A rational construction of Lie algebras of type E_{7}

Victor Petrov^{∗}

October 10, 2013

Abstract

We give an explicit construction of Lie algebras of type E_{7} out of a Lie algebra of type D_{6} with some restrictions. Up to odd degree extensions, every Lie algebra of type E_{7} arises this way. Some applications to Tits algebras and Rost invariant are mentioned.

1 Introduction

In [13] Jacques Tits wrote the following: “It might be worthwile trying to develop a similar theory for strongly inner groups of type E_{7}. For instance, can one give a general construction of such groups showing that there exist anisotropic strongly inner K-groups of type E_{7} as soon as there exist central division associative 16-dimensional K-algebras of order 4 in Br K whose reduced norm is not surjective?”

The goal of the present paper is to give such (and much more general) construction. We deal with Lie algebras; of course, the corresponding group is just the automorphism group of its Lie algebra. By rational constructions we mean those not appealing to the Galois descent, that is involving only terms defined over the base field.

∗This research is supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government grant 11.G34.31.0026. Research is supported by JSC “Gazprom Neft”. Partially supported by RFBR grants 12-01-31100, 13-01-00429, 13-01-00709, 13-01-91150, and 13-01-92699

Let us recall several milestones in the theory. Freudenthal in [5] gave an elegant explicit construction of the split Lie algebra of type E_{7}. On the language of maximal Lie subalgebras it is a particular case of A_{7}-construction. Another approach was proposed by Brown in [3] (see also [6] for a recent exposition); this is an E_{6}-construction. It gives only isotropic Lie algebras. In full generality A_{7}-construction was described by Allison and Faulkner in [1] as a particular case of a Cayley-Dickson doubling; generically it produces anisotropic Lie algebras of type E_{7}. Another construction with this property was discovered by Tits in [13]; in our terms it is an A_{3}+A_{3}+A_{1}-construction. On the other hand, some Lie algebras of type E_{7} can be obtained via the Freudenthal magic square, see [11] (or [7] for a particular case).

Our strategy is to define a Lie triple system structure on the (64-dimen-sional over F) simple module of the even Clifford algebra of a central simple algebra of degree 12 with an orthogonal involution under some restrictions. Then the embedding Lie algebra is of type E_{7}. Our construction is of type D_{6} + A_{1}.

The author is grateful to Ivan Panin, Anastasia Stavrova and Alexander Luzgarev for discussing earlier attempts to this work.

2 Lie triple systems and quaternionic gifts

Let F be a field of characteristic not 2. Recall that a Lie triple system is a vector space W over F together with a trilinear map

W × W × W → W
(u, v, w) ↦ [u, v, w] = D(u, v)w

satisfying the following axioms:

D(u, u) = 0

D(u, v)w + D(v, w)u + D(w, u)v = 0

D(u, v)[x, y, z] = [D(u, v)x, y, z] + [x, D(u, v)y, z] + [x, y, D(u, v)z].

A derivation is a linear map D: W → W such that

D[x, y, z] = [Dx, y, z] + [x, Dy, z] + [x, y, Dz].

The vector space of all derivations form a Lie algebra Der (W) under the usual commutator map.

The vector space Der (W) ⊕ W under the map

[D + u, E + v] = [D, E] + D(u, v) + Dv − Eu

form a ℤ /2-graded Lie algebra called the embedding Lie algebra of W. Conversely, degree 1 component of any ℤ /2-graded Lie algebra is a Lie triple system under the triple commutator map.

Consider a ℤ-graded Lie algebra

L = L_{−2} ⊕ L_{−1} ⊕ L_{0} ⊕ L_{1} ⊕ L_{2}

with one-dimensional components L_{−2} = F f, L_{2} = F e, such that each L_{i} is an eigenspace of the map [[e, f], ·] with the eigenvalue i. Then e, f and [e, f] form an 𝔰𝔩_{2}-triple, and the maps [e, ·] and [f, ·] are mutualy inverse isomporphisms of L_{−1} and L_{1}. Moreover, maps [x, ·] with x from ⟨e, f, [e, f]⟩ = 𝔰𝔩_{2} defines a structure of left M_{2}(F)-module on L_{1} ⊕ L_{−1}, that by inspection coincides with the usual structure on F^{2} ⊗ L_{1} (after identification of L_{−1} and L_{1} mentioned above).

Now L defines two kind of structures: one is a Lie triple structure on L_{1} ⊕ L_{−1}, and the other is a ternary system considered by Faulkner in [4] on L_{1} (roughly speaking, it is an asymmetric version of a Freudenthal triple system). Namely, define maps ⟨·, ·⟩ and ⟨·, ·, ·⟩ by formulas

[u, v] = ⟨u, v⟩e
⟨u, v, w⟩ = [[[f, u], v], w].

