# The Octonions

@inproceedings{Baez2001TheO, title={The Octonions}, author={John C. Baez}, year={2001} }

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.

## 977 Citations

A trip around octonions

- MathematicsJournal of Physics: Conference Series
- 2022

In these expository notes, after a contemplation on the dawn of octonions, we give proofs for the Frobenius theorem and the Hurwitz theorem, we review the basics of Clifford algebras and spin groups,…

Quaternions and Octonions

- Mathematics
- 2019

The quaternions and octonions are the two largest of the four normed division algebras. Despite their quirks of the quaternions being noncommutative and octonions even nonassociative, they continue…

Division Algebras; Spinors; Idempotents; The Algebraic Structure of Reality

- Mathematics, Physics
- 2010

A carefully constructed explanation of my connection of the real normed division algebras to the particles, charges and fields of the Standard Model of quarks and leptons provided to an interested…

Derivations of octonion matrix algebras

- MathematicsCommunications in Algebra
- 2019

Abstract It is well-known that the exceptional Lie algebras and arise from the octonions as the derivation algebras of the 3 × 3 hermitian and 1 × 1 antihermitian matrices, respectively. Inspired by…

Multiplication of integral octonions

- Mathematics
- 2016

The integral subsets of octonions are an analog of integers in real numbers and related to many interesting topics in geometry and physics via E8-lattices. In this paper, we study the properties of…

Quaternionic and octonionic spinors. A classification

- Mathematics
- 2003

Quaternionic and octonionic realizations of Clifford algebras and spinors are classified and explicitly constructed in terms of recursive formulas. The most general free dynamics in arbitrary…

Reality of non-Fock Spinors

- Mathematics
- 2003

The infinite dimensional Clifford Algebra has a maze of irreducible unitary representations. Here we determine their type -real, complex or quaternionic. Some, related to the Fermi-Fock…

Quaternionic and Octonionic Spinors

- Mathematics
- 2005

Quaternionic and octonionic spinors are introduced and their fundamental properties (such as the space‐times supporting them) are reviewed. The conditions for the existence of their associated Dirac…

The Sextonions and E 7

- Mathematics
- 2004

We fill in the “hole” in the exceptional series of Lie algebras that was observed by Cvitanovic, Deligne, Cohen and deMan. More precisely, we show that the intermediate Lie algebra between e7 and e8…

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