octonion in nLab

Context

Arithmetic

Algebra

Exceptional structures

• Idea
• Definition
• Properties
• General
• Automorphisms
• Left multiplication by imaginary octonions
• Basic triples
• Relation to quaternions
• Related concepts
• References
• General
• Relation to 10d/11d spin geometry

exceptional spinors and real normed division algebras

Definition

The octonions 𝕆\mathbb{O} are the elements of the non-associative star-algebra over the real numbers which is the Cayley-Dickson double of the star-algebra of quaternions ℍ\mathbb{H} (with (−)¯\overline{(-)} denoting the conjugation-operation).

This means (see there) that if i,j,k∈ℍi, j, k \in \mathbb{H} denote an orthonormal basis of imaginary unit-quaternions

i 2=j 2=k 2=−1 ij=k,ji=−kand cyclic \begin{aligned} & i^2 = j^2 = k^2 = -1 \\ & i j = k, \, j i = - k \;\;\;\text{and cyclic} \end{aligned}

then the algebra 𝕆\mathbb{O} of octonions is generated from these i,j,ki, j , k and from one more element ℓ\ell, subject to these relations:

ℓ 2=−1,AAAℓ¯=−ℓ \begin{aligned} \ell^2 \;=\; -1 \,, \phantom{AAA} \overline{\ell} \;=\; - \ell \end{aligned}

and

q(ℓq′)=ℓ(q¯q′) (qℓ)q′=(qq¯′)ℓ (ℓq)(q′ℓ)=−qq′¯ \begin{aligned} q (\ell q') = \ell (\overline{q} q') \\ (q \ell) q' = (q \overline{q}') \ell \\ (\ell q) (q' \ell) = - \overline{q q'} \end{aligned}

for all quaternions q 1,q 2∈ℍq_1, q_2 \in \mathbb{H}.

This gives the multiplication table on the right, where any two consecutive arrows a→b→ca \to b \to c mean that ab=ca b = c, ca=bc a = b, bc=ab c = a and ba=−cb a = -c.

Example

The following computation shows the operation of consecutive left multiplication by the generators e 4\mathrm{e}_4, e 5\mathrm{e}_5, e 6\mathrm{e}_6 e 7\mathrm{e}_7 (according to Def. ) on any octonion x=q+pℓx = q + p \ell (q,p∈ℍq,p \in \mathbb{H}) is by reversal of the sign of the ℓ\ell-component, hence has as fixed linear subspace the quaternions (see HSS 18, Lemma 4.13 for application of this fact to M-branes):

e 4(e 5(e 6(e 7x))) =ℓ((iℓ)((jℓ)((kℓ)x))) =ℓ((iℓ)((jℓ)((kx¯)ℓ))) =ℓ((iℓ)((xk)j)) =ℓ((i(j(kx¯))ℓ) =((x⏟q+pℓk)j)i =qkji+(((qℓ)k)j)i =qkji⏟=1−(pkji⏟=1)ℓ =q−pℓ. \begin{aligned} \mathrm{e}_4 \Big( \mathrm{e}_5 \big( \mathrm{e}_6 (\mathrm{e}_7 x) \big) \Big) & = \ell \bigg( (i \ell) \Big( (j \ell) \big( (k \ell) x \big) \Big) \bigg) \\ & = \ell \bigg( (i \ell) \Big( (j \ell) \big( (k \overline{x}) \ell \big) \Big) \bigg) \\ & = \ell \Big( (i \ell) \big( (x k) j \big) \Big) \\ & = \ell \bigg( \Big( i \big( j (k \overline{x} \big) \Big) \ell \bigg) \\ & = \big( ( \underset{ \mathclap{ q + p \ell } }{ \underbrace{ x } } k) j \big) i \\ & = q k j i + \Big( \big( (q \ell) k \big) j \Big) i \\ & = q \underset{ = 1 }{ \underbrace{ k j i } } - (p \underset{ = 1 }{ \underbrace{ k j i } } ) \ell \\ & = q - p \ell \,. \end{aligned}

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 i→e j→e 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

\,

This becomes a star-algebra with star involution

(1)(−)¯:𝕆⟶𝕆 \overline{(-)} \;\colon\; \mathbb{O} \longrightarrow \mathbb{O}

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

e i¯≔−e iAAAAi∈{1,⋯,7}. \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 11))))))=+1 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:

e 7(e 6(e 5(e 4(e 3(e 2e 1⏟−e 4))))) =−e 7(e 6(e 5(e 4(e 3e 4⏟e 6)))) =−e 7(e 6(e 5(e 4e 6⏟e 3))) =−e 7(e 6(e 5e 3⏟−e 2)) =+e 7(e 6e 2⏟−e 7) =−e 7e 7⏟=−1 =+1 \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∈𝕆a \in \mathbb{O} shuch that under the star involution (1) it is sent to its negative:

  a¯=−a \overline{a} = -a

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

  a¯=a \overline{a} = a

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

Re(a)≔12(a+a¯)AAIm(a)≔12(a−a¯). Re(a) \coloneqq \tfrac{1}{2}(a + \overline{a}) \phantom{AA} Im(a) \coloneqq \tfrac{1}{2}(a - \overline{a}) \,.

