Octonions

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The Octonions
John C. Baez

Department of Mathematics
University of California
Riverside CA 92521

May 16, 2001

Published in Bull. Amer. Math. Soc. 39 (2002), 145-205.

Errata in Bull. Amer. Math. Soc. 42 (2005), 213.

Also available in Postscript and PDF formats.

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:

Introduction
1. Preliminaries
Constructing the Octonions
Octonionic Projective Geometry
1. Projective Lines
2. OP¹ and Bott Periodicity
3. OP¹ and Lorentzian Geometry
4. OP² and the Exceptional Jordan Algebra
Exceptional Lie Algebras
1. G₂
2. F₄
3. The Magic Square
4. E₆
5. E₇
6. E₈
Conclusions
1. Acknowledgements
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.

Abstract:

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