to appear in Journal of Algebra

ON SOME ALGEBRAS RELATED TO SIMPLE LIE TRIPLE SYSTEMS

Pilar Benito^{∗} and Cristina Draper^{∗∗}

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

Alberto Elduque^{∗∗∗}

Departamento de Matemáticas, Universidad de Zaragoza, 50009 Zaragoza, Spain

Abstract:

The set Hom_{Der T} (T ⊗_{F} T, T) is determined for any simple finite dimensional Lie triple system T over a field F of characteristic zero. It turns out that it contains nontrivial elements if and only if T is related to a simple Jordan algebra. In particular this provides a new proof of the determination by Laquer of the invariant affine connections in the simply connected compact irreducible Riemannian symmetric spaces.

1. Introduction.

Given a Lie group G acting smoothly and transitively on a manifold M, the isotropy subgroup H at any point p ∈ M (so that M ≃ G/H), the Lie algebra 𝔤 of the group G and the Lie algebra 𝔥 of H (𝔥 ≤ 𝔤), the manifold M is called a reductive homogeneous space in case

𝔤 = 𝔥 ⊕ 𝔪  (♯)

for a subspace 𝔪 with (Ad h)(𝔪) ⊆ 𝔪 for any h ∈ H, which implies that [𝔥, 𝔪] ⊆ 𝔪 (and if H is connected, these are equivalent).

In this situation Nomizu ([19, Theorem 8.1]) established a bijection between the set of G–invariant affine connections on M and the set of bilinear maps

α : 𝔪 × 𝔪 ⟶ 𝔪

∗ Supported by the Spanish DGICYT (Pb 94-1311-C03-02) and by the Universidad de La Rioja (API-97/B05).

∗∗ Supported by a grant from the “Programa de Formación de Profesorado Universitario” (MEC)

and by the ‘Universidad de La Rioja’ (API-97/B05). ∗∗∗ Supported by the Spanish DGICYT (Pb 94-1311-C03-03) and by the Universidad de La Rioja

(API-97/B05).

with the property that

α((Ad h)X, (Ad h)Y) = (Ad h)α(X, Y) ∀h ∈ H, ∀X, Y ∈ 𝔪,

that is, with Ad H|_{𝔪} ⊆ Aut(𝔪, α) (the automorphism group of the nonassociative algebra (𝔪, α)). Again the condition Ad H|_{𝔪} ⊆ Aut(𝔪, α) implies the condition ad 𝔥|_{𝔪} ⊆ Der(𝔪, α), and they are equivalent if H is connected. The torsion, curvature, geodesics, holonomy, ..., of the connection associated to (𝔪, α) can then be expressed and studied in terms of this nonassociative algebra.

Notice too that the set of the multiplications α : 𝔪 × 𝔪 → 𝔪 such that ad 𝔥|_{𝔪} ⊆ Der(𝔪, α) is nothing else but the vector space of the homomorphisms of 𝔥-modules from 𝔪 ⊗_{ℝ} 𝔪 into 𝔪: Hom_{𝔥}(𝔪 ⊗_{ℝ} 𝔪, 𝔪), which is a purely algebraic object.

Thus, the problem of determining the invariant affine connections on reductive homogeneous spaces reduces to the problem of classifying algebras with a given subgroup of its automorphism group or a given subalgebra of its Lie algebra of derivations. The same problem has arisen in studying real division algebras ([1]), nonunital composition algebras ([2], [20]) or in the classification of some flexible Lie-admissible algebras (see [17] and the references therein), a subject we will come back to later on.

In some cases, the algebras that appear related to some reductive homogeneous spaces are already known algebras. This is what happens for the six and seven dimensional spheres, viewed as S^{6} = G_{2}/SU(3) and S^{7} = Spin(7)/G_{2}, where algebras related to a so called color algebra ([3]) and the simple non-Lie Malcev algebras ([4]) appear.

In 1992, Laquer ([13] and [14]) computed the sets Hom_{𝔥}(𝔪⊗_{ℝ}𝔪, 𝔪) associated to the simply connected compact irreducible Riemannian symmetric spaces (see [7] for definitions). For these spaces, the decomposition (♯) is a grading of 𝔤 over ℤ_{2} = ℤ/2ℤ, 𝔪 thus being a (simple) Lie triple system. Therefore the problem was to compute Hom_{S}(T ⊗_{ℝ}T, T) for some ℤ_{2}-graded simple (that is, without nontrivial homogeneous ideals) finite dimensional real Lie algebra L = S ⊕ T (S being the even part and T the odd one). This was done by extending scalars to the complex field and by decomposing there the tensor product T_{ℂ}⊗_{ℂ}T_{ℂ} as a direct sum of irreducible modules. The vector space Hom_{S}(T ⊗_{ℝ} T, T) turned out to be 0 (which amounts to say that the only invariant affine connections is the canonical one) with a few exceptions.

Our purpose in this paper is twofold:

i) we will determine the set

Hom_{S}(T ⊗_{F} T, T)

for any ℤ_{2}-graded simple finite dimensional Lie algebra L = S ⊕ T over any arbitrary field of characteristic zero, without actually decomposing T ⊗_{F} T (when restricted to the real field, this determines the invariant affine connections on the irreducible affine symmetric spaces), and

ii) we will relate the resulting nontrivial algebras with the simple finite dimensional Jordan algebras.

In a sense, we can say that the Jordan algebras are responsible for the existence of noncanonical invariant affine connections on the symmetric spaces. The close relationship of Jordan algebras with some symmetric spaces is well-known (see, for instance, [16, Chapter VIII]). Our results give another aspect of this relationship.

Actually, our determination of the sets Hom_{S}(T ⊗_{F} T, T) over algebraically closed fields will be very quick, using information on the ℤ_{2}-graded simple Lie algebras that is contained in the Dynkin diagrams associated to the corresponding Lie triple systems T by Faulkner [5], with no need of a case by case decomposition of T ⊗_{F} T. Our approach is inspired by [6, Theorem 1]. This allows to simplify some of the results in [13] and [14].

From the point of view of nonassociative algebras, a central role in what follows is played by the multiplication defined on the set J_{0} of (generic) trace zero elements in a Jordan algebra J, given by projecting the product in J onto J_{0} (see (∗) on 3.2). This procedure is not new; for example, when applied to Cayley-Dickson algebras it gives all the central simple non-Lie Malcev algebras; it appears too in the construction of the pseudo-octonion algebras (see [17] and the references there in).

