nLab
octonion

Octonions

Idea

The octonions form the largest of the four normed division algebras, denoted 𝕆\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)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. 1, forms a torsor over the automorphism group G2 =Aut(𝕆)= Aut(\mathbb{O}).

(e.g. Baez, 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)somesenseSL(2,𝕆)Spin(9,1) \underset{some\,sense}{\simeq} SL(2,\mathbb{O})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)