John Baez
Geometry of the exceptional Jordan algebra

Contents

Contents

This is a record of some calculations done by John Baez, Greg Egan and John Huerta around November 2015.

9+1-dimensional geometry

The behavior of spinors depends heavily on the dimension of space, or spacetime, modulo 8. For example, in spacetimes of any dimension that’s 2 more than a multiple of 8 there exist β€˜Majorana–Weyl spinors’: spin-1/2 particles that have an intrinsic handedness (that’s the β€˜Weyl’ part) and are their own antiparticles (that’s the β€˜Majorana’ part).

In 10d Minkowski spacetime something special happens: both kinds of Majorana–Weyl spinors can be described using octonions. Mathematically this originates from the fact that there are two representations of Spin(9,1)Spin(9,1), called the left-handed and right-handed Majorana–Weyl spinor representations, which can both be identified with 𝕆 2\mathbb{O}^2, the space of pairs of octonions. This in turn follows from a simpler fact: 10-dimensional Minkowski spacetime can be identified with π”₯ 2(𝕆)\mathfrak{h}_2(\mathbb{O}), the space of self-adjoint 2Γ—22 \times 2 octonionic matrices.

All this is very similar to something that happens in 4 dimensions. 4d Minkowski spacetime can be identified with π”₯ 2(β„‚)\mathfrak{h}_2(\mathbb{C}), the space of self-adjoint 2Γ—22 \times 2 complex matrices, and Spin(3,1)Spin(3,1) has two representations on β„‚ 2\mathbb{C}^2, the left-handed and right-handed Weyl spinor representations. All this should be familiar to students of particle physics. The 10d case works essentially the same way: we just replace complex numbers with octonions. Of course the octonions are noncommutative and nonassociative, but it doesn’t really cause a problem here.

For a careful treatment of all this, try Section 2 here:

  • John C. Baez, John Huerta, Division algebras and supersymmetry I, in Superstrings, Geometry, Topology, and C *{}^\ast-algebras, eds. R. Doran, G. Friedman and J. Rosenberg, Proc. Symp. Pure Math. 81, AMS, Providence, 2010, pp. 65–80.

Here we will start with four vector spaces, that will turn out to be representations of Spin(9,1)Spin(9,1):

  • ℝ\mathbb{R} is the real numbers: the trivial representation of Spin(9,1)Spin(9,1), which physicists call the scalar representation.

  • S βˆ’S_- is 𝕆 2\mathbb{O}^2, treated as the left-handed Majorana-Weyl spinor representation of Spin(9,1)Spin(9,1).

  • S +S_+ is 𝕆 2\mathbb{O}^2 treated as right-handed Majorana-Weyl spin representation, the dual of S βˆ’S_-.

  • V=π”₯ 2(𝕆)V = \mathfrak{h}_2(\mathbb{O}) is the space of 2Γ—22 \times 2 self-adjoint octonionic matrices: the vector representation of Spin(9,1)Spin(9,1).

Concretely, we can write v∈Vv \in V as

v=(Ξ² x x * Ξ³) v = \left( \begin{array}{cc} \beta & x \\ x^\ast & \gamma \end{array} \right)

with Ξ²,Ξ³βˆˆβ„\beta, \gamma \in \mathbb{R} and xβˆˆπ•†x \in \mathbb{O}. Playing a key supporting role in the following algebra is the operation of trace reversal:

v˜=vβˆ’tr(v)1\tilde{v} = v - tr(v)1

Concretely, this looks like:

v˜=(βˆ’Ξ³ x x * βˆ’Ξ²) \tilde{v} = \left( \begin{array}{cc} -\gamma & x \\ x^\ast & -\beta \end{array} \right)

The Minkowski metric on VV

g:VΓ—V→ℝ g : V \times V \to \mathbb{R}

is given by

g(v,w)=12Retr(vw˜)=12Retr(v˜w) g(v,w) = \frac{1}{2} Re \, tr(v \tilde{w}) = \frac{1}{2} Re \,tr(\tilde{v} w)

and we also have

vv˜=v˜v=βˆ’det(v)1 v \tilde{v} = \tilde{v} v = -det(v) 1

so

g(v,v)=βˆ’det(v)=xx *βˆ’Ξ²Ξ³ g(v,v) = -det(v) = x x^\ast - \beta \gamma

Thus, the metric gg has signature (9,1)(9,1), that is 9 plus signs and 1 minus.

Invariant bilinear maps

Besides the metric

g:VΓ—V→ℝ g : V \times V \to \mathbb{R}

there are various other important Spin(9,1)Spin(9,1)-invariant bilinear maps.

We have an invariant bilinear map

VΓ—S βˆ’β†’S + V \times S_- \to S_+

given by

(v,s βˆ’)↦v˜s βˆ’ (v, s_-) \mapsto \tilde{v} s_-

and also one

VΓ—S +β†’S βˆ’ V \times S_+ \to S_-

given by

(v,s +)↦vs + (v, s_+) \mapsto v s_+

We also have brackets:

[βˆ’,βˆ’]:S βˆ’Γ—S βˆ’β†’V=π”₯ 2(𝕆) [-,-]: S_- \times S_- \to V = \mathfrak{h}_2(\mathbb{O})

given by

[s βˆ’,t βˆ’]=s βˆ’t βˆ’ †+t βˆ’s βˆ’ † [s_-,t_-] = s_- t_-^\dagger + t_- s_-^\dagger

and

[βˆ’,βˆ’]:S +Γ—S +β†’V=π”₯ 2(𝕆) [-,-]: S_+ \times S_+ \to V = \mathfrak{h}_2(\mathbb{O})

given by

[s +,t +]=(s +t + †+t +s + †)˜ [s_+,t_+] = \widetilde{(s_+ t_+^\dagger + t_+ s_+^\dagger)}

In both of these formulas, s Β±s_\pm and t Β±t_\pm are column vectors consisting of pairs of octonions, like this:

(x y),x,yβˆˆπ•† \left( \begin{matrix} x \\ y \end{matrix} \right), \quad x, y \in \mathbb{O}

so the result of bracketing is a 2Γ—22 \times 2 hermitian matrix.

We also have pairings of left-handed with right-handed spinors:

βŸ¨βˆ’,βˆ’βŸ©:S +Γ—S βˆ’β†’β„ \langle -,- \rangle : S_+ \times S_- \to \mathbb{R}

given by:

⟨s +,t βˆ’βŸ©=Re(s + †t βˆ’) \langle s_+, t_- \rangle = Re(s_+^\dagger t_-)

and

βŸ¨βˆ’,βˆ’βŸ©:S βˆ’Γ—S +→ℝ \langle -,- \rangle : S_- \times S_+ \to \mathbb{R}

given by:

⟨s βˆ’,t +⟩=Re(s βˆ’ †t +) \langle s_-, t_+ \rangle = Re(s_-^\dagger t_+)

The exceptional Jordan algebra

π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}) consists of 3Γ—33 \times 3 self-adjoint matrices of octonions, thus matrices of the form

a=(Ξ± z y * z * Ξ² x y x * Ξ³) a = \left( \begin{array}{ccc} \alpha & z & y^\ast \\ z^\ast & \beta & x \\ y & x^\ast & \gamma \end{array} \right)

where x,y,zβˆˆπ•†x,y,z \in \mathbb{O} and Ξ±,Ξ²,Ξ³βˆˆβ„\alpha, \beta, \gamma \in \mathbb{R}.

