# nLab octonion

Octonions

### 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)

This becomes a star-algebra with star involution

(1)$\overline{(-)} \;\colon\; \mathbb{O} \longrightarrow \mathbb{O}$

which is the antihomomorphism $\overline{a b} = \overline{b} \overline{a}$ that is given on the above generators by

$\overline{e_i} \coloneqq - e_i \phantom{AAAA} i \in \{1, \cdots, 7\} \,.$
###### Example

The product of all the generators with each other, bracketed to the right, is

$e_7 (e_6 (e_5 (e_4 (e_3 ( e_2 (e_1 1 )))))) \;=\; + 1$
###### Proof

By iteratively using the multiplication table in def. we compute as follows:

\begin{aligned} & e_7 (e_6 (e_5 (e_4 (e_3 (\underset{-e_4}{\underbrace{e_2 e_1}}))))) \\ & = - e_7 (e_6 (e_5 (e_4 (\underset{e_6}{\underbrace{e_3 e_4}})))) \\ & = - e_7 (e_6 (e_5 (\underset{e_3}{\underbrace{e_4 e_6}}))) \\ & = - e_7 (e_6 (\underset{-e_2}{\underbrace{e_5 e_3}})) \\ & = + e_7 (\underset{-e_7}{\underbrace{e_6 e_2}}) \\ & = - \underset{= -1}{\underbrace{e_7 e_7}} \\ & = + 1 \end{aligned}
###### Definition

(real and imaginary octonions)

As for the complex numbers one says that

• an imaginary octonion is an $a \in \mathbb{O}$ shuch that under the star involution (1) it is sent to its negative:

$\overline{a} = -a$
• a real octonions is an $a \in \mathcal{O}$ shuch that under the star involution (1) it is sent to itself

$\overline{a} = a$

Accordingly every octonion decomposes into a real part and an imaginary part:

$Re(a) \coloneqq \tfrac{1}{2}(a + \overline{a}) \phantom{AA} Im(a) \coloneqq \tfrac{1}{2}(a - \overline{a}) \,.$

## Properties

### General

###### Proposition

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

###### Proposition

(octonions are alternative)

The octonions form an alternative algebra

###### Proof

By linearity it is sufficient to check this on generators. So let $e_i \to e_j \to e_k$ be a circle or a cyclic permutation of an edge in the Fano plane as in Def. . Then by definition of the octonion multiplication we have

\begin{aligned} (e_i e_j) e_j &= e_k e_j \\ &= - e_j e_k \\ & = -e_i \\ & = e_i (e_j e_j) \end{aligned}

and similarly

\begin{aligned} (e_i e_i ) e_j &= - e_j \\ &= - e_k e_i \\ &= e_i e_k \\ &= e_i (e_i e_j) \end{aligned} \,.
###### Proposition

The automorphism group of the octonions, as a real algebra, is the exceptional Lie group G2.

### Left multiplication by imaginary octonions

###### Definition

Given any octonion $o$, then the operation of left multiplication by $o$

$\array{ \mathbb{O} &\overset{L_o}{\longrightarrow}& \mathbb{O} \\ a &\mapsto& o a }$

is an $\mathbb{R}-$linear map. Under composition of linear maps, this defines an associative monoid acting linearly on $\mathbb{O}$.

###### Proposition

(Clifford action of imaginary octonions)

Consider the Clifford algebra

$Cl(Im(\mathbb{O}), -{\vert -\vert}^2)$

on the underlying real vector space of that of the imaginary octonions (Def. ) regarded as an inner product space via the quadratic form given by the negative square norm.

Then the operation of left multiplication on $\mathbb{O}$ (def. ) induces a representation of this Clifford algebra on $\mathbb{R}^8 \simeq_{\mathbb{R}} \mathbb{O}$.

###### Proof

By alternativity (Prop. ) we have for every $v \in Im(\mathbb{O})$ and every $a \in \mathbb{O}$

\begin{aligned} L_v L_v (a) & \coloneqq v (v a) \\ & = (v v) a \\ & = - {\vert v\vert}^2 a \\ & = L_{- {\vert v\vert}^2} (a) \end{aligned}
###### Proposition

(consecutive left action by imaginary generators is unity)

The consecutive left multiplication action (Def. ) by all the imaginary octonion generators $e_i$ (Def. ) is $\pm$ the identity function on the octonions. Specifically, if one acts in increasing order of the labels in Def. , then it is +1:

$L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1} \;=\; + Id_{\mathbb{O}}$
###### Proof

All the generators $e_i$ are imaginary octonions (Def. ). By Prop. their left action on $\mathbb{O}$ represents a Clifford algebra-action of $Cl(Im(\mathbb{O}), -{\vert-\vert}^2) \simeq Cl_{0,7}$ on $\mathbb{R}^{8} \simeq_{\mathbb{R}} \mathbb{O}$.

