STRUCTURABLE ALGEBRAS AND GROUPS OF TYPE E_{6} AND E_{7}

R. SKIP GARIBALDI

Abstract. It is well-known that every group of type F_{4} is the automorphism group of an exceptional Jordan algebra, and that up to isogeny all groups of type ^{1}E_{6} with trivial Tits algebras arise as the isometry groups of norm forms of such Jordan algebras. We describe a similar relationship between groups of type E_{6} and groups of type E_{7} and use it to give explicit descriptions of the homogeneous projective varieties associated to groups of type E_{7} with trivial Tits algebras. We also show that the kernel of the Rost invariant for quasi-split groups of type E_{6} and E_{7} is trivial.

It is well-known that over an arbitrary field F (which for our purposes we will assume has characteristic ≠ 2, 3) every algebraic group of type F_{4} is obtained as the automorphism group of some 27-dimensional exceptional Jordan algebra and that some groups of type E_{6} can be obtained as automorphism groups of norm forms of such algebras.

In [Bro63], R.B. Brown introduced a new kind of F -algebra, which we will call a Brown algebra. The automorphism groups of Brown algebras provide a somewhat wider class of groups of type E_{6}, specifically all of those with trivial Tits algebras. Allison and Faulkner [AF84] showed that there is a Freudenthal triple system (i.e., a quartic form and a skewsymmetric bilinear form satisfying certain relations) determined up to similarity by every Brown algebra. The automorphism group of this triple system is a simply connected group of type E_{7}, and we show that this provides a construction of all simply connected groups of type E_{7} with trivial Tits algebras (and more generally all Freudenthal triple systems). This is interesting because it allows one to relate properties of these algebraic groups over our ground field F, which are generally hard to examine, with properties of these algebras, which are relatively much easier to study.

Brown algebras are neither power-associative nor commutative, but they do belong to a wide class of algebras with involution known as central simple structurable algebras which were introduced in [All78], see [All94] for a nice survey. Other examples of such algebras are central simple associative algebras with involution and Jordan algebras. Brown algebras comprise the most poorly understood class of central simple structurable algebras. Although they are simple and so by definition have no 2-sided ideals which are stable under the involution, they do have so-called “inner ideals”, and we study them in Section 6. In particular, the largest inner ideal in a Brown algebra is 12-dimensional and the largest singular ideal is 7-dimensional.

1991 Mathematics Subject Classification. 17A30 (11E72 14L30 17A40 17B25 17C30 20G15).

Original: November 1998, Revised: April 1999.

In Section 7, we produce descriptions of the homogeneous projective (a.k.a. twisted flag) varieties associated to groups of type E_{7} with trivial Tits algebras. (These varieties are essentially the spherical building associated to the group, see [Bro89, Ch. V] or [Tit74, Ch. 5].) In another paper [Gar99a], I define objects called gifts (short for generalized Freudenthal triple systems) whose automorphism groups produce all groups of type E_{7} over an arbitrary field up to isogeny. The description of the flag varieties here immediately gives a description of the homogeneous projective varieties for arbitrary groups of type E_{7} in terms of gifts, which answers the question raised in [MPW98, p. 143].

Finally, in Section 8 we discuss cohomological invariants of quasi-split simply connected groups of type E_{6} and E_{7}. Specifically, for G simple simply connected, Rost has defined a canonical invariant

R_{G}: H^{1}(F, G) ⟶ H^{3}(F, ℚ/ℤ(2)) for ℚ/ℤ(2) := lim_{m∈ℕ^{⟶}}μμ_{m}^{⊗2}, (0.1)

where μμ_{m}^{⊗2} ⟶ μμ_{n}^{⊗2} whenever m divides n, see [KMRT98, §31] for more details. Determining the kernel of this invariant seems to be out of the question for general G at the moment. For example, the question of whether R_{G} has trivial kernel for arbitrary groups of type F_{4} is perhaps the principal open question in the study of Albert algebras. On the other hand, the question is tractable if we assume that G is quasi-split. For example, it is easy to find a field F over which the kernel of R_{G} is nontrivial for G split of type ^{1}D_{8}. On the other hand, the exceptional groups fall into a series

G_{2}, D_{4}, F_{4}, E_{6}, E_{7}, E_{8}

and the material in [KMRT98] is sufficient to show that the kernel of R_{G} is trivial if G is quasi-split of type G_{2}, D_{4}, or F_{4}. We show that the kernel is also trivial for G quasi-split of type E_{6} and E_{7}, which answers a question posed by Rost.

Notational conventions. All fields that we consider will have characteristic ≠ 2, 3. For a field F, we write F_{s} for its separable closure.

For g an element in a group G, we write Int (g) for the automorphism of G given by x ↦ gxg^{−1}.

For X a variety over a field F and K any field extension of F, we write X(K) for the K-points of X.

When we say that an affine algebraic group (scheme) G is simple, we mean that it is absolutely almost simple in the usual sense (i.e., G(F_{s}) has a finite center and no noncentral normal subgroups). For any simple algebraic group G over a field F, there is a unique minimal finite Galois field extension L of F such that G is of inner type over L (i.e., the absolute Galois group of L acts trivially on the Dynkin diagram of G). We call L the inner extension for G.

We write 𝔾_{m,F} for the algebraic group whose F -points are F^{∗} and μμ_{n} for the group of nth roots of unity.

We will also follow the usual conventions for Galois cohomology and write H^{i}(F, G) := H^{i}(Gaℓ (F_{s}/F), G(F_{s}) for G any algebraic group over F, and similarly for the cocycles Z^{1}(F, G). For more information about Galois cohomology, see [Ser79] and [Ser94].

For a, b ∈ F^{∗}, we write (a, b)_{F} for the (associative) quaternion F -algebra generated by skew-commuting elements i and j such that i^{2} = a and j^{2} = b, please see [Lam73] or [Dra83, §14] for more information.

One oddity of the presentation should be pointed out to the reader. We will be doing some explicit computations with Cayley algebras (see [Sch66, Ch. III, §4] or [KMRT98, §33.C] for a definition), but not with their usual multiplication. Instead, with juxtaposition denoting the usual product and π the standard involution, we will use the multiplication ⋆ defined by x ⋆ y := π(x)π(y). This ⋆ multiplication is not even power-associative, but it has some advantages when doing computations with exceptional groups. We will also make use of a particular basis u_{1}, u_{2}, …, u_{8} of the split Cayley algebra ℭ^{d}, which has the following multiplication table where each entry is x ⋆ y and “·” replaces zero for clarity of reading:

\begin{aligned}& & & & & & y \\ & & u_{1} & u_{2} & u_{3} & u_{4} & & u_{5} & u_{6} & u_{7} & u_{8} \\ & u_{1} & · & · & · & −u_{1} & & · & −u_{2} & u_{3} & −u_{4} \\ & u_{2} & · & · & u_{1} & · & & −u_{2} & · & −u_{5} & −u_{6} \\ & u_{3} & · & −u_{1} & · & · & & −u_{3} & −u_{5} & · & u_{7} \\ x & u_{4} & · & −u_{2} & −u_{3} & u_{5} & & · & · & · & −u_{8} \\ & u_{5} & −u_{1} & · & · & · & & u_{4} & −u_{6} & −u_{7} & · \\ & u_{6} & u_{2} & · & −u_{4} & −u_{6} & & · & · & −u_{8} & · \\ & u_{7} & −u_{3} & −u_{4} & · & −u_{7} & & · & u_{8} & · & · \\ & u_{8} & −u_{5} & u_{6} & −u_{7} & · & & −u_{8} & · & · & · \end{aligned} (0.2)

For more discussion about this multiplication, please see [Gar98, §1].

1. Background on Albert algebras

An Albert algebra over a field F is a 27-dimensional central simple exceptional Jordan algebra. (Some of these adjectives are redundant.) Good introductions to Albert algebras may be found in [PR94a] or [Jac68, Ch. IX], but we will recall what we need here.

Example 1.1. Let ℭ be a Cayley F -algebra and let γ ∈ GL_{3}(F) be a diagonal matrix. Let ∗ denote the conjugate transpose on M_{3}(ℭ). We write ℌ_{3}(ℭ, γ) for the subspace of M_{3}(ℭ) fixed by Int (γ) ∘ ∗ and endowed with a symmetrized product · given by

a · b := \frac{1}{2}(ab + ba),

where juxtaposition denotes the usual product in M_{3}(ℭ). Then ℌ_{3}(ℭ, γ) is an Albert F - algebra.

An Albert algebra is called split if it is isomorphic to ℌ_{3}(ℭ, 1) for ℭ the split Cayley algebra. It is called reduced if it is isomorphic to one as in the preceding example. We will want to do some explicit computations in reduced Albert aglebras ℌ_{3}(ℭ, γ) for γ = diag (γ_{0}, γ_{1}, γ_{2}). For simplicity of notation we will write

(1.2)  ( \begin{matrix}ε_{0} & c & · \\ · & ε_{1} & a \\ b & · & ε_{2} \end{matrix} ) instead of ( \begin{matrix}ε_{0} & c & γ_{0}^{−1}γ_{2}\overline{b} \\ γ_{1}^{−1}γ_{0}\overline{c} & ε_{1} & a \\ b & γ_{2}^{−1}γ_{1}\overline{a} & ε_{2} \end{matrix} ) ,

since the entries we have replaced with a · are forced by the fact that elements of ℌ_{3}(ℭ, γ) are fixed by the involution.

Every Albert F -algebra J is endowed with a cubic norm map N : J ⟶ F and a linear trace map T: J ⟶ F. We also use T to denote the map T: J × J ⟶ F given by

T (x, y) := T (xy).

This is a symmetric bilinear form on J, and it is nondegenerate [Jac68, p. 240, Thm. 5]. For any f ∈ End_{F} (J), we denote the adjoint of f with respect to T by f^{∗}, so T (fx, y) = T (x, f^{∗}y) for all x, y ∈ J. For simplicity, if f is invertible we write f^{†} for (f^{−1})^{∗} = (f^{∗})^{−1}.

Now if one extends scalars to F (t), expands N(x + ty) and considers the coefficient of t, then for a fixed x ∈ J this provides a linear map f_{x}: J ⟶ F given by substituting in for y. Since T is nondegenerate, there is an element x^{#} ∈ J such that T (x^{#}, y) = f_{x}(y) for all y ∈ J. Then # provides a quadratic map #: J ⟶ J, cf. [McC69, pp. 495, 496]. We define a linearization of # called the Freudenthal cross product by

x × y := (x + y)^{#} − x^{#} − y^{#},

as in [McC69, p. 496]. Note that x × x = 2x^{#}, which differs by a factor of 2 from the definition of × given in [Jac68].

Definition 1.3. Let J be an Albert algebra over F. We call an F -vector space map φ : J ⟶ J a norm similarity if there is some λ ∈ F^{∗} such that N(φ(j)) = λN(j) for every j ∈ J. In that case, we call λ the multiplier of φ. If φ is a norm similarity with multiplier 1, we call φ a norm isometry.

Algebraic groups. For J an Albert F -algebra, we define Inv (J) to be the algebraic group whose F -points are the norm isometries of J. (This is Freudenthal’s notation from [Fre54, §1].) This is a simple simply connected algebraic group of type ^{1}E_{6} over F.

Associated to any semisimple algebraic group are its Tits algebras, which are the endomorphism rings of irreducible representations, see [KMRT98, §27] or [Tit71]. In general, they are central simple algebras over finite separable extensions of F. We say that a group has trivial Tits algebras if all of them are split.

There is a strong connection between Albert F -algebras and groups of type ^{1}E_{6} over F with trivial Tits algebras. We summarize this in the following theorem. In 4.10 and 4.15 we will show that there is a similar connection between Brown F -algebras (defined in 2.7) and groups of type E_{7} over F with trivial Tits algebras.

Theorem 1.4. (1) Every simple simply connected group of type ^{1}E_{6} over F with trivial Tits algebras is isomorphic to Inv (J) for some Albert F -algebra J.

(2) For J_{1}, J_{2} Albert F -algebras, the following are equivalent: (a) Inv (J_{1}) ≅ Inv (J_{2})

(b) J_{1} and J_{2} have similar norm forms

(c) J_{1} ∼ J_{2} (i.e., J_{1} is isotopic to J_{2}, see below).

In the statement of the preceding theorem, we used the notion of isotopy of Jordan algebras which provides an equivalence relation for such algebras which is weaker than isomorphism. Specifically, for u ∈ J, we define a new Jordan algebra J^{⟨u⟩} which has the same underlying vector space as J and whose multiplication ·_{u} is given by

x ·_{u} y := {a, u, b} = (a · u) · b + (b · u) · a − (a · b) · u, (1.5)

where · denotes the usual multiplication in J. We say that another Jordan algebra J′ is isotopic to J (written J′ ∼ J) if J′ is isomorphic to J^{⟨u⟩} for some u ∈ J.

Proof: (1) follows from [Tit71, 6.4.2]. That (2b) implies (2a) is clear, and the converse is [Jac71, p. 38, Thm. 7]. Finally, (2a) is equivalent to (2c) by [Jac71, p. 55, Thm. 10].

Useful lemmas. A very useful fact for us is that if J is a reduced Albert F -algebra, then there is a norm similarity of J with multiplier λ for every λ ∈ F^{∗}. Such a similarity is given by ψ for

ψ ( \begin{matrix}ε_{0} & c & · \\ · & ε_{1} & a \\ b & · & ε_{2} \end{matrix} ) = ( \begin{matrix}λε_{0} & λc & · \\ · & λε_{1} & a \\ b & · & λ^{−1}ε_{2} \end{matrix} ) . (1.6)

Lemma 1.7. Suppose that φ is a norm similarity of an Albert F -algebra J with multiplier λ. Then φ^{†} is a norm similarity for J with multiplier 1/λ,

φ(j_{1}) × φ(j_{2}) = λφ^{†}(j_{1} × j_{2}), and φ^{†}(j_{1}) × φ^{†}(j_{2}) = \frac{1}{λ}φ(j_{1} × j_{2}).

Proof: Since these formulas hold if and only if they hold over a field extension of F, we may assume that F is algebraically closed. Let ℓ ∈ F^{∗} be such that ℓ^{3} = λ. Then φ_{1} := \frac{1}{ℓ}φ is a norm isometry of J. Since the conclusions hold for φ_{1} and φ_{1}^{†} by [Jac61, p. 76] and × is bilinear, we are done.

2. Brown algebras and groups of type E_{6}

Definition 2.1. [All78, p. 135], [AF84, 1.1] Suppose that (A, −) is a finite-dimensional (and perhaps nonassociative) F -algebra with F -linear involution. For x, y ∈ A, define V_{x,y} ∈ End_{F} (A) by

V_{x,y}z := \{x, y, z\} := (x\overline{y})z + (z\overline{y})x − (z\overline{x})y, (2.2)

for z ∈ A. One says that (A, −) is a structurable algebra if

[V_{x,y}, V_{z,w}] = V_{V_{x,y}z,w} − V_{z,V_{y,x}w}.

The multiplication algebra of (A, −) is the (associative) subalgebra of End_{F} (A) generated by the involution −, left multiplications by elements of A, and right multiplications by elements of A. If the center of the multiplication algebra of (A, −) is F, then (A, −) is said to be central.

We say that (A, −) is simple if it has no two-sided ideals which are stabilized by −.

