Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Octonions

John C. Baez

TL;DR

Baez surveys the octonions as the largest normed division algebra, emphasizing their nonassociativity and central role in connecting Clifford algebras, spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. It presents four constructions (multiplication table, Fano plane, Cayley–Dickson, and Clifford-spinor-triality), develops octonionic projective geometry ${\mathbb {OP}}^1$ and ${\mathbb {OP}}^2$, and derives octonionic descriptions of the exceptional Lie groups via the magic square, culminating in detailed accounts of ${\rm G}_2$, ${\rm F}_4$, ${\rm E}_6$, ${\rm E}_7$, and ${\rm E}_8$. The work highlights Bott periodicity, triality, and the appearance of octonions in higher-dimensional Lorentzian physics and quantum structures, while outlining foundational links to geometry, topology, and theoretical physics. It also notes that while octonions have rich theoretical significance, their physical role remains unresolved, inviting further exploration.

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.

The Octonions

TL;DR

Baez surveys the octonions as the largest normed division algebra, emphasizing their nonassociativity and central role in connecting Clifford algebras, spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. It presents four constructions (multiplication table, Fano plane, Cayley–Dickson, and Clifford-spinor-triality), develops octonionic projective geometry and , and derives octonionic descriptions of the exceptional Lie groups via the magic square, culminating in detailed accounts of , , , , and . The work highlights Bott periodicity, triality, and the appearance of octonions in higher-dimensional Lorentzian physics and quantum structures, while outlining foundational links to geometry, topology, and theoretical physics. It also notes that while octonions have rich theoretical significance, their physical role remains unresolved, inviting further exploration.

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.

Paper Structure

This paper contains 20 sections, 13 theorems, 242 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constructing the Octonions
  4. The Fano plane
  5. The Cayley--Dickson construction
  6. Clifford Algebras
  7. Spinors and Trialities
  8. Octonionic Projective Geometry
  9. Projective Lines
  10. ${\mathbb {OP}}^1$ and Bott Periodicity
  11. ${\mathbb {OP}}^1$ and Lorentzian Geometry
  12. ${\mathbb {OP}}^2$ and the Exceptional Jordan Algebra
  13. Exceptional Lie Algebras
  14. ${\rm G}_2$
  15. ${\rm F}_4$
  16. ...and 5 more sections

Key Result

Theorem 1

.${\mathbb R},{\mathbb C},{\mathbb H}$, and ${\mathbb O}$ are the only normed division algebras.

Figures (1)

  • Figure :

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 4
  • Theorem 5
  • ...and 3 more