2. Lie triple systems.

In this section, some facts concerning simple Lie triple systems and the Dynkin diagrams attached to them will be recalled.

2.1 A Lie triple system is a vector space T, over a ground field F, equipped with a trilinear product [xyz] satisfying

[xxy] = 0
[xyz] + [yzx] + [zxy] = 0
the map [xy−] : T → T, z ↦ [xyz] is a derivation of T

for any x, y, z ∈ T. If S = span ⟨[xy−] : x, y ∈ T ⟩, S is a Lie subalgebra of the Lie algebra gl(T) (actually of Der(T)); the ℤ_{2}-graded Lie algebra L = S ⊕ T (where the multiplication is given by the multiplication in S (even part) as a subalgebra of gl(T), the natural action of S on T and [t_{1}, t_{2}] = [t_{1}t_{2}−] for any t_{1}, t_{2} ∈ T) is called the standard imbedding of T. Thus any Lie triple system is nothing else but the odd part of a ℤ_{2}-graded Lie algebra with product [xyz] = [[x, y], z]. The Lie triple system is simple if and only if its standard imbedding is graded simple (see for these facts [15]).

2.2 Now assume that T is a simple Lie triple system over the field F and let Γ = {α ∈ End_{F} (T) : α[xyz] = [(αx)yz] = [x(αy)z] = [xy(αz)] ∀x, y, z ∈ T } be its centroid. Γ is a field extension of F and for any x, y ∈ T [xy−] ∈ End_{Γ}(T); therefore the standard imbedding L is a Γ-algebra. Moreover, the centroid Λ of L is a commutative ℤ_{2}-graded algebra: Λ = Λ¯0 ⊕ Λ¯1, with Λ¯0 = {α ∈ Λ : αS ⊆ S, αT ⊆ T } and Λ¯1 = {α ∈ Λ : αS ⊆ T, αT ⊆ S}; with any nonzero homogeneous element being invertible (by the graded simplicity of L) and it easily follows that Λ¯0 = Γ. There are two possibilities: 2.2.a) If there is an element 0 ≠ u ∈ Λ¯1, then Λ¯1 = Γu, so that Λ = Γ[u], with 0 ≠ u^{2} = λ ∈ Γ, T = uS and L = Γ[u] ⊗_{Γ} S. In this case S is a central simple Lie algebra over Γ and T = uS is the adjoint module for S; so we will refer to this case as T being of adjoint type. For any x, y, z ∈ S, [(ux)(uy)(uz)] = u^{2}(u[[x, y], z]) = λ(u[[x, y], z]); so T is isomorphic to the Lie triple system S under the product {xyz} = λ[[x, y], z]. Notice that if u^{2} ∈ Γ^{2}, then λ can be taken to be 1. 2.2.b) Λ = Γ. Then L is a simple algebra (ungraded): otherwise for any proper ideal I of L, by graded simplicity (S ∩ I) ⊕ (T ∩ I) = 0 and π_{S}(I) ⊕ π_{T} (I) = L (where π_{S} and π_{T} denote the projections on S and T respectively), therefore S ∩ I = T ∩ I = 0 and π_{S}(I) = S, π_{T} (I) = T. Hence for any s ∈ S there is a unique t_{s} ∈ T with s + t_{s} ∈ I, and also for any t ∈ T there is a unique s_{t} ∈ S with s_{t} + t ∈ I. Then the odd map given by s ↦ t_{s} and t ↦ s_{t} is easily checked to be an odd element in the centroid of L, a contradiction with Λ = Γ. Therefore in this case, for any extension field K/Γ, K ⊗_{Γ} L is a simple algebra (and ℤ_{2}-graded).

From now on all the algebras and systems considered will be assumed to be finite dimensional.

2.3 Lister showed in [15, Theorem 4.5] (see also [5, Proposition 1]) that given any simple Lie triple system T over an algebraically closed field of characteristic zero with standard imbedding L = S ⊕ T, either 2.3.i) S is semisimple and T is an irreducible selfdual S-module (this is always the situation for the adjoint type cases), or 2.3.ii) S = F z ⊕ [S, S], where z is a nonzero central element of S and [S, S] is semisimple, and T = T_{1} ⊕ T_{2}, where T_{1} and T_{2} are irreducible dual S-modules, so that z acts as a nonzero scalar on T_{1} and as its negative on T_{2}. This is the situation that occurs after complexifying the simple Lie triple systems associated to the hermitian symmetric spaces.

2.4 Faulkner [5] associated a diagram to any such simple Lie triple system, starting with the Dynkin diagram of the semisimple Lie algebra [S, S] and adding “marked” nodes representing the lowest weight of the representation of [S, S] on each irreducible summand of T. These diagrams thus contain the information on the structure of S and of T as an S-module (which is enough to determine L). The resulting diagrams are exactly the affine diagrams X_{N}^{(1)} and X_{N}^{(2)} (which are equipped with some numerical labels in the nodes representing the linear dependence of the roots and weights involved –see [12, Chapter 4]–) with one or two marked nodes. More precisely, the diagrams that appear are exactly:

2.4.I) The diagrams X_{N}^{(1)} with a node labeled by 1 marked (by the symmetry of these diagrams, in this case it does not matter which is the marked node). These correspond to the adjoint types.

2.4.II) The diagrams X_{N}^{(1)} with two nodes labeled by 1 marked. These correspond to case 2.3.ii) above. Here the same diagram may give different possibilities according to which two nodes are marked. The same comment applies to the next cases.

2.4.III) The diagrams X_{N}^{(1)} with a node labeled by 2 marked. In this case the highest weight λ of T as an S-module (which is the negative of the lowest weight since T is selfdual) is not in the root lattice, which is equivalent to 0 not being a weight of the representation of S on T (see [8, exercise 21.3]).

2.4.IV) The diagrams X_{N}^{(2)} (so X_{N} is either A_{N} (N ≥ 2), D_{N} or E_{6}) with a node labeled by 1 marked. In this case 0 is a weight of the representation of S on T. This implies that there are roots which are weights and, therefore, at least the short roots of every simple ideal of S are weights (notice that the action of S on T is faithful, so that for any simple ideal I of S and any 0 ≠ v ∈ T in the zero weight space, Iv ≠ 0).