We can also think of π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}) as consisting of matrices

a=(r s † s v) a = \left( \begin{array}{cc} r & s^\dagger \\ s & v \end{array} \right)

where rβˆˆβ„r \in \mathbb{R}, s∈S βˆ’s \in S_- and v∈Vv \in V. Concretely, we have

s=(z * y) s = \left( \begin{array}{cc} z^\ast \\ y \end{array} \right)

and

v=(Ξ² x x * Ξ³) v = \left( \begin{array}{cc} \beta & x \\ x^\ast & \gamma \end{array} \right)

So, we have a chosen isomorphism

π”₯ 3(𝕆)β‰…β„βŠ•VβŠ•S βˆ’ \mathfrak{h}_3(\mathbb{O}) \cong \mathbb{R} \oplus V \oplus S_-

which is equivariant under Spin(9,1)Spin(9,1).

The determinant as a cubic form on π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O})

The determinant of a matrix a∈π”₯ 3(𝕆)a \in \mathfrak{h}_3(\mathbb{O}) is given by

det(a)=Ξ±Ξ²Ξ³βˆ’(Ξ±|x| 2+Ξ²|y| 2+Ξ³|z| 2)+2Re(xyz) det(a) = \alpha \beta \gamma - (\alpha |x|^2 + \beta |y|^2 + \gamma |z|^2) + 2 Re(x y z)

If we write

a=(r s † s v) a = \left( \begin{array}{cc} r & s^\dagger \\ s & v \end{array} \right)

as above, then

det(a)=rdet(v)+Re(s †v˜s)=rdet(v)+g(v,[s,s]) det(a) = r det(v) + Re(s^\dagger \tilde{v} s) = r det(v) + g(v, [s, s])

where

v˜=vβˆ’tr(v)1\tilde{v} = v - tr(v)1

is the trace-reversed version of vv, and

det(v)=Ξ²Ξ³βˆ’xx *det(v) = \beta \gamma - x x^\ast

is the determinant of v∈π”₯ 2(𝕆)v \in \mathfrak{h}_2(\mathbb{O}).

The trilinear form on π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O})

There is a unique symmetric trilinear form

t:π”₯ 3(𝕆)Γ—π”₯ 3(𝕆)Γ—π”₯ 3(𝕆)→ℝ t : \mathfrak{h}_3(\mathbb{O}) \times \mathfrak{h}_3(\mathbb{O}) \times \mathfrak{h}_3(\mathbb{O}) \to \mathbb{R}

with

det(a)=t(a,a,a) det(a) = t(a,a,a)

Explicitly, if we have

a i=(Ξ± i z i y i * z i * Ξ² i x i y i x i * Ξ³ i) a_i = \left( \begin{array}{ccc} \alpha_i & z_i & y^\ast_i \\ z_i^\ast & \beta_i & x_i \\ y_i & x^\ast_i & \gamma_i \end{array} \right)

then

6t(a 1,a 2,a 3)=6 t(a_1,a_2,a_3) =
βˆ‘ all6permutationsof{1,2,3}Ξ± iΞ² jΞ³ k+2Re(x iy jz k)βˆ’2βˆ‘ i∈{1,2,3},with{j,k}={1,2,3}βˆ’i(Ξ± iRe(x j *x k)+Ξ² iRe(y j *y k)+Ξ³ iRe(z j *z k)) \sum_{all \; 6 \; permutations \; of \; \{1,2,3\}} \alpha_i \beta_j \gamma_k + 2 Re(x_i y_j z_k) - 2 \sum_{i \in \{1,2,3\}, with \; \{j,k\} = \{1,2,3\}-i} \left( \alpha_i Re(x_j^\ast x_k) + \beta_i Re(y_j^\ast y_k) + \gamma_i Re(z_j^\ast z_k) \right)

Alternatively, if we write a ia_i in terms of scalars, spinors and vectors:

a i=(r i s i † s i v i) a_i = \left( \begin{array}{cc} r_i & s^\dagger_i \\ s_i & v_i \end{array} \right)

then we have

t(a 1,a 2,a 3) = (1/3)βˆ‘ i∈{1,2,3},with{j,k}={1,2,3}βˆ’iRe(s j †v˜ is k)βˆ’r ig(v j,v k) = (1/3)βˆ‘ i∈{1,2,3},with{j,k}={1,2,3}βˆ’ig(v i,[s j,s k])βˆ’r ig(v j,v k) \begin{array}{rcl} t(a_1,a_2,a_3) & = & (1/3) \sum_{i \in \{1,2,3\}, with \; \{j,k\} = \{1,2,3\}-i } Re(s_j^\dagger \tilde{v}_i s_k) - r_i g(v_j, v_k) \\ & = & (1/3) \sum_{i \in \{1,2,3\}, with \; \{j,k\} = \{1,2,3\}-i } g(v_i, [s_j, s_k]) - r_i g(v_j, v_k) \end{array}

The cross product on π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O})

Dualizing the trilinear form

t:π”₯ 3(𝕆)Γ—π”₯ 3(𝕆)Γ—π”₯ 3(𝕆)→ℝ t : \mathfrak{h}_3(\mathbb{O}) \times \mathfrak{h}_3(\mathbb{O}) \times \mathfrak{h}_3(\mathbb{O}) \to \mathbb{R}

we get a symmetric bilinear map called the cross product:

Γ—:π”₯ 3(𝕆)Γ—π”₯ 3(𝕆)β†’π”₯ 3(𝕆) * \times : \mathfrak{h}_3(\mathbb{O}) \times \mathfrak{h}_3(\mathbb{O}) \to \mathfrak{h}_3(\mathbb{O})^\ast

We can think of π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast as consisting of 3Γ—33 \times 3 self-adjoint matrices of octonions, where the pairing

βŸ¨βˆ’,βˆ’βŸ©:π”₯ 3(𝕆) *Γ—π”₯ 3(𝕆)→ℝ \langle -,- \rangle : \mathfrak{h}_3(\mathbb{O})^\ast \times \mathfrak{h}_3(\mathbb{O}) \to \mathbb{R}

is given by

⟨hβ€²,h⟩=12Retr(hβ€²h) \langle h', h \rangle = \frac{1}{2} Re\;tr(h' h)

Furthermore, as representations of Spin(9,1)Spin(9,1) there is an isomorphism

π”₯ 3(𝕆) *β‰…β„βŠ•VβŠ•S + \mathfrak{h}_3(\mathbb{O})^\ast \cong \mathbb{R} \oplus V \oplus S_+

by which any element (r,v,s +)βˆˆβ„βŠ•VβŠ•S +(r,v,s_+) \in \mathbb{R} \oplus V \oplus S_+ corresponds to the matrix

