ON POLYNOMIAL INVARIANTS OF EXCEPTIONAL SIMPLE ALGEBRAIC GROUPS

A.ELDUQUE AND A.V.ILTYAKOV

Abstract. We study polynomial invariants of vector systems with respect to exceptional simple algebraic groups in their minimal linear representations. For each type we prove that the algebra of invariants is integral over the subalgebra of trace polynomials for a suitable algebraic system (cf.[26, 27, 11]).

1. Introduction

The ground field F is algebraically closed of characteristic zero.

Let V be a finite dimensional module over an algebraic group G (we denote it by G : V); then G acts on the algebra F [V] of polynomial functions from V to F by shifts: if ϕ ∈ G and f = f(y) ∈ F [V], then (f · ϕ)(v) = f(ϕ^{−1}(v)). The subalgebra of fixed functions F [V]^{G} is called the algebra of polynomial invariants; if V is the direct sum of several copies of a G-module W with the diagonal action of G (denote V = \underbrace{W ⊕ … ⊕ W}_{k} = kW), then

F [kW]^{G} is called the algebra invariants of the system of k vectors of G : V.

Although invariants of vector systems for classical linear groups are known for a long time [28], the case of exceptional simple algebraic groups has not been studied well enough: the generators (First Main Theorem) and relations between them (Second Main Theorem) are known only for the minimal (irreducible) module over a simple algebraic group of type G_{2} [26]. This group can be represented as the automorphism group G = Aut(O) of the Cayley-Dickson algebra O (e.g., [22]). Every element x ∈ O satisfies the equation

x^{2} − tr(x)x + n(x) = 0 ,

where tr(x) is the trace of a, and, as a G-module, O is the direct sum of a trivial one F · 1 and an irreducible module O′ = {a ∈ O| tr(a) = 0} ( dim_{F} O′ = 7). In particular, the generators of the algebra of invariants F [kO]^{G} (F [kO′]^{G}) are trace polynomials.

It is well-known that irreducible modules of minimal dimension for any type of exceptional simple algebraic group can be associated with a certain algebra or, more precisely, an (algebraic) system, that is, a vector space equipped with a finite number of multilinear operations. In particular, such a group turns out to be either the automorphism group of the associated system or at least its identity component.

Supported by the Spanish DGICYT (Pb 94-1311-C03 -03) and by the Universidad de la Rioja (96PYB07PBC).

Part of this work was done when the second author visited the University of Oviedo and was supported by the Ministry of Education and Science of Spain, Dirección General de Investigación Cientifica y Enseñanza Superior, Beca SAB95-0295.

Indeed, let O_{3} = O ⊗_{F} F_{3} be the algebra of 3 × 3-matrices over O. Define an involution a ⊗ b ↦ ā ⊗ b^{⊤}, where b ↦ b^{⊤} is the matrix transposition. The subspace H(O_{3}) of symmetric elements with the symmetrized multiplication a ∘ b = 1/2(ab + ba) is a central simple Jordan algebra A of dimension 27, which is called the Albert algebra [13].

Every x ∈ A satisfies a cubic equation:

x^{3} − tr(x)x^{2} + S(x)a − N(x) = 0 ,  (1)

where tr(x), S(x), N(x) ∈ F. The group G = Aut(A) is simple of type F_{4} (e.g.[13]) and A is the direct sum of a trivial G-submodule F · 1 and an irreducible one, A′ = {a ∈ A| tr(a) = 0}, dim_{F} A′ = 26.

The group M of non-singular transformations of A which preserve the norm N(x) is a simple group of type E_{6} and A is an irreducible M-module of minimal dimension [12]. Also, M has an outer automorphism μ of second order that defines the contragredient representation of M in End_{F} A; denote this module by A^{∗}.

In the next section we will consider an algebra A with involution where the identity component of Aut(A) is M; as an M-module, A is the direct sum of A, A^{∗} and two copies of a one-dimensional trivial M-module; we will call A the Chevalley algebra (cf.[25]).

In the final section we will consider the Freudenthal triple system T ; as a vector space it coincides with A and it is endowed with two 3-linear operations; the automorphism group of T is a simple algebraic group of type E_{7} which acts irreducibly on T [7, 5].

If L is a simple Lie algebra of type E_{8}, then G = Aut(L) is a simple group of type E_{8} [13, p.281] and L is a G-module of minimal dimension.

One of the main results of this work is about the null-cone of G : V; let us recall the definition. Let P be a subalgebra of F [V]^{G}; for a point y ∈ kA denote by π_{P} (y) the fibre of P at point y,

π_{P} (y) := {y′ ∈ V | f(y) = f(y′) ∀f ∈ P } .

If P = F [V]^{G}, then π_{P} (0) is called the null-cone of G : V [18, p.196].

Also, let B be a subsystem of A and let g(y_{1}, …, y_{r}) be an operation of A. If we replace r−1 variables with some elements in B, then we will get a linear operator in End_{F} A called a multiplication operator. For example, if A is an algebra and g is the multiplication, then in this way we get two operators L_{b} : y → by and R_{b} : y → yb. Denote by M^{A}(B) the subalgebra of End_{F} A generated by all such operators. In particular, M(A) = M^{A}(A) is called the multiplication algebra.

A subsystem B ⊆ A is said to be ad-nilpotent, if M^{A}(B) is nilpotent.

Theorem 1. Let A be the system associated with a minimal irreducible representation of an exceptional simple algebraic group, that is, A ∈ {O, A, A, T , L}, and let G = Aut(A).

A point x = (a_{1}, …, a_{k}) ∈ kA belongs to the null-cone G : kA iff the subsystem A_{x} generated by a_{1}, …, a_{k} in A is ad-nilpotent.

In fact, it is a new result only for the Chevalley algebra and the Freudenthal triple system; indeed, for A = O it follows from [26, 27], the case A = A was considered in [11] and for the adjoint representation of a linearly reductive algebraic group it was done in [21]. On the other hand, we will prove the similar statement for a wider class of simple algebras (Theorem 2).

The next proposition enables us to rewrite Theorem 1 in terms of trace polynomials.

