ON 14-DIMENSIONAL QUADRATIC FORMS, THEIR SPINORS, AND THE DIFFERENCE OF TWO OCTONION ALGEBRAS

MARKUS ROST

This text originates from an E-mail to in May 1994. It has been revised on September 18, 1996 and March 9, 1999 (but still may contain some inaccuracies).

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Let me give a short report on some thoughts on low dimensional quadratic forms. “low” means here up to dimension 14.

1. Introduction.

One may start with the following: Consider a quadratic form q of dimension d and with trivial e^{1}- and e^{2}-invariants. Then q comes from a torsor q̃ ∈ H^{1}(Spin(d)). Consider a fundamental spinor representation S of Spin(d) and let S(q̃) be the corresponding “twisted” representation.

It turns out that if d is smaller or equal to 14, then the dimension of S is smaller than the dimension of the group Spin(d). These dimensions are “exceptional” in this sense. The significance is as follows: If you take any nontrivial vector s ∈ S(q̃), it will have a nontrivial isotropy group, and one may hope that this gives an interesting reduction of the torsor q̃ ∈ H^{1}(Spin(d)) to H^{1} of a subgroup of Spin(d).

This is indeed the case: The isotropy group of a generic spinor s ∈ S has the 1-component

• G_{2} × G_{2} (for d = 14),

• SL(6) (for d = 12),

• semi-direct product of an additive group with Spin(7) (for d = 10),

• Spin(7) (for d = 8, 9),

• G_{2} (for d = 7). Moreover if d = 10, then the (1-component of the) isotropy group of a generic

vector in S ⊕ S, (i.e., the intersection of the isotropy group of two generic s) is G_{2}.

Let me note that for d = 9 the embedding of the isotropy group Spin(7) → Spin(9) is not the usual one: it is in Spin(8) twisted by a triality, so that the center of Spin(7) does not map to the center of Spin(9). Similarly for Spin(7) → Spin(10) in the case d = 10.

The computations (together with computations of the normalizers, e.g., in the 14-dimensional case the normalizer is (G_{2} × G_{2}) ⋊ μ_{8}) yield the following:

• for a 14-dimensional form q in I^{3} there is a quadratic extension L = F (\sqrt{a}) and a 3-fold Pfister form p over L, such that q is the trace of \sqrt{a}p′, where p′ is the pure subform of p. (This was not known to me before and I don’t know any other proof than the one described below.)

• a 12-dimensional form in I^{3} is the trace of a 6-dimensional hermitian form with trivial determinant. (A known fact.)

Date: March 9, 1999.

• a 10-dimensional form in I^{3} is isotropic. (A well-known fact.)

• etc. I will discuss here only the cases d = 10, 14. The case d = 12 is of different

nature and will not be considered.

2. A construction of “the difference of two composition algebras”

The goal of the following considerations is to give a certain description of the (64dimensional) spinor representation of Spin(14). The advantage of this description is, that it makes it easy to determine the dimension of the main orbit and to compute its isotropy group.

I assume char(F) ≠ 2, or even char(F) = 0.

Let C be a composition algebra over a field F with norm form N and unit e and let

C = eF ⊕ V

be the orthogonal decomposition.

Let D be another composition algebra over F with norm form M and unit f and let

D = fF ⊕ W

be the orthogonal decomposition.

Put

R = C ⊗ D

and

L = ⋀^{2}(V ⊕ W) = ⋀^{2}V ⊕ ⋀^{2}W ⊕ V ⊗ W.

I define a map

Θ: L ⟶ End(R)

by the following formulas:

Θ(v ∧ v̄)(c ⊗ d) = [v · (v̄ · c) − v̄ · (v · c)] ⊗ d,
Θ(w ∧ w̄)(c ⊗ d) = c ⊗ [w · (w̄ · d) − w̄ · (w · d)],
Θ(v ∧ w)(c ⊗ d) = (v · c) ⊗ (w · d).

Here x · y is the product in the composition algebras.

Lemma 1. The map Θ is injective and its image is a Lie subalgebra.

Proof. This follows from standard rules for composition algebras. □

Let

p: V ⟶ F, p(v) = N(v)

be the restriction of N. I identify ⋀^{2}V with the Lie algebra so(p). (In the natural way as subspaces of the even Clifford algebra of p.)

Let

J: ⋀^{2}V ⟶ V, J(v ∧ w) = v · w − w · v.

• Suppose dim C = 4. Then C is a quaternion algebra and J is bijective.

• Suppose dim C = 8. Then C is a octonion algebra. In this case J is surjective and its kernel is the Lie algebra of G_{2}.

Let

q: W ⟶ F, q(v) = M(v)

be the restriction of M. I identify ⋀^{2}W with the Lie algebra so(q).

Let

U = V ⊕ W

and consider the quadratic form

[p − q]: U ⟶ F,
[p − q](v, w) = p(v) − q(w).

Note that L = ⋀^{2}U. I identify L with the Lie algebra of so([p − q]). In this way the inclusion

⋀^{2}V ⊕ ⋀^{2}W ⟶ L

is identified with the inclusion

so(p) ⊕ so(q) ⟶ so([p − q]).

Lemma 2. With these identifications (with the Lie algebra structures eventually altered by a scalar factor), the map Θ is a homomorphism of Lie algebras.

