# nLab octonion

### Context

#### Algebra

higher algebra

universal algebra

# Octonions

## Idea

The octonions form the largest of the four normed division algebras, denoted $\mathbb{O}$.

## Properties

### Non-associativity

The octonions are not an associative algebra. The non-zero octonions and the unit octonions form Moufang loops.

### Automorphisms

The automorphism group of the octonions is G2.

### Basic triples

###### Definition

A special triple or basic triple is a triple $(e_1, e_2, e_3) \in \mathbb{O}^3$ of three octonions such that

• $e_i^2 = -1$

• $e_i e_j = - e_j e_i$.

###### Remark

The choice of $e_1$ identifies an inclusion of the complex numbers $\mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{O}$.

Then the choice of $e_2$ on top of that identifies a compatible inclusion of the quaternions $\mathbb{R} \hookrightarrow \mathbb{C}\hookrightarrow \mathbb{H} \hookrightarrow \mathbb{O}$.

Finally the choice of $e_3$ on top of that induces a basis for all of $\mathbb{O}$.

###### Proposition

The set of basic triples, def. 1, forms a torsor over the automorphism group G2 $= Aut(\mathbb{O})$.

(e.g. Baez, 4.1)

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
$3 = 2+1$$Spin(2,1) \simeq SL(2,\mathbb{R})$$\mathbb{R}$ the real numbers
$4 = 3+1$$Spin(3,1) \simeq SL(2, \mathbb{C})$$\mathbb{C}$ the complex numbers
$6 = 5+1$$Spin(5,1) \simeq SL(2, \mathbb{H})$$\mathbb{H}$ the quaternionslittle string
$10 = 9+1$$Spin(9,1) \underset{some\,sense}{\simeq} SL(2,\mathbb{O})$$\mathbb{O}$ the octonionsheterotic/type II string

## References

A survey is in

The concept of “special triples” or (“basic triples”) used above seems to go back to

Revised on February 26, 2016 10:00:26 by Urs Schreiber (194.210.233.5)