Proposition 1. Let V be a finite dimensional module over a linearly reductive algebraic group G and let P be a homogeneous affine subalgebra of F [V]^{G}. Then F [V]^{G} is a finitely generated P -module if and only if

π_{P} (0) = π_{F [V]^{G}}(0) ,  (2)

Proof. Suppose that F [V]^{G} is a finitely generated module over P; pick a homogeneous element f ∈ F [V]^{G} of positive degree, then for suitable h_{1}, …, h_{n} ∈ P we have

f^{n} + h_{1}f^{n−1} + … + h_{n−1}f + h_{n} = 0;

we may assume that h_{1}, …, h_{n} are homogeneous and deg(h_{i}) = i·deg(f) for all i. Hence, f(x) = 0 for all x ∈ π_{P} (0), therefore, π_{P} (0) ⊆ π_{F [V]^{G}}(0). The reciprocal inclusion is evident.

For a proof of the converse statement see, for example, [15, sec.4.3.F]. □

Now, let P be a subalgebra of F [kA]^{G}, where A and G are from the hypothesis of Theorem 1, then (2) means that x ∈ π_{P} (0) if and only if the subsystem A_{x} is ad-nilpotent. One of the examples of P satisfying this condition is the algebra of trace polynomials which can be defined for an arbitrary finite dimensional system A (cf. [11, 19]).

Let x_{i} be the projection of kA onto the i-th summand, hence, x_{1}, …, x_{k} can be considered as k independent identity mappings from A to A; in particular, they lie in the system Pol(kA, A) ≃ F [kA] ⊗_{F} A of polynomial mappings from kA to A (with pointwise operations). Hence, we may define the system of generic elements F_{k}(A) of rank k as the subsystem over F generated by x_{1}, …, x_{k}. In particular, if A is a simple algebra generated ≤ k elements, then F_{k}(A) is a free affine algebra of type A and rank k [17].

Now, let g(y_{1}, …, y_{r}) be a multilinear operation of A, for a fixed i ∈ {1, …, r} we get a multilinear mapping from A to End_{F} A:

E = E(y_{1}, …, y_{i−1}, y_{i+1}, …, y_{r}) : y_{i} → g(y_{1}, …, y_{r}) .

The composition of E with u_{1}, …, u_{i−1}, u_{i+1}, …, u_{r} ∈ F_{k}(A) gives a polynomial mapping

E_{1}(x_{1}, …, x_{k}) = E(u_{1}, …, u_{i−1}, u_{i+1}, …, u_{r})

from kA to End_{F} A. Denote by Mult_{k}(A) the subalgebra of Pol(kA, End_{F} A) generated over F by all such mappings. Obviously, any values of y_{1}, …, y_{k} in A define uniquely a homomorphism from Mult_{k}(A) to M(A). Moreover, if the values belong to a subsystem B, then the image of Mult_{k}(A) is contained in M^{A}(B).

Next, for every E ∈ Mult_{k}(A) the mapping Tr_{A}(E), where Tr_{A}(x) is the trace of a linear operator x ∈ End_{F} A, belongs to F [kA]. Moreover, it is easy to check that Tr_{A}(E) is an invariant of G; we call the subalgebra Tr[kA] generated by 1 and all such elements the algebra of trace polynomials of A of rank k (cf. [11, 19]).

Proposition 2. Let A be a finite dimensional system, then Tr[kA] is a finitely generated algebra over F.

Proof. Observe that P = Tr[kA], B = F_{k}(A), Mult_{k}(A) (⊆ Pol(kA, End_{F})) are homogeneous. Denote by P_{+} the sum of homogeneous components of P of positive degree, then P_{+} is an ideal of P spanned by all Tr_{A}(E), where E ranges over the sum C of homogeneous components of Mult_{k}(A) of positive degree.

Next, in Pol(kA, A) ≃ F [kA] ⊗_{F} A we consider the F -subsystem B̃ = P B = B + P_{+}B (P B is the P -submodule of Pol(kA, A) generated by B) and its ideal B̃_{+} = P_{+}B. Similarly, in Pol(kA, End_{F} A) we consider the subalgebra C̃ = C + P_{+}C and its ideal C̃_{+} = P_{+}C.

Since the number of operations of the system A is finite, there is r ∈ N such that every operation on A depends on ≤ r variables. By the induction on m ∈ N it is easy to prove that every homogeneous component B_{s} of B of degree s ≥ r^{m} is contained in the subspace spanned by elements E · b where b ∈ B and E ranges over the m-th power of C:

∑_{s≥r^{m}} B_{s} ⊆ C^{m} · B .

Indeed, if m = 1, then it is evident; next, B_{s} for s ≥ r^{m+1} is spanned by b = g(b_{1}, …, b_{l}), where g = g(y_{1}, …, y_{l}) is an operation of A and b_{1}, …, b_{l} are homogeneous elements in B. We need to show that b ∈ C^{m+1} · B; we may assume that l ≥ 2 and there are at least two elements among b_{1}, …, b_{l} with positive degree (otherwise we consider b_{i} with deg(b_{i}) = deg(b) instead of b). By definition, l ≤ r, hence, for some i ∈ {1, …, l} there is b_{i} with degree being not less than r^{m}; by induction, it belongs to C^{m} · B, hence, b ∈ C^{m+1} · B.

Denote n = dim_{F} A; by Cayley-Hamilton’s theorem (e.g., [20, p.18]), every element in C is integral over P_{+} of degree ≤ n; obviously, C̃ has the same property, hence, C̃/C̃_{+} is a nil algebra of index n, and by Nagata-Higman’s theorem [20, p.149], it is nilpotent. Let m be the nilpotency index, then C̃^{m} ⊆ C̃_{+} and, therefore, every homogeneous component B_{s} of degree s ≥ r^{m} is contained in the P -module Q generated by B_{t}, t < r^{m}:

B̃ = P ( ∑_{t<r^{m}} B_{t}) ,

so that

C̃ = P ( ∑_{t<(r−1)r^{m}} C_{t}) = ∑_{t<(r−1)r^{m}} C_{t} + P_{+}C̃ .

Taking traces, we prove that P is generated by the homogeneous elements in P of degree ≤ (r − 1)r^{m}. Since these form a finite dimensional F -subspace of P, we conclude that P is finitely generated. □

Obviously, Tr[kA] is homogeneous and, by definition, contains 1, therefore, it is a homogeneous affine subalgebra of F [kA]^{G}.