(r s + † s + v˜)∈π”₯ 3(𝕆) * \left( \begin{array}{cc} r & s_+^\dagger \\ s_+ & \tilde{v} \end{array} \right) \in \mathfrak{h}_3(\mathbb{O})^\ast

In these terms the cross product is:

(r A,v A,s A)Γ—(r B,v B,s B)=(1/3)(βˆ’2g(v A,v B),βˆ’r Av Bβˆ’r Bv A+[s A,s B],v˜ As B+v˜ Bs A) (r_A, v_A, s_A) \times (r_B, v_B, s_B) = (1/3) \big( -2g(v_A, v_B), -r_A v_B - r_B v_A + [s_A , s_B], \tilde{v}_A s_B + \tilde{v}_B s_A \big)

The dual of the exceptional Jordan algebra

The dual of the exceptional Jordan algebra, π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast, is inequivalent to π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}) as a representation of E 6\mathrm{E}_6. So, we must carefully distinguish between them. Nonetheless, every E 6\mathrm{E}_6-invariant structure carried by one is also carried by the other, since there’s an outer automorphism of E 6\mathrm{E}_6 that interchanges these two representations.

We shall think of π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast as consisting of 3Γ—33 \times 3 self-adjoint matrices of octonions, where the pairing

βŸ¨βˆ’,βˆ’βŸ©:π”₯ 3(𝕆) *Γ—π”₯ 3(𝕆)→ℝ \langle -,- \rangle : \mathfrak{h}_3(\mathbb{O})^\ast \times \mathfrak{h}_3(\mathbb{O}) \to \mathbb{R}

is given by

⟨hβ€²,h⟩=12Retr(hβ€²h) \langle h', h \rangle = \frac{1}{2} Re\;tr(h' h)

As a representation of Spin(9,1)Spin(9,1) there is an isomorphism

π”₯ 3(𝕆) *β‰…β„βŠ•VβŠ•S + \mathfrak{h}_3(\mathbb{O})^\ast \cong \mathbb{R} \oplus V \oplus S_+

under which (r,v,s +)βˆˆβ„βŠ•VβŠ•S +(r,v,s_+) \in \mathbb{R} \oplus V \oplus S_+ corresponds to

(r s + † s + v˜)∈π”₯ 3(𝕆) * \left( \begin{array}{cc} r & s_+^\dagger \\ s_+ & \tilde{v} \end{array} \right) \in \mathfrak{h}_3(\mathbb{O})^\ast

In these terms, the pairing

βŸ¨βˆ’,βˆ’βŸ©:π”₯ 3(𝕆) *Γ—π”₯ 3(𝕆)→ℝ \langle -,- \rangle : \mathfrak{h}_3(\mathbb{O})^\ast \times \mathfrak{h}_3(\mathbb{O}) \to \mathbb{R}

is given as follows:

⟨(rβ€²,vβ€²,s +β€²),(r,v,s βˆ’)⟩ = 12Retr((rβ€² s +β€² † s +β€² vΛœβ€²)(r s βˆ’ † s βˆ’ v)) = 12rβ€²r+g(vβ€²,v)+⟨s +β€²,s βˆ’βŸ©\begin{array}{rcl} \langle (r', v', s_+') , (r, v, s_-) \rangle & = & \frac{1}{2} Re\;tr\big( \left( \begin{array}{cc} r' & s_+'^\dagger \\ s_+' & \tilde{v}'\end{array} \right) \left( \begin{array}{cc} r & s_-^\dagger \\ s_- & v\end{array} \right) \big)\\ & = & \frac{1}{2} r' r + g(v',v) + \langle s_+', s_- \rangle \end{array}

The cubic form on π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast

There is a cubic form on π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast given by

detβ€²(r,s +,v)=rdet(v)+g(v,[s +,s +]) det'(r, s_+, v) = r det(v) + g(v, [s_+, s_+])

The trilinear form on π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast

There is a unique symmetric trilinear form

tβ€²:π”₯ 3(𝕆) *Γ—π”₯ 3(𝕆) *Γ—π”₯ 3(𝕆) *→ℝ t' : \mathfrak{h}_3(\mathbb{O})^\ast \times \mathfrak{h}_3(\mathbb{O})^\ast \times \mathfrak{h}_3(\mathbb{O})^\ast \to \mathbb{R}

with

detβ€²(a)=tβ€²(a,a,a) det'(a) = t'(a,a,a)

and this has an explicit form that looks very similar to tt on π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}):

tβ€²(a 1,a 2,a 3) = (1/3)βˆ‘ i∈{1,2,3},with{j,k}={1,2,3}βˆ’iRe(s j †v is k)βˆ’r ig(v˜ j,v˜ k) = (1/3)βˆ‘ i∈{1,2,3},with{j,k}={1,2,3}βˆ’ig(v i,[s j,s k])βˆ’r ig(v j,v k) \begin{array}{rcl} t'(a_1,a_2,a_3) & = & (1/3) \sum_{i \in \{1,2,3\}, with \; \{j,k\} = \{1,2,3\}-i } Re(s_j^\dagger v_i s_k) - r_i g(\tilde{v}_j, \tilde{v}_k) \\ & = & (1/3) \sum_{i \in \{1,2,3\}, with \; \{j,k\} = \{1,2,3\}-i } g(v_i, [s_j, s_k]) - r_i g(v_j, v_k) \end{array}

The second formula here appears identical to that for tt. However, the bracket operation on right-handed spinors differs from that on left-handed spinors, so strictly speaking it is not the same.

The cross product on π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast

Dualizing the trilinear form

tβ€²:π”₯ 3(𝕆) *Γ—π”₯ 3(𝕆) *Γ—π”₯ 3(𝕆) *→ℝ t' : \mathfrak{h}_3(\mathbb{O})^\ast \times \mathfrak{h}_3(\mathbb{O})^\ast \times \mathfrak{h}_3(\mathbb{O})^\ast \to \mathbb{R}

we get a symmetric bilinear map, another cross product:

Γ—:π”₯ 3(𝕆) *Γ—π”₯ 3(𝕆) *β†’π”₯ 3(𝕆) \times : \mathfrak{h}_3(\mathbb{O})^\ast \times \mathfrak{h}_3(\mathbb{O})^\ast \to \mathfrak{h}_3(\mathbb{O})

This has an explicit form similar to the original cross product above:

(r A,v A,s A)Γ—(r B,v B,s B)=(1/3)(βˆ’2g(v A,v B),βˆ’r Av Bβˆ’r Bv A+[s A,s B],v As B+v Bs A) (r_A, v_A, s_A) \times (r_B, v_B, s_B) = (1/3) \big( -2g(v_A, v_B), -r_A v_B - r_B v_A + [s_A , s_B], v_A s_B + v_B s_A \big)

The action of E 6\mathrm{E}_6 on π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O})

A certain noncompact real form of E 6\mathrm{E}_6, technically E 6(26)E_{6(26)}, is the group of collineations of 𝕆P 2\mathbb{O}\mathrm{P}^2, and also the group of determinant-preserving linear transformations of π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}). We call this group simply E 6\mathrm{E}_6.

