Octonions

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Next: Introduction

Abstract:

The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.

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Table of Contents:

1. Introduction
   1. Preliminaries
2. Constructing the Octonions
   1. The Fano Plane
   2. The Cayley-Dickson Construction
   3. Clifford Algebras
   4. Spinors and Trialities

3. Octonionic Projective Geometry
   1. Projective Lines
   2. OP¹ and Bott Periodicity
   3. OP¹ and Lorentzian Geometry
   4. OP² and the Exceptional Jordan Algebra

4. Exceptional Lie Algebras
   1. G₂
   2. F₄
   3. The Magic Square
   4. E₆
   5. E₇
   6. E₈

5. Conclusions
   1. Acknowledgements

6. Bibliography

This website also contains some extra stuff, namely:

• Octonions Online
  links to other websites containing material about the octonions.

• Brougham Bridge
  pictures of the bridge where Hamilton carved his definition of the quaternions.

• On Quaternions and Octonions: Their Geometry, Arithmetic and Symmetry
  my review of John Conway and Derek Smith's book.

And if all this is too hard for you, try this two-part feature in Plus Magazine:

• Curious Quaternions

and

• Ubiquitous Octonions

in which Helen Joyce and I have a fun nontechnical chat about the real numbers, complex numbers, quaternions and octonions.

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© 2004 John Baez
baez@math.removethis.ucr.andthis.edu

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