Now, pick x = (a_{1}, …, a_{k}) ∈ π_{Tr[kA]}(0), then the subsystem A_{x} generated by a_{1}, …, a_{k} is ad-nilpotent. Actually, the condition x = (a_{1}, …, a_{k}) ∈ π_{Tr[kA]}(0) forces that the trace of any element E(a_{1}, …, a_{k}), where E(x_{1}, …, x_{k}) ∈ Mult_{k}(A) has zero trace, hence, this implies that any element in M^{A}(A_{x}) is nilpotent, hence, this finite dimensional associative algebra is nilpotent, therefore, A_{x} is ad-nilpotent. So, we get the following statement which is equivalent to Theorem 1.

Corollary 1. Let A be the system associated with the minimal irreducible representation of a exceptional simple algebraic group, let G = Aut(A). Then F [kA]^{G} is a finitely generated module over Tr[kA].

In the same notations, let G^{∘} be the identity component of G, then G^{∘} is a simple algebraic group; since G/G^{∘} is finite, F [kA]^{G^{∘}} is integral over F [kA]^{G}, hence, since F [kA]^{G^{∘}} is an affine algebra (e.g.[18, sect.3.4]), it is a finitely generated module over Tr[kA].

Now, if we replace the generators x_{1}, …, x_{k} in trace polynomials with the projectors x′_{1}, …, x′_{k} onto the corresponding irreducible G^{∘}-submodule V ⊆ A, we will get a subalgebra Tr[kV] of F [kV]^{G^{∘}} and the latter one is a finitely generated module over it. Thus, Corollary 1 provides an effective way to construct a “significant” part of polynomial invariants of exceptional simple algebraic groups. The problem whether there are invariants of a different type is unsolved so far (except the cases mentioned above).

Next, if f(x_{1}, …, x_{k}) ∈ F_{k}(A), then for all u_{1}, …, u_{k} ∈ F_{k}(A) we have f(u_{1}, …, u_{k}) ∈ F_{k}(A). In other words, such a replacement of variables defines an algebra endomorphism of F_{k}(A) and, conversely, every endomorphism has a such a form (the F_{k}(A) is a free algebra of rank k in the variety of algebras generated by A).

Notice that an endomorphism of F_{k}(A) induces an endomorphism ϕ of F [kA] (the replacement of the variables x_{1}, …, x_{k} with elements in F_{k}(A)). Moreover, the subalgebra of invariants F [kA]^{G} is stable under ϕ, that is, F [kA]^{G} is a fully characteristic subalgebra of F [kA]^{G}. Therefore, it makes sense to describe generators of F [kA]^{G} as a fully characteristic algebra (in classical terms we are looking for typical generators [28]).

For example, by [26, 27], the algebra F [kO]^{G} is generated tr(x_{1}). In the case of the Albert algebra, the fully characteristic subalgebra P generated by tr(x_{1}) satisfies (2), therefore, the algebra F [kA]^{G} is a finitely generated module over P. If A is a simple Lie algebra, then the same holds for the fully characteristic subalgebra of F [kA]^{G} generated by tr(R_{x_{1}}^{j}), j ∈ {2, …, dim_{F} (A)}.

2. Structurable algebras and invariants of E_{6}

First of all, let us recall the definition of structurable algebra [1].

Let A be an algebra over F with an involution x ↦ x̄. Then it is decoposed into the direct sum of the subspaces of symmetric and skew-symmetric elements, that is, A = H(A) ˙+S(A). We call A a structurable algebra if it satisfies the following identities (∀x, y ∈ A, ∀a, b, c ∈ H(A), ∀s ∈ S(A)):

(s, x, y) = −(x, s, y);  (3)
(a, b, c) − (b, a, c) = (c, a, b) − (c, b, a);  (4)
2/3[[a^{2}, a], b] = (b, a^{2}, a) − (b, a, a^{2}) ,  (5)

where [a, b] = ab − ba and (a, b, c) = (ab)c − a(bc) are the commutator and associator respectively.

If A is unital (one can join 1 in the external way, putting ¯1 = 1), then (3-5) are equivalent to the following identity

[T_{z}, V_{x,y}] = V_{T_{z}x,y} − V_{x,Tz̄y} ,  (6)

where V_{x,y} ∈ End_{F} A, V_{x,y}z = (xȳ)z + (zȳ)x − (zx̄)y and T_{z} = V_{z,1}.

Note that the class of structurable algebras includes Jordan algebras where the involution is the identity mapping (in particular, the Albert algebra) and alternative algebras with an involution (for example, O).

Also, since the involution is regarded as an additional operation on A, subalgebras and ideals are (by definition) closed by the involution, automorphisms and derivations commute with the involution and so on. For example, the algebra of generic elements F_{k}(A) is generated as an algebra over F by x_{1}, …, x_{k}, x̄_{1}, …, x̄_{k}.

Now we will give an important example of structurable algebra. Let N(x, y, z) be the complete linearization of N(x), where N(x, x, x) = 6N(x); since tr(xy) is a Killing form of A, we can define a quadratic mapping from A to A^{∗}, x ↦ x^{#}, where tr(x^{#}y) = 1/2N(x, x, y) (here x^{#}y is the product of elements in the Albert algebra); also, put a×b = (a + b)^{#} − a^{#} − b^{#}. Consider the direct sum of A, A^{∗} and two copies of F written in the matrix form :

A = ( \begin{matrix}F & A \\ A^{∗} & F \end{matrix} ) .

Next, define the multiplication on A:

( \begin{matrix}α_{1} & a_{1} \\ b_{1} & β_{1} \end{matrix} ) · ( \begin{matrix}α_{2} & a_{2} \\ b_{2} & β_{2} \end{matrix} ) = ( \begin{matrix}α_{1}α_{2} + (a_{1}, b_{2}) & α_{1}a_{2} + β_{2}a_{1} + b_{1} × b_{2} \\ α_{2}b_{1} + β_{1}b_{2} + a_{1} × a_{2} & (b_{1}, a_{2}) + β_{1}β_{2} \end{matrix} )

and an involution

\overline{( \begin{matrix}α & a \\ b & β \end{matrix} )} = ( \begin{matrix}β & a \\ b & α \end{matrix} ) .

This algebra A is a structurable one (the algebra of an admisible triple [1]), we call it the Chevalley algebra (cf.,[25]). From the definition of A it follows easily that the automorphism group of A consists of two components: M and τM, where τ is an automorphism of period 2 which swops the lines and columns and for all a ∈ A and g ∈ M we have τ(g(a)) = μ(g)(τ(a)), where μ is the outer automorphism of M (see Introduction).