E 6\mathrm{E}_6 is 78-dimensional, so we have

dim(E 8) = dim(Spin(9,1))+dim(S +)+dim(S βˆ’)+dim(ℝ) 78 = 45+16+16+1 \begin{array}{ccc} dim(E_8) &=& dim(Spin(9,1)) + dim(S_+) + dim(S_-) + dim(\mathbb{R}) \\ 78 &=& 45 + 16 + 16 + 1 \end{array}

This suggests that perhaps as vector spaces we have

e 6β‰…so(9,1)βŠ•S +βŠ•S βˆ’βŠ•β„ e_6 \cong so(9,1) \oplus S_+ \oplus S_- \oplus \mathbb{R}

This is true, but even better, Spin(9,1)Spin(9,1) and abelian Lie groups isomorphic to S +S_+, S βˆ’S_- and ℝ\mathbb{R} show up as Lie subgroups of E 6\mathrm{E}_6, in a way that gives rise to this direct sum decomposition.

1) First, Spin(9,1)Spin(9,1) is a Lie subgroup of E 6\mathrm{E}_6: if we use our identification

π”₯ 3(𝕆)β‰…β„βŠ•VβŠ•S βˆ’ \mathfrak{h}_3(\mathbb{O}) \cong \mathbb{R} \oplus V \oplus S_-

then each summand is a representation of Spin(9,1)Spin(9,1), and its action preserves the determinant

det(r s βˆ’ † s βˆ’ v)=rdet(v)+g(v,[s βˆ’,s βˆ’]) det \left( \begin{array}{cc} r & s_-^\dagger \\ s_- & v \end{array} \right) = r det(v) + g(v, [s_-, s_-])

2) Any u +u_+ in S +S_+ acts on (r,v,s βˆ’)∈π”₯ 3(𝕆)(r,v,s_-) \in \mathfrak{h}_3(\mathbb{O}) by

r ↦ r+g(v,[u +,u +])+2⟨u +,s βˆ’βŸ© v ↦ v s βˆ’ ↦ s βˆ’+vu +\begin{array}{ccl} r & \mapsto & r + g(v, [u_+, u_+]) + 2 \langle u_+,s_-\rangle \\ v & \mapsto & v \\ s_- & \mapsto & s_- + v u_+ \end{array}

Here the angle brackets denote the dual pairing between S +=S βˆ’ *S_+ = S_-^\ast and S βˆ’S_-. We can check that these formulas define a transformation that preserves the determinant rdet(v)+g(v,[s βˆ’,s βˆ’])r det(v) + g(v, [s_-, s_-]). By adding vu +v u_+ to s βˆ’s_-, g(v,[s βˆ’,s βˆ’])g(v, [s_-, s_-]) gains these 3 extra terms:

g(v,[s βˆ’,vu +])=βˆ’det(v)⟨u +,s βˆ’βŸ© g(v, [s_-, v u_+]) = -det(v) \langle u_+,s_- \rangle
g(v,[vu +,s βˆ’])=βˆ’det(v)⟨u +,s βˆ’βŸ© g(v, [v u_+, s_-]) = -det(v) \langle u_+,s_- \rangle
g(v,[vu +,vu +])=βˆ’det(v)g(v,[u +,u +]) g(v, [v u_+, v u_+]) = -det(v) g(v, [u_+, u_+])

These cancel out the extra terms in rdet(v)r det(v), so the determinant is unchanged.

3) Similarly, any u βˆ’u_- in S βˆ’S_- acts on (r,v,s βˆ’)∈π”₯ 3(𝕆)(r,v,s_-) \in \mathfrak{h}_3(\mathbb{O}) by

r ↦ r v ↦ v+(1/2)r[u βˆ’,u βˆ’]+[s βˆ’,u βˆ’] s βˆ’ ↦ s βˆ’+ru βˆ’ \begin{array}{ccl} r & \mapsto & r \\ v & \mapsto & v + (1/2) r [u_-, u_-] + [s_-, u_-] \\ s_- & \mapsto & s_- + r u_- \end{array}

4) Any positive real number tt acts on (r,v,s βˆ’)∈π”₯ 3(𝕆)(r,v,s_-) \in \mathfrak{h}_3(\mathbb{O}) by

r ↦ t 4r v ↦ t βˆ’2v s βˆ’ ↦ ts βˆ’ \begin{array}{ccl} r & \mapsto & t^4 r \\ v & \mapsto & t^{-2} v \\ s_- & \mapsto & t s_- \end{array}

These rescalings clearly preserve the determinant rdet(v)+g(v,[s βˆ’,s βˆ’])r det(v) + g(v, [s_-, s_-]). Note that tt can be negative, too, so in fact the multiplicative group ℝ *=β„βˆ’{0}\mathbb{R}^\ast = \mathbb{R} - \{0\} appears as a subgroup of E 6\mathrm{E}_6.

5) It is possible to implement the three actions described here by transformations on the matrix:

h=(r s βˆ’ † s βˆ’ v)h = \left( \begin{array}{cc} r & s_-^\dagger \\ s_- & v \end{array} \right)

that all take the form:

h↦12((gh)g †+g(hg †)) h \mapsto \frac{1}{2}((g h)g^\dagger + g (h g^\dagger))

For the action of a right-handed spinor u +u_+, we set:

g=(1 u + 0 1) g = \left( \begin{array}{cc} 1 & u_+ \\ 0 & 1 \end{array} \right)

where the 1 in the bottom-right corner of the matrix is a 2Γ—22 \times 2 identity matrix. For the action of a left-handed spinor u βˆ’u_-, we set:

g=(1 0 u βˆ’ 1) g = \left( \begin{array}{cc} 1 & 0 \\ u_- & 1 \end{array} \right)

And for the action of a scalar tt, we set:

g=(t 2 0 0 0 t βˆ’1 0 0 0 t βˆ’1) g = \left( \begin{array}{ccc} t^2 & 0 & 0\\ 0 & t^{-1} & 0\\ 0 & 0 & t^{-1} \end{array} \right)

Orbits of the E 6\mathrm{E}_6 action on π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O})

Any element in π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}) can be diagonalized by an element of F 4βŠ†E 6\mathrm{F}_4 \subseteq \mathrm{E}_6. So, when computing the orbit of any element, we may assume without loss of generalize that it has the form

(Ξ± 0 0 0 Ξ² 0 0 0 Ξ³) \left( \begin{array}{ccc} \alpha & 0 & 0 \\ 0 & \beta & 0 \\ 0 & 0 & \gamma \end{array} \right)

with Ξ±,Ξ²,Ξ³βˆˆβ„\alpha, \beta, \gamma \in \mathbb{R}. Then its determinant is Ξ±Ξ²Ξ³\alpha \beta \gamma, and this is E 6\mathrm{E}_6-invariant.

We can use transformations in F 4\mathrm{F}_4 (or even O(3)βŠ†F 4O(3) \subseteq \mathrm{F}_4) to permute Ξ±,Ξ²,\alpha, \beta, and Ξ³\gamma, so their order doesn’t matter.