Proof

By linearity it is sufficient to check this on generators. So let e i→e j→e ke_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

(e ie j)e j =e ke j =−e je k =−e i =e i(e je j) \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

(e ie i)e j =−e j =−e ke i =e ie k =e i(e ie j). \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} \,.

Definition

Given any octonion oo, then the operation of left multiplication by oo

𝕆 ⟶L o 𝕆 a ↦ oa \array{ \mathbb{O} &\overset{L_o}{\longrightarrow}& \mathbb{O} \\ a &\mapsto& o a }

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

Proposition

(Clifford action of imaginary octonions)

Consider the Clifford algebra

Cl(Im(𝕆),−|−| 2) 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 𝕆\mathbb{O} (def. ) induces a representation of this Clifford algebra on ℝ 8≃ ℝ𝕆\mathbb{R}^8 \simeq_{\mathbb{R}} \mathbb{O}.

Proof

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

L vL v(a) ≔v(va) =(vv)a =−|v| 2a =L −|v| 2(a) \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 ie_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 7L e 6L e 5L e 4L e 3L e 2L e 1=+Id 𝕆 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 ie_i are imaginary octonions (Def. ). By Prop. their left action on 𝕆\mathbb{O} represents a Clifford algebra-action of Cl(Im(𝕆),−|−| 2)≃Cl 0,7Cl(Im(\mathbb{O}), -{\vert-\vert}^2) \simeq Cl_{0,7} on ℝ 8≃ ℝ𝕆\mathbb{R}^{8} \simeq_{\mathbb{R}} \mathbb{O}.

By the classification of real Clifford algebras, Cl 0,7Cl_{0,7} has, up to isomorphism, two different irreducible modules. Their underlying vector space is ℝ 8\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” Γ 7Γ 6⋯Γ 1\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∈𝕆1 \in \mathbb{O}. Then the claim follows with the computation in Example :

L e 7L e 6L e 5L e 4L e 3L e 2L e 1(1) =e 7(e 6(e 5(e 4(e 3(e 2(e 11)))))) =+1. \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}

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 for i≠ji \ne j

• e i(e je k)=−(e je k)e ie_i (e_j e_k) = - (e_j e_k) e_i for i,j,ki,j,k pairwise distinct.

Proposition

Let

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

be the quaternions equipped with canonical basis elements, and let

(2)𝕆=ℍ⊕ℓℍ \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 ℓL ℓiL ℓjL ℓk:𝕆⟶𝕆 L_{\ell} L_{\ell i} L_{\ell j} L_{\ell k} \;\colon\; \mathbb{O} \longrightarrow \mathbb{O}

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

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

Proof

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

a(ℓb)=ℓ(a¯b),AAa(ℓb)=(ab¯)ℓ,AA(ℓa)(bℓ −1)=ab¯ 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 𝕆\mathbb{O} for all a,b∈ℍa,b \in \mathbb{H}, as well as

ℓe=−eℓ \ell e = - e \ell

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

With this we compute

L ℓL ℓiL ℓjL ℓk(a) =ℓ((ℓi)((ℓj)((ℓk)a))) =−ℓ((ℓi)((ℓj)((kℓ)a))) =−ℓ((ℓi)((ℓj)((ka¯)ℓ))) =+ℓ((ℓi)((ℓj)((ka¯)ℓ −1))) =+ℓ((ℓi)(j(ka¯)¯)) =+ℓ((ℓi)(ia¯¯)) =−ℓ((iℓ)(ia¯¯)) =−ℓ((iia¯)ℓ) =+ℓ(a¯ℓ) =−ℓ(a¯ℓ −1) =−a \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

L ℓL ℓiL ℓjL ℓk(ℓa) =ℓ((ℓi)((ℓj)((ℓk)(ℓa)))) =ℓ((ℓi)((ℓj)((ℓk)(a¯ℓ)))) =−ℓ((ℓi)((ℓj)(ka¯¯))) =+ℓ((ℓi)((jℓ)(ka¯¯))) =+ℓ((ℓi)((jka¯)ℓ)) =−ℓ((ℓi)((jka¯)ℓ −1)) =−ℓ(ijka¯¯) =+ℓ(a¯¯) =+ℓa \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