Let A be a unital structurable algebra; denote by Inder(A) the algebra of inner derivations of A (spanned by derivations D_{a,b}, see [1]), then Instrl(A) = T_{A} + Inder(A), where T_{A} = {T_{a}| a ∈ A}, is a subalgebra of a Lie algebra End_{F} (A)^{(−)}. Also, on the vector space

K(A) = S(A) ˙+A ˙+Instrl(A) ˙+A^{∼} ˙+S(A)^{∼} ,

where A^{∼} ≃ A, S(A)^{∼} ≃ S(A), one can define the structure of a Z-graded Lie algebra of length 5 [1] (the Kantor-Koecher-Tits construction for structurable algebras). In particular, the component K_{1}(A) (K_{−1}(A)) is equal to A(A^{∼}); we will also make use of the following properties of the multiplication [a, b] on K(A) :

(a) K(A) is generated by A ∪ A^{∼};

(b) for all a ∈ A and b ∈ A^{∼}

[a, b] = V_{a,b} ∈ Instrl(A) = K_{0}(A) .

Let ϕ be an automorphism of the graded Lie algebra K(A), then its restriction to K_{1}A) = A is an autotopy of the structurable algebra A [2]. It means that there exists ϕ̂ ∈ End_{F} (A) such that for all x, y, z ∈ A we have

ϕ(V_{x,y}z) = V_{ϕ(x),ϕ̂(y)} ϕ(z) .  (7)

The set of all autotopies of A is a group Γ(A) called the structure group of A (a closed subgroup of GL(A))[2, p.135]; ϕ ↦ ϕ̂ is an automorphism of Γ(A) of period 2; besides, the automorphism group of A is the subgroup of all elements in Γ(A) which fix the identity element of A [2, p.134]:

Aut(A) = {ϕ ∈ Γ(A)| ϕ(1) = 1} ,  (8)

Conversely, by [2, Prop. 12.3, Cor.8.6], every autotopy ϕ of A can be extended uniquely to an automorphism ϕ̃ of the graded Lie algebra K(A) and this defines an isomorphism of Γ(A) onto the automorphism group of the Z-graded Lie algebra K(A).

Observe that the elements 1 ∈ L_{1} = A, 2 ∈ L_{−1} = A^{∼} and 2T_{1} ∈ L_{0} = Instrl(A) form a standard basis of a simple 3-dimensional Lie subalgebra H of K(A) (in particular, the Z-grading is defined by the action of the semisimple element 2T_{1}). Summarizing, we get the following statement.

Lemma 1. Let Aut_{H}(K(A)) be the subgroup of all automorphisms of the Lie algebra K(A) which fix the elements in H (the group of automorphisms over H), then for every ψ ∈ Aut_{H}K(A) the restriction ψ|_{A} to the graded component K_{1}(A) = A is an automorphism of the structurable algebra A and every automorphism of A has such a form.

Now we are going to prove an analog of Theorem 1 for structurable algebras.

Theorem 2. Let A be a finite dimensional simple structurable algebra and let G = Aut(A). Then a point x = (a_{1}, …, a_{k}) ∈ kA is in the null-cone G : kA iff the subalgebra A_{x} = alg_{F} {a_{1}, …, a_{k}} ⊆ A generated by a_{1}, …, a_{k} is nilpotent.

Proof. If x is an element in the null-cone, then for every f ∈ Tr[kA] we have f(a_{1}, …, a_{k}) = 0, hence, A_{x} is a nilpotent subalgebra of A.

Conversely, suppose that A_{x} is nilpotent.

Lemma 2. Let B be a nilpotent subalgebra of a structurable algebra A, then

M^{B}(A) = alg_{F} {R_{b}, L_{b}| b ∈ B} ⊆ End_{F} (A)

is nilpotent.

Proof. Consider the algebra A as a structurable bimodule N over B, then the split null extension C = B ˙+N [23, p.485] is a solvable structurable algebra; by [24] it is nilpotent, hence, M^{B}(C) is nilpotent. Notice that M^{B}(A) ≃ M^{B}(N) is a homomorphic image of M^{B}(C), therefore, M^{B}(A) is also nilpotent. □

We continue the proof of Theorem 2. Denote L = K(A); by [1, cor.6], the algebra L is simple. Next, let B_{x} be the subalgebra of A generated by A_{x} and the identity element 1, consider a subalgebra of L

K̃(B_{x}) = S(B_{x}) ˙+B_{x} ˙+(T_{B_{x}} ˙+alg{D_{a,b}| a, b ∈ B_{x}}) ˙+B_{x}^{∼} ˙+S(B_{x})^{∼} ;

it is equal to the sum of H and the ideal I of K̃(B_{x}) generated by A_{x} ∪ A_{x}^{∼}. By the previous lemma every element in I is ad-nilpotent (that is, for all a ∈ I the operator of adjoint multiplication ad(a) : x ↦ [x, a] in End_{F} L is nilpotent).

Let K be the automorphism group Aut(L) of the algebra L without grading; note that K is a closed subgroup of GL(L) and its Lie algebra L(K) is the derivation algebra Der(L)

[9, 13.2], where elements in K act by conjugation (the adjoint representation of K). Since L is simple, all derivations of L are inner, i.e.,

Der(L) = ad(L) = {ad(a)| a ∈ L} ≃ L .

Denote by N the normalizer N_{K}(I) = {ϕ ∈ K ϕ(I) ⊆ I} of I in K; notice that the normalizer in GL(L) of any subspace is a closed subgroup in GL(L), hence, N_{K}(I) is closed in K:

N_{K}(I) = K ∩ N_{GL(L)}(I) .

Next, let S be the subgroup generated by exp(ad(a)), for all a ∈ I; observe that S is a closed connected subgroup of Aut(L) [9, 7.5]; also, S ⊆ N, hence, S lies in the identity component N^{∘} of N. Moreover, S is a normal subgroup of N consisting of unipotent linear transformations of L. hence, by [9, 19.5], S lies in the unipotent radical R = R_{u}(N) = R_{u}(N^{∘}). Thanks to [9, 13.1] the Lie algebra L(S) contains ad(I), hence, ad(I) ⊆ L(R_{u}(N^{∘})).