We can use transformations in Spin 0(9,1)βŠ†E 6Spin_0(9,1) \subseteq \mathrm{E}_6 to multiply Ξ²\beta by any positive constant and divide Ξ³\gamma by that same constant. Thanks to our ability to permute, the same is true of Ξ±\alpha and Ξ²\beta, or Ξ±\alpha and Ξ³\gamma.

Thus, if Ξ±,Ξ²,Ξ³>0\alpha, \beta, \gamma \gt 0, their product is a complete invariant for the action of E 6\mathrm{E}_6. We thus get one E 6\mathrm{E}_6 orbit for each value of Ξ΄>0\delta \gt 0:

+++ Ξ΄+++{}_\delta: the orbit containing the matrices diag(Ξ±,Ξ²,Ξ³)diag(\alpha, \beta, \gamma) with Ξ±,Ξ²,Ξ³>0\alpha, \beta, \gamma \gt 0, Ξ±Ξ²Ξ³=Ξ΄\alpha \beta \gamma = \delta.

We cannot, it seems, use a transformation in E 6\mathrm{E}_6 to multiply two of Ξ±,Ξ²,Ξ³\alpha, \beta, \gamma by βˆ’1-1 and leave the third alone. Thus, there is a separate family of E 6\mathrm{E}_6 orbits, one for each Ξ΄>0\delta \gt 0:

+βˆ’βˆ’ Ξ΄+--{}_\delta: the orbit containing the matrices diag(Ξ±,Ξ²,Ξ³)diag(\alpha, \beta, \gamma) with Ξ±,Ξ²<0\alpha, \beta \lt 0, Ξ³>0\gamma \gt 0, Ξ±Ξ²Ξ³=Ξ΄\alpha \beta \gamma = \delta.

By the same reasoning, there are two more one-parameter families of orbits with Ξ΄<0\delta \lt 0:

++βˆ’ Ξ΄++-{}_\delta: the orbit containing the matrices diag(Ξ±,Ξ²,Ξ³)diag(\alpha, \beta, \gamma) with Ξ±,Ξ²>0\alpha, \beta \gt 0, Ξ³<0\gamma \lt 0, Ξ±Ξ²Ξ³=Ξ΄\alpha \beta \gamma = \delta.

+βˆ’βˆ’ Ξ΄+- -{}_\delta: the orbit containing the matrices diag(Ξ±,Ξ²,Ξ³)diag(\alpha, \beta, \gamma) with Ξ±β‰₯0\alpha \ge 0, Ξ²,Ξ³<0\beta, \gamma \lt 0, Ξ±Ξ²Ξ³=Ξ΄\alpha \beta \gamma = \delta.

We’re left with the case Ξ±Ξ²Ξ³=0\alpha \beta \gamma = 0. This case gives 6 orbits:

++0++0: The orbit of the matrix diag(1,1,0)diag(1,1,0).

+βˆ’0+-0: The orbit of the matrix diag(1,βˆ’1,0)diag(1,-1,0).

βˆ’βˆ’0--0: The orbit of the matrix diag(βˆ’1,βˆ’1,0)diag(-1,-1,0).

+00+00: The orbit of the matrix diag(1,0,0)diag(1,0,0).

βˆ’00-00: The orbit of the matrix diag(βˆ’1,0,0)diag(-1,0,0).

000000: The orbit of the matrix diag(0,0,0)diag(0,0,0).

So, there are 6 orbits and 4 one-parameter families of orbits where the parameter takes values in an open half-line. We can organize these by rank: every element of π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}) has a rank, which is the number of nonzero entries in the matrix after it has been diagonalized. This is an E 6\mathrm{E}_6-invariant concept.

Rank 3: a rank-3 element h∈π”₯ 3(𝕆)h \in \mathfrak{h}_3(\mathbb{O}) is one where hΓ—h∈π”₯ 3(𝕆) *h \times h \in \mathfrak{h}_3(\mathbb{O})^\ast doesn’t annihilate any nonzero element of π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}). There are 4 one-parameter families of rank-3 orbits.

For any value of Ξ΄>0\delta \gt 0, there are 2 orbits of matrices with determinant Ξ΄\delta:

  • +++ Ξ΄+++{}_\delta: the orbit containing the matrices diag(Ξ±,Ξ²,Ξ³)diag(\alpha, \beta, \gamma) with Ξ±,Ξ²,Ξ³>0\alpha, \beta, \gamma \gt 0, Ξ±Ξ²Ξ³=Ξ΄\alpha \beta \gamma = \delta. This orbit is 26-dimensional. The stabilizer of any point in this orbit is a 52-dimensional group isomorphic to the compact real form of F 4\mathrm{F}_4.

  • +βˆ’βˆ’ Ξ΄+--{}_\delta: the orbit containing the matrices diag(Ξ±,Ξ²,Ξ³)diag(\alpha, \beta, \gamma) with Ξ±>0\alpha \gt 0, Ξ²,Ξ³<0\beta, \gamma \lt 0, Ξ±Ξ²Ξ³=Ξ΄\alpha \beta \gamma = \delta. This orbit is 26-dimensional. The identity component of the stabilizer of any point in this orbit is probably a 52-dimensional group isomorphic to the noncompact real form of F 4\mathrm{F}_4 called F 4(20)F_{4(20)}, which is diffeomorphic to Spin(9)×ℝ 16Spin(9) \times \mathbb{R}^{16}.

For any value of Ξ΄<0\delta \lt 0, there are 2 orbits of matrices with determinant Ξ΄\delta:

  • ++βˆ’ Ξ΄++-{}_\delta: the orbit containing the matrices diag(Ξ±,Ξ²,Ξ³)diag(\alpha, \beta, \gamma) with Ξ±,Ξ²>0\alpha, \beta \gt 0, Ξ³<0\gamma \lt 0, Ξ±Ξ²Ξ³=Ξ΄\alpha \beta \gamma = \delta. This orbit is 26-dimensional. The identity component of the stabilizer of any point in this orbit is probably a 52-dimensional group isomorphic to the noncompact real form of F 4\mathrm{F}_4 called F 4(20)F_{4(20)}, which is diffeomorphic to Spin(9)×ℝ 16Spin(9) \times \mathbb{R}^{16}.

  • βˆ’βˆ’βˆ’ Ξ΄---{}_\delta: the orbit containing the matrices diag(Ξ±,Ξ²,Ξ³)diag(\alpha, \beta, \gamma) with Ξ±,Ξ²,Ξ³<0\alpha ,\beta, \gamma \lt 0, Ξ±Ξ²Ξ³=Ξ΄\alpha \beta \gamma = \delta. This orbit is 26-dimensional. The stabilizer of any point in this orbit is a 52-dimensional group isomorphic to the compact real form of F 4\mathrm{F}_4.