In case I) L = S ⊕ T is isomorphic (as ungraded algebra) to the direct sum of two copies of the simple algebra of type X_{N}. In the remaining cases L is a simple (ungraded) Lie algebra of type X_{N} and the semisimple Lie algebra [S, S] (which is the whole S if only one marked node appears) is the semisimple Lie algebra associated to the finite Dynkin diagram obtained by removing the marked nodes.

Remark: Faulkner obtained these diagrams just from the Dynkin diagram of [S, S] plus the lowest weights of the irreducible subrepresentations of [S, S] on T. V. Kac had previously arrived to the same situation in his research on automorphisms of finite order in semisimple Lie algebras over an algebraically closed field of characteristic zero (see [10], [11], [12, Chapter 8] or [7, Chapter X]), since a ℤ_{2}–graded Lie algebra is just a Lie algebra with an automorphism of order 2 (characteristic not 2).

2.5 Among the simple Lie triple systems, the ones obtained from Jordan algebras will be of particular interest to us. Given any simple Jordan algebra J of degree n ≥ 3 with generic trace t ([9, Chapter VI]) and multiplication x ∘ y over a field of characteristic zero, its set of trace zero elements J_{0} = {x ∈ J : t(x) = 0} becomes a simple Lie triple system under the trilinear multiplication [xyz] = (y ∘ z) ∘ x − y ∘ (z ∘ x) (the associator of y, z and x).

Since [xy−] = [R_{x}, R_{y}], where R_{u}v = u ∘ v = v ∘ u, it follows that in this case S = [R_{J}, R_{J}] = Der J is the Lie algebra of derivations of the Jordan algebra J and the standard imbedding is L = Der J ⊕ J_{0} ≅ Der J ⊕ R_{J_{0}} which is the derived subalgebra of the Lie multiplication algebra of J ([9, Theorem 8.3]).

Over an algebraically closed field F of characteristic zero there are the following possibilities for the simple Jordan algebras of degree n ≥ 3: 2.5.i) J = Mat_{n}(F)^{+} is the algebra of n × n matrices over F under the product x ∘ y = \frac{1}{2}(xy + yx). In this case J_{0} = sl(n, F) is a simple Lie triple system of adjoint type since Der J = ad J_{0}, so that the associated diagram is

A_{n−1}^{(1)}  • ◯ ◯  ◯ ◯ ◯^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ........_{n − 1 ≥ 2}

2.5.ii) J is the algebra of symmetric n × n matrices over F: J = H(Mat_{n}(F), t) (t the transposition). In this case Der J is isomorphic to the set of skew-symmetric matrices (see [9, Theorem 6.9]) and the Lie algebra L = Der J ⊕ J_{0} is isomorphic to sl(n, F). The associated diagram is

A_{2}^{(2)}  • ◯ ◯^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}^{2}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  if n = 3,

A_{2l}^{(2)}  • ◯ ◯  ◯ ◯^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}^{2}  ^{2}  ^{2}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ....  if n = 2l + 1, l ≥ 2,

D_{3}^{(2)}  • ◯ ◯ ◯^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}  ^{1}  ^{1}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  if n = 4,

A_{2l−1}^{(2)}  • ◯ ◯ ◯  ◯ ◯^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}^{1}_{1}  ^{2}  ^{2} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ....  if n = 2l, l ≥ 3.

2.5.iii) J is the algebra of symmetric 2n × 2n matrices with respect to the standard symplectic involution. Again in this case Der J is isomorphic to the skew-symmetric matrices with respect to this involution ([9, Theorem 6.9]), L is isomorphic to sl(2l, F) and the associated diagram is

A_{2l−1}^{(2)}  • ◯ ◯ ◯  ◯ ◯^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}^{1}_{1}  ^{2}  ^{2} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ....  (l ≥ 3)

2.5.iv) J is the exceptional simple Jordan algebra, whose derivation algebra is the Lie algebra of type F_{4} ([9, Section 9.11]). The associated diagram is then

E_{6}^{(2)}  • ◯ ◯ ◯ ◯ ◯_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}  ^{1}^{2}  ^{2}^{3}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}

3. Algebraically closed fields.

We intend to determine in this section the vector space Hom_{S}(T ⊗_{F} T, T) for any simple Lie triple system T with standard imbedding L = S ⊕ T, over an algebraically closed field F of characteristic zero, an assumption we will keep throughout the section.

3.1 In case T = T_{1} ⊕ T_{2} is a sum of two irreducible S-modules (diagrams 2.4.II)), the center of S is one-dimensional: Z(S) = F z, with ad z|_{T_{1}} = 1, ad z|_{T_{2}} = −1. Therefore T ⊗_{F} T is the sum of the eigenspaces 2, 0, −2 for ad z, which forces Hom_{S}(T ⊗_{F} T, T) = 0.

3.2 In case T is of adjoint type, then Hom_{S}(T ⊗_{F} T, T) ≅ Hom_{S}(S⊗_{F} S, S) and there is a distinguished (skew-symmetric) element in this latter space: the Lie multiplication in S. So at least dim Hom_{S}(T ⊗_{F} T, T) ≥ 1. Actually, by [6, Theorem 1 and Corollary 2]

dim Hom_{S}(T ⊗_{F} T, T) = \{ \begin{matrix}2 & if S is of type A_{n} (n ≥ 2) \\ 1 & otherwise. \end{matrix}

On the other hand, if T = J_{0} is the triple system associated to a simple Jordan algebra of degree n ≥ 3 with multiplication x ∘ y, then S = Der J and we obtain a nonzero (symmetric) element of Hom_{S}(T ⊗_{F} T, T) by means of

\begin{matrix}J_{0} ⊗_{F} J_{0} & ⟶ J_{0} \\ x ⊗ y & ↦ x · y = x ∘ y − \frac{1}{n}t(x ∘ y)1 \end{matrix}  (∗)

(the projection of the element x ∘ y onto J_{0}). Notice that if the degree were two, this would be zero. Besides, if J = Mat_{n}(F)^{+}, then the triple system J_{0} is both of adjoint type (A_{n−1}) and of “Jordan type”.

Therefore, for the adjoint type (diagrams 2.4.I)), Hom_{S}(T ⊗_{F} T, T) is spanned by the “Lie bracket in S” for types other that A_{n} (n ≥ 2) and by the “Lie bracket in S” and the “Jordan product (∗)” for types A_{n} (n ≥ 2).

