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The Geometry of the Octonions cover

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.

Sample Chapter(s)
Chapter 1: Introduction (90 KB)

    • 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; non-experts with moderately sophisticated mathematics background.
Free Access
  • Pages:i–xvii

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Chapter 1: Introduction
  • Pages:1–2

I. Number Systems

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Chapter 2: The Geometry of the Complex Numbers
  • Pages:5–7

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Chapter 3: The Geometry of the Quaternions
  • Pages:9–14

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Chapter 4: The Geometry of the Octonions
  • Pages:15–22

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Chapter 5: Other Number Systems
  • Pages:23–29

II. Symmetry Groups

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Chapter 6: Some Orthogonal Groups
  • Pages:33–44

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Chapter 7: Some Unitary Groups
  • Pages:45–55

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Chapter 8: Some Symplectic Groups
  • Pages:57–60

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Chapter 9: Symmetry Groups over Other Division Algebras
  • Pages:61–76

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Chapter 10: Lie Groups and Lie Algebras
  • Pages:77–83

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Chapter 11: The Exceptional Groups
  • Pages:85–105

III. Applications

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Chapter 12: Division Algebras in Mathematics
  • Pages:109–131

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Chapter 13: Octonionic Eigenvalue Problems
  • Pages:133–162

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Chapter 14: The Physics of the Octonions
  • Pages:163–188

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Chapter 15: Magic Squares
  • Pages:189–198

Free Access
  • Pages:199–210

“This is an attractive book, with many thought provoking and novel ideas. It is also very well presented with good quality paper and typography, which make it very pleasant to handle.”

David B Fairlie
Emeritus Professor

“This interesting book is recommended to advanced physics and mathematics students and to scientists working in differential geometry. It has a great methodological value because of the multitude of unusual advanced concepts and applications.”

Journal of Geometry and Symmetry in Physics

“Anybody interested in the topics covered here should find this book to be a valuable reference.”

Mathematical Association of America

"The book offers an abundance of explicit constructions that are valuable for both beginners and experts, who often tend to underestimate such an approach in the pursuit of abstract ideas … It is an interesting compilation of an approachable introductory level review on division algebras and their connection to Lie groups."

Journal of Geometry and Symmetry in Physics

"The book under review is a nice addition and will serve as an introduction to more advanced material."

Mathematical Reviews Clippings

"To sum up, this is a really interesting book. It has encouraged me to further explore this fascinating topic."

London Mathematical Society Newsletter

"The book presents a nice addition to the literature and may serve as a primer for more advanced texts."

Zentralblatt MATH