Rank 2: a rank-2 element h∈π”₯ 3(𝕆)h \in \mathfrak{h}_3(\mathbb{O}) is one where hΓ—h∈π”₯ 3(𝕆) *h \times h \in \mathfrak{h}_3(\mathbb{O})^\ast annihilates some nonzero element of π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}) but hΓ—hβ‰ 0h \times h \ne 0. There are 3 orbits of rank-2 elements:

  • ++0++0: The orbit of the matrix diag(1,1,0)diag(1,1,0). This orbit is 26-dimensional. The identity component of the stabilizer of any point in this orbit is a 52-dimensional group isomorphic to Spin(9)⋉S +Spin(9) \ltimes S_+.

  • +βˆ’0+-0: The orbit of the matrix diag(1,βˆ’1,0)diag(1,-1,0). This orbit is 26-dimensional dimensional. The identity component of the stabilizer of any point in this orbit is a 52-dimensional group isomorphic to Spin(8,1)⋉S βˆ’\mathrm{Spin}(8,1) \ltimes S_-.

  • βˆ’βˆ’0--0: The orbit of the matrix diag(βˆ’1,βˆ’1,0)diag(-1,-1,0). This orbit is 26-dimensional. The identit component of the stabilizer of any point in this orbit is a 52-dimensional group isomorphic to Spin(9)⋉S +Spin(9) \ltimes S_+.

Rank 1: a rank-1 element h∈π”₯ 3(𝕆)h \in \mathfrak{h}_3(\mathbb{O}) is one where hΓ—h=0h \times h = 0 but hβ‰ 0h \ne 0. There are 2 orbits of rank-1 elements:

  • +00+00: The orbit of the matrix diag(1,0,0)diag(1,0,0). This orbit is 17-dimensional. The identity component of the stabilizer of any point in this orbit is a 61-dimensional group isomorphic to Spin(9,1)⋉S +Spin(9,1) \ltimes S_+.

  • βˆ’00-00: The orbit of the matrix diag(βˆ’1,0,0)diag(-1,0,0). This orbit is 17-dimensional. The identity component of the stabilizer of any point in this orbit is a 61-dimensional group isomorphic to Spin(9,1)⋉S +Spin(9,1) \ltimes S_+.

Rank 0: The only rank-0 element of π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}) is zero, so there is just one orbit:

  • 000000: The orbit of the matrix diag(0,0,0)diag(0,0,0). This orbit is 0-dimensional. The stabilizer of the unique point in this orbit is all of E 6\mathrm{E}_6.

Some of these orbits, or unions of these orbits, have alternate descriptions. Most notably we have the large lightcone:

{a∈π”₯ 3(𝕆):det(a)=0}=++0βˆͺ+βˆ’0βˆͺβˆ’βˆ’0βˆͺ+00βˆͺβˆ’00βˆͺ000 \{ a \in \mathfrak{h}_3(\mathbb{O}) : \; det(a) = 0 \} = +\!+0 \; \cup +\!-0 \;\cup \; -\!-0 \; \cup \; +00 \; \cup \; -00 \; \cup\; 000

and the small lightcone:

{a∈π”₯ 3(𝕆):aΓ—a=0}=+00βˆͺβˆ’00βˆͺ000 \{ a \in \mathfrak{h}_3(\mathbb{O}) : \; a \times a = 0 \} = +00 \;\cup\; -00 \;\cup\; 000

We also have the forwards small lightcone:

{a∈π”₯ 3(𝕆):aΓ—a=0,tr(a)>0}=+00 \{ a \in \mathfrak{h}_3(\mathbb{O}) : \; a \times a = 0, \; \tr(a) \gt 0 \} = +00

and the backwards small lightcone:

{a∈π”₯ 3(𝕆):aΓ—a=0,tr(a)<0}=βˆ’00 \{ a \in \mathfrak{h}_3(\mathbb{O}) : \; a \times a = 0, \; \tr(a) \lt 0 \} = -00

The forwards small lightcone is diffeomorphic to 𝕆P 2×ℝ +\mathbb{O}\mathrm{P}^2 \times \mathbb{R}^+, and so is the backwards small lightcone.

Points in the small lightcone can be explicitly described using the identification π”₯ 3(𝕆)β‰…β„βŠ•VβŠ•S βˆ’\mathfrak{h}_3(\mathbb{O}) \cong \mathbb{R} \oplus V \oplus S_- as follows. It is the union of these two sets:

  • (1a): Points of the form a=(r,12r[s,s],s)a = (r, \frac{1}{2r} [s,s], s) where we can choose ss freely, including the origin, and choose any rβ‰ 0r \ne 0. This set is 17-dimensional. It does not include points with r=0r = 0.

  • (1b): Points of the form a=(0,v LL,0)a = (0, v_{LL}, 0) where v LLv_{LL} is a nonzero lightlike vector. This set is 9-dimensional. We can think of its elements as the limit points of points in set (1a) where rβ†’0r \to 0 and sβ†’0s \to 0 together.

In this parametrization we get the forwards small lightcone from (1a) with r>0r \gt 0 and from (1b) with v LLv_{LL} on the forwards lightcone in ℝ 9,1\mathbb{R}^{9,1}.

Here is another parametrization of the small lightcone. Again, it is the union of two sets:

  • (2a): Points of the form a=(12tr([n v,n v])/tr(v LL),v LL,n v)a = (\frac{1}{2} tr([n_v,n_v]) / tr(v_{LL}), v_{LL}, n_v) where v LLv_{LL} is a nonzero lightlike vector and n vn_v belongs to the 8-dimensional kernel of v˜ LL\tilde{v}_{LL}. This set is 17-dimensional. It does not include points with v LL=0v_{LL} = 0.

  • (2b): Points of the form a=(r,0,0)a = (r, 0, 0) where rβ‰ 0r \ne 0. This set is 1-dimensional. We can think of its elements as the limit points of points in set (2a) where v LLβ†’0v_{LL} \to 0 and n vβ†’0n_v \to 0 together.

In this parametrization we get the forwards small lightcone from (2a) with v LLv_{LL} on the forwards lightcone in ℝ 9,1\mathbb{R}^{9,1}, and from (2b) with r>0r \gt 0.

Stabilizer computation: +00 orbit

If we look at the stabilizer of the rank-1 element h=diag(1,0,0)h = diag(1,0,0), with pieces (r,v,s βˆ’)=(1,0,0)(r,v,s_-)=(1,0,0), this will be fixed by:

1) any element gg of Spin(9,1)Spin(9,1), since v=0v=0

2) any element u +u_+ of S +S_+, since that action is given by:

  • r↦r+g(v,[u +,u +])+2⟨u +,s βˆ’βŸ© r \mapsto r + g(v, [u_+,u_+]) + 2\langle u_+,s_- \rangle , which takes r=1r = 1 to r=1r = 1 since v=0v=0 and s βˆ’=0s_- = 0,

  • v↦v v \mapsto v which clearly preserves v=0v = 0,

and

  • s βˆ’β†¦s βˆ’+vu + s_- \mapsto s_- + v u_+ which takes s βˆ’=0s_- = 0 to s βˆ’=0s_- = 0 since v=0v = 0.