3.3 The situation for the simple Lie triple systems with diagrams 2.4.III) can be settled immediately with the following Lemma (inspired in [6, Theorem 1]):

Lemma Let S be a semisimple Lie algebra, H a Cartan subalgebra of S, V an irreducible selfdual module with highest weight λ relative to H, v_{λ} and v_{−λ} nonzero weight vectors for λ and −λ. For any root α of S relative to H let S_{α} be the corresponding root space. If V_{0} is the weight 0-space in V, then the linear map

\begin{matrix}ϕ : Hom_{S}(V ⊗_{F} V, V) & ⟶ \{v ∈ V_{0} : S_{α}v = 0 ∀α ⊥ λ\} \\ φ & ↦ φ(v_{λ} ⊗ v_{−λ}) \end{matrix}

is well defined and one-to-one.

Proof: Since V is selfdual, −λ is the lowest weight of V and v_{λ} ⊗ v_{−λ} generates V ⊗_{F} V. Hence any φ ∈ Hom_{S}(V ⊗_{F} V, V) is determined by φ(v_{λ} ⊗ v_{−λ}) ∈ V_{0}. But for any root α orthogonal to the weight λ and any x_{α} ∈ S_{α}, x_{α}v_{λ} = 0 = x_{α}v_{−λ}, so x_{α}(v_{λ} ⊗ v_{−λ}) = 0 and x_{α}φ(v_{λ} ⊗ v_{−λ}) = 0.

As a direct consequence, Hom_{S}(T ⊗_{F} T, T) = 0 for the simple Lie triple systems with diagrams 2.4.III), for which 0 is not a weight of the representation of S on T.

3.4 From the paragraphs above, only the simple Lie triple systems with diagrams 2.4.IV) remain to be studied. According to Lemma 3.3, we will try to bound the dimension of {t ∈ T_{0} : S_{α}t = 0 ∀α ⊥ λ} for a triple system T which is an irreducible S-module of highest weight λ.

Lemma Let T be a simple Lie triple system which is an irreducible module for S = [T T −]. Let H be a Cartan subalgebra of the semisimple Lie algebra S, Φ the associated root system, Δ a base of Φ and Λ the set of weights of T relative to H. For any ∅ ≠ Δ′ ⊆ Δ, let S_{Δ′} be the subalgebra of S generated by the root spaces S_{α}, α ∈ Δ′ ∪ −Δ′, T_{Δ′} be the sum of the weight spaces T_{μ} for μ ∈ Λ ∩ ℤΔ′ and L_{Δ′} = S_{Δ′} ⊕ T_{Δ′}.

Then L_{Δ′} is a graded subalgebra of L = S ⊕ T that decomposes as

L_{Δ′} = Z_{Δ′} ⊕ [L_{Δ′}, L_{Δ′}]

where Z_{Δ′} = {v ∈ T_{0} : S_{α}v = 0 ∀α ∈ Δ′ ∪ −Δ′} is its center and [L_{Δ′}, L_{Δ′}] is semisimple (hence a direct sum of graded simple algebras).

In particular, the codimension in T_{0} of Z_{Δ′} equals the dimension of the zero weight space in [S_{Δ′}, T_{Δ′}] (the odd part of [L_{Δ′}, L_{Δ′}]).

Proof: Since S_{Δ′} = ∑_{α∈Δ′} F h_{α} + ∑_{β∈Φ∩ℤΔ′} S_{β} (the h_{α}’s as in [8, page 37]), S_{Δ′} is a semisimple subalgebra of S with Cartan subalgebra H_{Δ′} = H ∩ S_{Δ′} = ∑_{α∈Δ′} F h_{α}. Besides, Δ′ is a base of the root system of S_{Δ′}: Φ ∩ ℤΔ′. Moreover, L_{Δ′} is clearly a graded subalgebra of L. By complete reducibility T_{Δ′} = Z_{Δ′} ⊕ [S_{Δ′}, T_{Δ′}], and if 0 ≠ W is an irreducible S_{Δ′}-module in [S_{Δ′}, T_{Δ′}], its highest weight belongs to ℤΔ′, so that 0 is a weight in W ([8, exercise 21.3]). Now, for any 0 ≠ w ∈ T_{0} ∩ W, [Z_{Δ′}, w] ⊆ [T_{0}, T_{0}] = 0, since H + T_{0} is a Cartan subalgebra (hence abelian) of the semisimple Lie algebra L = S ⊕ T ([15, Lemma 4.12] or [12, Lemma 8.1.b)]). Since W is irreducible, it follows that [Z_{Δ′}, W] = 0, so that Z_{Δ′} is precisely the center of L_{Δ′}. Besides, any abelian ideal of L_{Δ′} is easily shown to be contained in Z_{Δ′}, so that [L_{Δ′}, L_{Δ′}] is semisimple, as required.

3.5 Let T be a simple Lie triple system, L = S ⊕ T be its standard imbedding, and assume that T is an irreducible S-module. As before, we take H a Cartan subalgebra of S, so that the sum of H and the zero weight space T_{0} of T is a Cartan subalgebra of L. Let Φ be the root system of S and Λ the set of weights of T with respect to H. A look at the marked diagrams shows that: 3.5.i) If S is simple of type A_{1} and 0 is a weight of T, then dim T_{0} = 1.

Actually, either T is of adjoint type or the diagram is the one associated to A_{2}^{(2)}, so L ≅ sl(3) and dim T_{0} = rank L − rank S = 2 − 1 = 1.

3.5.ii) If S is simple of type A_{2} and 0 is a weight, then T is of adjoint type and dim T_{0} = 2.

3.5.iii) If S is simple of type B_{l} (l ≥ 2) and Φ ⊆ Λ then dim T_{0} = l.

In fact Φ ⊆ Λ if and only if the difference between the highest weight and the longest root is a sum of positive roots. In this case either T is of adjoint type or its diagram is obtained from A_{2l}^{(2)}, hence the result (2l − l = l).

3.5.iv) There is no simple Lie triple system where S is a direct sum of two simple ideals and all nonzero weights are roots; because for such S, T is a tensor product of an irreducible module for the first ideal times an irreducible module for the second, and the maximal weight cannot be a root.