Proof. By carefully comparing both Lie brackets. □

3. The main orbit

We have now established R as Spin([p − q])-module. The tangent space of the Spin([p − q])-orbit of some vector r of R is Θ(L)(r).

We now assume that dim C, dim D are equal to 4 or to 8. This means that the maps J are in both cases surjective.

Let us consider the specific element r = e ⊗ f. One finds for the tangent space of the orbit through r:

Θ(L)(e ⊗ f) = e ⊗ W ⊕ V ⊗ f ⊕ V ⊗ W.

This is a subspace of codimension 1, transversal to r. So, if we extend the group by scalar multiplication, we see that the orbit of r is open. Therefore r generates a generic line and the orbit of r has codimension 1.

Let us compute the Lie algebra of the isotropy group of r. This is the sum of the Lie algebras of the isotropy groups of e and f. These are the kernels of the maps J on so(p) = ⋀^{2}V, and on so(q) = ⋀^{2}W. These are trivial if C, D have dimension 4 and are the Lie algebras of G_{2} if C, D have dimension 8.

Since the main orbit has codimension 1 it follows that there is an invariant form

ψ: R ⟶ F

on R such that the varieties

{ ψ = nonzero constant }

are the main orbits.

4. On the case dim C = dim D = 8

Consider the above construction for split C and D of dimension 8. Then we are given a representation

split- Spin(14) → End(R).

Now given a 14-dimensional quadratic form h: X ⟶ F with trivial e^{1}- and e^{2}-invariants, we may twist this representation to a representation

Spin(h) ⟶ End(R).

leaving invariant a certain form ρ: R ⟶ F, with ρ the twist of ψ.

If you take a vector r in R such that ρ(r) is nontrivial, then there comes along a decomposition X = Y ⊕ Z, such that h restricted to Y and h restricted to Z are similar to the pure subform of 8-dimensional composition algebras C and D, respectively. Moreover R = C ⊗ D.

Well, this is not quite true: The decomposition X = Y ⊕ Z might be defined only after passing to a quadratic extension E/F, whose Galois group interchanges the two factors. In fact, there are elements in the normalizer of G_{2} × G_{2} which interchange the two factors.

To make this more precise we compute the normalizer of G_{2} × G_{2}:

Lemma 3. The normalizer of G_{2} × G_{2} in Spin(7, 7) is (G_{2} × G_{2}) ⋊ μ_{8}. Here a generator ω ∈ μ_{8} acts on G_{2} × G_{2} via (g, h) ↦ (h, g).

Proof. Every automorphism of G_{2} is inner and G_{2} → GL(7) is an irreducible representation. Therefore the normalizer of G_{2} × G_{2} in GL(14) is

((G_{2} × G_{m}) × (G_{2} × G_{m})) ⋊ Z/2.

Intersecting this group with SO(7, 7) yields the normalizer of G_{2} × G_{2} in SO(7, 7):

(G_{2} × G_{2}) ⋊ μ_{4}

where μ_{4} is considered as a subgroup of SO(7, 7) via

ζ ↦ τ(ζ) := ζ (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) .

The extension μ_{2} → Spin(7, 7) → SO(7, 7) is nontrivial on the subgroup τ(μ_{4}) ⊂ SO(7, 7), whence the claim.

To be more specific, we describe μ_{8} as a subgroup of Spin(7, 7) explicitly. Let v_{1}, …, v_{7}, w_{1}, …, w_{7} be an orthogonal basis with ⟨v_{i}, v_{i}⟩ = 1 and ⟨w_{i}, w_{i}⟩ = −1. Consider the element

ω = ∏_{i=1}^{7} (\frac{1 + ζv_{i}w_{i}}{\sqrt{2}} ) ∈ Spin(7, 7) ⊂ C_{0}(7, 7)

where ζ is a primitive 4-th root of unity. Then ω is of order 8 and its image in SO(7, 7) is τ(ζ). □

One concludes that the map

H^{1}(F, (G_{2} × G_{2}) ⋊ μ_{8}) → H^{1}(F, Spin(7, 7))

is surjective.

This gives the mentioned result on 14-dimensional quadratic forms.

5. On the cases dim D = 4, dim C = 4, 8.

5.1. The case dim C = 8 and dim D = 4. Here the isotropy group is G_{2} with normalizer G_{2} × Spin(3) · μ_{4}.

The associativity of D shows that the group action is compatible with the right D-module structure of R = C ⊗ D. Moreover R = S ⊕ S, where S is a fundamental spinor representation of Spin([p − q]).

Suppose that D is split, D = End(K) where K is a 2-dimensional vector space. Then S is the tensor product over D of R with K so that S = C ⊗ K. A computation similar as above shows that the Spin([p − q])-group action on S has a dense orbit.

5.2. The case dim C = dim D = 4. Here the isotropy group is trivial. If dim D = dim C = 4, then R is an algebra and the group action is given by multiplication from the left with Spin([p − q]) = SL(R). In this case the form ψ is the reduced norm on R.

NWF I - Mathematik, Universität Regensburg, D-93040 Regensburg, Germany

   E-mail address: markus.rost@mathematik.uni-regensburg.de
   URL: http://www.physik.uni-regensburg.de/~rom03516