ON LOCALLY FINITE SPLIT LIE TRIPLE SYSTEMS

A. J. Calderón Martín Departamento de Matemáticas.

Universidad de Cádiz. 11510 Puerto Real, Cádiz, Spain. e-mail: ajesus.calderon@uca.es ^{∗}

M. Forero Piulestán Departamento de Matemáticas.

Universidad de Cádiz. 11510 Puerto Real, Cádiz, Spain. e-mail: ForeroManuel@hotmail.com

Abstract

Lie triple systems appear as the natural ternary extension of Lie algebras. The classification in the finite-dimensional setup (over an algebraically closed field of characteristic zero) is well-known (see [1]). In order to suggest a possible approach to a structure theory of infinite-dimensional Lie triple systems, we introduce and study split and locally finite Lie triple systems, stating that under certain conditions the standard embedding of a split Lie triple system is a split Lie algebra and that the standard embedding of a locally finite Lie triple system is a locally finite Lie algebra. We also give a description of certain locally finite simple split Lie triple systems.

2000 MSC: 17B40, 17A40, 17B65.

Key words: Lie triple systems, split Lie algebras, locally finite Lie algebras.

1 INTRODUCTION

The main tool used by Lister ([1]) to study finite-dimensional Lie triple systems is the standard embedding, this one being a two-graded Lie algebra L = L^{0} ⊕ L^{1}, with L^{0} the span of {L(x, y) : x, y ∈ T }, where L(x, y) denotes the left multiplication operator in T, L^{1} := T, and where the product is given by

∗Supported by the PCI of the spanish Junta de Andalucía ‘Estudio analítico-algebraico de sistemas triples y de pares en diferentes clases de estructuras no asociativas’, by the PAI of the spanish Junta de Andalucía with project number FQM-298 and by the spanish DGICYT with project number PB97-1497

[(L(x, y), z), (L(u, v), w)] :=
= (L([u, v, y], x) − L([u, v, x], y) + L(z, w), [x, y, w] − [u, v, z]).

Later, Faulkner gives in [2] an alternative approach to the classification of Lie triple systems, also in the finite-dimensional setup, by introducing a Cartan subalgebra H^{0} of the even part of the standard embedding, and a decomposition of T as the direct sum of certain root spaces relative to H^{0}.

In the framework of infinite-dimensional Lie algebras, K. H. Neeb, N. Stumme and other authors have successfully developed over the recent years a theory of split and locally finite Lie algebras (c. f. [3, 4, 5, 6, 7, 8, 9, 10]).

In this paper, we combine all the above ideas in order to introduce and study split Lie triple systems of arbitrary dimension in sections 2 and 3, the main result is Theorem 3.1 which states that under certain conditions the standard embedding of a split Lie triple system is a split Lie algebra. In section 4 we also state that the standard embedding of a locally finite Lie triple system is a locally finite Lie algebra (Proposition 4.1), and obtain in Theorem 4.1 a description of an important class of locally finite split simple Lie triple systems.

2 BASIC DEFINITIONS

Let 𝕂 be a field of characteristic zero and let T be a vector space over 𝕂. We say that T is a triple system if it is endowed with a trilinear map

< ·, ·, · >: T × T × T → T,

called the triple product of T.

A triple system T is called a Lie triple system if its triple product, denoted by [·, ·, ·], satisfies

1. [x, x, y] = 0

2. [x, y, z] + [y, z, x] + [z, x, y] = 0 (Jacobi identity)

3. [x, y, [a, b, c]] − [a, b, [x, y, c]] = [[x, y, a], b, c] + [a, [x, y, b], c]

for any x, y, z, a, b, c ∈ T.

An ideal of a Lie triple system T is a subspace I for which [I, T, T] ⊆ I. Let us observe that [I, T, T] ⊆ I implies that [T, I, T] ⊆ I and [T, T, I] ⊆ I. A Lie triple system T is called simple if the product is nonzero and its only ideals are {0} and T.