This definition of a structurable algebra in terms of this V operator may seem unmotivated. There is, however, an alternative (partial) characterization which works for the case that we are interested in. Suppose that (A, −) is an F -algebra with F -linear involution which is generated as an F -algebra by its space of symmetric elements. Then by [All78, p. 144] (A, −) is structurable if and only if it is skew-alternative (i.e., [s, x, y] = −[x, s, y] for all x, y ∈ A and s skew in A where [x, y, z] := (xy)z − x(yz)) and it supports a symmetric bilinear form ⟨,⟩ which satisfies

⟨x̄, ȳ⟩ = ⟨x, y⟩ and ⟨zx, y⟩ = ⟨x, z̄y⟩

for all x, y, z ∈ A.

Basic examples of structurable algebras are Jordan algebras (with involution the identity) and central simple algebras with involution. For Jordan algebras, the ternary product {, , } given in (2.2) is the usual triple product as in (1.5) or [Jac68, p. 36, (58)] and the symmetric bilinear form is the trace form T.

Example 2.3. [All90, 1.9] Let J be an Albert F -algebra and ζ ∈ F^{∗}. We define a structurable algebra (B, −) := B(J, F × F, ζ) by setting B to be the vector space

( \begin{matrix}F & J \\ J & F \end{matrix} )

with multiplication given by

( \begin{matrix}α_{1} & j_{1} \\ j′_{1} & β_{1} \end{matrix} ) ( \begin{matrix}α_{2} & j_{2} \\ j′_{2} & β_{2} \end{matrix} ) = ( \begin{matrix}α_{1}α_{2} + ζT (j_{1}, j′_{2}) & α_{1}j_{2} + β_{2}j_{1} + ζ(j′_{1} × j′_{2}) \\ α_{2}j′_{1} + β_{1}j′_{2} + j_{1} × j_{2} & β_{1}β_{2} + ζT (j_{2}, j′_{1}) \end{matrix} ) .

Endow B with the involution − given by

\overline{( \begin{matrix}α & j \\ j′ & β \end{matrix} )} = ( \begin{matrix}β & j \\ j′ & α \end{matrix} ) .

We use the abbreviation B(J, F × F) for B(J, F × F, 1).

This is a central simple structurable algebra, and is denoted by M(ζT, ζN, ζ^{2}N) in [AF84] for T and N the trace and norm of J respectively.

The study of such algebras precedes the notion of structurable algebras significantly: structurable algebras were introduced in [All78] and these algebras are a special case of those discussed in [Bro64] and [Bro63]. To be precise, the algebras that Brown studied involved parameters μ, ν, ω_{1}, ω_{2}, δ_{1}, and δ_{2}. If one sets μ = ν = ω_{1} = 1 and ω_{2} = δ_{1} = δ_{2} = ζ, the algebra B(J, F × F, ζ) is obtained.

Example 2.4. If Δ is a quadratic field extension of F, we define a structurable algebra B(J, Δ). There is an “outer” automorphism ϖ of B(J, F × F) given by

ϖ ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) := ( \begin{matrix}β & j′ \\ j & α \end{matrix} ) . (2.5)

Let ι denote the unique nontrivial F -automorphism of Δ and set B(J, Δ) to be the F - subalgebra of the Δ-algebra B(J, F ×F)⊗_{F} Δ fixed by ϖ ⊗ι. Then B(J, Δ) is a structurable algebra over F and

B(J, Δ) ⊗_{F} Δ ≅ B(J ⊗_{F} Δ, Δ × Δ). (2.6)

Thus the algebra B(J, Δ) is also a central simple structurable algebra by descent.

Definition 2.7. For J^{d} the split Albert algebra over F, we call B^{d} := B(J^{d}, F × F) the split Brown algebra over F and B_{Δ/F}^{q} := B(J^{d}, Δ) the quasi-split Brown algebra with inner extension Δ.

We say that an F -algebra with involution (B, −) is a Brown algebra if (B, −) ⊗_{F} F_{s} ≅ B^{d} ⊗_{F} F_{s} for F_{s} a separable closure of F.

By the classification of central simple structurable algebras due to Smirnov (see [Smi90] or [Smi92]) and Allison, if F has characteristic ≠ 5 (and, as always, ≠ 2, 3), we could have equally well defined a Brown F -algebra to be a central simple structurable algebra over F of dimension 56 and skew-dimension 1 (i.e. the space of skew-symmetric elements is 1-dimensional). I do not know of a classification theorem for central simple structurable algebras in characteristic 5.

In any event, any Brown algebra (B, −) has a 1-dimensional space of skew-symmetric elements. If s_{0} ∈ B spans this space, then s_{0}^{2} ∈ F^{∗}, which one can see by descent from the split case or see [AF84, 2.1(b)] for a different argument. We say that (B, −) is of type 1 if s_{0}^{2} is a square in F and that it is of of type 2 otherwise. We call Δ := F [s_{0}] the discriminant algebra of (B, −).

It is worth mentioning that not all Brown algebras of type 2 are as in Example 2.4, see Remark 5.13.

Lemma 2.8. (1) [AF84, 4.5] Any Brown algebra of type 1 is isomorphic to some algebra

of the form B(J, F × F, ζ).

(2) For Albert F -algebras J_{1} and J_{2}, B(J_{1}, F ×F, ζ_{1}) ≅ B(J_{2}, F ×F, ζ_{2}) if and only if there is a norm similarity J_{1} ⟶ J_{2} which has multiplier ζ_{1}/ζ_{2} or ζ_{1}/ζ_{2}^{2}.

(3) If J is reduced then B(J, F × F, ζ) ≅ B(J, F × F) for all ζ ∈ F^{∗}.

Before we proceed with the proof, it is should be noted that any obvious analogue of statement (2) for Brown algebras of type 2 is false. Specifically, let J_{1} := ℌ_{3}(ℭ, 1) and J_{2} := ℌ_{3}(ℭ, γ) for ℭ the Cayley division algebra over the reals and γ := diag (1, −1, 1). Then J_{1} and J_{2} have isometric norms. However, for B_{i} := B(J_{i}, ℂ), the Lie algebra of Aut^{+} (B_{i}) is classically denoted by L(J_{i})_{−1}. The two Lie algebras L(J_{i})_{−1} for i = 1, 2 are not isomorphic since they have different signatures, see [Jac71, pp. 119, 120].

Proof: (1) boils down to the main results of the paper [Spr62] of Springer, see [AF84, 4.5]. (3): We have an isomorphism π: B(J, F × F, ζ) ⟶ B(J, F × F, ζ^{2}) given by (2), ⟹ : Suppose that τ: B(J_{1}, F ×F, ζ_{1}) ⟶ B(J_{2}, F ×F, ζ_{2}) is an algebra isomorphism.

We may think of

s_{0} := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )

as a skew-symmetric element in B(J_{i}, F × F) for i = 1, 2. Since τ(s_{0}) must also be skewsymmetric and τ(s_{0})^{2} = τ(s_{0}^{2}) = 1, we must have that τ(s_{0}) = ±s_{0}.

Suppose first that τ(s_{0}) = s_{0}. Then τ fixes the diagonal matrices elementwise and so τ is given by

τ ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}α & φ(j) \\ φ′(j′) & β \end{matrix} )

for some linear maps φ, φ′: J_{1} ⟶ J_{2}. Since τ is an algebra isomorphism, ζ_{2}T_{2}(φ(j_{1}), φ′(j′_{1})) = ζ_{1}T_{1}(j_{1}, j′_{1}) for T_{i} the trace form on J_{i} and j_{1}, j′_{1} ∈ J_{1}. Let φ^{†}: J_{1} ⟶ J_{2} be the unique linear map such that T_{2}(φ(j_{1}), φ^{†}(j′_{1})) = T_{1}(j_{1}, j′_{1}) for all j_{1}, j′_{1} ∈ J_{1}. Then φ′ = (ζ_{2}/ζ_{1})φ^{†}. Since

φ(j_{1}) × φ(j′_{1}) = \frac{ζ_{1}}{ζ_{2}} φ^{†}(j_{1} × j′_{1}),

we have

N_{2}(φ(j)) = \frac{ζ_{1}}{ζ_{2}} N_{1}(j).

Now suppose that τ(s_{0}) = −s_{0}. We have an isomorphism π: B(J_{2}, F ×F, ζ_{2}) ⟶^{∼} B(J_{2}, F × F, ζ_{2}^{2}) given by

π ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}β & j′ \\ ζ_{2}^{−1}j & α \end{matrix} ) .

Composing τ with π, we get a new isomorphism πτ : B(J_{1}, F × F, ζ_{1}) ⟶ B(J_{2}, F × F, ζ_{2}^{2}) such that πτ(s_{0}) = s_{0}. By our previous result, there must be a norm similarity between J_{1} and J_{2} with multiplier ζ_{1}/ζ_{2}^{2}.

(2), ⟸ : Conversely, if one is given such a norm similarity φ, then one can run the argument just given for the other direction backwards to produce the desired algebra isomorphism.

(3) is clear from (2), since for any ζ ∈ F^{∗}, J has a norm similarity with multiplier ζ as in (1.6).

Theorem 2.9. (1) If (B, −) is a Brown algebra of type t over F, then Aut^{+} (B, −) is a simply connected group of type ^{t}E_{6} over F with trivial Tits algebras. Every simply connected group of type E_{6} with trivial Tits algebras arises in this way.

(2) The automorphism group of the split Brown algebra, Aut^{+} (B^{d}), is the split simply connected group of type E_{6}. For Δ a quadratic field extension of F, Aut^{+} (B_{Δ/F}^{q}) is the quasi-split simply connected group of type ^{2}E_{6} with inner extension Δ.

Proof: (2): Let f be an F -algebra automorphism of B^{d}. Since f respects the involution − on B^{d}, it must map the 1-dimensional subspace S of skew-symmetric elements to itself. Since S and F · 1 span the diagonal elements of B^{d}, f must preserve these. Set u := ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ). Then the diagonal matrices are spanned by u and \overline{u}. Set ( \begin{matrix}a & 0 \\ 0 & b \end{matrix}) := f(u). Then

( \begin{matrix}ab & 0 \\ 0 & ab \end{matrix} ) = f(u)\overline{f(u)} = f(u)f(\overline{u}) = 0.

Thus a = 0 or b = 0, but not both.

Suppose for the moment that b = 0. Then since f is a Brown algebra automorphism,

f ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}α & φ(j) \\ φ^{†}(j′) & β \end{matrix} ) (2.10)

for φ some norm isometry of J^{d}.

Otherwise a = 0 and ϖf is of the form just described in the “b = 0” case.

Thus Aut^{+} (B^{d}) is isomorphic to a semidirect product of ℤ_{2} and the group of norm isometries of J^{d}, which is known to be split simply connected of type E_{6}.

Now consider B^{q} := B(J^{d}, Δ). Since B^{q} ⊗_{F} Δ ≅ B^{d} ⊗_{F} Δ, Aut^{+} (B^{q}) is a simply connected of type E_{6}. It has trivial Tits algebras since any Tits algebra would have exponent a power of 3 and would be split by Δ. The center of Aut^{+} (B^{d} ⊗_{F} Δ) consists of the maps f_{ω}, where

f_{ω} ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}α & ωj \\ ω^{2}j′ & β \end{matrix} )

for ω a cube root of unity. To see that Aut^{+} (B^{q}) is of type 2, we need to see that the Galois action on the center of Aut^{+} (B^{d} ⊗_{F} Δ) is not the same as the action induced by the ι-semilinear automorphism ϖ ⊗ ι of B^{d} ⊗_{F} Δ which defines B^{q}:

(Id ⊗ ι)f_{ω}(Id ⊗ ι) ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) ≠ ( \begin{matrix}α & ι(ω)^{2}j \\ ι(ω)j′ & β \end{matrix} ) = (ϖ ⊗ ι)f_{ω}(ϖ ⊗ ι) ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) ≠

This shows that Aut^{+} (B^{q}) is simply connected and Tits-trivial of type ^{2}E_{6} with inner extension Δ.

We have an injection Aut (J^{d}) ↪ Aut^{+} (B^{q}) given by φ ↦ f as in (2.10) which produces a rank four F -split torus in Aut^{+} (B^{q}). The Galois group of Δ over F acts nontrivially on the set of simple roots of the Dynkin diagram of Aut^{+} (B^{q}) which is of type E_{6}, so it has precisely four orbits. Thus Aut^{+} (B^{q}) is quasi-split [Gar98, Lem. 4.2].

(1): Since Aut^{+} (B, −) is a form of Aut^{+} (B^{d}) which is simply connected of type E_{6}, so is Aut^{+} (B, −). Suppose that Δ is the discriminant algebra of (B, −) so that Δ = F × F if (B, −) is of type 1 and (B, −) ⊗_{F} Δ is of type 1 otherwise. Let K be a field extension of F which generically quasi-splits Aut^{+} (B, −) as in [KR94]. Then Aut^{+} (B, −) ×_{F} K ≅ Aut^{+} (B(J^{d} ⊗_{F} K, Δ ⊗_{F} K)), which verifies the type of Aut^{+} (B, −) since F is algebraically closed in K. That one obtains every group of type E_{6} in this manner follows by the usual Galois cohomology argument.

Remark 2.11. Zinovy Reichstein suggested to me that there should be an invariant

g_{4} : H^{1}(F, G) ⟶ H^{4}(F, μμ_{3})

defined for G simply connected of type E_{6}. The theorem we just proved allows us to sketch the definition of such an invariant here in the case where G is quasi-split. By [Rei98, 12.13], this shows that the 3-primary essential dimensional (see [Rei98, 3.1] for a definition) of a simply connected group of type E_{6} is ≥ 4, hence that the overall essential dimension of such a group is ≥ 4. This strengthens [Rei98, 12.11(a)]. (This result has also been obtained by Reichstein and Youssin by other means, see [RY99, 8.12].)

We first observe that as in the construction of the Serre-Rost invariant

g_{3} : H^{1}(F, Aut (J^{d})) ⟶ H^{3}(F, ℤ/3) (2.12)

in [Ros91] or [PR96], since 2 and 3 are coprime we can extend scalars up to a quadratic extension, define g_{4} up there, and then use the corestriction to define g_{4} over the ground field. In particular, this reduces us to considering the case where G is actually split. By Theorem 2.9(2), H^{1}(F, G) classifies pairs (B, ϕ) where B is a Brown F -algebra of type 1 and ϕ is an algebra isomorphism from the discriminant algebra of B to F × F. By Lemma 2.8, we can write any such algebra B as B(J, F × F, ζ) where ϕ is given by ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) ↦ (α, β). Then we define

g_{4}(B, ϕ) := g_{3}(J) ∪ (ζ),

where (ζ) ∈ H^{1}(F, μμ_{3}) and we are identifying ℤ/3 ⊗ μμ_{3} = μμ_{3}.