3.5.v) If S is simple of type C_{l} (l ≥ 2) and Λ = {short roots}∪{0}, then dim T_{0} = l−1. This is because the only possibilities are given by the affine diagrams A_{2l−1}^{(2)} (l ≥ 3) and D_{3}^{(2)} (for l = 2).

3.5.vi) If S is simple of type D_{l} (l ≥ 3 with D_{3} = A_{3}) and 0 is a weight, then dim T_{0} = l − 1 (the only possibility is given by the affine diagrams A_{2l−1}^{(2)}).

3.6 With this in mind we can tackle the problem of bounding the dimension of

T_{0}^{λ} = {t ∈ T_{0} : S_{α}t = 0 ∀α ⊥ λ}

(λ being the highest weight) for the triple systems with diagrams in 2.4.IV). In each case we will find a subset Δ′ of the base Δ such that all the roots in Δ′ are orthogonal to λ, so that T_{0}^{λ} ⊆ Z_{Δ′} and will apply Lemma 3.4.

• A_{2}^{(2)}. Here dim T_{0} = 1 and we already know that dim Hom_{S}(T ⊗_{F} T, T) ≥ 1 by 3.2 and 2.5.ii), hence dim Hom_{S}(T ⊗_{F} T, T) = dim T_{0} = 1.

• A_{2l}^{(2)} (l ≥ 2):

• ◯ ◯  ◯ ◯^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}^{2}  ^{2}  ^{2}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} .... _{α_{1}}_{α_{l−1}}_{α_{l}}

Here all the roots are weights. With Δ′ = {α_{2}, …, α_{l}}, S_{Δ′} is simple of type B_{l−1} if l ≥ 3 or A_{1} if l = 2. From 3.5.iii) (l ≥ 3) or 3.5.i) (l = 2) and Lemma 3.4 it follows that codim Z_{Δ′} = l−1 so, since dim T_{0} = 2l−l = l, it follows that dim T_{0}^{λ} ≤ 1 and we already know that dim Hom_{S}(T ⊗_{F} T, T) ≥ 1 in this case (2.5.ii)), so that again dim Hom_{S}(T ⊗_{F} T, T) = 1.

• A_{2l−1}^{(2)} (l ≥ 3) marked as

• ◯ ◯ ◯  ◯ ◯^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}^{1}_{1}  ^{2}  ^{2} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ....^{α_{1}} _{α_{2}}  _{α_{l−1}}  _{α_{l}}

Here the nonzero weights are the short roots. We take Δ′ = {α_{1}, α_{3}, …, α_{l}} if l ≥ 4, then because of 3.5.iv) [L_{Δ′}, L_{Δ′}] is the sum of two graded simple ideals and by 3.5.i) and 3.5.v) codim Z_{Δ′} = 1 + l − 3 = l − 2 so dim T_{0}^{λ} ≤ 1 (dim T_{0} = l − 1). Also, if l = 3 we take Δ′ = {α_{1}} (since α_{l} is not a weight) and then S_{Δ′} is of type A_{1} and by 3.5.i) codim Z_{Δ′} = 1 and dim T_{0}^{λ} ≤ 1.

Again by 2.5 we get that dim Hom_{S}(T ⊗_{F} T, T) = 1 in this case.

• A_{2l−1}^{(2)} (l ≥ 3) marked as

• ◯ ◯ ◯  ◯ ◯^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}^{1}_{1}  ^{2}  ^{2} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ....^{α_{l−1}}_{α_{l}} _{α_{l−2}}  _{α_{1}}

Here dim T_{0} = 2l − 1 − l = l − 1. If l = 3 we take Δ′ = {α_{3}}, and by 3.5.i) codim Z_{Δ′} = 1, so dim T_{0}^{λ} ≤ 1. Also, for l > 3 we take Δ′ = {α_{2}, …, α_{l}} and by 3.5.vi) codim Z_{Δ′} = (l − 1) − 1 = l − 2, so again dim T_{0}^{λ} ≤ 1. Now by 2.5 we get dim Hom_{S}(T ⊗_{F} T, T) = 1.

• D_{l+1}^{(2)} (l ≥ 2) marked at one end:

• ◯ ◯  ◯ ◯^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}  ^{1}  ^{1}  ^{1}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ...._{α_{1}}  _{α_{l−1}}  _{α_{l}}

Since dim T_{0} = rank D_{l+1} − rank B_{l} = 1, the short roots are weights and it is enough to take Δ′ = {α_{l}} and apply 3.5.i) to conclude that Hom_{S}(T ⊗_{F} T, T) = 0.

• D_{l+1}^{(2)} (l ≥ 2) marked in the middle:

• ◯ ◯ ◯ ◯ ◯^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}  ^{1}  ^{1}  ^{1}  ^{1}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} .... ...._{α_{r}} _{α_{r−1}}  _{β_{s−1}}  _{β_{s}}

Here L is simple of type D_{l+1} (D_{3} = A_{3}) and S is a sum of two simple ideals of type B_{r} and B_{s} respectively, with r, s ≥ 1 (B_{1} = A_{1}) and r+s = l. Hence dim T_{0} = 1. If r = s = 1 we conclude by 2.5.ii) that dim Hom_{S}(T ⊗_{F} T, T) = 1. If, for instance, r > 1 we can take Δ′ = {α_{r}} and conclude from 3.5.i) that Hom_{S}(T ⊗_{F} T, T) = 0.

• E_{6}^{(2)} marked as

• ◯ ◯ ◯ ◯ ◯_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{1}  ^{1}^{2}  ^{2}^{3}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{α_{1}}  _{α_{2}}  _{α_{3}}  _{α_{4}}

Then S is of type F_{4}, so that dim T_{0} = 2. With Δ′ = {α_{3}} (α_{3} is a short root and hence a weight) we get codim Z_{Δ′} = 1 by 3.5.i), so by 2.5.iv) we get dim Hom_{S}(T ⊗_{F} T, T) = 1.

• Finally, E_{6}^{(2)} marked as

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With Δ′ = {α_{1}, α_{2}} and because of 3.5.ii) we get codim Z_{Δ′} = 2 = dim T_{0}, so Hom_{S}(T ⊗_{F} T, T) = 0.