Next, consider the subgroup P generated by exp(ad(a)) where a ranges over strongly nilpotent elements in H; by [8, p.55], P is a closed connected subgroup of K and its Lie algebra is equal to H [6, p.175], hence, P is a simple algebraic subgroup of K. Moreover, its generators keep I being invariant, hence, P ⊆ N^{∘} and, therefore, P lies in a Levi factor Q of N^{∘} [9, 30.2].

By [21, 2.6 and 2.3], in K^{∘} there is a one-parametric subgroup λ_{t}, t ∈ F^{∗} = F \{0} such that:

(a) lim_{t→0} λ_{t}(a) exists for all a ∈ L(N^{∘});

(b) lim_{t→0} λ_{t}(a) = 0 for all a ∈ L(R);

(c) λ_{t}(a) = a for all a ∈ L(Q).

Recall that ad(H) ⊆ L(Q), hence, by (c), ∀t ∈ F^{∗} λ_{t} fixes all elements in ad(H) and, therefore, it fixes all elements in H. It means that λ_{t} is an automorphism of the graded Lie algebra L and, by Lemma 1, the restriction of λ_{t} onto A = L_{1} is an automorphism of A for all t ∈ F^{∗}. From (a)-(b), it follows that x lies in the null-cone of G : kA. The theorem is proven. □

Proposition 1 yields immediately the following statement.

Corollary 2. Let A be a finite dimensional simple structurable algebra and let G = Aut(A). Then F [kA]^{G} is a finitely generated module over Tr[kA].

It is known [4] that a finite-dimensional structurable algebra A has a Killing form ⟨x, y⟩ = Tr_{A}(L_{x̄y+ȳx}) and the radical (the maximal solvable ideal) of A coincides with the radical of the form. Also, by [24] the radical of A is nilpotent. Observe that for A = A we have Tr_{A}(L_{x}) = 2tr_{A}(x), where tr_{A}(x) is the sum of the diagonal entries of x (this is precisely tr(x) in (1). Hence, for every subalgebra B ⊆ A if tr_{A}(b) = 0 for all b ∈ B, then B is nilpotent. Indeed, suppose that B is not nilpotent, then B is an ideal of the structurable algebra F · 1 ˙+B, so by [4, theorem 7] there is a nonzero idempotent e = ē in B. Now by [4, theorem 11] we have Tr_{A}(L_{e}) ≠ 0, hence, tr_{A}(e) ≠ 0, a contradiction.

Let tr[kA] be the subalgebra of F [kA]^{G} generated by 1 and tr_{A}(u), where u ∈ F_{k}(A). Hence, Corollary 2 can be transformed into the following statement.

Corollary 3. Let G = Aut(A), then F [kA]^{G} is a finitely generated module over tr[kA].

Now, let us consider the M-module

kA ⊕ kA^{∗} ≃ k(A ⊕ A^{∗}) ⊆ kA .

Recall that the connected component of Aut(A) is M, hence, F [kA]^{M} is also a finitely generated module over tr[kA]. Let y_{1}, … y_{k} (z_{1}, …, z_{k}) be the projectors onto the direct summands of kA (respectively, kA^{∗}); if we replace the generic elements x_{i} of A in the trace polynomials in tr[kA] with

( \begin{matrix}0 & y_{i} \\ 0 & 0 \end{matrix} ) or ( \begin{matrix}0 & 0 \\ z_{i} & 0 \end{matrix} )

for all i ∈ {1, …, k}, we will get invariants in F [kA⊕kA^{∗}]^{M}. Denote by P the subalgebra generated by them; obviously, F [kA ⊕ kA^{∗}]^{M} is a finitely generated module over P.

Also, from the definition of the multiplication of A it easily follows that the generators of P are obtained in the following way.

Let S^{+}, S^{−} be the minimal sets of polynomials satisfying

1. y_{1}, …, y_{k} ∈ S^{+}, z_{1}, …, z_{k} ∈ S^{−};

2. if a_{σ}, b_{σ} ∈ S^{σ}, then a_{σ} × b_{σ} ∈ S^{−σ} for all

σ ∈ {+, −}. Then the set of polynomials tr_{A}(ab), where a ∈ S^{+}, b ∈ S^{−}, generates P.

3. Invariants of E_{7}

Let us recall first the definition of Freudenthal triple system (cf. [5]). On the vector space A define two forms (cf. [3, p.192],[5, p.87]):

ν(x) = 4αN(a) + 4βN(b) − 4tr_{A}(a^{#}b^{#}) + (αβ − tr_{A}(ab))^{2},  (9)
⟨x, y⟩ = tr_{A}((sx)ȳ) = αδ − βγ + tr_{A}(ac) − tr_{A}(bd)

where

x = ( \begin{matrix}α & a \\ b & β \end{matrix} ) , y = ( \begin{matrix}γ & c \\ d & δ \end{matrix} )

and s is the generator of the subspace S(A):

s = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

It is known [7, 5] that the group of invertible linear transformations E ⊆ End_{F} A preserving ν and ⟨∗, ∗⟩ is a simple algebraic group of type E_{7}.

Denote by ν(x, y, z, t) the complete linearization of ν(x), where ν(x, x, x, x) = 24ν(x). Since ⟨x, y⟩ is nondegenerate, we may define a 3-linear operation (x, y, z)_{1} on A :

⟨x, (y, z, t)_{1}⟩ = \frac{1}{2}ν(x, y, z, t) .

The vector space A with the operation (x, y, z)_{1} and the form ⟨x, y⟩ is a Freudenthal triple system. For us it is more convenient to replace the form ⟨x, y⟩ with a 3-linear operation

(x, y, z)_{2} = ⟨x, y⟩z .

Denote by T the vector space A with the operations (x, y, z)_{1}, (x, y, z)_{2}.

Lemma 3. The automorphism group Aut(T ) is equal to E.

Proof. Let ϕ ∈ Aut(T ), then ϕ((x, y, z)_{2}) = (ϕ(x), ϕ(y), ϕ(z))_{2} = ⟨ϕ(x), ϕ(y)⟩ · ϕ(z), on the other hand, since ϕ is linear, ϕ((x, y, z)_{2}) = ⟨x, y⟩ϕ(z) for all x, y, z ∈ T . Hence, ⟨ϕ(x), ϕ(y)⟩ = ⟨x, y⟩.