We must check that this is well-defined. Suppose that (B, ϕ) ≅ (B, ϕ′) where B′ ≅ B(J′, F × F, ζ′). Then by Lemma 2.8(2), there is a norm similarity J ⟶ J′ with multiplier ζ/ζ′. (The other possibility of having multiplier ζ/(ζ′)^{2} doesn’t occur since this requires a switch in the identification of the diagonal matrices with F × F.) Again we may extend scalars to a quadratic extension so that J is a first Tits construction (a quadratic extension is sufficient by, for example, [KMRT98, 39.19]). Then since J and J′ are isotopic, they are even isomorphic by [PR84, 4.9], so

(g_{3}(J) ∪ (ζ)) − (g_{3}(J′) ∪ (ζ′)) = g_{3}(J) ∪ (ζ/ζ′). (2.13)

However, J has a norm similarity with multiplier ζ/ζ′, so its norm represents ζ/ζ′. By [PR84, 4.6], J ≅ J(A, λ) for some central simple F -algebra A of degree 3 and some λ ∈ F^{∗} such that ζ/ζ′ ∈ Nrd_{A}(A^{∗}), so (A) ∪ (ζ/ζ′) = 0. Since g_{3}(J) = (A) ∪ (λ), it follows that the difference in (2.13) of our two possibilities is zero, so our map g_{4} is well-defined.

3. Background on Freudenthal triple systems

Multiple authors have studied Freudenthal triple systems as a means to understanding groups of type E_{7}. Axiomatic treatments appear in [Bro69], [Mey68], and [Fer72], for example. These authors considered a general sort of Freudenthal triple system, but we are only interested in a particular kind.

Definition 3.1. (Cf. [Fer72, p. 314]) A (simple) Freudenthal triple system is a 3-tuple (V, b, t) such that V is a 56-dimensional vector space, b is a nondegenerate skew-symmetric bilinear form on V, and t is a trilinear product t: V × V × V ⟶ V.

We define a 4-linear form q(x, y, z, w) := b(x, t(y, z, w)) for x, y, z, w ∈ V, and we require that

FTS1: q is symmetric,

FTS2: q is not identically zero, and

FTS3: t(t(x, x, x), x, y) = b(y, x)t(x, x, x) + q(y, x, x, x)x for all x, y ∈ V. We say that such a triple system is nondegenerate if the quartic form v ↦ q(v, v, v, v) on V is absolutely irreducible (i.e., irreducible over a separable closure of the base field) and degenerate otherwise.

Note that since b is nondegenerate, FTS1 implies that t is symmetric.

Example 3.2. For J an Albert F -algebra and ζ ∈ F^{∗}, we can construct a Freudenthal triple system as follows. Set

V := ( \begin{matrix}F & J \\ J & F \end{matrix} ) .

As in [All90, 1.10] or [Bro69, (5), (6), p. 87], for

x_{1} = ( \begin{matrix}α_{1} & j_{1} \\ j′_{1} & β_{1} \end{matrix} ) and x_{2} = ( \begin{matrix}α_{2} & j_{2} \\ j′_{2} & β_{2} \end{matrix} ) ,

set

b(x_{1}, x_{2}) = (α_{1}β_{2} − α_{2}β_{1}) + ζ(T (j_{1}, j′_{2}) − T (j′_{1}, j_{2})). (3.3)

Let q be as in the definition of a Freudenthal triple system. Since b is nondegenerate, if we know b and q then we can determine t, at least in principle. In this case, for

x = ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) ,

we define

q(x, x, x, x) = 12 (4αζN(j) + 4βζ^{2}N(j′) − 4ζ^{2}T (j′^{#}, j^{#}) + (αβ − ζT (j, j′))^{2}) . (3.4)

Then (V, b, t) is a nondegenerate Freudenthal triple system. When ζ = 1, it is denoted by 𝔐(J).

By [Bro69, §4], a Freudenthal triple system is nondegenerate if and only if it is a form of a triple system 𝔐(J) for some J (i.e., it becomes isomorphic to 𝔐(J) when one extends scalars to a separable closure). There is a way to distinguish between the two sorts of triple systems over the ground field, see [Gar99a, §2]. Meyberg [Mey68, §7] uses a different terminology; he says that the the nondegenerate ones are “of main type”. It is the nondegenerate ones which are relevant to groups of type E_{7}. (It is not clear what the automorphism group of a degenerate triple system is, but it is not connected and it contains — at least over a separably closed field — a torus of rank 28, see [Gar99a, §1].)

For 𝔐 a Freudenthal triple system over F, we follow Freudenthal’s notation [Fre54, §3] and write Inv (𝔐) for the algebraic group whose F -points are the isomorphisms of 𝔐.

Theorem 3.5. For 𝔐 a nondegenerate Freudenthal triple system over F, Inv (𝔐) is a simple simply connected algebraic group of type E_{7} with trivial Tits algebras. This construction produces all such groups. Moreover, Inv (𝔐(J^{d})) is split.

Proof: The group Inv (𝔐) is simple by [Bro69, p. 100, Thm. 6] and it is of type E_{7} by [Fre54, §5]. Since it has center μμ_{2}, it is simply connected. A simply connected group of type E_{7} is isomorphic to Inv (𝔐) for some 𝔐 if and only if it has trivial Tits algebras by [Tit71, 6.5.2].

Finally we show that Inv (𝔐^{d}) is split for 𝔐^{d} := 𝔐(J^{d}). For an arbitrary Albert F -algebra J and a norm similarity φ of J with multiplier λ ∈ F^{∗}, we have an element f_{φ} ∈ Inv (𝔐(J)) given by

f_{φ} ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) := ( \begin{matrix}λ^{−1}α & φ(j) \\ φ^{†}(j′) & λβ \end{matrix} ) . (3.6)

This restricts to an injection Inv (J) ↪ Inv (𝔐(J)).

Now, Inv (J^{d}) is split of type E_{6}, so let S_{6} denote the image in Inv (𝔐^{d}) of a rank 6 split torus in Inv (J^{d}). Let S_{1} be the image of 𝔾_{m,F} in Inv (𝔐^{d}) under the map x ↦ f_{L_{x}} where L_{x} denotes left multiplication by x. Then S := S_{1}S_{6} is a rank 7 F -split torus in Inv (𝔐^{d}), so the group is split.

Another popular approach to constructing groups of type E_{7} is by realizing the groups as automorphism groups of a particular quartic form which is not the one used here, see [Asc88, §§7, 8] and [Coo95]. The precise relationship between their approach and ours is not clear.

4. Brown algebras and groups of type E_{7}

Let (B, −) be a Brown algebra over F. By definition, the space S of skew-symmetric elements of B has dimension 1, so pick some nonzero s_{0} ∈ B such that S = F s_{0}. There is a natural map ψ: B × B ⟶ S given by

ψ(x, y) := x\overline{y} − y\overline{x}. (4.1)

It is known that for any x ∈ B,

s_{0}(s_{0}x) = (xs_{0})s_{0} = x(s_{0}^{2}) and s_{0}^{2} = μ (4.2) for some μ ∈ F^{∗} by descent or by [All78, p. 135, Prop. 1]. Thus the map

b(x, y) := ψ(x, y)s_{0} (4.3)

provides a skew-symmetric bilinear form on B. We also have a trilinear map t: B×B×B ⟶ B given by

t(y, z, w) := 2{y, s_{0}z, w} − b(z, w)y − b(z, y)w − b(y, w)z. (4.4)

Then b and t give B the structure of a (Freudenthal) triple system by [AF84, 2.18], and we say that (B, b, t) is a triple system associated to (B, −). Of course, this triple system is not uniquely determined: if we choose some other element s′_{0} ∈ S such that S = F s′_{0}, then s′_{0} = λs_{0} for some λ ∈ F^{∗}, and this choice of s′_{0} would give us a skew-symmetric bilinear form b′ and a trilinear form t′ such that b = λb′ and t = λt′.

The definitions of b and t given above may appear to be ad hoc, but in fact they are not. For any structurable algebra with involution −, there is a natural symmetric bilinear trace form given by setting ⟨x, y⟩ to be the trace of left multiplication by x\overline{y} + y\overline{x}. This trace form was instrumental in the classification of central simple structurable algebras in [All78]. In the special case where the structurable algebra is a Brown algebra, any nonzero skew-symmetric element s_{0} spans the skew elements, and b is a scalar multiple of the map (x, y) ↦ ⟨s_{0}x, y⟩. There is also a norm form on any central simple structurable algebra defined up to a scalar [AF92]. This norm specializes to the usual norm in the case where the structurable algebra is Jordan and to the reduced norm when it is an associative central simple algebra. If we write ν for this norm on a Brown algebra, then by [AF84, 2.17] the trilinear map t is given by

b(x, t(x, x, x)) = 12μν(x)

where μ := s_{0}^{2} ∈ F^{∗} as above.

Example 4.5. Consider the Brown algebra B := B(J, F × F, ζ) and let s_{0} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). Then a triple system associated to B is given by the formulas (3.3) and (3.4) [AF84, pp. 192–195]. Since this triple system is a form of 𝔐(J) and all Brown algebras are a form of B, every Freudenthal triple system associated to a Brown algebra is nondegenerate.

It is these triple systems which will provide us with our groups of type E_{7}, and so we need to understand when two Brown algebras provide us with the same group.

Definition 4.6. Let (B_{1}, −) and (B_{2}, −) be two Brown algebras over F, and let (B_{i}, b_{i}, t_{i}) be a triple system related to B_{i} for i = 1, 2. We say that (B_{1}, −) and (B_{2}, −) are similar if there is some F -vector space map f: B_{1} ⟶ B_{2} and some λ ∈ F^{∗} such that

b_{2}(f(x), f(y)) = λb_{1}(x, y) and t_{2}(f(x), f(y), f(z)) = λf(t_{1}(x, y, z)) (4.7)

for all x, y, z ∈ B_{1}. Such an f is said to be a similarity.

If (B_{1}, −) = (B_{2}, −), (B_{1}, b_{1}, t_{1}) = (B_{2}, b_{2}, t_{2}), and f satisfies (4.7), then we say that f is a similarity of (B_{i}, −) with multiplier λ. If λ = 1, f is called an isometry.

Note that these definitions are independent of the choice of triple systems for our Brown algebras.

We define three algebraic groups associated to a Brown algebra B. Set GInv (B) to be the algebraic group with F -points

GInv (B)(F) = {f ∈ End_{F} (B) | f is a similarity}

and let Inv (B) be the algebraic group with F -points

Inv (B)(F) = {f ∈ End_{F} (B) | f is an isometry}.

Set PInv (B) to be the quotient of GInv (B) by its center.

Two Albert algebras have similar norms if and only if they are isotopic. There is a natural notion of isotopy for structurable algebras, and we recall the definition here.

Definition 4.8. [AH81, p. 132] Let (A, −) and (A′, −) be two structurable algebras. They are said to be isotopic (abbreviated (A, −) ∼ (A′, −)) if there are F -linear maps α, β: A ⟶ A′ such that

α{x, y, z} = {α(x), β(y), α(z)}′. (4.9)

In the special case where A = A′ as vector spaces and the two structurable algebras are actually Jordan, this reduces to the standard notion of isotopy. One direction is clear enough: substituting y = 1 in (4.9), we see that this definition specializes to that given in (1.5) with u = β(1). Conversely, any isotopy as classically defined for Jordan algebras induces an isotopy as we have just defined them, see [AH81, p. 83].

It turns out that the notions of similarity and isotopy are equivalent for Brown algebras, so that we have a situation analogous to that with Albert algebras.

Proposition 4.10. Two Brown algebras are similar if and only if they are isotopic.

Proof: Let B := (B, −) and B′ := (B′, −′) denote our two Brown algebras, and let (B, b, t) and (B′, b′, t′) provide triple systems associated to them.

( ⟹ ): Let f : B ⟶ B′ and λ be as in Definition 4.6. Let s_{0}, s′_{0} span the skewsymmetric elements in B and B′. We may assume that b, t and b′, t′ are given by the formulas in (4.3) and (4.4). Then plugging this into (4.7) provides

{f(y), s′_{0}f(z), f(w)}′ = λf{y, s_{0}z, w}

for all x, y, z ∈ B.

We replace z by (λs_{0})^{−1}z and f by g defined by g(z) := s′_{0}f((λs_{0})^{−1}z) to obtain

{f(y), g(z), f(w)}′ = f{y, z, w}.

Thus f is an isotopy.

( ⟸ ): Since B and B′ are isotopic, there is some element u ∈ B such that B′ is isomorphic to a new Brown algebra denoted by B^{⟨u⟩} [AH81, p. 134, Prop. 8.5]. This algebra is a Brown algebra over F with the same underlying vector space as B. Fix some s_{0} which spans the space of skew-symmetric elements of B. Then s_{0}^{⟨u⟩} := s_{0}u ≠ 0 spans the space of skew-symmetric elements of B^{⟨u⟩} by [AF84, 1.16]. Let (B, b, t) and (B, b^{⟨u⟩}, t^{⟨u⟩}) be the triple systems associated to B and B^{⟨u⟩} determined by s_{0} and s_{0}^{⟨u⟩} by the formulas (4.3) and (4.4).

Let ψ^{⟨u⟩} be a map on B^{⟨u⟩} defined as in (4.1), and let ν denote the conjugate norm on B, so ν(x) = \frac{1}{12μ}q(x, x, x, x) for μ := s_{0}^{2} ∈ F^{∗} by [AF84, 2.17]. (Note that this definition of ν is independent of the choice of s_{0}.) For x, y ∈ B, let λ ∈ F^{∗} be such that ψ(x, y) = λs_{0}. Then

b^{⟨u⟩}(x, y) = ψ^{⟨u⟩}(x, y)s_{0}^{⟨u⟩}
= (ψ(x, y)u)s_{0}^{⟨u⟩} by [AF84, 1.17]
= λ(s_{0}^{⟨u⟩})^{2}
= λν(u)μ  by [AF84, 3.2]
= ν(u)b(x, y).

(4.11)

Let μ^{⟨u⟩} := (s_{0}^{⟨u⟩})^{2} = ν(u)μ and let ν^{⟨u⟩} be the conjugate norm on B^{⟨u⟩}. By [AF84, 3.7], ν^{⟨u⟩}(x) = ν(u)ν(x) for all x ∈ B. We can linearize ν^{⟨u⟩} to get a unique symmetric 4-linear form such that ν^{⟨u⟩}(x, x, x, x) = 24ν^{⟨u⟩}(x). Then

b^{⟨u⟩}(x, t^{⟨u⟩}(y, z, w)) = \frac{μ^{⟨u⟩}}{2} ν^{⟨u⟩}(x, y, z, w)
= \frac{ν(u)^{2}μ}{2}ν(x, y, z, w)
= ν(u)^{2}b(x, t(y, z, w)).

By the nondegeneracy of b and (4.11), the identity map on B is a similarity with multiplier ν(u).

Lemma 4.12. (1) B(J, F × F, ζ) ∼ B(J, F × F).

(2) B(J_{1}, F × F) ∼ B(J_{2}, F × F) if and only if J_{1} ∼ J_{2}.

Proof: (1): Define an F -vector space isomorphism f: B(J, F × F) ⟶ B(J, F × F, ζ) via

f ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}ζα & j \\ j′ & β \end{matrix} ) . (4.13)

A quick check of formulas (3.3) and (3.4) show that this is a similarity of the triple systems.

(2): One of the triple systems associated to B(J_{i}, F × F) is 𝔐(J_{i}). Thus B(J, F × F) ∼ B(J′, F × F) if and only if 𝔐(J_{1}) is similar to 𝔐(J_{2}). By [Fer72, 6.8], this occurs if and only if J_{1} ∼ J_{2}.

The construction of Freudenthal triple systems from Brown algebras is already interesting, and the following lemma makes it a useful tool.