A two-graded 𝕂-algebra A is a 𝕂-algebra which splits into the direct sum A = A^{0} ⊕ A^{1} of subspaces (called the even and the odd part respectively) satisfying A^{α}A^{β} ⊂ A^{α+β} for any α, β in ℤ_{2}. A homomorphism f between twograded algebras A and B is a homomorphism which preserves gradings i.e.,

f(A^{α}) ⊂ B^{α}

for all α ∈ ℤ_{2}. A graded ideal of a two-graded algebra A = A^{0} ⊕ A^{1} is an ideal I of A such that there exist two subspaces I^{0} and I^{1} with I = I^{0} ⊕ I^{1} and I^{α} ⊆ A^{α} for any α ∈ ℤ_{2}. A two-graded algebra A is graded-simple if the product is nonzero and its only graded ideals are {0} and A.

The standard embedding of a Lie triple system T is the two-graded Lie algebra L = L^{0} ⊕ L^{1}, L^{0} being the 𝕂-span of {L(x, y) : x, y ∈ T }, where L(x, y) denotes the left multiplication operator in T, L(x, y)(z) := [x, y, z]; L^{1} := T and where the product is given by

[(L(x, y), z), (L(u, v), w)] :=
= (L([u, v, y], x) − L([u, v, x], y) + L(z, w), [x, y, w] − [u, v, z]).

Let us observe that L^{0} with the product induced by the one in L = L^{0} ⊕ L^{1} becomes a Lie algebra.

By using the same arguments as in the finite dimensional case, (see [1, Theorem 2.13 ]), we can prove the following propositions

Proposition 2.1 A Lie triple system T is simple if and only if its standard embedding is graded-simple.

Proposition 2.2 Let L = L^{0} ⊕ L^{1} be a graded-simple two-graded Lie algebra. Then either

1. L is simple (in ungraded sense) and the grading is given by

L = Sym(L, ξ) ⊕ Skw(L, ξ)

where ξ is an involutive automorphism of L, or

2. L is isomorphic to L′ ⊕L′ where L′ is a simple Lie algebra and the product

is

[(a, b), (c, d)] = ([a, c] + [b, d], [a, d] + [b, c]).

3 SPLIT LIE TRIPLE SYSTEMS

3.1 Definitions and Basic Properties

Given an element x of a Lie algebra L, we denote by ad(x) the adjoint mapping defined as ad(x)(y) = [x, y] for any y ∈ L.

Following [7, 10, 11, 12], we recall that a splitting Cartan subalgebra H of a Lie algebra L is defined as a maximal abelian subalgebra, (MASA), of L satisfying that the adjoint mappings ad(h) for h ∈ H are simultaneously diagonalizable. If L contains a splitting Cartan subalgebra H, then L is called a split Lie algebra. This means that we have a root decomposition L = H ⊕ ( ⨁_{α∈Λ} L_{α}) where L_{α} = {v_{α} ∈ L : [h, v_{α}] = α(h)v_{α} for any h ∈ H} for a linear

functional α ∈ H^{∗} and Λ := {α ∈ H^{∗}\{0} : L_{α} ≠ 0}. The subspaces L_{α} for α ∈ H^{∗} are called root spaces of L (respect to H) and the elements α ∈ Λ ∪ {0} are called roots of L respect to H.

Definition 3.1 Let T be a Lie triple system, let L = L^{0} ⊕ L^{1} be its standard embedding, and let H^{0} be a MASA of L^{0}. Following the ideas in [2], for a linear functional α ∈ (H^{0})^{∗} we define the root space of T (respect to H^{0}) associated to α as the subspace T_{α} := {t_{α} ∈ T : [h, t_{α}] = α(h)t_{α} for any h ∈ H^{0}}. The elements α ∈ (H^{0})^{∗} satisfying T_{α} ≠ 0 are called roots of T respect to H^{0} and we denote Λ^{1} := {α ∈ (H^{0})^{∗}\{0} : T_{α} ≠ 0}.

