nLab
octonion

Context

Arithmetic

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 𝕆\mathbb{O}, form the largest of the four normed division algebras over the real numbers.

Definition

Definition

The octonions 𝕆\mathbb{O} is the nonassociative algebra over the real numbers which is generated from seven generators {e 1,,e 7}\{e_1, \cdots, e_7\} subject to the relations

  1. for all ii

    e i 2=1e_i^2 = -1

  2. for e ie je ke_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 ie j=e ke_i e_j = e_k

    2. e je i=e ke_j e_i = -e_k

octonion multiplication table

(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)𝕆 3(e_1, e_2, e_3) \in \mathbb{O}^3 of three octonions such that

  • e i 2=1e_i^2 = -1

  • e ie j=e je ie_i e_j = - e_j e_i.

(Whitehead 71, p. 691)

Remark

The choice of e 1e_1 identifies an inclusion of the complex numbers 𝕆\mathbb{R} \hookrightarrow \mathbb{C} \hookrightarrow \mathbb{O}.

Then the choice of e 2e_2 on top of that identifies a compatible inclusion of the quaternions 𝕆\mathbb{R} \hookrightarrow \mathbb{C}\hookrightarrow \mathbb{H} \hookrightarrow \mathbb{O}.

Finally the choice of e 3e_3 on top of that induces a basis for all of 𝕆\mathbb{O}.

Proposition

The set of basic triples, def. 2, forms a torsor over the automorphism group G2 =Aut(𝕆)= Aut(\mathbb{O}).

(e.g. Baez 02, 4.1)

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})\mathbb{R} the real numbers
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})\mathbb{C} the complex numbers
6=5+16 = 5+1Spin(5,1)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})\mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1)"SL(2,𝕆)"Spin(9,1) {\simeq} \text{"}SL(2,\mathbb{O})\text{"}𝕆\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)