Lemma 4.14. Every Freudenthal triple system is associated to some Brown algebra.

Proof: Let 𝔐 = (V, b, t) be a Freudenthal triple system. If 𝔐 ≅ 𝔐(J) for some Albert F -algebra J, then 𝔐 is obtained from B(J, F × F) and we are done. Otherwise, we pick some v ∈ V such that q(v) ≠ 0. Since 𝔐 ≅ 𝔐̸(J) for any J, d := q(v)/12 is not a square in F^{∗} by [Fer72, 3.4]. Set Δ := F (\sqrt{d}). Then 𝔐 ⊗_{F} Δ ≅ 𝔐(J) for some Albert Δ-algebra J.

Moreover, for ι the nontrivial F -automorphism of Δ, 𝔐 is F -isomorphic to the fixed points of 𝔐(J) under the ι-semilinear automorphism given by

η ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}ι(β/δ) & φ^{†}ι(j′) \\ −φι(j) & δι(α) \end{matrix} )

for δ ∈ Δ^{∗} such that δ^{2} = d and φ a norm similarity of J over Δ with multiplier −1/δ.

Let b, t and b′, t′ be triple systems associated to B := B(J, Δ×Δ) and B′ := B(J, Δ×Δ, −δ) given by equations (3.3) and (3.4). Then define

b′_{δ} := −\frac{1}{δ}b′ and t′_{δ} := −\frac{1}{δ}t′.

These also define a triple system associated to B′.

We have a similarity f: B ⟶ B′ as given in (4.13). Thus η induces an ι-semilinear map π := fηf^{−1} of B′ given by

π ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}ι(β) & φ^{†}ι(j′) \\ −φι(j) & ι(α) \end{matrix} ) .

Note that b′_{δ}(πx, πy) = ιb′_{δ}(x, y) and similarly for t′_{δ}. Thus f is an isometry between the triple systems b, t on B and b′_{δ}, t′_{δ} on B′ and so these triple systems restrict to be F -isomorphic on B^{η} and B′^{π}, where B^{η} and B′^{π} denote the F -subspaces fixed by η and π respectively.

A direct check (making use of Lemma 1.7) then shows that π is actually an ι-semilinear automorphism of B′ as a Brown algebra. Therefore, B′^{π} has the structure of a Brown algebra over F with associated triple system 𝔐.

Theorem 4.15. (1) For B a Brown algebra over F, Inv (B) is a simply connected group of type E_{7} over F with trivial Tits algebras. Every simply connected group of type E_{7} with trivial Tits algebras is obtained in this way.

(2) Inv (B) ≅ Inv (B′) if and only if B ∼ B′.

Proof: (1): Since Inv (B) = Inv (𝔐) for some nondegenerate Freudenthal triple system 𝔐 over F, it is a simply connected group of type E_{7} with trivial Tits algebras by 3.5, which finishes the first statement. Also by 3.5, if one has a simply connected group of type E_{7} with trivial Tits algebras, then it is isomorphic to Inv (𝔐) for some nondegenerate Freudenthal triple system over F and we are done by Lemma 4.14.

(2): ⟸ is clear, so we show ⟹ . Brown algebras are classified up to similarity by the set H^{1}(F, GInv (B)) and by (1) simply connected groups of type E_{7} are classified by H^{1}(F, PInv (B)). The short exact sequence

1 ⟶ 𝔾_{m,F} ⟶ GInv (B) ⟶ PInv (B) ⟶ 1

induces an exact sequence on cohomology

H^{1}(F, 𝔾_{m,F}) ⟶ H^{1}(F, GInv (B)) ⟶ H^{1}(F, PInv (B))

where the last map sends B′ to Inv (B′). By Hilbert’s Theorem 90, this second map has trivial kernel. Thus B ∼ B′.

Several authors have studied Freudenthal triple systems “in disguise” as ternary algebras or something similar, as in [Fau71], [Fau72], [FF77, §5], [FF72], [All73], and [Hei75]. It follows from [FF72, §3] that the ternary product that they are concerned with is in fact the ternary product on a Brown algebra given by (x, y, z) ↦ {y, s_{0}z, x}. However, those authors study Lie algebras associated to the ternary algebras which are not the Lie algebras of the automorphism groups of the triple systems, which are the groups that we are interested in.

5. Singular elements in Brown algebras

For ease of notation, for an element e in a Brown algebra B := (B, −), we define a vector space endomorphism U_{e} of B given by

U_{e} x := {e, x, e} for all x ∈ B.

Definition 5.1. We say that an element e in a Brown algebra (B, −) is singular if e ≠ 0 and U_{e} B ⊆ F e.

In [AF84, p. 196] and [Fer72], such elements were called “strictly regular”. We are following the (shorter) terminology from [Coo] and [Coo95] in that these elements are singular with respect to the quartic form q associated to the Brown algebra (i.e., the radical of the quadratic form v ↦ q(e, e, v, v) is a hyperplane of B and contains e).

We will produce many examples of singular elements in a moment.

Following Freudenthal [Fre54, 1.18] and [SV68, p. 250], we define a pairing ⟨ , ⟩: J × J ⟶ End_{F} (J) by

⟨x, y⟩j := \frac{1}{2} (y × (x × j) − T (j, y)x − \frac{1}{3}T (x, y)j) . (5.2)

We immediately note that ⟨x, y⟩^{∗} = ⟨y, x⟩ and that for for ψ ∈ Inv (J), we have

⟨ψ(x), ψ^{†}(y)⟩ = ψ⟨x, y⟩ψ^{−1}. (5.3)

Lemma 5.4. Suppose that j, j′ ∈ J satisfy j^{#} = (j′)^{#} = 0. Then the following are equivalent:

(1) j′ ∈ j × J.

(2) ⟨j, j′⟩ = 0.

(3) ⟨j, j′⟩ has image in F j.

In the case where j and j′ have trace zero, the equivalence of (1) and (2) goes back to [Fre59, 27.14].

Proof: (1) ⟹ (2): Suppose that j′ = j × u for u ∈ J. Then T (j, j′) = T (j × j, u) = 0 and j′ × (j × v) = T (u × v, j)j = T (j′, v)j for all v ∈ J by [McC69, (12)], so ⟨j, j′⟩ = 0.

(3) ⟹ (1): We may certainly extend scalars so that we may assume that our base field is separably closed and that we are in fact working inside the split Albert F -algebra. By (5.3) and Lemma 1.7, we may replace j by anything in its Inv (J^{d})(F_{s})-orbit. In particular, we may suppose that j is the primitive idempotent e_{0} := ( \begin{matrix}1 & 0 & · \\ · & 0 & 0 \\ 0 & · & 0 \end{matrix} ). Then if we write j′ = ( \begin{matrix}ε_{0} & c & · \\ · & ε_{1} & a \\ b & · & ε_{2} \end{matrix} ), we observe that

⟨e_{0}, j′⟩ ( \begin{matrix}0 & 0 & · \\ · & 0 & a′ \\ 0 & · & 0 \end{matrix} ) = ( \begin{matrix}0 & −b ⋆ a′ & · \\ · & 0 & \frac{2}{3}ε_{0}a′ \\ −a′ ⋆ c & · & 0 \end{matrix} )

for all a′ ∈ ℭ^{d}. Thus ε_{0} = b = c = 0 and so j′ ∈ ( \begin{matrix}0 & 0 & · \\ · & F & ℭ^{d} \\ 0 & · & F \end{matrix} ). This set is precisely e_{0} × J^{d}, as one can calculate from the explicit formula for × given in [Jac68, p. 358].

Lemma 5.5. (Cf. [Fer72, 6.1]) Let J be any Albert algebra and set e := ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) in B(J, F × F). Then e is singular if and only if

(1) T (j, j′) = 3αβ,

(2) (j′)^{#} = αj,

(3) j^{#} = βj′, and

(4) ⟨j, j′⟩ = 0

If α or β is nonzero, then conditions (1) through (3) imply (4).

Proof: Direct computation shows that if (1) through (4) hold, then e is singular, so we suppose that e is singular and show the converse.

If α or β is nonzero, then by symmetry we may assume that α ≠ 0. We will deal with the α = β = 0 case at the end.

Since e is singular,

U_{e} ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}2α^{2} & 2(j′)^{#} \\ 2αj′ & T (j, j′) − αβ \end{matrix} ) , (5.6)

lies in F e, so it must be 2αe. Hence conditions (1) and (2) hold. Also,

U_{e} ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}2αβ & 2βj \\ 2j^{#} & 2β^{2} \end{matrix} )

lies in F e, so it must be 2βe. Thus condition (3) holds.

Now we show that conditions (1) through (3) imply (4). For any x, y ∈ J,

x^{#} × (x × y) = N(x)y + T (x^{#}, y)x

by [McC69, p. 496, (10)]. Since N(j′) = \frac{1}{3}T (j′, (j′)^{#}) = α^{2}β,

j × (j′ × k′) = \frac{1}{α}(j′)^{#} × (j′ × k′) = αβk′ + T (j, k′)j′

for all k′ ∈ J. Then since (3) holds, we have 0 = ⟨j′, j⟩ = ⟨j, j′⟩^{∗}, hence (4).

Suppose now that α = β = 0. Then

U_{e} ( \begin{matrix}0 & k \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}0 & 2j′ × (j × k) − kT (j, j′) \\ ∗ & 2T (k, j^{#}) \end{matrix} ) (5.7)

for all k ∈ J. Since T is nondegenerate, j^{#} = 0, hence (3). The analogous observation with U_{e} ( \begin{matrix}0 & 0 \\ k′ & 0 \end{matrix} ) proves (2) and considering (5.6) demonstrates (1). Finally, looking at equation (5.7) again, we see that ⟨j, j′⟩ has image contained in F j, so by Lemma 5.4 we have (4).

Example 5.8. In B(J, F ×F), the elements ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ), ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ), and ( \begin{matrix}0 & j \\ 0 & 0 \end{matrix} ) and ( \begin{matrix}0 & 0 \\ j & 0 \end{matrix} ) for j ∈ J such that j^{#} = 0 are all singular.

Example 5.9. Examining the proof of Lemma 5.4, we see that for j′ := ( \begin{matrix}0 & 0 & · \\ · & 0 & u_{1} \\ 0 & · & 0 \end{matrix} ), the

element ( \begin{matrix}0 & e_{0} \\ j′ & 0 \end{matrix} ) satisfies conditions (1) through (3) but not (4) of Lemma 5.5. Now, [Fre54, 2.7] appears to assert that (1) through (3) imply (4) when α = β = 0, which would be a contradiction. However, Freudenthal only considered the case where J = ℌ_{3}(ℭ, 1) for ℭ a Cayley division algebra, so that J has no nonzero nilpotent elements. For this particular algebra, conditions (1) through (3) do imply (4), regardless of α and β.

We have the following application.

Proposition 5.10. For every Albert F -algebra J and every quadratic étale F -algebra Δ, B(J, Δ) ∼ B(J, F × F).

Proof: Fix some δ ∈ Δ^{∗} such that δ^{2} ∈ F^{∗} and Δ = F (δ). Set 𝔐 to be the triple system associated to B(J, Δ) by the formulas (4.3) and (4.4) with s_{0} = δ ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ). By the lemma,

f_{1} := ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) and f_{2} := δ ( \begin{matrix}−1 & −1 \\ 1 & 1 \end{matrix} )

are singular elements in 𝔐.

The map h: 𝔐 ⊗_{F} Δ ⟶ 𝔐(J) ⊗_{F} Δ given by

h ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) := ( \begin{matrix}α/δ & δj \\ j′ & δ^{2}β \end{matrix} )

is an isomorphism of triple systems. As described in [Bro69, p. 95], for k ∈ J we have automorphisms of 𝔐(J) given by

φ_{k} ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}α + βN(k) + T (j′, k) + T (j, k^{#}) & j + βk \\ j′ + j × k + βk^{#} & β \end{matrix} ) (5.11)

and

ψ_{k} ( \begin{matrix}α & j \\ j′ & β \end{matrix} ) = ( \begin{matrix}α & j + j′ × k + αk^{#} \\ j′ + αk & β + αN(k) + T (j, k) + T (j′, k^{#}) \end{matrix} ) , (5.12)

where φ_{k}^{−1} = φ_{−k} and ψ_{k}^{−1} = ψ_{−k}. Then

m := φ_{− \frac{1}{2δ}} ψ_{δ} h: 𝔐 ⊗_{F} Δ ⟶ 𝔐(J) ⊗_{F} Δ

is an isomorphism such that

m(f_{1}) = ( \begin{matrix}0 & 0 \\ 0 & 8δ^{2} \end{matrix} ) and m(f_{2}) = ( \begin{matrix}−1 & 0 \\ 0 & 0 \end{matrix} ) .

Now for ι the nontrivial F -automorphism of Δ, consider the 1-cycle z ∈ Z^{1}(Δ/F, Inv (𝔐(J))) given by z_{ι} := m(ϖ ⊗ ι)m^{−1}(1 ⊗ ι)^{−1}. It fixes the diagonal matrices in 𝔐(J) elementwise, so by [Fer72, 7.5] it is equal to f_{ϕ} for some ϕ ∈ Inv (J)(Δ), in the notation of (3.6). Then the obvious computation shows that ϕ is the identity, so that z is the trivial cocycle and 𝔐 ≅ 𝔐(J).

Remark 5.13. Not all Brown algebras contain singular elements. Having a singular element corresponds to one (equivalently, all) of the triple systems associated to it being of the form 𝔐(J) for some J. Ferrar gives an example in [Fer72, p. 330] and [Fer69, pp. 64, 65] of a nondegenerate Freudenthal triple system 𝔐 over a field of transcendence degree 4 over ℚ such that 𝔐 is not of such a form. By Lemma 4.14, 𝔐 arises from some Brown algebra B which then contains no singular elements, so in particular is of type 2 and not of the form B(J, Δ) for any J or Δ by 5.10.

6. Inner ideals

Definition 6.1. A vector subspace I of a structurable algebra (A, −) is said to be an inner ideal if U_{e}(A) ⊆ I for all e ∈ I. We say that I is proper if I ≠ A.

Example 6.2 (McCrimmon). One says that an element d in an Albert F -algebra J is of rank one if d^{#} = 0. We say that an F -subspace V of J is totally singular if every element of V is of rank one. By [McC71, p. 467, Thm. 8], the proper inner ideals in an Albert algebra are the totally singular subspaces and the subspaces of the form d×J for some d of rank one. We will call this last sort of subspace a hyperline, following Tits’ terminology from [Tit57].

Example 6.3. For V a totally singular subspace as in the preceding example, ( \begin{matrix}F & 0 \\ V & 0 \end{matrix} ) is an inner ideal in B(J, F × F) since every element is singular by Lemma 5.5.

We say that an inner ideal in a Brown algebra is singular if it consists of singular elements. We call such inner ideals singular ideals for short. (Of course, any subspace of a Brown algebra consisting of singular elements is automatically a singular ideal.)

Example 6.4. Suppose that d ∈ J satisfies d^{#} = 0. Then

I := ( \begin{matrix}F & F d \\ d × J & 0 \end{matrix} )

is an inner ideal of B(J, F × F). Moreover, it is not singular, since the element ( \begin{matrix}1 & d \\ 0 & 0 \end{matrix} ) is in I and it is not singular by Lemma 5.5. (After we have proven Lemma 6.6, we will also know that I is 12-dimensional.)