These interact nicely so that the semidirect product Spin(9,1)⋉S +Spin(9,1) \ltimes S_+ stabilizes diag(1,0,0)diag(1,0,0). By dimension counting, this is the whole stabilizer, since

dim(Spin(9,1)⋉S +)=45+16=61 dim (Spin(9,1) \ltimes S_+) = 45 + 16 = 61

and the orbit of diag(1,0,0)diag(1,0,0) is 17-dimensional, and 61+17=7861 + 17 = 78.

Stabilizer computation: ++- orbit

The stabilizer of the rank-3 element h=diag(βˆ’1,1,1)h = diag(-1,1,1), with pieces (r,v,s βˆ’)=(βˆ’1,diag(1,1),0)(r,v,s_-)=(-1,diag(1,1),0) is a 52-dimensional group. Page 67 of this book:

describes all 3 real forms of F 4\mathrm{F}_4, and I believe this can be used to prove the stabilizer of hh is F 4(20)F_{4(20)}, the real form whose Killing form has signature 20, meaning that it’s positive definite on a 36-dimensional subspace and negative definite on a 16-dimensional subspace (or the other way around if you use the other sign convention). Among the real forms of F 4\mathrm{F}_4, this one is characterized by having Spin(9)Spin(9) as its maximal compact subgroup, so it is diffeomorphic to Spin(9)×ℝ 16Spin(9) \times \mathbb{R}^{16}.

It is easy to see that Spin(9)Spin(9) appears as a subgroup of the stabilizer of hh, so there is only a bit left to prove here.

The action of E 6\mathrm{E}_6 on π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast

The action of E 6\mathrm{E}_6 on π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast is very much like its action on π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}), but with the roles of S +S_+ and S βˆ’S_- reversed.

Explicitly, we act on the matrices:

hβ€²=(rβ€² s +β€² † s +β€² vΛœβ€²)h' = \left( \begin{array}{cc} r' & s_+'^\dagger \\ s_+' & \tilde{v}' \end{array} \right)

with the transformation:

h′↦12((gβ€²hβ€²)gβ€² †+gβ€²(hβ€²gβ€² †)) h' \mapsto \frac{1}{2}((g' h')g'^\dagger + g' (h' g'^\dagger))

where:

gβ€²=(g βˆ’1) †g' = (g^{-1})^\dagger

Acting this way on hβ€²h' while acting with gg on hh preserves the pairing:

⟨hβ€²,h⟩=12Retr(hβ€²h)\langle h',h \rangle = \frac{1}{2} Re\;tr(h'h)

The orbits of E 6\mathrm{E}_6 on π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast are classified almost exactly as for π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}). The differences are all routine: for example, in set (2a), where we had demanded that n vn_v belong to the 8-dimensional null space of v˜ LL\tilde{v}_{LL}, we should now use the kernel of v LLv_{LL}. This is the appropriate action of VV on S +S_+. We also need to switch the roles of S +S_+ and S βˆ’S_-. For example, since a point in the forwards small lightcone in π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}), i.e. the +00+00 orbit in π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}), is stabilized by a group conjugate to Spin(9,1)⋉S +Spin(9,1) \ltimes S_+, so a β€˜null momentum vector’, i.e. a point in the +00+00 orbit in π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast, is stabilized by a group conjugate to Spin(9,1)⋉S βˆ’Spin(9,1) \ltimes S_-.

The E 6\mathrm{E}_6 action on 𝕆P 2\mathbb{O}\mathrm{P}^2

We can think of the octonionic projective plane 𝕆P 2\mathbb{O}\mathrm{P}^2 as the projectivization of the small lightcone

{a∈π”₯ 3(𝕆):aΓ—a=0}=+00βˆͺβˆ’00βˆͺ000 \{ a \in \mathfrak{h}_3(\mathbb{O}) : \; a \times a = 0 \} = +00 \;\cup\; -00 \;\cup\; 000

or in other words, +00βˆͺβˆ’00+00 \cup -00 modulo the ℝ *\mathbb{R}^\ast action given by rescaling. Since the small lightcone is 17-dimensional this makes 𝕆P 2\mathbb{O}\mathrm{P}^2 16-dimensional, as it must be.

E 6\mathrm{E}_6 acts transitively on the small lightcone and thus on 𝕆P 2\mathbb{O}\mathrm{P}^2. The stabilizer of a point in the forward small lightcone is Spin(9,1)⋉S +Spin(9,1) \ltimes S_+. Thus, the stabilizer of a point in 𝕆P 2\mathbb{O}\mathrm{P}^2 is the 62-dimensional group (Spin(9,1)×ℝ *)⋉S +(Spin(9,1) \times \mathbb{R}^\ast) \ltimes S_+.

E 6\mathrm{E}_6 also acts transitively on the space of lines in 𝕆P 2\mathbb{O}\mathrm{P}^2. The space of lines is diffeomorphic to 𝕆P 2\mathbb{O}\mathrm{P}^2, but with a different action of E 6\mathrm{E}_6. The stabilizer of any is the 62-dimensional group (Spin(9,1)×ℝ *)⋉S βˆ’(Spin(9,1) \times \mathbb{R}^\ast) \ltimes S_-.

The group E 6\mathrm{E}_6 also acts transitively on the space of antiflags: pairs consisting of a point and line where the line does not contain the point. The space of antiflags is a 32-dimensional manifold, so the stabilizer of any antiflag must be a group of dimension 78βˆ’32=4678 - 32 = 46. This group is isomorphic to Spin(9,1)×ℝ *Spin(9,1) \times \mathbb{R}^\ast.

The last fact is easiest to see if we treat 𝕆P 2\mathbb{O}\mathrm{P}^2 as the union of 𝕆 2\mathbb{O}^2 and the β€˜line at infinity’, a copy of 𝕆P 1\mathbb{O} P^1. The subgroup stabilizing the line at infinity is (Spin(9,1)×ℝ *)⋉S βˆ’(Spin(9,1) \times \mathbb{R}^\ast) \ltimes S_-, so this group acts on the complement, 𝕆 2\mathbb{O}^2. If we identify 𝕆 2\mathbb{O}^2 with S βˆ’S_- the action is easy to understand. Spin(9,1)Spin(9,1) acts on S βˆ’S_- via its spinor representation, ℝ *\mathbb{R}^\ast acts on S βˆ’S_- via dilations, and the additive group S βˆ’S_- acts on S βˆ’S_- by translations. The subgroup of (Spin(9,1)×ℝ *)⋉S βˆ’(Spin(9,1) \times \mathbb{R}^\ast) \ltimes S_- that also fixes the origin in S βˆ’S_- is thus Spin(9,1)×ℝ *Spin(9,1) \times \mathbb{R}^\ast.

Lines in 𝕆P 2\mathbb{O}\mathrm{P}^2

We can identify the space of lines in 𝕆P 2\mathbb{O}\mathrm{P}^2 either with:

  • the projectivization of the small lightcone in π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast, or
  • equivalence classes of rank-2 elements in π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}).