3.7 As a conclusion we get that in all cases dim Hom_{S}(T ⊗_{F} T, T) = dim T_{0}^{λ} and this is nonzero only in the cases contemplated in 3.2. So, summarizing all the work done, we get:

Theorem Let T be a simple Lie triple system over an algebraically closed field F of characteristic zero and let L = S ⊕ T be its standard imbedding. Then Hom_{S}(T ⊗_{F} T, T) = 0 unless either:

a) T is of adjoint type other than A_{n} (n ≥ 2). In this case Hom_{S}(T ⊗_{F} T, T) ≅ Hom_{S}(S ⊗_{F} S, S), which is spanned by the Lie multiplication in S, or

b) there exists a simple Jordan algebra J of degree n ≥ 3 such that T is the Lie

triple system J_{0}. In this case either:

i) J is not of type A (that is, J is not isomorphic to Mat_{n}(F)^{+}), then Hom_{S}(T ⊗_{F} T, T) = Hom_{Der J}(J_{0} ⊗_{F} J_{0}, J_{0}) is spanned by the product in (∗).

ii) J is of type A, then J_{0} is of adjoint type S ≅ sl(n, F) and Hom_{S}(T ⊗_{F} T, T) = Hom_{Der J}(J_{0} ⊗_{F} J_{0}, J_{0}) is spanned by the product in (∗) and the Lie multiplication in J_{0} ≅ sl(n, F).

As mentioned in the Introduction, this Theorem provides alternative proofs to [13, Theorem 8.1] and [14, Theorem 2.1].

4. Arbitrary fields.

This section will be devoted to show that the hypothesis of the field being algebraically closed in 3.7 can be avoided with only some minor modifications.

4.1 Given a Lie algebra L over a field K, two L-modules M and N and a subfield F of K, the F -descent of L is just L but considered as a Lie algebra over F. Then M and N are also modules for the F -descent of L. We can form M ⊗_{F} N, which is a module for the F -descent of L, and also M ⊗_{K} N, which is a module for L and, a fortiori, also a module for its F -descent.

We have then a natural homomorphism of modules (for the F -descent of L)

\begin{matrix}p : M ⊗_{F} N & ⟶ M ⊗_{K} N \\ m ⊗ n & ↦ m ⊗ n \end{matrix}

(with different meanings of the same sign ⊗).

Lemma Let L be a Lie algebra over a field K, let M, N and P be three irreducible nontrivial L-modules, and let F be a subfield of K such that K/F a finite separable field extension. Then the linear map (of vector spaces over K)

\begin{matrix}p^{∗} : Hom_{L}(M ⊗_{K} N, P) & ⟶ Hom_{L}(M ⊗_{F} N, P) \\ φ & ↦ φ ∘ p \end{matrix}

(p as above) is a bijection.

Proof: Notice that Hom_{L}(M ⊗_{F} N, P) is a vector space over K because so is P. Let Ω be an algebraic closure of F and σ_{1}, …, σ_{r} : K → Ω the r = [K : F] different F -homomorphisms. Then by separability

Ω ⊗_{F} K = Ωe_{1} ⊕ ⋯ ⊕ Ωe_{r}

with e_{1}, …, e_{r} orthogonal idempotents. Besides, as a right K-vector space, e_{i}α = σ_{i}(α)e_{i} for any i = 1, …, r and any α ∈ K.

Now,

L_{Ω} = Ω ⊗_{F} L ≅ (Ω ⊗_{F} K) ⊗_{K} L
= (Ωe_{1} ⊗_{K} L) ⊕ ⋯ ⊕ (Ωe_{r} ⊗_{K} L)
= L_{1} ⊕ ⋯ ⊕ L_{r} ,

where each L_{i} = Ωe_{i} ⊗_{K} L is the algebra obtained from L by extending scalars from K to Ω by means of σ_{i}. Similarly

M_{Ω} = Ω ⊗_{F} M ≅ (Ωe_{1} ⊗_{K} M) ⊕ ⋯ ⊕ (Ωe_{r} ⊗_{K} M) = M_{1} ⊕ ⋯ ⊕ M_{r} ,

and the same for N_{Ω} and P_{Ω}. By orthogonality, L_{i}M_{j} = 0 if i ≠ j.

By scalar extension

Ω ⊗_{F} Hom_{L}(M ⊗_{F} N, P) ≅ Hom_{L_{Ω}}(M_{Ω} ⊗_{Ω} N_{Ω}, P_{Ω}) ,

but

M_{Ω} ⊗_{Ω} N_{Ω} = ⨁_{i,j=1}^{r} M_{i} ⊗_{Ω} N_{j} .

For any ϕ ∈ Hom_{L_{Ω}}(M_{i} ⊗_{Ω} N_{j}, P_{k}), if i, j ≠ k

L_{k}ϕ(M_{i} ⊗_{Ω} N_{j}) ⊆ ϕ(L_{k}M_{i} ⊗_{Ω} N_{j}) + ϕ(M_{i} ⊗_{Ω} L_{k}N_{j}) = 0 ,

so that

ϕ(M_{i} ⊗_{Ω} N_{j}) = 0

since

{x ∈ P_{k} : L_{k}x = 0} = Ωe_{k} ⊗ {x ∈ P : Lx = 0} = 0 ;

while if i = k ≠ j

ϕ(M_{i} ⊗_{Ω} N_{j}) = ϕ(M_{i} ⊗_{Ω} L_{j}N_{j}) = ϕ(L_{j}(M_{i} ⊗_{Ω} N_{j})) = L_{j}ϕ(M_{i} ⊗_{Ω} N_{j}) ⊆ L_{j}P_{k} = 0 .

Therefore

Hom_{L_{Ω}}(M_{Ω} ⊗_{Ω} N_{Ω}, P_{Ω}) = ⨁_{i=1}^{r} Hom_{L_{i}}(M_{i} ⊗_{Ω} N_{i}, P_{i})

= ⨁_{i=1}^{r} Ωe_{i} ⊗_{K} Hom_{L}(M ⊗_{K} N, P) .

Hence

dim_{F} Hom_{L}(M ⊗_{K} N, P) = r dim_{K} Hom_{L}(M ⊗_{K} N, P)

= ∑_{i=1}^{r} dim_{Ω} Hom_{L_{i}}(M_{i} ⊗_{Ω} N_{i}, P_{i})

= dim_{Ω} Hom_{L_{Ω}}(M_{Ω} ⊗_{Ω} N_{Ω}, P_{Ω}) = dim_{F} Hom_{L}(M ⊗_{F} N, P) .

