nLab octonion

Context

Algebra

higher algebra

universal algebra

Octonions

Idea

An octonion or Cayley number is a kind of number similar to a quaternion number but with seven instead of just three square roots of unity adjoined, and satisfying certain relations.

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

Definition

Definition

The octonions $\mathbb{O}$ is the nonassociative algebra over the real numbers which is generated from seven generators $\{e_1, \cdots, e_7\}$ subject to the relations

1. for all $i$

$e_i^2 = -1$

2. for $e_i \to e_j \to e_k$ an edge or circle in the following diagram (a labeled version of the Fano plane) the relations

1. $e_i e_j = e_k$

2. $e_j e_i = -e_k$

(graphics grabbed from Baez 02)

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. 2, forms a torsor over the automorphism group G2 $= Aut(\mathbb{O})$.

(e.g. Baez 02, 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$$Spin(9,1) {\simeq} \text{"}SL(2,\mathbb{O})\text{"}$$\mathbb{O}$ the octonionsheterotic/type II string

References

A survey is in

• John Baez, The Octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205. (web)

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

Revised on May 24, 2017 10:32:24 by Urs Schreiber (131.220.184.222)