Note that ⟨·, ·⟩ allows to identify the dual L_{1}^{∗} with L_{1}, and so the map

L_{1} ⊗ L_{1} → End (L_{1})

corresponding to ⟨·, ·, ·⟩ produces a linear map

π: End (L_{1}) → End (L_{1}),

namely

π(⟨·, u⟩v) = ⟨u, v, w⟩.

By the Morita equivalence, we can consider π as a map

End_{M_{2}(F)}(F^{2} ⊗ L_{1}) → End_{M_{2}(F)}(F^{2} ⊗ L_{1}).

Also, the same equivalence gives rise to a Hermitian (with respect to the canonical symplectic involution on M_{2}(F)) form

ϕ ((\begin{matrix}u_{1} \\ u_{2} \end{matrix}) , (\begin{matrix}v_{1} \\ v_{2} \end{matrix})) = (\begin{matrix}⟨u_{1}, v_{2}⟩ & −⟨u_{1}, v_{1}⟩ \\ ⟨u_{2}, v_{2}⟩ & −⟨u_{2}, v_{1}⟩ \end{matrix}) .

Now we want to relate the two structures on V_{1} ⊕ V_{−1} ≃ F^{2} ⊗ V_{1}. Direct calculation shows that

D(u, v) = \frac{1}{2}(π(ϕ(·, u)v − ϕ(·, v)u) + ϕ(v, u) − ϕ(u, v)).  (*)

This description admits a Galois descent. Namely, let Q be a quaternion algebra over F, W be a left Q-module equipped with a Hermitian (with respect to the canonical involution on Q) form ϕ and a linear map

π: End_{Q}(W) → End_{Q}(W).

Assume that ϕ and π become maps as above over a splitting field of Q. In terms of [7] this means that End_{Q}(W) together with π and the symplectic involution adjoint to ϕ form a gift (an abbreviation for a generalized Freudenthal (or Faulkner) triple); one can state the conditions on π and ϕ as a list of axioms not appealing to the descent (Garibaldi assumes that W is of dimension 28 over Q, but this really doesn’t matter, at least under some additional restrictions on the characteristic of F). Then equation (*) defines on W a structure of a Lie triple system, hence the embedded Lie algebra Der (W) ⊕ W.

3 D_{6} + A_{1}-construction

We say that a map of functors A → B from fields to sets is surjective at 2 if for any field F and b ∈ B(F) there exists an odd degree separable extension E/F and a ∈ A(E) such that the images of a and b in B(E) coincide.

We enumerate simple roots as in [2]. Erasing vertex 1 from the extended Dynkin diagram of E_{7} we see that the simply connected split group E_{7}^{sc} contains a subgroup of type D_{6} + A_{1}, namely (Spin_{12} × SL_{2})/μ_{2}. Its image in the adjoint group E_{7}^{ad} is (HSpin_{12} × SL_{2})/μ_{2}, which we denote by H for brevity.

Theorem 1. The map H^{1}(F, H) → H^{1}(F, E_{7}^{ad}) is surjective at 2.

Proof. Note that W (D_{6} + A_{1}) and W (E_{7}) has the same Sylow 2-subgroup. Then the result follows by repeating the argument from the proof of Proposition 14.7, Step 1 in [8] (this argument is a kind of folklore).

The long exact sequence

H^{1}(F, μ_{2}) → H^{1}(F, H) → H^{1}(PGO_{12}^{+} × PGL_{2}) → H^{2}(F, μ_{2})

shows that the orbits of H^{1}(F, H) under the action of H^{1}(F, μ_{2}) are the isometry classes of central simple algebras of degree 12 with orthogonal involutions (A, σ) and fixed isomorphism Cent(C_{0}(A, σ)) ≃ F ×F, with [C_{0}^{+}(A, σ)] = [Q] in Br(F) for some quaternion algebra Q, where C_{0} stands for the Clifford algebra and C_{0}^{+} for its first component (see [9, § 8] for definitions). Now

C_{0}^{+}(A, σ) ≃ End_{Q}(W)

for a 16-dimensional space W over Q, and the canonical involution on C_{0}^{+}(A, σ) induces a Hermitian form ϕ on W up to a scalar factor. It is not hard to see that H^{1}(F, H) parametrizes all the mentioned data together with ϕ (and not only its similarity class), and H^{1}(F, μ_{2}) multiplies ϕ by the respective constant. Over a splitting field of Q the 32-dimensional half-spin representation carries a structure of Faulkner ternary system, so we are in the situation of Section 2. The resulting embedding 66 + 3 + 64 = 133-dimensional Lie algebra Der (W) ⊕ W is the twist of the split Lie algebra of type E_{7} obtained by a cocycle representing the image in H^{1}(F, E_{7}^{ad}). Theorem 1 shows that any Lie algebra of type E_{7} over F arises this way up to an odd degree extension.