But p : M ⊗_{F} N → M ⊗_{K} N is onto, so p^{∗} is one-to-one and hence, because of the same dimension, it is a bijection.

As a consequence, if T is a simple Lie triple system over a field F of characteristic zero, Γ is its centroid and S = [T T −] ⊆ gl(T), to determine Hom_{S}(T ⊗_{F} T, T) it is enough to determine Hom_{S}(T ⊗_{Γ} T, T). That is, it is enough to deal with central simple Lie triple systems. Moreover, for these systems we can extend scalars up to an algebraically closed field and then use Theorem 3.7.

4.2 Let T be a central simple Lie triple system over a field F of characteristic zero with standard imbedding L = S ⊕T, and let Ω be an algebraic closure of F. Assume there is a simple Jordan algebra of degree n ≥ 3 over Ω such that T_{Ω} = Ω⊗_{F} T is the triple system J_{0} constructed in 2.5. Then by Theorem 3.7, dim_{F} Hom_{S}(S^{2}(T), T) = 1, where S^{2}(T) denotes the space of symmetric tensors in T ⊗_{F} T.

Let ω : T × T → T be symmetric, bilinear and such that the map determined by u ⊗ u ↦ ω(u, u) (u ∈ T) spans Hom_{S}(S^{2}(T), T). Then denoting by the same symbol ω its extension to a bilinear map T_{Ω} × T_{Ω} → T_{Ω}, it follows that there is a 0 ≠ μ ∈ Ω such that x · y = μω(x, y) for any x, y ∈ T_{Ω} = J_{0} (x · y as in (∗)). Thus the multiplication in the Jordan algebra J = Ω1 ⊕ J_{0} is given by

(α1 + x) ∘ (β1 + y) = (αβ + \frac{1}{n}t(x, y)) 1 + (αy + βx + x · y)  (⋄)

where t(x, y) = t(x ∘ y) defines an invariant symmetric bilinear form on J_{0}. Since the algebra J is isomorphic (by means of the map a ↦ μa) to the algebra defined on the same J but with the new multiplication given by μ^{−1}a ∘ b, we can assume that x · y = ω(x, y) for any x, y ∈ J_{0}.

Now, the Jordan identity (x^{∘2} ∘ y) ∘ x = x^{∘2} ∘ (y ∘ x) gives for x, y ∈ J_{0} that

\frac{1}{n}t(x · x, y)x + ((x · x) · y) · x = \frac{1}{n}t(y, x)x · x + (x · x) · (y · x) .  (†)

On the other hand, t ∈ Hom_{Der J}(J_{0} ⊗_{Ω} J_{0}, Ω) ≅ Ω ⊗_{F} Hom_{S}(T ⊗_{F} T, F), and by irreducibility of T and Schur’s Lemma the dimension of these spaces is 1; hence there is a λ : T × T → F bilinear, symmetric and nondegenerate and a 0 ≠ α ∈ Ω such that t(x, y) = αλ(x, y) for any x, y ∈ T.

If α ∉ F, for any x ∈ T, take y ∈ T with λ(x, y) = 1 and (†) allows us to conclude that x · x ∈ Ωx for any x ∈ T. But this implies that the degree of J is at most 2, a contradiction. Hence α ∈ F, t(x, y) ∈ F for any x, y ∈ T and A = F 1 ⊕ T is closed under the Jordan product in J. In other words, A = F 1 ⊕ T is a Jordan algebra over F which is a form of J. Moreover S = Der A since S_{Ω} = Der J. But we cannot conclude that T is the Lie triple system associated to the simple Jordan algebra A (because of the scaling we did of the product in J); we can only conclude that there is a nonzero scalar μ ∈ F such that [xyz] = μ((y ∘ z) ∘ x − y ∘ (z ∘ x)) for any x, y, z ∈ A_{0} = T. We will denote this triple system by (A_{0}, μ).

Let us put all this another way. For any central simple Jordan algebra J over F of degree ≥ 3 let L(J) = Der J ⊕ J_{0} be the ℤ_{2}-graded simple Lie algebra which is the standard imbedding of the Lie triple system J_{0}. Also, for any ℤ_{2}-graded Lie algebra L = S ⊕ T and any 0 ≠ μ ∈ F, let L_{μ} be the new ℤ_{2}-graded Lie algebra constructed on the same vector space as L with the same grading and the same product of elements in S and of elements in S times elements in T, but with the new product of elements in T given by

[t_{1}, t_{2}]^{μ} := μ[t_{1}, t_{2}] .

Then the arguments above prove the first part of:

Proposition Let L be a ℤ_{2}-graded simple Lie algebra over a field F and let Ω be an algebraic closure of F. Assume that there is a simple Jordan algebra J over Ω of degree ≥ 3 such that L_{Ω} ≅ L(J). Then:

i) There exists a a nonzero scalar μ ∈ F and a Jordan algebra A over F, unique up to isomorphism, with A_{Ω} ≅ J and L ≅ L(A)_{μ} (as graded algebras).

ii) L(A)_{μ} ≅ L(A)_{ν} (as graded algebras) if and only if μ^{−1}ν ∈ F^{2}.

Proof: Assume that A and B are central simple Jordan algebras with A_{Ω} ≅ J ≅ B_{Ω} and let 0 ≠ μ, ν ∈ F and ϕ : L(A)_{μ} → L(B)_{ν} an isomorphism of graded algebras. Then it is immediate to check that ϕ is an isomorphism from L(A) to L(B)_{μ^{−1}ν}, so that we may assume μ = 1. Let S = Der A, then the restriction ϕ|_{S} : Der A → Der B is an isomorphism which makes B_{0} an irreducible S-module. Let φ = ϕ|_{A_{0}} : A_{0} → B_{0}, then since Hom_{S}(S^{2}(A_{0}), A_{0}) and Hom_{S}(S^{2}(B_{0}), B_{0}) = Hom_{Der B}(S^{2}(B_{0}), B_{0}) have dimension 1, there is a nonzero scalar 0 ≠ η ∈ F such that φ(a_{1} · a_{2}) = ηφ(a_{1}) · φ(a_{2}) for any a_{1}, a_{2} ∈ A_{0}. Therefore (ηφ)(a_{1} · a_{2}) = (ηφ)(a_{1}) · (ηφ)(a_{2}) and ηφ is an isomorphism between the algebras (A_{0}, ·) and (B_{0}, ·). By (†) and since the degree of the Jordan algebras A and B is ≥ 3 it follows that the traces of A and B are determined by the algebras (A_{0}, ·) and (B_{0}, ·) respectively. From the expression in (⋄) ηφ extends to an isomorphism of Jordan algebras ψ : A → B (with ψ(1) = 1 and ψ(a) = ηφ(a) for any a ∈ A_{0}). This shows the uniqueness in i). Besides, ψ induces an isomorphism Ψ : L(A) → L(B) by means of Ψ(s) = ψ ∘ s ∘ ψ^{−1} for any s ∈ S and Ψ(a) = ψ(a) for any a ∈ A_{0}. Now, ϕ ∘ Ψ^{−1} : L(B) → L(B)_{ν} is an isomorphism of simple graded Lie algebras with ϕ ∘ Ψ^{−1}(b) = η^{−1}b for any b ∈ B_{0} (the odd part). But for any b_{1}, b_{2} ∈ B_{0}