We want to classify the inner ideals in a Brown algebra B enough to describe the homogeneous projective varieties associated to groups of type E_{7} in terms of such ideals. Our classification is going to rest upon first understanding the inner ideals in Albert algebras a bit better.

Lemma 6.5. The set {j^{#} | j ∈ J^{d}} spans J^{d} over any separably closed field.

Proof: In the notation of (1.2), if one sets precisely one entry to be nonzero (and the entry to be u_{i} for some i if it is a, b, or c) then one gets a rank one element and these elements span J^{d}. Thus it suffices to prove that for every x of rank one, there is some j ∈ J^{d} with j^{#} ∈ F x. As discussed in [Jac61, p. 70], for x ≠ 0 of rank one, x is either (1) a primitive idempotent or it satisfies (2) T (x, 1) = 0 and x^{2} = 0.

(1): Let e_{0}, e_{1}, and e_{2} be a triple of orthogonal idempotents in J^{d} such that 1 = e_{0} + e_{1} + e_{2}. The group Aut (J^{d})(F) acts transitively on the primitive idempotents of J^{d} by the Coordinatization Theorem [Jac68, p. 137], so we may assume that x = e_{0}. Then (e_{1} +e_{2})^{#} = e_{0}.

(2): The group Aut (J^{d})(F_{s}) acts transitively on such rank one elements by [Fre59, 28.22], so we may assume that

x = ( \begin{matrix}0 & 0 & · \\ · & 0 & 0 \\ u_{1} & · & 0 \end{matrix} ), and so j := ( \begin{matrix}0 & −u_{4} & · \\ · & 0 & u_{1} \\ 0 & · & 0 \end{matrix} )

satisfies j^{#} = x.

Lemma 6.6. Any hyperline in an Albert algebra is 10-dimensional.

Proof: We may certainly extend scalars to assume that our base field is separably closed and that the Albert algebra is quasi-split. Suppose our hyperline is d × J^{d}. Then there is some norm isometry φ such that φ(d) is the primitive idempotent e_{0} := ( \begin{matrix}1 & 0 & · \\ · & 0 & 0 \\ 0 & · & 0 \end{matrix} ). Of course,

dim(x × J) = dim φ^{†}(x × J) = dim(e_{0} × J),

But we already know what e_{0} × J is from the proof of Lemma 5.4, and that it is 10dimensional.

What follows is our last preparatory lemma concerning Albert algebras. It may seem mysterious now, but after Lemma 7.1 we will see that it is precisely a simple algebraic interpretation of the nontrivial automorphism on the Dynkin diagram of type E_{6}, or, if you prefer, of the natural duality in the spherical building of type E_{6} a.k.a. a Hjelmslev-Moufang plane. It is stronger than what we actually need for the rest of this section, but it will all be of use later.

Duality Lemma 6.7. Let J be an Albert F -algebra. The map on subspaces of J which takes a subspace W of J to

{j ∈ J | ⟨w, j⟩ = 0 for all w ∈ W }

induces one-to-one correspondences

3-dim’l t. singular subspaces ↔ 3-dim’l t. singular subspaces
2-dim’l t. singular subspaces ↔ 5-dim’l maximal t. singular subspaces
1-dim’l t. singular subspaces ↔ hyperlines

The bottom correspondence is given by

F d ↔ d × J.

In particular this gives us the simple fact that if d × J = d′ × J for rank one elements d and d′, then F d = F d′.

Proof: We may certainly assume that our base field F is separably closed and that our Albert algebra is split. Then by [SV68, 3.2, 3.12, 3.14] the group of norm isometries acts transitively on each of the six kinds of subspaces specified. So all we really need to do is produce an example of a pair of subspaces (W, W′) which are sent to each other by the specified map. For if V is another subspace of the same kind as W, there is some norm isometry ψ with ψ(V) = W. By (5.3), our correspondence map will then send V to ψ^{∗}(W′), which in turn is itself sent to V, proving that the correspondence is indeed a bijection.

Consider the primitive idempotent e_{0} from the proof of Lemma 6.6. We will show that (F e_{0}, e_{0} ×J) provides a pair which is an example of the last correspondence. By Lemma 5.4, F e_{0} is sent to e_{0} × J, so we must check the other direction. We can define other primitive idempotents e_{1} and e_{2} such that e_{i} has all entries zero except for a one in the (i + 1, i + 1)- position. Since e_{1}, e_{2} ∈ e_{0} × J, by Lemma 5.4 e_{0} × J is sent to a subspace W which is contained in

(e_{1} × J) ∩ (e_{2} × J) = ( \begin{matrix}F & 0 & · \\ · & 0 & 0 \\ ℭ^{d} & · & F \end{matrix} ) ∩ ( \begin{matrix}F & ℭ^{d} & · \\ · & F & 0 \\ 0 & · & 0 \end{matrix} ) = F e_{0}.

But W certainly contains F e_{0}, so e_{0} × J is sent to W = F e_{0}.

Note that we have now even proven the claim about the form of the last correspondence, since for a norm isometry ψ such that F ψ(d) = F e_{0}, F d is sent to ψ^{∗}(e_{0}×J) = ψ^{−1}(e_{0})×J = d × J.

Now set

W := ( \begin{matrix}0 & 0 & · \\ · & 0 & F u_{1} \\ 0 & · & F \end{matrix} ) and W′ := ( \begin{matrix}F & ℭ ⋆ u_{1} & · \\ · & 0 & 0 \\ 0 & · & 0 \end{matrix} ) .

It is quickly checked that W is a 2-dimensional totally singular subspace and that W′ is a 5-dimensional maximal totally singular subspace. Similarly to what we just did for the last correspondence, we can apply Lemma 5.4 to rewrite the correspondence map as

W ↦ ∩_{w∈W} w × J.

Then it is just a tedious computation to see that the pair (W, W′) we have just defined are indeed sent to each other.

Finally,

W := W′ := ( \begin{matrix}0 & F u_{5} & · \\ · & 0 & F u_{1} \\ F u_{2} & · & 0 \end{matrix} ) (6.8)

provide a pair (W, W′) of 3-dimensional totally singular subspaces which are mapped to each other by the correspondence map. (There are other choices for (W, W′) which would be easier to check here, but we will use this subspace again at the end of Section 7. We also caution the reader that this pair is atypical in that a 3-dimensional totally singular space is not necessarily sent to itself by our correspondence map.)

Remark 6.9. Incidentally, the other sort of maximal totally singular subspaces are 6-dimensional and Inv (J^{d}) acts transitively on the set of such subspaces [SV68, 3.14]. A particular example is given by

( \begin{matrix}0 & 0 & · \\ · & 0 & ℭ^{d} ⋆ u_{1} \\ F u_{1} & · & F \end{matrix} ) .

Since this subspace is sent to zero by the correspondence map above, so are all 6-dimensional totally singular subspaces.

Lemma 6.10. Let I be an inner ideal in B(J, F × F). If I contains ( \begin{matrix}0 & 0 \\ J & 0 \end{matrix} ) or ( \begin{matrix}0 & J \\ 0 & 0 \end{matrix} ), then I is all of B.

Proof: Since the property of being proper is invariant under scalar extension, we may assume that our base field is separably closed and J is split (i.e., J = J^{d}).

In the first case, I contains

U_{( \begin{matrix}0 & 0 \\ j′ & 0 \end{matrix} )} ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}0 & 2(j′)^{#} \\ 0 & 0 \end{matrix} )

for all j′ ∈ J^{d}. Since (J^{d})^{#} spans J^{d} by Lemma 6.5, I contains ( \begin{matrix}0 & J^{d} \\ J^{d} & 0 \end{matrix} ).

Then I also contains

U_{( \begin{matrix}0 & j \\ j′ & 0 \end{matrix} )} ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}0 & 2(j′)^{#} \\ 0 & T (j, j′) \end{matrix} ) ,

so I contains ( \begin{matrix}0 & J^{d} \\ J^{d} & F \end{matrix} ). Then

U_{( \begin{matrix}0 & j \\ j′ & 0 \end{matrix} )} ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}T (j, j′) & 0 \\ 2j^{#} & 0 \end{matrix} ) ∈ I,

so we are done with the first case.

The proof in the second case is symmetric to the first case.

We also observe in the following proposition that whether or not a subspace is a singular or an inner ideal is already determined by its characteristics in an associated triple system, so that Inv (B) takes inner (resp. singular) ideals to inner (resp. singular) ideals.

Proposition 6.11. Let B = (B, −) be a Brown algebra with b, t an associated triple system. Then a vector subspace I is an inner ideal of B if and only if

t(I, I, B) ⊆ I.

It is singular if and only if

t(u, v, z) = b(z, v)u + b(z, u)v for all u, v ∈ I and z ∈ B.

Proof: Observe that

t(u, v, z) = \frac{1}{2} (t(u, z, v) + t(v, z, u)) = \{u, s_{0}z, v\} + \{v, s_{0}z, u\} − b(z, v)u − b(z, u)v.

I is an inner ideal if and only if the sum of the two brace terms on the left-hand side is in I for all u, v ∈ I, so this proves the first equivalence.

By [Fer72, p. 317], an element u ∈ B is strictly regular if and only if t(u, u, y) = 2b(y, u)u for all y ∈ B. Since

t(u + v, u + v, z) = t(u, u, z) + t(v, v, z) + 2t(u, v, z),

if u, v, and u + v are strictly regular, then

t(u, v, z) = b(y, u + v)(u + v) − b(y, u)u − b(y, v)v = b(y, u)v + b(y, v)u. (6.12)

for all z ∈ B. Conversely, if (6.12) holds for all u, v ∈ I, then every element of I is singular.

Theorem 6.13. If a proper inner ideal of a Brown algebra B contains a singular element, then it is in the Inv (B)(F_{s})-orbit of an inner ideal as in Example 6.3 or Example 6.4. In particular, it has dimension ≤ 7 and is singular or is 12-dimensional and is not singular.

Proof: We may assume that our base field F is separably closed and so that our algebra is split. Since Inv (B)(F_{s}) acts transitively on singular elements [Fer72, 7.7], we may assume that the ideal I contains e := ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ).

Since e ∈ I, we may extend e to a basis x_{2}, …, x_{n} of I such that

x_{i} := ( \begin{matrix}0 & j_{i} \\ j′_{i} & β_{i} \end{matrix} )

for 2 ≤ i ≤ n = dim_{F} I.

First, consider

\begin{aligned}(U_{e+x_{i}} − U_{e} − U_{x_{i}}) ( \begin{matrix}0 & 0 \\ k′ & 0 \end{matrix} ) & = & \{e, ( \begin{matrix}0 & 0 \\ k′ & 0 \end{matrix} ) , x_{i}\} + \{x_{i}, ( \begin{matrix}0 & 0 \\ k′ & 0 \end{matrix} ) , e\} \\ & = & ( \begin{matrix}0 & 0 \\ β_{i}k′ & 0 \end{matrix} ) ∈ I. \end{aligned}

Thus, if any β_{i} is nonzero, the ideal would have to contain ( \begin{matrix}F & 0 \\ J & 0 \end{matrix} ), and so by Lemma 6.10 it would not be proper. Thus β_{i} = 0 for all i.

If I is singular, then take x := ( \begin{matrix}0 & j \\ j′ & 0 \end{matrix} ) in I. Since x + αe ∈ I for all α ∈ F, it is singular. By Lemma 5.5, (j′)^{#} = αj for all α ∈ F^{∗}, so j = 0. Then I is one of the ideals described in Example 6.3.

Otherwise,

(U_{x_{i}+x_{ℓ}} − U_{x_{i}} − U_{x_{ℓ}}) ( \begin{matrix}0 & k \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}0 & ∗ \\ ∗ & 2T (k, (j_{i} × j_{ℓ})) \end{matrix} ) ∈ I.

Since T is nondegenerate, j_{i} × j_{ℓ} = 0 for all 2 ≤ i, ℓ ≤ n. So if we write I ⊆ ( \begin{matrix}F & W \\ V & 0 \end{matrix} ) where W and V are the projections of I into J on the off-diagonal entries, we have shown that w^{#} = 0 for all w ∈ W.

Let x := ( \begin{matrix}α & w \\ v & 0 \end{matrix} ) represent an arbitrary element of I. Then

U_{x−αe} ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}0 & 2v^{#} \\ 0 & T (v, w) \end{matrix} ) ∈ I,

so T (v, w) = 0 and v^{#} ∈ W. Also,

U_{x} ( \begin{matrix}0 & k \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}∗ & ∗ \\ 2αw × k − 2v^{#} × k + 2vT (v, k) & 0 \end{matrix} ) . (6.14)

Setting α = 0, we see that V^{#} × J ⊆ V. (We will use the α ≠ 0 case in a moment.) Thus V is an inner ideal of J by [McC71, p. 467]. We consider the following cases: (1) V = J, (2) V is a hyperline, or (3) V^{#} = 0.

(1) cannot occur because in that case V^{#} = J by Lemma 6.5, and we already know that V^{#} ⊆ W and W^{#} = 0.

In case (2), V = d × J for some d of rank one. However, letting α be arbitrary in (6.14) and keeping in mind that V^{#} × J ⊆ V, we see that in fact W × J ⊆ V = d × J. Since W^{#} = 0, w × J = w′ × J if and only if w ∈ F w′ by the Duality Lemma 6.7. The fact that hyperlines are maximal proper inner ideals of J [McC71, p. 467, Thm. 8] tells us that W ⊆ F d. Since F d = V^{#} ⊆ W, the ideal is as in Example 6.4.

Finally, we examine case (3). There, since V^{#} = 0, V is at most 6-dimensional by [SV68, 3.14]. However, W^{#} = 0 and W × J ⊆ V. Since hyperlines are 10-dimensional, W = 0. Thus I is as in Example 6.3.

Theorem 6.15. Any proper inner ideal in a Brown algebra has dimension at most 12. If it is 12-dimensional, then it is in the Inv (B)(F_{s})-orbit of an inner ideal as in Example 6.4.

Proof: Clearly we may assume that our base field is separably closed and so that B = B(J, F × F) for F = F_{s}. Suppose that dim_{F} I ≥ 12 and pick a basis x_{1}, x_{2}, …, x_{n} of I such that

x_{i} := ( \begin{matrix}α_{i} & j_{i} \\ j′_{i} & β_{i} \end{matrix} )

where β_{i} = 0 for 2 ≤ i ≤ n. Set c := 1 if β_{1} = 0, otherwise set c := 2. Finally, set I′ to be the span of x_{i} for c ≤ i ≤ n. Let W denote the span of j_{i} for c ≤ i ≤ n.

For x := ( \begin{matrix}α & j \\ j′ & 0 \end{matrix} ) an arbitrary element of I′,

U_{x} ( \begin{matrix}0 & 0 \\ k′ & 0 \end{matrix} ) = ( \begin{matrix}∗ & 2jT (j, k′) − 2k′ × j^{#} \\ ∗ & 0 \end{matrix} ) ∈ I.

In fact, it lies in I′. Thus W^{#} × J ⊆ W and W is an inner ideal of J, so (1) W = J, (2) W is a hyperline, or (3) W^{#} = 0.