Next,

ν(ϕ(x), ϕ(y), ϕ(z), ϕ(t)) = 2⟨ϕ(x), (ϕ(y), ϕ(z), ϕ(t))_{1}⟩ = 2⟨ϕ(x), ϕ((y, z, t)_{1})⟩ =

= 2⟨x, (y, z, t)_{1}⟩ = ν(x, y, z.t) ,

hence, ϕ ∈ E.

Conversely, since ⟨x, y⟩ is nondegenerate, we get E ⊆ Aut(T ), therefore, E = Aut(T ). □

Lemma 4. The group E lies in Γ(A). Also, let F^{∗} be the multiplicative group of the ground field F ⊆ End_{F} (A), then Γ(A) = F^{∗} · E.

Proof. By (2.16) in [3], for all x, y, z ∈ A

(x, y, z)_{1} = 2V_{x,sy}(z) − ⟨y, z⟩x − ⟨y, x⟩z − ⟨x, z⟩y .  (10)

Also, L_{s} ∈ Γ, there L̂_{s} = −L_{s} [2, sect.11]. Hence, if ϕ ∈ E, then we put ϕ̂ = L_{s}ϕL_{s} and equality (7) follows from (10), therefore, ϕ ∈ Γ.

Next, pick ϕ ∈ Γ(A) and put a = ϕ(1); then, by [5, theorem 3], there is ψ ∈ E such that ψ(a) = β1, where β^{4} = ν(a). Hence, the composition of autotopies 1/β ∈ F^{∗}, ψ and ϕ fixes 1, hence, it is an automorphism of A [2, cor.8.6] and, therefore, belongs to E. □

The multiplication operators of T have the following form:

(∗, y, z)_{i} : x ↦ (x, y, z)_{i};
(x, ∗, z)_{i} : y ↦ (x, y, z)_{i};  (11)
(x, y, ∗)_{i} : z ↦ (x, y, z)_{i} ,

where i ∈ {1, 2}. So, the subalgebra Mult_{k}(T ) ⊆ Pol(kT , End_{F} T ) is generated by {(∗, a, b)_{i}, (a, ∗, b)_{i}, (a, b, ∗)_{i}| i = 1, 2; a, b ∈ F_{k}(T )} and the algebra of trace polynomials Tr[kT ] is defined as

alg_{F} {1, Tr_{T} (E)| E ∈ Mult_{k}(T )} ⊆ F [kT ]^{E} .

We will prove first Corollary 1 for A = T ; by Proposition 1 it will complete the proof of Theorem 1.

Theorem 3. The algebra F [kT ]^{E} is a finitely generated module over the algebra of trace polynomials Tr[kT ].

We divide the proof into several steps.

Let A be a simple structurable algebra and let Γ(A) be the structure group of A; also, denote by A^{∼} the Γ(A)-module, where every ϕ ∈ Γ(A) acts as ϕ̂. First of all, we need to describe the generators of the algebra F [k(A ⊕ A^{∼})]^{Γ(A)}; it is convenient to do it in terms of pairs .

Definition 1. (cf. [16]) The direct sum of two vector spaces P = P^{+} ⊕ P^{−} equipped with two 3-linear operations: σ ∈ {+, −}

P^{σ} × P^{−σ} × P^{σ} ⟶ P^{σ} ,

(a^{σ}, b^{−σ}, c^{σ}) ↦ {a^{σ} b^{−σ} c^{σ}} is called a pair (−σ means +, if σ = − and −σ = −, otherwise).

The subpair F_{k}(P) of generic elements of a pair P of rank k is generated in the pair

Pol(kP, P) = (F [kP] ⊗_{F} P^{+}) ⊕ (F [kP] ⊗_{F} P^{−})

by 2k projectors x_{i}^{σ} (i ∈ {1, …, k}, σ ∈ {+, −}). Next, the algebra Mult_{k}(P) is generated by multiplication operators (σ ∈ {+, −}):

{∗ b^{−σ} c^{σ}} , {a^{σ} ∗ c^{σ}} , {a^{σ} b^{−σ} ∗} ,

where a^{σ}, c^{σ} ∈ P^{σ}, b^{−σ} ∈ P^{−σ}.

Finally, the algebra of trace polynomials Tr[kP] is generated by 1 and Tr_{P} (E), where E ∈ Mult_{k}(P). Notice that for any ϕ ∈ End_{F} P we have a decomposition

ϕ(a^{+}, a^{−}) = (ϕ_{+}(a^{+}) + ϕ_{−,+}(a^{−}) , ϕ_{+,−}(a^{+}) + ϕ_{−}(a^{−})) ,

where ϕ_{σ} ∈ End_{F} P^{σ} and ϕ_{σ,−σ} ∈ Hom_{F} (P^{σ}, P^{−σ}). Therefore, by taking traces,

Tr_{P} (ϕ) = Tr_{P^{+}}(ϕ_{+}) + Tr_{P^{−}}(ϕ_{−}) .  (12)

Also, for any E ∈ Mult_{k}(P), both E_{+} and E_{−} belong to Mult_{k}(P) (we identify End_{F} P^{σ} with a subalgebra of End_{F} P in the usual way).

Like in Proposition 2 one can show that Tr[kP] is affine.

For a structurable algebra A define a pair P = P^{+} ⊕ P^{−} letting P^{+} = A, P^{−} = A^{∼} and

{a^{+} b^{−} c^{+}} = V_{a^{+},b^{−}}c^{+} , {a^{−} b^{+} c^{−}} = −V_{b^{+},a^{−}}c^{−}

for a^{σ}, b^{σ}, c^{σ} ∈ P^{σ}, σ ∈ {+, −}. By definition, the automorphism group of P consists of all pairs (ϕ, ϕ̂), where ϕ ∈ Γ(A), and, therefore, it is isomorphic to Γ(A).

Proposition 3. Let A be a simple structurable algebra, then F [k(A⊕A^{∼})]^{Γ(A)} = F [kP]^{Γ(A)} is a finitely generated module over Tr[kP].

Proof. Let x = (a_{1}, b_{1}, …, a_{k}, b_{k}) ∈ kP and let M^{+}⊕M^{−} be the subpair of P generated by the elements a_{1}, …, a_{k} ∈ P^{+}, b_{1}, …, b_{k} ∈ P^{−}. Since P = A ⊕ A^{∼} is contained in the Z-graded Lie algebra L = K(A), we may consider the subalgebra M generated by a_{1}, …, a_{k} ∈ A = L_{1} and b_{1}, …, b_{k} ∈ A^{∼} = L_{−1}. From the definition of multiplication [a, b] in L [1] it follows that M = ∑_{i=−2}^{2} M_{i}, where M_{1} = M^{+}, M_{−1} = M^{−}, M_{0} = [M_{1}, M_{−1}] and M_{±2} = [M_{±1}, M_{±1}].