4 Tits algebras and Rost invariant

Recall that the class of Tits algebra of a cocycle class from H^{1}(F, E_{7}^{ad}) is its image under the connecting map of the long exact sequence

H^{1}(F, E_{7}^{sc}) → H^{1}(F, E_{7}^{ad}) → H^{2}(F, μ_{2}).

The sequence fits in the following diagram:

[figure: H^{1}(F, (Spin_{12} × SL_{2})/μ_{2}); H^{1}(F, H); H^{2}(F, μ_{2}); H^{1}(F, E_{7}^{sc}); H^{1}(F, E_{7}^{ad}); H^{2}(F, μ_{2}).]

Since the middle vertical arrow is surjective at 2, we obtain the following result:

Theorem 2. The class of the Tits algebra in Br(F) of the class in H^{1}(F, E_{7}^{ad}) corresponding to Der (W) ⊕ W is [A] + [Q]. For any cocycle class from H^{1}(F, E_{7}^{ad}) there is an odd degree extension E/F such that the class of the Tits algebra in Br(E) is a sum of three symbols.

Proof. Indeed, by the fundamental relation for groups of type D_{6} (see [9, 9.14]) the image of a cocycle class from H^{1}(F, H) in H^{2}(F, μ_{2}) is [A] + [Q]. Here Q is a quaternion algebra, and A over an odd degree extension is Brauerequivalent to an algebra of degree 4 and exponent 2, that is to a biquaternion algebra. The second claim follows from Theorem 1.

Now we reproduce a construction from [13] in our terms. Let D be an algebra of degree 4 and μ be a constant from F^{×}. By the exceptional isomorphism A_{3} = D_{3} the group PGL_{1}(D) defines a 3-dimensional anti-Hermitian form h over Q up to a constant, where [Q] = 2[D] in Br(F). Consider the algebra M_{6}(Q) with the orthogonal involution σ adjoint to the 6-dimensional form h ⊥ −μh. One of the component of C_{0}(M_{6}(Q), σ) is trivial in Br(F) and the other is Brauer-equivalent to Q. Choose ϕ on W = Q^{1}6; by Theorem 2 the class of the Tits algebra of the corresponding cocycle class in H^{1}(F, E_{7}^{ad}) is trivial, so the cocycle class comes from some ξ ∈ H^{1}(F, E_{7}^{sc}). Let us compute the Rost invariant of ξ (see [8] or [9, § 31] for definitions).

Theorem 3. For D and μ as above, there is a cocycle from H^{1}(F, E_{7}^{sc}) whose Rost invariant is (μ) ∪ [D]. In particular, if this element cannot be written as a sum of two symbols from H^{3}(F, ℤ /2) with a common slot, then there is a strongly inner anisotropic group of type E_{7} over F.

Proof. Consider ξ as above. Over the function field F (SB(Q)) the image of the Rost invariant of ξ equals to the Arason invariant of the 12-dimensional quadratic form Morita-equivalent to h − μh. Explicitly, over F (SB(Q)) the algebra D becomes a biquaternion algebra (a, b) ⊗ (c, d), and the quadratic form is Witt equivalent to ⟨⟨μ⟩⟩(⟨⟨a, b⟩⟩ − ⟨⟨c, d⟩⟩), so the Arason invariant equals the image of (μ)∪[D] over F (SB(Q)). It follows that the Rost invariant is (μ) ∪ [D] + (λ) ∪ [Q] for some Q. Changing ϕ to λϕ adds (λ) ∪ [Q] (cf. [9, p. 441]), so there is a cocycle class from H^{1}(F, E_{7}^{sc}) whose Rost invariant is (μ) ∪ [D]. The last claim follows from the easy computation of the Rost invariant of cocycles corresponding to isotropic groups of type E_{7} (cf. [8, Appendix A, Proposition]).

References

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[12] J. Tits, Classification of algebraic semisimple groups, Algebraic groups and discontinuous subgroups, Proc. Sympos. Pure Math. 9, Amer. Math. Soc., Providence RI, 1966, 33–62.

[13] J. Tits, Strongly inner anisotropic forms of simple algebraic groups, J. of Algebra 131 (1990), 648–677.

Victor Petrov PDMI RAS, Nab. Fontanki, 27, Saint Peterburg, 191023 Russia Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia

victorapetrov@googlemail.com