In case (1),

U_{x} ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}0 & 0 \\ 2j^{#} & 0 \end{matrix} ) ∈ I′,

so by Lemma 6.5 I contains ( \begin{matrix}0 & 0 \\ J & 0 \end{matrix} ). By Lemma 6.10, I is not proper, so this is a contradiction.

In case (2) W is 10-dimensional and in case (3) W is at most 6-dimensional by [SV68, 3.14]. Thus we may rewrite the basis so that j_{i} = 0 for i ≥ c + 10 (which is ≤ 12). Consider

x := x_{c+10} = ( \begin{matrix}α & 0 \\ v & 0 \end{matrix} ) ,

for v := j′_{c+10}. If v = 0, then x ≠ 0 would imply that I contains a singular element and so we would be done by Theorem 6.13. So we may assume that v ≠ 0. If v^{#} = 0 then x is singular and we are done. Otherwise, consider

x′ := \frac{1}{2}U_{x} ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) − αx = ( \begin{matrix}0 & v^{#} \\ 0 & 0 \end{matrix} ) ∈ I.

If v^{##} = 0 then x′ is singular and we are again done, so we may assume that N(v)v = v^{##} ≠ 0. Then

x′′_{k} := \frac{1}{2}U_{x′} ( \begin{matrix}0 & k \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}0 & 0 \\ 0 & T (k, v)N(v) \end{matrix} ) ∈ I

for all k ∈ J. Since T is nondegenerate, x′′_{k} ∈ I is nonzero and singular for some choice of k.

7. Flag varieties for groups of type E_{7}

In Section 6 I promised that the homogeneous projective varieties associated to the group Inv (B) of type E_{7} can be described in terms of the inner ideals of B. We will fulfill this promise in 7.4, but first wee must set up some notation.

Background on flag varieties. Given a maximal torus T in a simple affine algebraic group G, we fix a set of simple roots Δ = {α_{1}, …, α_{r}} of G with respect to T. For each α_{i}, there is a uniquely defined root group U_{α_{i}} lying in G [Bor91, 13.18]. For any subset Θ = {i_{1}, i_{2}, …, i_{n}} of Δ, we define a parabolic subgroup of G by

P (Θ) := ⟨T, {U_{α} | α ∈ Δ}, {U_{−α} | α ∉ Δ}⟩

and an associated flag variety by

X(Θ) := G/P (Θ).

Thus P (Δ) is a Borel subgroup of G and P (∅) = G. This notation is similar to [MPW96] and [Gar99c] and is opposite that of [KR94].

We are interested in the group G = Inv (A) for A an Albert or Brown F -algebra. Each variety X(i) is going to have F -points corresponding to certain subspaces of A which we will call i-spaces. We will also define a symmetric and reflexive binary relation called incidence between i-spaces and j-spaces for all i and j. Two i-spaces will be incident if and only if they are the same. Then the other flag varieties will be of the form X(Θ) and have F -points

{(V_{1}, …, V_{n}) ∈ X(i_{1})(F) × ⋯ × X(i_{n})(F) | V_{i} is incident to V_{j} for all 1 ≤ i, j ≤ n } .

Flag varieties for E_{6}. As a prelude to describing the flag varieties for Inv (B), we recall the description of the flag varieties of Inv (J) for J an Albert algebra. We number the Dynkin diagram of Inv (J) as

[figure: 1; 2; 3; 4; 6; 5]

Theorem 7.1. Let J be an Albert F -algebra. The i-spaces for Inv (J) are the i-dimensional totally singular subspaces of J for i = 1, 2, 3. The 4-spaces are the 5-dimensional maximal totally singular subspaces of J. The 5-spaces are the 6-dimensional totally singular subspaces, and the 6-spaces are the subspaces of the form d × J for d ∈ J of rank one.

Incidence is defined by inclusion except for the following: A 4-space and a 5-space are incident if and only if their intersection is 3-dimensional. A 6-space and a 5-space are incident if and only if their intersection is 5-dimensional (in which case it is necessarily a nonmaximal totally singular subspace of J).

Proof: For the purposes of the proof we define an i-space to be a subspace of J as specified in the theorem statement, and we will show that one can identify X(i)(F) with the set of i-spaces.

Let X_{i} denote the functor mapping field extensions of F to sets of subspaces of J such that X_{i}(K) is the set of i-spaces of J ⊗_{F} K. Then certainly X_{i} is a projective variety for i = 1, 2, 3, 5. Duality for points and hyperlines (6.7) shows that X_{6} and X_{4} are projective varieties since X_{1} and X_{2} are.

The transitivity of the natural Inv (B)-action on X_{i} is given by [SV68, 3.2] for i = 1, by [SV68, 3.12] for i = 2, 3, and by [SV68, 3.14] for i = 4, 5. The transitivity of the action on X_{6} follows by the definition of a hyperline and the transitivity of the action on X_{1}. The incidence relations and the associations with simple roots are in [Tit57, 3.2], but we will produce a set of simple roots explicitly because we will need them later when we consider E_{7}. All of the material we develop here will also see use in Section 8.

In order to describe the associations with simple roots, we extend scalars to split J.

Consider the split Cayley algebra ℭ^{d} with hyperbolic norm form 𝔫 and multiplication ⋆ as described in the introduction. We will define two algebraic groups Rel (ℭ^{d}, 𝔫) and Spin (ℭ^{d}, 𝔫) associated to (ℭ^{d}, 𝔫). First, consider the (connected, reductive) algebraic group GO^{+}(ℭ^{d}, 𝔫) whose F -points are

GO^{+}(ℭ^{d}, 𝔫)(F) := {f ∈ End_{F} (ℭ^{d}) | σ(f)f ∈ F^{∗} and (det f)^{4} = σ(f)f},

where σ is the involution on End_{F} (ℭ^{d}) which is adjoint to 𝔫. For f ∈ GO^{+}(ℭ^{d}, 𝔫)(F), we write μ(f) := σ(f)f, the multiplier of f. We define Rel (ℭ^{d}, 𝔫) to be the algebraic group whose F -points are the related triples in GO^{+}(ℭ^{d}, 𝔫)^{×3}, i.e., those triples \underline{t} := (t_{0}, t_{1}, t_{2}) such that

μ(t_{i})^{−1}t_{i}(x ⋆ y) = t_{i+2}(x) ⋆ t_{i+1}(y)

for i = 0, 1, 2 (subscripts taken modulo 3) and all x, y ∈ ℭ^{d}. We write O^{+}(ℭ^{d}, 𝔫) for the closed subgroup of GO^{+}(ℭ^{d}, 𝔫) consisting of those f’s with det f = 1 and define Spin (ℭ^{d}, 𝔫) to be the algebraic group consisting of the related triples in O^{+}(ℭ^{d}, 𝔫)^{×3}. Now Rel (ℭ^{d}, 𝔫) injects into Inv (J^{d}) by sending \underline{t} to g\underline{t}, which is defined by

g\underline{t} ( \begin{matrix}ε_{0} & c & · \\ · & ε_{1} & a \\ b & · & ε_{2} \end{matrix} ) := ( \begin{matrix}μ(t_{0})^{−1}ε_{0} & t_{2}c & · \\ · & μ(t_{1})^{−1}ε_{1} & t_{0}a \\ t_{1}b & · & μ(t_{2})^{−1}ε_{2} \end{matrix} ) . (7.2)

This map restricts to an injection of Spin (ℭ^{d}, 𝔫) into Aut (J^{d}). The subset of Spin (ℭ^{d}, 𝔫) consisting of triples \underline{t} such that t_{i} is diagonal for all i forms a rank 4 split torus in Spin (ℭ^{d}, 𝔫) [Gar98, 1.6], and the image of this torus under the composition Spin (ℭ^{d}, 𝔫) ↪ Aut (J^{d}) ↪ Inv (J^{d}) provides a rank 4 split torus S_{4} in Inv (J^{d}).

For \underline{λ} = (λ_{0}, λ_{1}, λ_{2}) ∈ (F^{∗})^{×3} such that λ_{0}λ_{1}λ_{2} = 1, we have a map S\underline{λ} ∈ Inv (J^{d}) given by

S\underline{λ} ( \begin{matrix}ε_{0} & c & · \\ · & ε_{1} & a \\ b & · & ε_{2} \end{matrix} ) = ( \begin{matrix}λ_{0}^{−2}ε_{0} & λ_{2}c & · \\ · & λ_{1}^{−2}ε_{1} & λ_{0}a \\ λ_{1}b & · & λ_{2}^{−2}e_{2} \end{matrix} ) .

Let S_{2} denote the rank 2 split torus in Inv (J^{d}) generated by such maps. Then S_{6} := S_{2} × S_{4} is a rank 6 split torus in Inv (J^{d}). We have characters χ_{i,j} defined for 0 ≤ i ≤ 2, 1 ≤ j ≤ 8 by setting χ_{i,j} to be trivial on S_{2} and to take the value of the (j, j)-entry of t_{i} on (t_{0}, t_{1}, t_{2}). Define ρ_{i} to be the character which is trivial on S_{4} and such that ρ_{i}(S_{(λ_{0},λ_{1},λ_{2})}) = λ_{i}.

A set of simple roots is given by the following, where we have written ω_{j} := χ_{0,j} for short:

α_{1} = −ω_{1} − (ρ_{2} − ρ_{1}), (7.3)

α_{2} = ω_{1} − ω_{2}, α_{3} = ω_{2} − ω_{3}, α_{4} = ω_{3} + ω_{4}, α_{5} = ω_{3} − ω_{4},

α_{6} = χ_{2,8} − (ρ_{1} − ρ_{0}) = −\frac{1}{2}(ω_{1} + ω_{2} + ω_{3} + ω_{4}) − (ρ_{1} − ρ_{0})

The root groups corresponding to the root subsystem of type D_{4} spanned by α_{2} through α_{5} are given explicitly in [Gar98, 4.3]. The 1-dimensional root Lie algebras corresponding to the roots α_{1} and α_{6} are S_{(u_{8})_{12}} and S_{(u_{8})_{23}} respectively, in the notation of [Jac71, p. 35].

Consider the following spaces:

V_{1} := ( \begin{matrix}0 & 0 & · \\ · & 0 & 0 \\ 0 & · & F \end{matrix} ) , V_{2} := ( \begin{matrix}0 & 0 & · \\ · & 0 & F u_{1} \\ 0 & · & F \end{matrix} ) , V_{3} := ( \begin{matrix}0 & 0 & · \\ · & 0 & F u_{1}+F u_{2} \\ 0 & · & F \end{matrix} ) ,

V_{4} := ( \begin{matrix}0 & 0 & · \\ · & 0 & u_{1}⋆ℭ^{d} \\ 0 & · & F \end{matrix} ) , V_{5} := ( \begin{matrix}0 & 0 & · \\ · & 0 & ℭ^{d}⋆u_{1} \\ F u_{1} & · & F \end{matrix} ) , and V_{6} := e_{0} × J = ( \begin{matrix}0 & 0 & · \\ · & F & ℭ^{d} \\ 0 & · & F \end{matrix} ) ,

for e_{0} the primitive idempotent as in the proof of 5.4. Then V_{i} is the sort of space which is claimed to be an i-space.

We leave it to the reader to verify that the stabilizer of V_{i} in Inv (J^{d}) is precisely the parabolic P (i), so that in fact X_{i} = X(i).

Flag varieties for E_{7}. We label the Dynkin diagram for E_{7} as follows

[figure: 1; 2; 3; 4; 5; 7; 6]

The key idea here is that nodes 2 through 7 span a Dynkin diagram of type E_{6}, and that this corresponds to the subgroup Inv (J) of Inv (B(J)).

Theorem 7.4. Let B be a Brown F -algebra. The i-spaces for Inv (B) are the i-dimensional singular ideals for i = 1,2,3,4. The 5-spaces are the 6-dimensional maximal singular ideals. The 6-spaces are the 7-dimensional singular ideals, and the 7-spaces are the 12-dimensional inner ideals.

Incidence is defined by inclusion except for the following: A 5-space and a 6-space are incident if and only if their intersection is 4-dimensional. A 7-space and a 6-space are incident if and only if their intersection is 6-dimensional (in which case it is necessarily a nonmaximal singular ideal of B).

Proof: Define X_{i} as in the proof of 7.1. By Proposition 6.11, there is a natural action of Inv (B)(F_{s}) on X_{i}(F_{s}), and it is transitive for all i by Theorems 6.13 and 6.15 and the description of the flag varieties for Inv (J) in 7.1.

We now show that the roots are associated to the i-spaces as claimed. We extend scalars so that B is split. Let S be a rank 7 split torus in Inv (B) as in the proof of Theorem 3.5, where S_{6} is the torus from the proof of Theorem 7.1.

We may extend the characters χ_{i,j} and ρ_{i} to S by setting them to be trivial on S_{1}. We get a new character τ defined by

τ|_{S_{6}} = 1 and τ(f_{L_{x}}) = x.

Then a set of simple roots for Inv (B) with respect to S is given by

β_{1} := 2ρ_{2} + 2τ and β_{j} := α_{j−1} for 2 ≤ j ≤ 7.

For 1 ≤ j ≤ 6, a j-space is given by

W_{j} := ( \begin{matrix}0 & V_{j−1} \\ 0 & F \end{matrix} )

for V_{i} as in the proof of 7.1 and V_{0} := 0. We also have a 7-space

W_{7} := ( \begin{matrix}0 & V_{6} \\ F e_{0} & F \end{matrix} )

where e_{i} is the idempotent of J whose only nonzero entry is the (i + 1, i + 1)-entry, which is 1.

The root group for β_{1} is generated by ψ_{e_{2}} for ψ as in (5.12), and the root group for −β_{1} is generated by φ_{e_{2}} for φ as in (5.11).

Since Inv (B)(F_{s}) acts transitively on X_{i}(F_{s}) and X_{i} is clearly a projective variety for i ≠ 5, the stabilizer of W_{i} in Inv (B) is a parabolic subgroup. For X_{5}, the stabilizer of W_{5} is a closed subgroup of Inv (B) which contains the Borel subgroup P (∅) determined by S and our choice of a set of simple roots. Thus it is a parabolic subgroup [Bor91, 11.2]. Given the result for Inv (J), it is now an easy check to see that the stabilizer of W_{i} in Inv (B) is precisely P (i). Thus X_{i} = X(i). It only remains to confirm the incidence relations, which follow easily from the corresponding claims for E_{6}, making use of the strong restrictions on the form of an inner ideal containing a singular element observed in the proof of Theorem 6.13.

It can be hard to visualize what the singular ideals in B(J, Δ) look like. We mention the particular example

I_{6} := ( \begin{matrix}0 & W \\ W & 0 \end{matrix} ) ⊗_{F} Δ (7.5)

for W as in (6.8). Then by Lemma 5.5, I_{6} is a 6-dimensional singular ideal in B(J^{d}, Δ). Since it is stable under ϖ ⊗ ι, it is even defined over F. We end this section with an application of our results.

8. Cohomological invariants

The purpose of this section is to prove the following:

Theorem 8.1. If G is a quasi-split simply connected group of type E_{6} or E_{7}, then the kernel of the Rost invariant from (0.1) is trivial.

