray, Department of Mathematics, Oregon State University, USA, Corinne A. Manogue, Department of Physics, Oregon State University, USA.

Format:

Book

Author/Creator:

Dray, Tevian, author.

Manogue, Corinne A., author.

Language:

English

Subjects (All):

Cayley numbers (Algebra).

Cayley algebras.

Nonassociative algebras.

Geometry, Algebraic.

Physical Description:

1 online resource (229 p.)

Place of Publication:

[Hackensack,] New Jersey : World Scientific, [2015]

Language Note:

English

Summary:

"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, and in fact, all 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."-- Provided by publisher.

Contents:

""Contents""; ""Preface""; ""Acknowledgments""; ""List of Figures""; ""List of Tables""; ""1. Introduction""; ""I. Number Systems""; ""2. The Geometry of the Complex Numbers""; ""2.1 Complex Numbers""; ""2.2 History""; ""2.3 Algebra""; ""2.4 Geometry""; ""3. The Geometry of the Quaternions""; ""3.1 Quaternions""; ""3.2 History""; ""3.3 Algebra""; ""3.4 Geometry""; ""4. The Geometry of the Octonions""; ""4.1 Octonions""; ""4.2 History""; ""4.3 Algebra""; ""4.4 Geometry""; ""5. Other Number Systems""; ""5.1 The Cayley-Dickson Process""; ""5.2 Sedenions""; ""5.3 The Hurwitz Theorem""

""5.4 Split Complex Numbers""""5.5 Split Quaternions""; ""5.6 Split Octonions""; ""5.7 Subalgebras of the Split Octonions""; ""II. Symmetry Groups""; ""6. Some Orthogonal Groups""; ""6.1 Rotations""; ""6.2 The Geometry of SO(2)""; ""6.3 The Geometry of SO(3)""; ""6.4 The Geometry of SO(4)""; ""6.5 Lorentz Transformations""; ""6.6 The Geometry of SO(3,1)""; ""6.7 The Geometry of SO(4,2)""; ""7. Some Unitary Groups""; ""7.1 Unitary Transformations""; ""7.2 The Geometry of U(1)""; ""7.3 The Geometry of SU(2)""; ""7.4 The Geometry of SU(3)""; ""7.5 The Geometry of SU(2,2)""

""8. Some Symplectic Groups""""8.1 Symplectic Transformations""; ""8.2 The Geometry of Sp(4,R)""; ""8.3 The Geometry of Sp(6,R)""; ""9. Symmetry Groups over Other Division Algebras""; ""9.1 Some Orthogonal Groups over Other Division Algebras""; ""9.1.1 A Quaternionic Description of SO(3)""; ""9.1.2 A Quaternionic Description of SO(4)""; ""9.1.3 An Octonionic Description of SO(7)""; ""9.1.4 An Octonionic Description of SO(8)""; ""9.2 Some Unitary Groups over Other Division Algebras""; ""9.3 Some Lorentz Groups over Other Division Algebras""

""9.4 Some Symplectic Groups over Other Division Algebras""""10. Lie Groups and Lie Algebras""; ""10.1 Lie Groups""; ""10.2 Lie Algebras""; ""10.3 The Classification of Lie Groups""; ""10.4 Real Forms""; ""11. The Exceptional Groups""; ""11.1 The Geometry of G2""; ""11.2 The Albert Algebra""; ""11.3 The Geometry of F4 ""; ""11.4 The Geometry of E6""; ""11.5 The Geometry of E7""; ""11.5.1 The Symplectic Structure of so(k + 2, 2)""; ""11.5.2 Cubies""; ""11.5.3 The Symplectic Structure of e7""; ""11.5.4 Further Properties""; ""11.6 The Geometry of E8""; ""III. Applications""

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

981-4401-82-X