Let us observe that T_{0} = {t_{0} ∈ T : [h, t_{0}] = 0 for any h ∈ H^{0}}. In the following, we shall denote by Λ^{0} the set of all nonzero α ∈ (H^{0})^{∗} such that L_{α}^{0} := {v_{α}^{0} ∈ L^{0} : [h, v_{α}^{0}] = α(h)v_{α}^{0} for any h ∈ H^{0}} ≠ 0.

Lemma 3.1 Let T be a Lie triple system, let L = L^{0} ⊕ L^{1} be its standard embedding, and let H^{0} be a MASA of L^{0}. If α, β, γ ∈ Λ^{1} ∪{0} and δ ∈ Λ^{0} ∪{0}. Then,

1. If [T_{α}, T_{β}] ≠ 0 then α + β ∈ Λ^{0} ∪ {0} and [T_{α}, T_{β}] ⊆ L_{α+β}^{0}.

2. If [L_{δ}^{0}, T_{α}] ≠ 0 then δ + α ∈ Λ^{1} ∪ {0} and [L_{δ}^{0}, T_{α}] ⊆ T_{δ+α}

3. If [T_{α}, T_{β}, T_{γ}] ≠ 0 then α + β + γ ∈ Λ^{1} ∪ {0} and [T_{α}, T_{β}, T_{γ}] ⊆ T_{α+β+γ.}

Proof. 1. For any x ∈ T_{α}, y ∈ T_{β} and h ∈ H^{0}, we have [h, [x, y]] = −[[h, y], x] + [[h, x], y] = (α + β)(h)[x, y]. Therefore [T_{α}, T_{β}] ⊆ L_{α+β}^{0}.

2. and 3. The proof is similar to 1.□

Let us observe that if [T_{α}, T_{0}] ≠ 0 then α is a root of L^{0} relative to H^{0} and if [L_{α}^{0}, T_{0}] ≠ 0 then α is a root of T relative to H^{0}.

Definition 3.2 Let T be a Lie triple system, let L = L^{0} ⊕ L^{1} be its standard embedding, and let H^{0} be a MASA of L^{0}. We shall call that T is a split Lie triple system (respect to H^{0}) if:

1. T = T_{0} ⊕ ( ⨁_{α∈Λ^{1}} T_{α}),

2. [T_{0}, T_{0}, T] = 0.

Proposition 3.1 Let T be a split Lie triple system respect to H^{0}, then L^{0} is a split Lie algebra respect to the splitting Cartan subalgebra H^{0}.

Proof. We first prove that ∑_{β∈Λ^{0}} L_{β}^{0} is a direct sum, (let us note that we can also apply [13, 2.1 Prop.1]). If x ∈ L_{α}^{0} ∩ ∑_{β≠α} L_{β}^{0}, we can write x = ∑_{β≠α} x_{β} with x_{β} ∈ L_{β}^{0}, and x = x_{α} with x_{α} ∈ L_{α}^{0}. From Lemma 3.1, we deduce for any y ∈ T_{γ} and γ ∈ Λ^{1} ∪ {0}, that

[x, y] ∈ T_{α+γ} ∩ ∑_{β≠α} T_{β+γ}

and therefore [x, y] = 0 for any y ∈ T. As x is a linear map on T, satisfying x(y) = [x, y], we deduce that x = 0. Let us also check that H^{0} + ( ⨁_{β∈Λ^{0}} L_{β}^{0}) is a

direct sum. If x ∈ H^{0} ∩ ( ⨁_{β∈Λ^{0}} L_{β}^{0}) we can write x = ∑_{β∈Λ^{0}} x_{β}, with x_{β} ∈ L_{β}^{0},

and x = h^{0} ∈ H^{0}. For any β ∈ Λ^{0}, we can take h_{β} ∈ H^{0}, such that β(h_{β}) ≠ 0, and then

0 = [h_{β}, h^{0}] = [h_{β}, x] = ∑_{β∈Λ^{0}}[h_{β}, x_{β}] = ⨁_{β∈Λ^{0}} β(h_{β})x_{β},

therefore x_{β} = 0. We conclude x = 0. Thus H^{0} ⊕ ( ⨁_{β∈Λ^{0}} L_{β}^{0}) ⊆ L^{0}.