The Rost invariant and standard embeddings. Suppose that we have a simple split simply connected group G defined over F. In [KMRT98, §31], the H^{3} invariants of H^{1}(F, G) are analyzed with the help of a group Q(G). For any group H, Q(H) is defined [KMRT98, p. 432] to be an abelian group of maps q : X_{∗}(H) ⟶ ℤ satisfying a couple of identities, where X_{∗}(H) is the group of loops (= cocharacters) in H (i.e., the maps 𝔾_{m,F} ⟶ H). The important thing is that for S a split maximal torus of G, Q(S) is the group of rdimensional integral quadratic forms on X_{∗}(S) and Q(G) is infinite cyclic [KMRT98, 31.27] with a canonical generator such that the restriction map Q(G) ⟶ Q(S) is injective and sends the canonical generator of Q(G) to the “minimal” positive definite Weyl-invariant quadratic form q on X_{∗}(S). There is up to scalar multiple precisely one definite Weylinvariant quadratic form, so by the minimal one we mean the one which is positive definite and not a ℤ-multiple of any other such form. (That there is precisely one such form is because the same statement holds for forms on the root lattice for G with respect to S, which is [Bou68, VI.1.2, Prop. 7]. The connection will be clear from the proof of Lemma 8.2.)

Now suppose that G′ is another simple split simply connected group with split maximal torus S′ so that there is an embedding of G in G′ which restricts to be an embedding of S in S′. Then we obtain a commutative diagram

Q(G′) ⟶ Q(G)
↓  ↓
Q(S′) ⟶ Q(S)

and in particular, for q′ the image in Q(S′) of the canonical generator of Q(G), the image of q′ in Q(S) is equal to nq for some positive integer n. The embedding of G in G′ induces a map on the H^{1} level which allows us to induce an H^{3} invariant of H^{1}(F, G) from the Rost invariant R_{G′} of G′ via the composition

H^{1}(F, G) ⟶ H^{1}(F, G′) ⟶ H^{3}(F, ℚ/ℤ(2)).

This composition is precisely nR_{G} by [KMRT98, p. 434].

We have a sequence of embeddings of split groups

Spin (ℭ^{d}, 𝔫) ↪ Aut (J^{d}) ↪ Inv (J^{d}) ↪ Inv (𝔐(J^{d}))

of types D_{4}, F_{4}, E_{6}, and E_{7} and we have described the corresponding restrictions to embeddings of split maximal tori in the proofs of 3.5 and 7.1. We refer to these as the standard embeddings.

Lemma 8.2. (Rost) The Rost invariant for the split simply connected group of type E_{7} induces via the standard embeddings above precisely the Rost invariants for the split simply connected groups of type E_{6}, F_{4}, and G_{2}.

Proof: We treat only the case of the embedding of F_{4} in E_{6}, as the other two cases are similar and in a general sense easier since there the embedding on the group level arises from an embedding of the root systems.

So we assume that G = Aut (J^{d}), and we denote the character group of the split maximal torus S by X(S), which can be identified with the weight lattice Λ since G is simply connected. It contains the root lattice Λ_{r} as a sublattice. We have the usual Weyl-invariant dual pairing

⟨ , ⟩: X_{∗}(S) × X(S) ⟶ ℤ (8.3)

as described in [Bor91, 8.6]. We also have a minimal symmetric bilinear positive definite Weyl-invariant bilinear form b on Λ_{r}, where the minimality is forced by assuming that b(α, α) = 2 for short roots α. We fix a basis γ_{1}, …, γ_{4} of simple roots for Λ_{r} identical to the one in [Bou68, p. 272] (so that, for example, γ_{1} and γ_{2} are the long roots). Let Λ_{c} denote the ℤ-sublattice of Λ_{r} ⊗_{ℤ} ℚ generated by the coroots, which are the elements α̌ := 2α/b(α, α) for α a root. Then b restricts to be a Weyl-invariant dual pairing

b: Λ_{c} × Λ ⟶ ℤ. (8.4)

For γ a root, we write γ^{∗} for the element of X_{∗}(S) such that ⟨γ^{∗}, ϖ⟩ = b(γ̌, ϖ) for all ϖ ∈ Λ and we get a minimal positive definite symmetric bilinear Weyl-invariant form b_{∗} on X_{∗}(S) by setting b_{∗}(γ_{i}^{∗}, γ_{j}^{∗}) := 2b(γ̌_{i}, γ̌_{j}). The factor of 2 is required to ensure that b_{∗} always takes values in ℤ, since one can compute with the help of the Cartan matrix that b(γ̌_{1}, γ̌_{2}) = b(γ_{1}, γ̌_{2})/2 = −1/2.

To compute the image of the generator of Q(G) in Q(S) is now easy. We know that the simple root γ_{1} is a long root, so it has b(γ_{1}, γ_{1}) = 4. Thus b_{∗}(γ_{1}^{∗}, γ_{1}^{∗}) := 2. In order to compare the quadratic form on X_{∗}(S) with the corresponding one for G′, we must describe γ_{1}^{∗} explicitly. In [Bou68], the root γ_{1} is given as γ_{1} = ω_{2} − ω_{3}, and consulting the list of corresponding fundamental weights given there we see that γ_{1}^{∗}: 𝔾_{m,F} ⟶ S must be precisely γ_{1}^{∗}(r) = g_{(t_{0}(r),t_{1}(r),t_{2}(r))} where

t_{0}(r) = diag (1, r, r^{−1}, 1, 1, r, r^{−1}, 1)

and g is as in (7.2). (The map t_{0} determines t_{1} and t_{2}, see [Gar98, 1.5].)

We can now perform an analogous study for G′ = Inv (J^{d}), denoting the corresponding bilinear forms as we did for G, but with an added ′. Using the set of simple roots for G′ from (7.3), we compute the corresponding fundamental weights using the formulas for them in terms of the simple roots given in [Bou68, p. 261]:

w_{1} = 2ρ_{0}, w_{2} = \frac{1}{2}(ω_{1} + ω_{2} + ω_{3} − ω_{4}) + ρ_{0} − ρ_{2}, w_{3} = \frac{1}{2}(ω_{1} + ω_{2} + ω_{3} + ω_{4}) + 2ρ_{0} − ρ_{2},

w_{4} = ω_{1} + ω_{2} + 2(ρ_{0} − ρ_{2}), w_{5} = ω_{1} + ρ_{0} − 2ρ_{2}, and w_{6} = −2ρ_{2}. (8.5)

The minimal quadratic form on X_{∗}(S′) is given by b′_{∗}(α^{∗}, α^{∗}) := b′(α̌, α̌). (No normalizing factor is required here because all of the roots have the same length.) Using the embedding of S in S′ to consider γ_{1}^{∗} as a loop in S′, we have that w_{i}γ_{1}^{∗}(r) = r^{δ_{i4}}, so α_{4}^{∗} is just this image of γ_{1}^{∗} in X_{∗}(S′). Since b′_{∗}(α_{4}^{∗}, α_{4}^{∗}) = b′(α̌_{4}, α̌_{4}) = 2 = b_{∗}(γ_{1}^{∗}, γ_{1}^{∗}), we are done.

Invariants of ^{1}D_{4}. We must also make a side trip to discuss the Rost invariant of non-split groups of type ^{1}D_{4}.

For (A, σ) a central simple F -algebra with orthogonal involution, there is a simple group Spin (A, σ) associated to it which is the simply connected cover of P O^{+}(A, σ) := Aut^{+} (A, σ). (Please see [KMRT98] for more information about algebras with involution in general, and [KMRT98, p. 187] for a more constructive definition of Spin (A, σ) in particular.) For A of even degree, we have a short exact sequence

1 ⟶ μμ_{2} ⟶ Spin (A, σ) ⟶ O^{+}(A, σ) ⟶ 1, (8.6) where O(A, σ) is the algebraic group with F -points

O(A, σ)(F) = {a ∈ A | σ(a)a = 1}

and identity component O^{+}(A, σ).

Lemma 8.7. Suppose that (A, σ) is a central simple F -algebra of index ≤ 2 with isotropic orthogonal involution with trivial discriminant such that deg A = 8 (so that Spin (A, σ) is isotropic of type ^{1}D_{4}). The kernel of the Rost invariant on Spin (A, σ) is contained in the image of H^{1}(F, μμ_{2}) under the map induced from (8.6).

For the proof of this lemma, we will need another result which we state carefully here.

Lemma 8.8. (Parimala, Sridharan, and Suresh [PSS]) Suppose that (A, σ) is a central simple F -algebra of index 2 with orthogonal involution, and let K be the function field of the Severi-Brauer variety of the quaternion algebra underlying A. Then the natural map H^{1}(F, O(A, σ)) ⟶ H^{1}(K, O(A, σ)) is an injection.

Proof of Lemma 8.7: Suppose that α ∈ H^{1}(F, Spin (A, σ)) is in the kernel of the Rost invariant.

We first show that the conclusion holds if A is split. In that case, σ is adjoint to a quadratic form q and H^{1}(F, O^{+}(A, σ)) classifies quadratic forms q′ with the same dimension and discriminant as q. If we let q′ denote the quadratic form corresponding to the image of α, then since q′ comes from H^{1}(F, Spin (A, σ)), it has the same even Clifford algebra as q. Thus the quadratic form q′′ := q ⊥ −q′ has even Clifford algebra with both components split so it lies in I^{3}F by Merkurjev’s Theorem. Then the Rost invariant of Spin (q) ≅ Spin (A, σ) factors through H^{3}(F, ℤ/2), so that it is the composition

H^{1}(F, Spin (q)) ⟶^{f_{3}} H^{3}(F, ℤ/2) ⟶ H^{3}(F, ℚ/ℤ(2)).

Here f_{3}(α) = e_{3}(q′′) as described in [KMRT98, p. 437], where e_{3} : I^{3}F ⟶ H^{3}(F, ℤ/2) is the Arason invariant. Since the kernel of e_{3} is I^{4}F (which is a highly nontrivial fact due independently to Merkurjev-Suslin and Rost), q′′ lies in I^{4}F. However, q′′ is isotropic (since q is) and dim_{F} q′′ = 2 dim_{F} q = 16, so the anisotropic part of q′′ has dimension < 16. Thus by the Arason-Pfister Hauptsatz [Lam73, X.3.1] q′′ must be hyperbolic. Ergo q ≅ q′ and we are done with the case where A is split.

We assume from now on that A is nonsplit, hence that it has index 2. There is a natural embedding O^{+}(A, σ) ⟶ O(A, σ), and this induces a map on cohomology. We first show that α has trivial image in H^{1}(F, O(A, σ)). Let K be the function field of the Severi-Brauer variety of the quaternion algebra underlying A. By Lemma 8.8, the natural map H^{1}(F, O(A, σ)) ⟶ H^{1}(K, O(A, σ)) is an injection. Since the image of α in H^{1}(K, O(A, σ)) is trivial by the split case, we have shown that α has trivial image in H^{1}(F, O(A, σ)) in general.

We conclude by showing that α has trivial image in H^{1}(F, O^{+}(A, σ)). As described in [GTW97, §1] or [KMRT98, pp. 447, 448], the kernel of the map

H^{1}(F, O^{+}(A, σ)) ⟶ H^{1}(F, O(A, σ))

consists of two elements. By the main theorem of [GTW97], the nontrivial element of the kernel has nontrivial image [A] ∈ Br_{2} F ≅ H^{2}(F, μμ_{2}), where the map into H^{2}(F, μμ_{2}) is the connecting homomorphism induced by sequence (8.6). Since α comes from H^{1}(F, Spin (A, σ)), it cannot map to the nontrivial element of the kernel and so must have trivial image in H^{1}(F, O^{+}(A, σ)).

Back to the exceptional groups. We are interested in a special subclass of the algebras with involution which were considered in Lemma 8.7, specifically in those which can be constructed in the following way: Consider a triple \underline{Q} := (Q_{0}, Q_{1}, Q_{2}) of quaternion algebras over F such that [Q_{0}] + [Q_{1}] + [Q_{2}] = 0 in the Brauer group of F. Then for γ_{i} the unique symplectic involution on Q_{i},

(M_{2}(Q_{i}), σ′_{i}) ≅ (Q_{i+1}, γ_{i+1}) ⊗_{F} (Q_{i+2}, γ_{i+2})

for i = 0, 1, 2 where the subscripts are taken modulo 3, and σ′_{i} is an orthogonal involution with trivial discriminant. If Q_{i} is split for some i, then σ′_{i} is adjoint to the quadratic form which is the norm of Q_{i+1} ≅ Q_{i+2}. In any event, we can extend σ′_{i} by “adding on a hyperbolic plane” (i.e., by taking a hyperbolic extension as defined in [Gar98, 5.2], which is the appropriate analogue for central simple algebras with involution of adding hyperbolic planes onto a hermitian form) to an orthogonal involution σ_{i} on M_{4}(Q_{i}) with trivial discriminant. Then (M_{4}(Q_{i}), σ_{i}) is an algebra with involution as in the statement of Lemma 8.7, and

C_{0}(M_{4}(Q_{i}), σ_{i}) ≅ M_{4}(Q_{i+1}) × M_{4}(Q_{i+2}) (8.9)

by [MPW96, pp. 584–585, Lemmas 5.8 and 5.9]. The involution \underline{σ_{i}} on C_{0}(M_{4}(Q_{i}), σ_{i}) induced by σ_{i} restricts to be an orthogonal involution with trivial discriminant on each component [KMRT98, 42.3] which is isotropic since σ_{i} is by [Gar99c, 1.5]. Then because σ_{i+1} and σ_{i+2} and (8.9) holds for all i, \underline{σ_{i}} restricts to be σ_{i+1} × σ_{i+2}.

For such a triple \underline{Q}, we define the group Spin (\underline{Q}) := Spin (M_{4}(Q_{i}), σ_{i}). (The three possibilities for i all give isomorphic groups by [KMRT98, 42.5].)

Lemma 8.10. For every triple of quaternion algebras \underline{Q} as above which are split by a quadratic extension Δ, there is a map

H^{1}(F, Spin (\underline{Q})) ⟶ H^{1}(F, Aut^{+} (B_{Δ/F}^{q}))

such that

(1) the composition H^{1}(F, Z(Spin (\underline{Q}))) ⟶ H^{1}(F, Spin (\underline{Q})) ⟶ H^{1}(F, Aut^{+} (B_{Δ/F}^{q})) is trivial, and

(2) the invariant induced on H^{1}(F, Spin (\underline{Q})) by the Rost invariant on H^{1}(F, Aut^{+} (B_{Δ/F}^{q})) is the usual Rost invariant or is 3 times it.

Moreover, for every element α ∈ H^{1}(Δ/F, Aut^{+} (B_{Δ/F}^{q})) ⊆ H^{1}(F, Aut^{+} (B_{Δ/F}^{q})), there is some triple \underline{Q} of quaternion algebras which is split by Δ such that α is in the image of this map.

Before we proceed with the proof, we introduce another algebraic group, which is connected and reductive with semisimple type ^{1}D_{4}. For Δ a quadratic field extension of F with nontrivial F -automorphism ι, we set Rel_{Δ/F} (ℭ^{d}, 𝔫) to be the algebraic group Rel (ℭ^{d}, 𝔫) where the action of ι is twisted so that

ι\underline{t} := ι(σ(t_{0})^{−1}, σ(t_{1})^{−1}, σ(t_{2})^{−1})ι,

where \underline{t} = (t_{0}, t_{1}, t_{2}) is a related triple and juxtaposition denotes the usual ι-action on ℭ^{d}. Now consider the map Rel_{Δ/F} (ℭ^{d}, 𝔫) ⟶ Aut^{+} (B_{Δ/F}^{q}) given by \underline{t} ↦ f_{g\underline{t}} for f as in (3.6) and g as in (7.2). Since g† ; \underline{t} = g_{σ(\underline{t})^{−1}}, this map is defined over F, and will be a key tool in our proof.