With the first identification, a point pp in 𝕆P 2\mathbb{O}\mathrm{P}^2, viewed as a 1-dimensional subspace contained in the small lightcone of π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}), is incident on a line LL in 𝕆P 2\mathbb{O}\mathrm{P}^2, viewed as a 1-dimensional subspace contained in the small lightcone of π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast, iff

L 0(p 0)=0L_0(p_0) = 0

for any p 0∈pp_0 \in p and L 0∈LL_0 \in L.

The second identification is a bit more complicated. Given any rank-2 element vv of π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}), if we translate the small lightcone by vv, it will intersect the untranslated small lightcone in an 8-dimensional set. The span of this 8-dimensional set will be a 10-dimensional subspace V(v)V(v) of π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}) containing elements of ranks 0, 1 and 2, including vv.

V(v)=span({p|p and pβˆ’v are both on the small lightcone})V(v) = span (\{ p | p\,\text{ and }\, p-v\, \text{ are both on the small lightcone}\})

The intersection of the translated and untranslated small lightcones will be precisely the intersections of the ordinary Minkowski light cones centred at the origin and at vv within V(v)V(v) (where the conformal structure on V(v)V(v) is derived from the trilinear form on π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O})). We can view all the rank-2 elements in V(v)V(v) as belonging to an equivalence class with vv, since the same construction performed with any of them will yield the same subspace.

If we identify this equivalence class of rank-2 elements with a line in 𝕆P 2\mathbb{O}\mathrm{P}^2, then the 1-dimensional spaces of rank-1 elements of V(v)V(v), which lie on the Minkowski lightcone, correspond to points that are incident on the line.

We can also identify elements LL of the projectivized small lightcone in π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast with 10-dimensional Minkowski subspaces of π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}), by defining:

V(L)=span({p|p is on the small lightcone and L 0(p)=0})V(L) = span(\{p | p\, \text{ is on the small lightcone and }\, L_0(p) = 0\})

where as before L 0∈LL_0 \in L.

We can connect the two ways of thinking about lines by noting that, for rank-2 elements v 2v_2:

v 2∈V(L) iff v 2Γ—v 2∈Lv_2 \in V(L)\,\text{ iff }\,v_2 \times v_2 \in L

This also gives us a new way of describing the equivalence relationship between rank-2 elements that describe the same line:

v 2~v 2β€² iff v 2Γ—v 2 is a multiple of v 2β€²Γ—v 2β€²v_2 ~ v_2'\,\text{ iff }\, v_2 \times v_2\,\text{ is a multiple of }\,v_2' \times v_2'

We’ve seen that the choice of an antiflag in 𝕆P 2\mathbb{O}\mathrm{P}^2 (a point, and a line not containing that point) gives rise to a choice of a 1-dimensional subspace ℝ\mathbb{R} of elements with ranks 0 and 1 in π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}), and a 10-dimensional subspace VV of elements with ranks 0, 1 and 2 in π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}), which intersect only at the origin.

Additionally, this choice singles out a 16-dimensional subspace SS of elements with ranks 0 and 2 only, defined by:

S={s|t(v,v,s)=0βˆ€v∈V and sΓ—r=0βˆ€rβˆˆβ„}S = \{s | t(v,v,s) = 0 \forall v \in V\,\text{ and }\, s \times r = 0 \forall r \in \mathbb{R}\}

In fact, if we pick any elements of maximum rank in each subspace, v 2∈Vv_2 \in V and r 1βˆˆβ„r_1 \in \mathbb{R}, we can obtain SS from the same conditions applied to that single choice of elements:

S={s|t(v 2,v 2,s)=0 and sΓ—r 1=0}S = \{s | t(v_2,v_2,s) = 0 \,\text{ and }\, s \times r_1 = 0 \}

The first condition means that ss belongs to the tangent space to the orbit of v 2v_2 at v 2v_2, and the second condition means that ss belongs to the tangent space to the orbit of r 1r_1 at r 1r_1, where we treat these tangent spaces as subspaces of π”₯ 3(𝕆)\mathfrak{h}_3(\mathbb{O}). So we can write:

S=T v 2(orbit(v 2))∩T r 1(orbit(r 1))S = T_{v_2}(orbit(v_2)) \cap T_{r_1}(orbit(r_1))

The rank-2 elements of SS all belong to the orbit +βˆ’0+-0.

We can describe the spaces ℝ\mathbb{R} and VV as:

ℝ=cl{r|T r(orbit(r))=T r 1(orbit(r 1))}\mathbb{R} = cl\{r | T_{r}(orbit(r)) = T_{r_1}(orbit(r_1)) \}
V=cl{v|T v(orbit(v))=T v 2(orbit(v 2))}V = cl\{v | T_{v}(orbit(v)) = T_{v_2}(orbit(v_2)) \}

The kernel of the cross product map

For any vector p∈π”₯ 3(𝕆) *p \in \mathfrak{h}_3(\mathbb{O})^\ast there is a cross product map

pΓ—:π”₯ 3(𝕆) *β†’π”₯ 3(𝕆) p\times : \mathfrak{h}_3(\mathbb{O})^\ast \to \mathfrak{h}_3(\mathbb{O})

mapping any vector vv to pΓ—vp \times v. The dimension of the kernel of this map depends on which orbit pp lies in:

  • If pp has rank 3 then the kernel of this map is 0-dimensional.

  • If pp has rank 2 then the kernel of this map is 9-dimensional. In this case the point pp lies in a copy of ℝ 9,1\mathbb{R}^{9,1} inside π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast, and as such it is either a timelike vector (if p∈++0p \in ++0 or βˆ’βˆ’0--0) or a spacelike vector (if p∈+βˆ’0p \in +-0). The subgroup of Spin(9,1)Spin(9,1) stabilizing this vector is thus either Spin(9)Spin(9) or Spin(8,1)Spin(8,1). In either case it acts on kerpΓ—ker p \!\times, giving a representation isomorphic to the 9-dimensional β€˜vector’ representation of this group.

  • If pp has rank 1 then the kernel of this map is 17-dimensional. In this case the kernel can be identified with the tangent space of the small lightcone at pp. The point pp lies in a copy of ℝ 9,1\mathbb{R}^{9,1} inside π”₯ 3(𝕆) *\mathfrak{h}_3(\mathbb{O})^\ast, and as such it is a lightlike vector. The subgroup of Spin(9,1)Spin(9,1) stabilizing this vector is isomorphic to Spin(8)⋉ℝ 8Spin(8) \ltimes \mathbb{R}^8, and it acts on kerpΓ—ker p\!\times, giving a representation isomorphic to the direct sum of the 1-dimensional β€˜scalar’ representation of Spin(8)Spin(8) on ℝ\mathbb{R} and the 16-dimensional β€˜left-handed plus right-handed spinor’ representation of Spin(8)Spin(8) on π•†βŠ•π•†\mathbb{O} \oplus \mathbb{O}.

  • If pp has rank 0 then the kernel of this map is 27-dimensional.

Last revised on December 22, 2020 at 15:10:36. See the history of this page for a list of all contributions to it.