Let us observe that the maximal abelian character of H^{0} in L^{0} implies H^{0} = L_{0}^{0}.

Finally, as

L^{0} = [T, T] = ∑_{α,γ∈Λ^{1}∪{0}}[T_{α}, T_{γ}]

⊆ ∑_{α,γ∈Λ^{1}∪{0}} L_{α+γ}^{0} ⊆ H^{0} ⊕ ( ⨁_{β∈Λ^{0}} L_{β}^{0}),

the proof is complete.□

3.2 Types of Roots. Integrable Roots

In this subsection we distinguish the various types in the root system of a split Lie triple system, taking particular notice of the integrable roots.

Lemma 3.2 For nonzero root vectors t_{±α} ∈ T_{±α} the subalgebra of the standard embedding,

L(t_{α}, t_{−α}) := span𝕂{t_{α}, t_{−α}, [t_{α}, t_{−α}]}

is of one of the following types:

1. If [t_{α}, t_{−α}] = 0, then L(t_{α}, t_{−α}) is two dimensional abelian. We say that L(t_{α}, t_{−α}) is of abelian type.

2. If [t_{α}, t_{−α}] ≠ 0 but α([t_{α}, t_{−α}]) = 0, then L(t_{α}, t_{−α}) is a three dimensional algebra. We say that L(t_{α}, t_{−α}) is of nilpotent type.

3. If α([t_{α}, t_{−α}]) ≠ 0, then L(t_{α}, t_{−α}) ≈ sl(2, 𝕂). We say that L(t_{α}, t_{−α}) is of simple type.

Proof. This is an easy verification (cf. [10, Lemma I.2]).□

Definition 3.3 For a root α ∈ Λ^{1} the subalgebras L(t_{α}, t_{−α}) are called test algebras associated to α. We say that a root α ∈ Λ^{1} is of nilpotent type if all test algebras associated to α are of abelian or nilpotent type. Note that a root α ∈ Λ^{1} with -α ∉ Λ^{1} is of nilpotent type. We call a root α ∈ Λ^{1} of simple type if there exists an associated test algebra of simple type. A root α ∈ Λ^{1} of simple type is called integrable if there exists an associated test algebra L(t_{α}, t_{−α}) of simple type such that the middle multiplication operators of T,

M(t_{ϵα}) : T → T,

M(t_{ϵα})(t) := [t_{ϵα}, t, t_{ϵα}], ϵ ∈ {+, −}, are locally nilpotent. We denote by Λ_{i}^{1} the set of all integrable nonzero roots of T relative to H^{0}.

Lemma 3.3 Let T be a split Lie triple system, let L = L^{0} ⊕ L^{1} be its standard embedding and let α ∈ Λ^{1}. Then we have:

1. M(t_{α})(t) = −ad^{2}(t_{α})(t), for any t_{α} ∈ T_{α} and for any t ∈ T.

2. ad^{3+2n}(t_{α})([t, t′]) = (−1)^{n}M^{n+1}(t_{α})([t, t′, t_{α}]), for any n ∈ ℕ, t_{α} ∈ T_{α} and t, t′ ∈ T.

Proof.

1. It is clear.

2. By induction on n:

For n = 0, we have ad^{3}(t_{α})([t, t′]) = [t_{α}, [t_{α}, [t_{α}, [t, t′]]]] = [t_{α}, [t, t′, t_{α}], t_{α}] = M(t_{α})[t, t′, t_{α}].