Proof of Lemma 8.10: For brevity, we will write G := Rel_{Δ/F} (ℭ^{d}, 𝔫).

Since \underline{Q} is split by Δ, we can write Δ := F (\sqrt{d}) for some d ∈ F^{∗} and then there are some a_{i} ∈ F^{∗} such that Q_{i} ≅ (a_{i}, d)_{F}. We may assume that the product of the a_{i}’s is 1, so that they form a related triple. We define a related triple \underline{t} ∈ G(Δ) as in [Gar98, 4.7] by

t := m_{i}(\underline{a}) d P ((1 2)(3 6)(4 5)(7 8)) (8.11)

where d := diag (1, 1, −1, 1, 1, −1, 1, 1),

m_{i}(\underline{a}) := diag (1, a_{i}, a_{i}, a_{i+2}^{−1}, a_{i+1}^{−1}, 1, 1, a_{i}), (8.12)

and for π ∈ S_{8}, P (π) is defined to be the matrix which sends u_{i} ↦ u_{π(i)}. All of this is with respect to the basis u_{1}, u_{2}, …, u_{8}, so that σ is given by

Int ( ..._{1}^{1} ) ∘ t.

Then σ(t_{i})t_{i} = a_{i} and σ(t_{i}) = t_{i}. This provides us with a 1-cocycle τ ∈ Z^{1}(Δ/F, G) given by τ_{ι} := \underline{t}.

We can identify Spin (\underline{Q}) with the group Spin (ℭ^{d}, 𝔫) where the Galois action is twisted by the inflation of the cocycle τ. This provides a map Spin (\underline{Q}) ⟶ _{τ}G defined over F, where τG is the group G with Galois action twisted by the inflation of τ as in [Ser94, I.5.3], so that γ ∈ Gaℓ (F_{s}/F) acts by sending x to τ_{γ} ^{γ}xτ_{γ}^{−1}, where ^{γ}x denotes the usual Galois action on G. Combined with the map H^{1}(F,_{τ}G) ⟶^{∼} H^{1}(F, G) induced by the map on 1-cocycles given by z ↦ z′ where z′_{γ} = z_{γ}τ_{γ} [Ser94, §I.5] and the map on the H^{1} level induced by the inclusion G ↪ Aut^{+} (B_{Δ/F}^{q}), we have our desired map.

We observe that the definition of our map is independent of our choice of \underline{a}. For if we had picked some other triple \underline{a′} such that Q_{i} ≅ (a′_{i}, d)_{F}, a_{i}/a′_{i} would be a norm from Δ. So we could set \underline{δ} = (δ_{0}, δ_{1}, δ_{2}) ∈ (Δ^{∗})^{×3} where N_{Δ/F} (δ_{i}) = a_{i}/a′_{i} for i = 0, 1 and δ_{2} = (δ_{0}δ_{1})^{−1}. Then N_{Δ/F} (δ_{2}) = a_{2}/a′_{2} as well and \underline{δ} is a related triple. Set \underline{s} ∈ G(Δ) to be the related triple with s_{i} := m_{i}(\underline{δ}) for m_{i} as in (8.12). (The triple \underline{s} is indeed related by [Gar98, 1.6].) Then

\underline{s}^{−1} \underline{t} ^{ι}\underline{s} = \underline{t}′,

so the 1-cocycle τ′ given as the inflation of τ′_{ι} = \underline{t}′ is cohomologous to τ, which proves the well-definition of our map.

(1): We will show that the image of H^{1}(F, Z(Spin (\underline{Q}))) in H^{1}(F,_{τ}G) is trivial. We can identify Z(Spin (\underline{Q})) with the related triples in μμ_{2}^{×3}, so projection on the ith component defines a morphism π_{i}: Z(Spin (\underline{Q})) ⟶ μμ_{2}.

Suppose that c ∈ Z^{1}(F, Z(Spin (\underline{Q}))). Then H^{1}(π_{i})(c) ∈ H^{1}(F, μμ_{2}) defines a square class ℓ_{i}F^{∗2} in F^{∗}/F^{∗2}. Since c_{γ} is a related triple for all γ ∈ Gaℓ (F_{s}/F), we may assume that ℓ_{0}ℓ_{1}ℓ_{2} = 1. We may further choose λ_{i} ∈ F_{s}^{∗} such that λ_{0}λ_{1}λ_{2} = 1 and λ_{i}^{2} = ℓ_{i} ∈ F^{∗}. Then \underline{λ} = (λ_{0}, λ_{1}, λ_{2}) is in _{τ}G(F_{s}) and the 1-cocycle z given by z_{γ} := \underline{λ}^{−1}c_{γ} ^{γ}\underline{λ} = c_{γ}\underline{λ}^{−1} ^{γ}\underline{λ} is the inflation of the 1-cocycle in Z^{1}(Δ/F,_{τ}G) given by z_{ι} := \underline{ℓ}^{−1} for \underline{ℓ} := (ℓ_{0}, ℓ_{1}, ℓ_{2}).

Setting \underline{s} ∈ _{τ}G(Δ) to be the related triple given by s_{i} := m_{i}(\underline{ℓ}^{−1}), we have \underline{s}^{−1}z^{ι}\underline{s} = 1, so the image of c in H^{1}(F,_{τ}G) is cohomologically trivial.

(2): The set of possible invariants H^{1}(F, Spin (\underline{Q})) ⟶ H^{3}(F, ℚ/ℤ(2)) forms a group isomorphic to ℤ/n with n = 4 if none of the Q_{i} are split and n = 2 otherwise [KMRT98, pp. 436, 442]. For any extension field L of F the restriction map gives a homomorphism from the group of invariants over F to the invariants over L, so we may assume that \underline{Q} has all components split and show that the induced invariant is nontrivial.

By the independence of choice of \underline{a}, we may take a_{i} := 1 in our choice of descent data \underline{t}. Now Spin (ℭ^{d}, 𝔫) embeds in G, and \underline{t} is in its image, so τ lies in the image of H^{1}(Δ/F, Spin (ℭ^{d}, 𝔫)). However, we also have an exact sequence

1 ⟶ Z ⟶ Spin (ℭ^{d}, 𝔫) ⟶ P O^{+}(ℭ^{d}, 𝔫) ⟶ 1,

where P O^{+}(ℭ^{d}, 𝔫) := Aut^{+} (End(ℭ^{d}, 𝔫)) is the adjoint group associated to Spin (ℭ^{d}, 𝔫), and this induces an exact sequence

H^{1}(F, Z) ⟶ H^{1}(F, Spin (ℭ^{d}, 𝔫)) ⟶ H^{1}(F, P O^{+}(ℭ^{d}, 𝔫)).

This last set classifies “trialitarian triples” over F, see [KMRT98, §42.A] for a definition. The image of τ in it is the class of \underline{Q}, which is trivial. Thus τ comes from H^{1}(F, Z). However, since Spin (ℭ^{d}, 𝔫) is split over F, the image of H^{1}(F, Z) is trivial [Gar99b, 2.3]. Thus our twist of G by τ is the trivial twist, and our map H^{1}(F, Spin (\underline{Q})) ⟶ H^{1}(F, Aut^{+} (B_{Δ/F}^{q})) is the one induced by the standard embedding of split groups D_{4} in E_{6}, so the induced invariant is the Rost invariant by Lemma 8.2.

Finally suppose that α ∈ H^{1}(Δ/F, Aut^{+} (B_{Δ/F}^{q})). The map G ↦ Aut^{+} (B_{Δ/F}^{q}) discussed just before the proof induces a map

H^{1}(Δ/F, G) ⟶ H^{1}(Δ/F, Aut^{+} (B_{Δ/F}^{q})). (8.13)

That this map is a surjection is the content of [Fer69, p. 65, Lem. 3]. Thus α must be the image of a 1-cocycle η′ ∈ Z^{1}(Δ/F, G). Set a_{i} to be the multiplier of (η′_{ι})_{i} and fix d ∈ F^{∗} such that Δ := F (\sqrt{d}). Then if we set \underline{Q} to be a triple of quaternion algebras such that Q_{i} ≅ (a_{i}, d)_{F} and define \underline{t} as before, η′ will be the image of η ∈ Z^{1}(Δ/F, Spin (\underline{Q})) where η_{ι} := η′_{ι}\underline{t}^{−1}.

The astute reader will have noticed that the image of Rel_{Δ/F} (ℭ^{d}, 𝔫) in Aut^{+} (B_{Δ/F}^{q}) is the reductive part (= Levi subgroup) of a parabolic subgroup of type P (1, 6) in the notation of Section 7. Now, there are some results in [PR94b, Ch. 6] (specifically, Lemmas 6.28 and 6.17′ there) which would seem to generalize directly to give a result analogous to the surjectivity of (8.13) for arbitrary semisimple groups and parabolic subgroups, so avoiding the direct computations in groups of type E_{6} that Ferrar makes. However, the results from [PR94b] aren’t suitable for our purposes as it is not clear how to generalize their proofs to imperfect fields and instead of the reductive part of our chosen parabolic subgroup we would get a surjection as in (8.13) where the domain is H^{1} of the reductive part of a parabolic which is a Δ-conjugate of our chosen one.

Proof of Theorem 8.1: First we reduce the problem to considering groups of type E_{6}. Suppose that our quasi-split simply connected group G is of type E_{7} (so G = Inv (𝔐(J^{d}))) and α ∈ H^{1}(F, G) has trivial Rost invariant. The class α corresponds to some nondegenerate Freudenthal triple system 𝔐 over F which arises from some Brown F -algebra B. If B is of type 1 then by Lemmas 2.8(1) and 4.12(1), we may assume that B = B(J, F × F) and so also that 𝔐 = 𝔐(J). (Since 𝔐(J) is one triple system associated to B and one can produce similarities of 𝔐(J) with any desired multiplier as in [Fer72, p. 327], 𝔐(J) is the unique triple system associated to B.) Then we have a commutative diagram

\begin{matrix}H^{1}(F, H) & ⟶^{mR_{H}} & H^{3}(F, ℚ/ℤ(2)) \\ ↓ & & ∥ \\ H^{1}(F, G) & ⟶^{R_{G}} & H^{3}(F, ℚ/ℤ(2)), \end{matrix} (8.14)

where H = Aut (J^{d}), G = Inv (𝔐(J^{d})), and the class α of 𝔐 is the image of the class of J. The left-hand map is induced by the standard embedding of split groups F_{4} ↪ E_{6} ↪ E_{7}, so m = 1. Then the conclusion follows since the kernel of R_{H} is trivial.

Thus we may assume that B is of type 2 so that the discriminant algebra Δ of B is a quadratic field extension of F. By Proposition 5.10, 𝔐(J^{d}) is a triple system associated to B_{Δ/F}^{q}, and this provides an embedding of H := Aut^{+} (B_{Δ/F}^{q}) in G. Then we have another commutative diagram as in (8.14). To use (8.14) as an inductive tool, we must show that m is relatively prime to the order of R_{H} (which is 12) and show that α is in the image of the left-hand map. The H^{3} invariants of H form a cyclic group of order 12 over F which restricts to a group of order 6 over Δ. Over Δ, the embedding of H in G is the standard embedding, so m = 1. Thus over F we must have m = 1 or 7, which settles the first issue.

Now fix a class in H^{1}(F, H) representing B. (There may be two possibilities.) This class maps to the class of some triple system 𝔐′ in H^{1}(F, G) such that 𝔐′ is also associated to B, i.e., there is some λ ∈ F^{∗} such that there is a similarity 𝔐′ ⟶ 𝔐 with multiplier λ. The center of G is isomorphic to μμ_{2}, and we can consider z ∈ Z^{1}(F, G) whose class lies in the image of λF^{∗2} ∈ F^{∗}/F^{∗2} ≅ H^{1}(F, μμ_{2}). Then we get a diagram

\begin{matrix}H^{1}(F, G) & ⟶^{R_{G}} & H^{1}(F, ℚ/ℤ(2)) \\ ↓ & & ∥ \\ H^{1}(F,_{z}G) & ⟶^{R_{G}} & H^{1}(F, ℚ/ℤ(2)). \end{matrix} (8.15)

The class of z in H^{1}(F, G) is the class of a triple system 𝔐′′ which is similar to 𝔐(J^{d}) via a similarity with multiplier λ. Since (as observed earlier in the proof) 𝔐(J^{d}) has a similarity with itself of multiplier λ, 𝔐′′ ≅ 𝔐(J^{d}). Thus z is cohomologically trivial in H^{1}(F, G) and (8.15) is commutative by [Gil98, p. 16, Lem. 8]. Moreover, the class of 𝔐′ is sent by the left-hand map to the class of 𝔐. Since _{z}G ≅ G, we may simply attach this diagram to the bottom of (8.14) and so obtain a commutative diagram as in (8.14) with m = 1 or 7 and α in the image of the left-hand map. Then the conclusion for groups of type ^{2}E_{6} finishes the proof of the E_{7} case.

Suppose now that G is of type ^{1}E_{6}. Then any class α ∈ H^{1}(F, G) corresponds to some Brown algebra B ≅ B(J, F × F, ζ) for some J and ζ. We have another commutative diagram as in (8.14) with H = Aut (J^{d}), m = 1, and G = Aut^{+} (B(J^{d}, F × F, ζ)) ≅ Aut^{+} (B(J^{d}, F × F)) where the left-hand map sends the class of J to the class of B. Since the kernel of the Rost invariant on the split group of type F_{4} is trivial, we are done with this case.

Thus we may assume that our group G is of type ^{2}E_{6} with inner extension Δ, so it is isomorphic to Aut^{+} (B^{q}) for B^{q} := B(J^{d}, Δ). Then α ∈ H^{1}(F, Aut^{+} (B^{q})) corresponds to some Brown F -algebra B of type 2 with inner extension Δ. By the argument in the type 1 case and the fact that these invariants are compatible with scalar extension, B is split by Δ, so α actually comes from H^{1}(Δ/F, Aut^{+} (B^{q})). So by Lemma 8.10, we have a diagram as in (8.14) with H = Spin (\underline{Q}), m = 1 or 3, and α in the image of the left-hand map. Since the image of R_{H} lies in the image of H^{3}(F, μμ_{n}^{⊗2}) in H^{3}(F, ℚ/ℤ(2)) for n = 2 or 4 [KMRT98, p. 442] and m is relatively prime to n, we are done by Lemma 8.7 since the image of μμ_{2} in Spin (\underline{Q}) lies in the center.

Acknowledgements

I would like to thank Markus Rost for answering my many questions about his H^{3} invariants. Thanks are also due to thank Kevin McCrimmon, Bruce Allison, John Faulkner, and Oleg Smirnov for providing some references to material on Albert algebras, Parimala, Sridharan, and Suresh for giving me a copy of their paper [PSS], and Ph. Gille for referring me to the results in Chapter 6 of [PR94b].

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R. Skip Garibaldi Dept. of Mathematics ETH Zentrum, HG G66.4 CH-8092 Zürich Switzerland

e-mail: skip@member.ams.org
web: http://math.ucsd.edu/~skip/