By the classification of real Clifford algebras, $Cl_{0,7}$ has, up to isomorphism, two different irreducible modules. Their underlying vector space is $\mathbb{R}^8$ in both cases, and so the left action of imaginary octonions we have must be one of the two. The two irreps may be distinguished by the action of the “volume element” $\Gamma_7 \Gamma_6 \cdots \Gamma_1$: On one of the two it acts as the identity, on the other as minus the identity.

Hence we may check the remaining sign by acting on any one octonion, for instance on the unit $1 \in \mathbb{O}$. Then the claim follows with the computation in Example :

\begin{aligned} L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1} (1) & = e_7 (e_6 (e_5 (e_4 (e_3 ( e_2 (e_1 1 )))))) \\ & = + 1 \,. \end{aligned}

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

(e.g. Baez 02, 4.1)

### Relation to quaternions

###### Proposition

Let

$\mathbb{H} = \langle 1, i, j, k\rangle$

be the quaternions equipped with canonical basis elements, and let

(2)$\mathbb{O} = \mathbb{H} \oplus \ell \mathbb{H}$

be the octonions equipped with the linear basis induced by the Cayley-Dickson construction (via this def.).

Then the linear map

$L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k} \;\colon\; \mathbb{O} \longrightarrow \mathbb{O}$

is an involution whose +1 eigenspace is $\ell \mathbb{H}$ and whose -1 eigenspace is $\mathbb{H}$, under the above identification (2).

(Here $L_{(-)}$ denotes the linear map on $\mathbb{O}$ given by left multiplication in $\mathbb{O}$.)

###### Proof

We use the Cayley-Dickson relations (this def.)

$a (\ell b) = \ell (\overline{a} b) \,, \phantom{AA} a(\ell b) = (a \overline{b}) \ell \,, \phantom{AA} (\ell a) (b \ell^{-1}) = \overline{a b}$

that hold in $\mathbb{O}$ for all $a,b \in \mathbb{H}$, as well as

$\ell e = - e \ell$

for all imaginary elements $e \in \mathbb{H}$.

With this we compute

\begin{aligned} & L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k} ( a) \\ & = \ell( (\ell i) ( (\ell j) ( (\ell k) a ) ) ) \\ & = - \ell( (\ell i) ( (\ell j) ( (k \ell) a ) ) ) \\ & = - \ell( (\ell i) ( (\ell j) ( (k \overline{a}) \ell ) ) ) \\ & = + \ell( (\ell i) ( (\ell j) ( (k \overline{a}) \ell^{-1} ) ) ) \\ & = + \ell( (\ell i) ( \overline{j (k \overline{a})} ) ) \\ & = + \ell( (\ell i) ( \overline{i \overline{a}} ) ) \\ & = - \ell( (i \ell) ( \overline{i \overline{a}} ) ) \\ & = - \ell( ( i i \overline{a} ) \ell ) \\ & = + \ell( \overline{a} \ell ) \\ & = - \ell( \overline{a} \ell^{-1} ) \\ & = - a \end{aligned}

and

\begin{aligned} & L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k} ( \ell a) \\ & = \ell( (\ell i) ( (\ell j) ( (\ell k) (\ell a) ) ) ) \\ & = \ell( (\ell i) ( (\ell j) ( (\ell k) ( \overline{a} \ell) ) ) ) \\ & = - \ell( (\ell i) ( (\ell j) ( \overline{k {\overline{a}}} ) ) ) \\ & = + \ell( (\ell i) ( (j \ell) ( \overline{k {\overline{a}}} ) ) ) \\ & = + \ell( (\ell i) ( (j k {\overline{a}}) \ell ) ) \\ & = - \ell( (\ell i) ( (j k {\overline{a}}) \ell^{-1} ) ) \\ & = - \ell( \overline{ i j k {\overline{a}} } ) \\ & = + \ell( \overline{ {\overline{a}} } ) \\ & = + \ell a \end{aligned}

exceptional spinors and real normed division algebras

Lorentzian
spacetime
dimension
$\phantom{AA}$spin groupnormed division algebra$\,\,$ brane scan entry
$3 = 2+1$$Spin(2,1) \simeq SL(2,\mathbb{R})$$\phantom{A}$ $\mathbb{R}$ the real numberssuper 1-brane in 3d
$4 = 3+1$$Spin(3,1) \simeq SL(2, \mathbb{C})$$\phantom{A}$ $\mathbb{C}$ the complex numberssuper 2-brane in 4d
$6 = 5+1$$Spin(5,1) \simeq SL(2, \mathbb{H})$$\phantom{A}$ $\mathbb{H}$ the quaternionslittle string
$10 = 9+1$$Spin(9,1) {\simeq}$SL(2,O)$\phantom{A}$ $\mathbb{O}$ the octonionsheterotic/type II string

## References

Textbook account:

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

Relation to the Leech lattice:

• Robert A. Wilson, Octonions and the Leech lattice, Journal of Algebra

Volume 322, Issue 6, 15 September 2009, Pages 2186-2190, (pdf, slides)

Last revised on May 17, 2019 at 09:34:58. See the history of this page for a list of all contributions to it.