Assume 2 holds for n, let us prove it for n + 1. We have

ad^{3+2(n+1)}(t_{α})([t, t′]) = ad^{2}(t_{α})(ad^{3+2n}(t_{α})[t, t′]) =
= ad^{2}(t_{α})((−1)^{n}M^{n+1}(t_{α})([t, t′, t_{α}])),

and by applying 1,

ad^{2}(t_{α})((−1)^{n}M^{n+1}(t_{α})([t, t′, t_{α}])) = −M(t_{α})((−1)^{n}M^{n+1}(t_{α})([t, t′, t_{α}])) =

= (−1)^{n+1}M^{n+2}(t_{α})([t, t′, t_{α}]).

The proof is complete.□

Proposition 3.2 Let T be a split Lie triple system, let L = L^{0} ⊕ L^{1} be its standard embedding and let α ∈ Λ^{1}. Then, for any t_{α} ∈ T_{α}, M(t_{α}) is locally nilpotent on T if and only if ad(t_{α}) is locally nilpotent on L.

Proof. Suppose that M(t_{α}) is locally nilpotent on T. Let be x = x^{0}+x^{1} ∈ L, with x^{0} ∈ L^{0} and x^{1} ∈ L^{1}. There exists m ∈ ℕ such that M^{m}(t_{α})(x^{1}) = 0. By Lemma 3.3-1 we also have ad^{2m}(t_{α})(x^{1}) = 0. As x^{0} = ∑_{i=1}^{q}[t_{i}, t′_{i}] with t_{i}, t′_{i} ∈ T, if we consider for each [t_{i}, t′_{i}] the element [t_{i}, t′_{i}, t_{α}] ∈ T, there exists m_{i} ∈ ℕ such that M^{m_{i}}([t_{i}, t′_{i}, t_{α}]) = 0, and by Lemma 3.3-2, ad^{3+2(m_{i}−1)}(t_{α})([t_{i}, t′_{i}]) = 0. By denoting r = max{2m, ⋃_{i=1,...,q} 3 + 2(m_{i} − 1)}, we conclude that ad^{r}(t_{α})(x) = 0

and therefore ad(t_{α}) is locally nilpotent on L.

By Lemma 3.3-1, it is clear that if ad(t_{α}) is locally nilpotent on L, then M(t_{α}) is locally nilpotent on T.□

Proposition 3.3 Let T be a split Lie triple system, let L = L^{0} ⊕ L^{1} be its standard embedding and let L(t_{α}, t_{−α}) be a test algebra associated to α ∈ Λ^{1} such that M(t_{±α}) are locally nilpotent. Then L is a locally finite L(t_{α}, t_{−α})−module with respect to the adjoint representation.

Proof. By Proposition 3.2, the adjoint mappings ad(t_{±α}) are locally nilpotent on L. By using now the same arguments as for sl(2, 𝕂) (cf. [13, Proposition 2.4.7]) we complete the proof.□

Proposition 3.4 Let T be a split Lie triple system, let L = L^{0} ⊕ L^{1} be its standard embedding and let α ∈ Λ_{i}^{1}. Then we have:

1. dimT_{α} = 1.

2. T_{±(2n−1)α} = L_{±2mα}^{0} = 0 for n > 1 and m ≥ 1.

3. If α ∈ Λ_{0}, then −α ∈ Λ_{0} and dimL_{α}^{0} = dimL_{−α}^{0}.

4. If α ∉ Λ_{0}, then ℤα ∩ Λ^{1} = ±α and ℤα ∩ Λ^{0} = ∅.

5. There exists a unique element h_{α} = [t_{α}, t_{−α}] with α(h_{α}) = 2.

6. For each β ∈ Λ^{1} we have β(h_{α}) ∈ ℤ.

7. 𝕂α ∩ Λ^{1} ⊆ (\frac{1}{2}ℤ)α.

8. 𝕂α ∩ Λ_{i}^{1} ⊆ \{±\frac{1}{2}α, ±α, ±2α\}.