Now, suppose that x ∈ π_{Tr[kP]}(0), let us prove that M is ad-nilpotent; because of Engel-Jacobson’s theorem [14, theorem 2.1] it suffices to show that ad(a) is nilpotent for all a ∈ M_{0}, since the homogeneous elements in nonzero components are clearly ad-nilpotent. But

[M_{1}, M_{−1}] ⊆ vect_{F} {V_{a,b}| a ∈ M^{+}, b ∈ M^{−}} ,

therefore, by our hypothesis on a, for all u ∈ M_{0}, Tr_{A}(u^{n}) = Tr_{A^{∼}}(u^{n}) = 0. Hence, ad(u) acts nilpotently on L_{±1} and, since L is generated by L_{1} ∪ L_{−1}, by the Jacobi identity, it acts nilpotently on the whole L.

So, M is a homogeneous nilpotent subalgebra (invariant under T_{1}), Also, A is simple, therefore, by [1], L is also simple; like in the proof of Theorem 2 we can show that x belongs to the null-cone of G_{0} : L, where G_{0} is the subgroup of Aut(L) which keeps T_{1} being fixed, that is, the automorphism group of the Z-graded Lie algebra L. Recall that G_{0} is precisely the structure group Γ(A) [2], so, the Proposition 1 completes the proof. □

The linear mapping L_{s} ∈ End_{F} A is invertible, where (L_{s})^{−1} = L_{s}; taking a ∈ A^{∼} to sa ∈ T , we identify the vector spaces A^{∼} and T . It defines an isomorphism ϕ ↦ L_{s}ϕL_{s} from End_{F} T onto End_{F} A^{∼}.

Now let P = A ⊕ A^{∼} and consider a linear bijection ψ from 2kT to kP ≃ kA ⊕ kA^{∼} letting

ψ : (a_{1}, …, a_{2k}) ↦ (a_{1}, …, a_{k}, sa_{k+1}, …, sa_{2k}) ;

it induces corresponding identifications:

ψ̂ : F [kP] ⟶ F [2kT ]
f ↦ f ∘ ψ
ψ̃^{+} : Pol(kP, A) ⟶ Pol(2kT , T )
f ↦ f ∘ ψ
ψ̃^{−} : Pol(kP, A^{∼}) ⟶ Pol(2kT , T )
f ↦ L_{s} ∘ f ∘ ψ
ψ̄^{+} : Pol(kP, End_{F} (A)) ⟶ Pol(2kT , End_{F} (T ))
f ↦ f ∘ ψ
ψ̄^{−} : Pol(kP, End_{F} (A^{∼})) ⟶ Pol(2kT , End_{F} (T ))
f ↦ L_{s} ∘ (f ∘ ψ) ∘ L_{s}  (13)

where the latter line should be understood as follows:

ψ̄^{−}(f)(a_{1}, …, a_{2k}) = L_{s} ∘ f(a_{1}, …, a_{k}, sa_{k+1}, …, sa_{2k}) ∘ L_{s} .

Notice that for any f ∈ Pol(kP, End_{F} A) and g ∈ Pol(kP, End_{F} A^{∼}),

ψ̂(Tr_{A}(f)) = Tr_{T} (ψ̄^{+}(f)) and ψ̂(Tr_{A^{∼}}(g)) = Tr_{T} (ψ̄^{−}(g)) .  (14)

The isomorphism kP ≃ kA ⊕ kA^{∼} defines an isomorphism F [kP] ≃ F [kA] ⊗_{F} F [kA^{∼}], which in turn provides F [kP] the structure of a bigraded algebra, so that we can speak of the degree in kA and the degree in kA^{∼}. Let Δ_{k} be the subspace of F [kP] spanned by the elements f(x)g(y) (≃ f(x) ⊗ g(y)), where x (respectively, y) is the projector of kP onto kA (kA^{∼}), for homogeneous f and g of the equal degree. That is, Δ_{k} is the subspace of F [kP] spanned by all homogeneous polynomials f = f(x, y) such that deg_{x}f = deg_{y}f. Obviously, it is a subalgebra of F [kP]; moreover, F [kP]^{Γ(A)} ⊆ Δ_{k} because F [kP]^{Γ(A)} is homogeneous, the scalars F^{∗} lie in Γ(A) and for all α ∈ F^{∗}, a ∈ A and b ∈ A^{∼} the mapping α takes a to αa and b to α^{−1}b.

Let Δ̂_{k} be the image of Δ_{k} under ψ̂, then Δ̂_{k} is spanned by all homogeneous polynomials with the degree in the first k projectors being equal to that of the last k ones. Put Λ_{k} = Δ̂_{k} ∩ F [2kT ]^{E}.

Proposition 4. The mapping ψ̂ restricts to an algebra isomorphism from F [kP]^{Γ(A)} onto Λ_{k}.

Proof. For all f ∈ F [kP]^{Γ(A)} and ϕ ∈ E (⊆ Γ(A))

(ψ̂(f) · ϕ) (a_{1}, …, a_{2k}) = ψ̂(f)(ϕ(a_{1}), …, ϕ(a_{2k}))
= f(ϕ(a_{1}), …, ϕ(a_{k}), sϕ(a_{k+1}), …, sϕ(a_{2k}))
= f(ϕ(a_{1}), …, ϕ(a_{k}), ϕ̂(sa_{k+1}), …, ϕ̂(sa_{2k}))
= (f · ϕ) (a_{1}, …, a_{k}, sa_{k+1}, …, sa_{2k})
= f(a_{1}, …, a_{k}, sa_{k+1}, …, sa_{2k})
= ψ̂(f)(a_{1}, …, a_{2k})  (15)

so ψ̂(f) ∈ Λ_{k}. In the other way, if f ∈ Λ_{k} and g ∈ Δ_{k} with ψ̂(g) = f, since g ∈ Δ_{k}, it is invariant under the action of F^{∗} ⊆ Γ. Also, for all ϕ ∈ E:

(g · ϕ) (a_{1}, …, a_{2k}) = g(ϕ(a_{1}), …, ϕ(a_{k}), ϕ̂(a_{k+1}), …, ϕ̂(a_{2k}))
= g(ϕ(a_{1}), …, ϕ(a_{k}), sϕ(sa_{k+1}), …, sϕ(sa_{2k}))
= f(ϕ(a_{1}), …, ϕ(a_{k}), ϕ(sa_{k+1}), …, ϕ(sa_{2k}))
= (f · ϕ) (a_{1}, …, a_{k}, sa_{k+1}, …, sa_{2k})
= f(a_{1}, …, a_{k}, sa_{k+1}, …, sa_{2k})
= ψ̂(g)(a_{1}, …, a_{k}, sa_{k+1}, …, sa_{2k})
= g(a_{1}, …, a_{k}, a_{k+1}, …, a_{2k}) since L_{s}^{2} = id,  (16)

as required. □

Now, pick a homogeneous h ∈ F [kT ]^{E} and define f in F [2kT ]^{E} letting

f = h(x_{1}, …, x_{k})h(x_{k+1}, …, x_{2k}) .

Obviously f ∈ Λ_{k}, hence there is a g ∈ F [kP]^{Γ(A)} such that ϕ̂(g) = f. By Proposition 3, g is integral over Tr[kP], so if we show that ψ̂(Tr[kP]) ⊆ Tr[2kT ], then we will have f to be integral over Tr[2kT ], which yields (replacing x_{k+i} with x_{i} for all i ∈ {1, …, k}) h^{2}, and hence h, to be integral over Tr[kT ]. Therefore, this will complete the proof of Theorems 3 and 1.

The isomorphisms above obtained from ψ allow us to define a linear mapping

ψ̃ : Pol(kP, P) ⟶ Pol(2kT , T )

by means of

ψ̃(f, g) = ψ̃^{+}(f) + ψ̃^{−}(g)

for all f ∈ Pol(kP, A) and all g ∈ Pol(kP, A^{∼}), and in exactly the same way we define a linear mapping

ψ̄ : Pol(kP, End_{F} (A) ⊕ End_{F} (A^{∼})) ⟶ Pol(2kT , End_{F} (T )) .

Lemma 5. ψ̃(F_{k}(P)) ⊆ F_{2k}(T ) and ψ̄(Mult_{k}(P)) ⊆ Mult_{2k}(T ).

Proof. Let z_{1}, …, z_{2k} be the projections (generic elements) in 2kT . Then since L_{s}^{2} is the identity, it follows that ψ̃(x_{i}) = z_{i} and ψ̃(y_{i}) = z_{k+i} for all i ∈ {1, …, k}.

Hence, by (10), if f, g, h ∈ F_{2k}(T ), then V_{f,L_{s}∘g}h ∈ F_{2k}(T ). Therefore, for all f_{1}, f_{2} ∈ F_{k}^{+} and all g ∈ F_{k}^{−} (where F_{k}(A ⊕ A^{∼}) = F_{k}^{+} ⊕ F_{k}^{−}), if ψ̃(f_{1}), ψ̃(f_{2}) and ψ̃(g) belong to F_{2k}(T ), then

ψ̃(V_{f_{1},g}f_{2}) = ψ̃^{+}(V_{f_{1},g}f_{2})
= (V_{f_{1},g}f_{2}) ∘ ψ = V_{f_{1}∘ψ,g∘ψ}f_{2} ∘ ψ
= V_{ψ̃(f_{1}),L_{s}∘ψ̃(g)}ψ̃(f_{2}) ∈ F_{2k}(T ) .  (17)

In the same way, for all f ∈ F_{k}^{+}, g_{1}, g_{2} ∈ F_{k}^{−}, if ψ̃(f), ψ̃(g_{1}) and ψ̃(g_{2}) belong to F_{2k}(T ) then

ψ̃(V_{g_{1},f}g_{2}) = ψ̃^{−}(V_{g_{1},f}g_{2})
= L_{s} ∘ (V_{g_{1},f}g_{2}) ∘ ψ = L_{s} ∘ V_{g_{1}∘ψ,f∘ψ}g_{2} ∘ ψ
= −V_{L_{s}∘g_{1}∘ψ,L_{s}∘f∘ψ}L_{s} ∘ g_{2} ∘ ψ since L_{s} ∈ Γ and L̂_{s} = −L_{s}
= −V_{ψ̃(g_{1}),L_{s}∘ψ̃(f)}ψ̃(g_{2}) ∈ F_{2k}(T ) by (10).  (18)

So, the first part is proven. Next, from the above it follows that for all a^{σ}, c^{σ} ∈ F_{k}^{σ} and b^{−σ} ∈ F_{k}^{−σ} (σ = ±):

ψ̃({a^{σ}, b^{−σ}, c^{σ}}) = σ V_{ψ̃(a^{σ}),L_{s}∘ψ̃(b^{−σ})}ψ̃(c^{σ}) ,

so by (10), the operators ψ̄({a^{σ}, b^{−σ}, ∗}), ψ̄({a^{σ}, ∗, c^{σ}}), and ψ̄({∗, b^{−σ}, c^{σ}}) belong to Mult_{2k}(T ), therefore, ψ̄(Mult_{k}(P)) ⊆ Mult_{2k}(T ).

Recall that P^{+} = A and P^{−} = A^{∼}; by (12) for any x ∈ End_{F} P

Tr_{P} (x) = Tr_{A}(x_{+}) + Tr_{A^{∼}}(x_{−}) = Tr_{T} (x_{+}) + Tr_{T} (x_{−}) ,

hence, for all E ∈ Mult_{k}(P)

ψ̂(Tr_{P} (E)) = ψ̂(Tr_{A}(E_{+})) + ψ̂(Tr_{A^{∼}}(E_{−})) =

Tr_{T} (ψ̄^{+}(E_{+}))) + Tr_{T} (ψ̄^{−}(E_{−})) ∈ Tr[2kT ] (19)

where we have used (14). Hence, ϕ̂(Tr[kP]) ⊆ Tr[2kT ], as required and this concludes the proof of lemma 5 and, therefore, Theorems 3 and 1.

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Departamento de Matemáticas y Computación, Universidad de la Rioja, 26004, Logroño, Spain

School of Mathematics and Statistics, University of Sydney, N.S.W., 2006, Australia (current address); Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, Novosibirsk, 630090, Russia