ϕ ∘ Ψ^{−1}([b_{1}, b_{2}]) = [ϕ ∘ ψ^{−1}(b_{1}), ϕ ∘ ψ^{−1}(b_{2})]^{ν} = η^{−2}ν[b_{1}, b_{2}]

and for any b_{1}, b_{2}, b_{3} ∈ B_{0}

\begin{aligned}η^{−1}[[b_{1}, b_{2}], b_{3}] & = ϕ ∘ ψ^{−1}([[b_{1}, b_{2}], b_{3}]) = [ϕ ∘ Ψ^{−1}([b_{1}, b_{2}]), ϕ ∘ ψ^{−1}(b_{3})] \\ & = η^{−2}νη^{−1}[[b_{1}, b_{2}], b_{3}] . \end{aligned}

Thus η^{−2}ν = 1, giving the converse in ii).

Conversely, if μ^{−1}ν = η^{2} for 0 ≠ η ∈ F, the map ϕ : L(A)_{ν} → L(A)_{μ} given by ϕ(s) = s for any s ∈ S and ϕ(t) = ηt for any t ∈ A_{0} is easily seen to be an isomorphism.

Remarks: i) The Proposition above can be deduced too from [18, Theorem 4.5]. The proof above is more on the spirit of the rest of the paper. ii) Over the real field, attached to any central simple Jordan algebra A of degree ≥ 3 there appear two nonisomorphic triple systems with standard imbeddings L(A) and L(A)_{−1}. One is the dual of the other ([7, page 235]). The irreducible compact simply connected Riemannian symmetric spaces of type I with noncanonical affine connections (see [14, Theorem 2.1]) appear as the duals of the ones associated to the Jordan algebras of symmetric real matrices, symmetric matrices of quaternions with respect to the standard involution and the simple exceptional Jordan algebra H_{3}(𝕆).

4.3 In order to state the result that summarizes most of the work done throughout the paper, we need one more thing. Let J be a central simple Jordan algebra of type A (that is, after extension of scalars it becomes isomorphic to Mat_{n}(K)^{+}) over a field Γ of characteristic zero. Then (see [9, Chapters IV and V]) either

i) there is a central simple associative algebra B over Γ such that J is the Jordan algebra B^{+}, or

ii) there is a central simple associative algebra A over a quadratic field extension P = Γ[q] (q^{2} ∈ Γ) equipped with an involution of second kind j such that J is the Jordan algebra of symmetric elements H(A, j) = {x ∈ A : j(x) = x}.

In the first case put P = Γ⊕Γ = Γ[q] with q^{2} = 1, A = B⊕B^{op} (where op denotes the opposite algebra) and j the involution of second kind given by j(b_{1}, b_{2}) = (b_{2}, b_{1}). So in both cases J = H(A, j) for a central simple involutorial algebra (A, j) of the second kind.

Then, Der J = S(A, j)_{0} = [S(A, j), S(A, j)] acting on H(A, j) by means of the Lie bracket in A, where S(A, j) = {x ∈ A : j(x) = −x} (see [9, Theorem 6.9]). Besides, H(A, j) = qS(A, j), H(A, j)_{0} is the adjoint module for S(A, j)_{0} and under the isomorphism of S(A, j)_{0}-modules given by H(A, j)_{0} → S(A, j)_{0}, x ↦ qx, the Lie bracket on S(A, j)_{0} becomes the S(A, j)_{0}-invariant product on H(A, j)_{0}

x ⋆ y = q(xy − yx) .  (∗∗)

Now, Theorem 3.7 plus the arguments in 4.1 and 4.2 prove the following:

Theorem Let T be a simple Lie triple system over a field F of characteristic zero with centroid Γ and let L = S ⊕ T be its standard imbedding. Then Hom_{S}(T ⊗_{F} T, T) = 0 unless either:

a) T is of adjoint type with S being a central simple Lie algebra over Γ of type different from A_{n} (n ≥ 2). In this case Hom_{S}(T ⊗_{F} T, T) ≅ Hom_{S}(S ⊗_{F} S, S), which is spanned over Γ by the Lie multiplication in S.

b) There exists a central simple Jordan algebra J of degree n ≥ 3 over Γ and a nonzero scalar μ ∈ Γ (which can be taken modulo Γ^{2}) such that T is the Lie triple system (J_{0}, μ). In this case either:

i) J is not of type A over Γ, then Hom_{S}(T ⊗_{F} T, T) = Hom_{Der J}(J_{0} ⊗_{F} J_{0}, J_{0}) is spanned over Γ by the product in (∗),

ii) J = H(A, j) for some central simple associative involutorial algebra (A, j) of the second kind over Γ. Let P = Γ[q] (0 ≠ q^{2} ∈ Γ) be the center of A. Then

Hom_{S}(T ⊗_{F} T, T) = Hom_{Der J}(J_{0} ⊗_{F} J_{0}, J_{0})
= Hom_{S(A,j)_{0}}(H(A, j)_{0} ⊗_{Γ} H(A, j)_{0}, H(A, j)_{0})

is spanned over Γ by the products in (∗) and (∗∗).

This theorem may be considered as an extension of the classification of the flexible Lie-admissible algebras over a field of characteristic zero with an underlying simple Lie algebra (see [17, Chapter III] and the references therein), since this classification is subsumed in cases a) and b)ii) of the above result.

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