Proof. Since α is integrable, we find t_{±α} ∈ T_{±α} such that

L(t_{α}, t_{−α}) ≈ sl(2, 𝕂)

and M(t_{±α}) are locally nilpotent. We may w.l.o.g. assume that α([t_{α}, t_{−α}]) = 2, and denote h_{α} := [t_{α}, t_{−α}].

1. and 2. We consider the L(t_{α}, t_{−α})−submodule of L

V := 𝕂t_{−α} + H^{0} + ∑_{n=1}^{∞} T_{(2n−1)α} + ∑_{n=1}^{∞} L_{2nα}^{0}.

By Proposition 3.3, L is a locally finite L(t_{α}, t_{−α})−module and, as a submodule of a locally finite module, V is also locally finite. In particular V is a sum of finite dimensional simple L(t_{α}, t_{−α})−submodules by Weyl’s theorem. Hence the representation theory of sl(2, 𝕂) implies that the set of h_{α}-eigenvalues on V is symmetric with

dimV^{μ}(h_{α}) = dimV^{−μ}(h_{α})

for each μ ∈ 𝕂. Now V^{−2}(h_{α}) = 𝕂t_{−α} implies that dimV^{2}(h_{α}) = dimT_{α} = 1 and furthermore that

dimV^{2(2n−1)}(h_{α}) = dimT_{(2n−1)α} = 0

for n > 1, and dimV^{2(2n)}(h_{α}) = dimL_{2nα}^{0} = 0 for n ≥ 1. Likewise we obtain dimT_{−(2n−1)α} = 0 for n > 1, and dimL_{−2nα}^{0} = 0 for n ≥ 1.

3. We have

V′ := T_{α} + T_{−α} + H_{0} + ∑_{n=0}^{∞} T_{±2nα} + ∑_{n=1}^{∞} L_{±(2n−1)α}^{0}

is an L(t_{α}, t_{−α})−submodule of L. By arguing as in 1., we prove 3.

4. Since by 1., dimT_{α} = dimT_{−α} = 1, and by 3. L_{α}^{0} = L_{−α}^{0} = 0, we can check that

V′′ := T_{α} + T_{−α} + H_{0} + ∑_{n=0}^{∞} T_{−2nα} + ∑_{n=2}^{∞} L_{−(2n−1)α}^{0}

is an L(t_{α}, t_{−α})−submodule of L. By arguing as in 1., we obtain

T_{±2nα} = L_{±(2n−1)α}^{0} = 0

for n ≥ 1. From 2. we complete the proof.

5. Since both spaces T_{±α} are one dimensional and do not commute, the space [T_{α}, T_{−α}] is one dimensional. Hence the element h_{α} is uniquely determined by α(h_{α}) = 2.

6. Since L is a locally finite L(t_{α}, t_{−α})−module and β(h_{α}) is the eigenvalue of ad(h_{α}) on the root space T_{β}, this is a consequence of the finite dimensional representation theory of sl(2, 𝕂)

7. Let β = cα ∈ Λ^{1} with c ∈ 𝕂. Then 5. and 6. imply that c ∈ \frac{1}{2}ℤ.

8. If, in addition, β is integrable, then we also have that \frac{1}{c} ∈ \frac{1}{2}ℤ and thus \frac{4}{2c} ∈ ℤ. Since 2c divides 4 it equals 1, 2, or 4, so that we may have c ∈ \{±\frac{1}{2}, ±1, ±2\}.□

We have seen that for integrable roots the root spaces T_{±α} are one dimensional, showing that the test algebras L(t_{α}, t_{−α}) do not depend on the choice of t_{±α}.

Definition 3.4 We say that a simple root α of T is abelian simple if L_{α}^{0} = 0. If L_{α}^{0} ≠ {0}, (and by Proposition 3.4-3, L_{−α}^{0} ≠ {0}), α is called strongly simple if there exist nonzero elements e_{±α}^{0} ∈ L_{±α}^{0} such that, for any ϵ ∈ {+, −}, [t_{ϵα}, e_{ϵα}^{0}] = 0 for some 0 ≠ t_{ϵα} ∈ T_{ϵα}, and [e_{α}^{0}, e_{−α}^{0}] ≠ 0. Finally, we say that a simple root α is weakly simple if L_{α}^{0} ≠ {0} and it is not a strongly simple root. We call an abelian simple root α abelian integrable if α is also an integrable root. A strongly simple root α is said to be strongly integrable if α is also an integrable root and the adjoint mappings ad(e_{±α}^{0}) are locally nilpotent on L^{0}. We denote by Λ_{a,i}^{1} (resp. Λ_{s,i}^{1}) the set of all abelian integrable (resp. strongly integrable) nonzero roots of T relative to H^{0}.

It is clear that if α ∈ Λ_{a,i}^{1} (resp. α ∈ Λ_{s,i}^{1}), then −α ∈ Λ_{a,i}^{1} (resp. −α ∈ Λ_{s,i}^{1}).

Proposition 3.5 Let α ∈ Λ_{s,i}^{1}. Then there exist nonzero elements e_{±α}^{0} ∈ L_{±α}^{0} such that α([e_{α}^{0}, e_{−α}^{0}]) ≠ 0 and the adjoint mappings ad(e_{±α}^{0}) are locally nilpotent on L^{0}.

Proof. Let us fix nonzero elements e_{±α}^{0} ∈ L_{±α}^{0} such that, for any ϵ ∈ {+, −}, [t_{ϵα}, e_{ϵα}^{0}] = 0 for some 0 ≠ t_{ϵα} ∈ T_{ϵα}, [e_{α}^{0}, e_{−α}^{0}] ≠ 0, and the adjoint mappings ad(e_{±α}^{0}) are locally nilpotent on L^{0}. By applying Jacobi identity to

[[e_{−α}^{0}, t_{α}], [e_{α}^{0}, t_{−α}]] ∈ [T_{0}, T_{0}] = 0,

we have

0 = [[[t_{α}, t_{−α}], e_{α}^{0}], e_{−α}^{0}] − [[[t_{α}, e_{α}^{0}], t_{−α}], e_{−α}^{0}]+
+[[[t_{−α}, e_{−α}^{0}], e_{α}^{0}], t_{α}] + [[[e_{−α}^{0}, e_{α}^{0}], t_{−α}], t_{α}].

Therefore,

α([t_{α}, t_{−α}])[e_{α}^{0}, e_{−α}^{0}] = α([e_{α}^{0}, e_{−α}^{0}])[t_{α}, t_{−α}],

and from here the proposition follows.□

Proposition 3.6 If α ∈ Λ_{s,i}^{1}, then

1. dimL_{±α}^{0} = 1.

2. ℤα ∩ Λ^{1} = {±α}.

3. ℤα ∩ Λ^{0} = {±α}.

Proof. 1. From Proposition 3.5, α is an integrable root of the split Lie algebra L^{0}, (see Proposition 3.1), in the sense of [10, Definition I.3]. Then, by [10, Proposition I.6] we have dimL_{±α}^{0} = 1.

2. and 3. As by 1. and Proposition 3.4-1, dimT_{±α} = dimL_{±α}^{0} = 1, we have [T_{α}, L_{α}^{0}] = [T_{−α}, L_{−α}^{0}] = 0.

Let us consider

V := T_{0} + H^{0} + T_{α} + T_{−α} + L_{α}^{0} + L_{−α}^{0} + ∑_{n=1}^{∞} T_{−2nα} + ∑_{n=2}^{∞} L_{−(2n−1)α}^{0}.

We can verify that V is an L(t_{α}, t_{−α})-module, and by arguing as in Proposition 3.4-1, we conclude T_{−2nα} = L_{−(2m−1)α}^{0} = 0, for n ≥ 1 and m ≥ 2. Proposition 3.4-2 completes the proof.□

Corollary 3.1 Let α ∈ Λ_{s,i}^{1}. Then we have:

1. [T_{α}, L_{−α}^{0}] ≠ 0

2. [T_{α}, T_{0}] ≠ 0

3. [L_{α}^{0}, T_{0}] ≠ 0

Proof. 1. and 2. Let us fix t_{±α} ∈ T_{±α} such that α([t_{α}, t_{−α}]) = 1, and 0 ≠ e_{α}^{0} ∈ L_{α}^{0} satisfying [t_{α}, e_{α}^{0}] = 0.

Let us consider

V := 𝕂[[e_{α}^{0}, t_{−α}], t_{−α}] ⊕ 𝕂[e_{α}^{0}, t_{−α}] ⊕ 𝕂e_{α}^{0}.

We assert that V is an L(t_{α}, t_{−α})-submodule of L. Indeed, this can be verified taking into account the following identities:

Since [H_{0}, T_{0}] = 0,

[[[e_{α}^{0}, t_{−α}], t_{−α}], t_{α}] = −[[t_{−α}, t_{α}], [e_{α}^{0}, t_{−α}]] − [[t_{α}, [e_{α}^{0}, t_{−α}]], t_{−α}] =

= −[[t_{α}, [e_{α}^{0}, t_{−α}]], t_{−α}] = [[[t_{α}, t_{−α}], e_{α}^{0}], t_{−α}] − [[[t_{α}, e_{α}^{0}], t_{−α}], t_{−α}] =

= [[[t_{α}, t_{−α}], e_{α}^{0}], t_{−α}] = [e_{α}^{0}, t_{−α}].

As [t_{α}, e_{α}^{0}] = 0,

[[e_{α}^{0}, t_{−α}], t_{α}] = e_{α}^{0},

and finally, by Proposition 3.6-2,

[[[e_{α}^{0}, t_{−α}], t_{−α}], t_{−α}] ⊂ T_{−2α} = 0.

By arguing as in Proposition 3.4-1, we conclude that [[e_{α}^{0}, t_{−α}], t_{−α}] ≠ 0, and from here 1. and 2.

3. From Proposition 3.5, we can fix e_{±α}^{0} ∈ L_{±α}^{0} such that [t_{ϵα}, e_{ϵα}^{0}] = 0 for some 0 ≠ t_{ϵα} ∈ T_{ϵα}, ϵ ∈ {+, −}, and satisfying α([e_{α}^{0}, e_{−α}^{0}]) ≠ 0. By Jacobi identity we obtain [[e_{−α}^{0}, t_{α}], e_{α}^{0}] = −α([e_{α}^{0}, e_{−α}^{0}])t_{α} ≠ 0, and 3. is proved.□

Lemma 3.4 Let α ∈ Λ_{s,i}^{1}. Then there exist nonzero elements t_{α} ∈ T_{α}, t_{0} ∈ T_{0}, e_{α}^{0} ∈ L_{α}^{0} such that [e_{α}^{0}, t_{0}] = t_{α} and [t_{α}, t_{0}] = e_{α}^{0}.

Proof. By Corollary 3.1 there exist t′_{α} ∈ T_{α}, t′_{0}, t′′_{0} ∈ T_{0} and f_{α}^{0} ∈ L_{α}^{0} such that [t′_{α}, t′_{0}] ≠ 0 and [f_{α}^{0}, t′′_{0}] ≠ 0. If [f_{α}^{0}, t′_{0}] ≠ 0 or [t′_{α}, t′′_{0}] ≠ 0 the proof is clear taking into account that dimT_{α} = dimL_{α}^{0} = 1.

If [f_{α}^{0}, t′_{0}] = 0 and [t′_{α}, t′′_{0}] = 0, let consider t′′′_{0} := t′_{0} + t′′_{0}. As [t′_{α}, t′′′_{0}] = λf_{α}^{0} and [f_{α}^{0}, t′′′_{0}] = βt′_{α} with α, β nonzero elements of 𝕂, it is easy to check that

t_{0} := \sqrt{\frac{1}{λβ}} t′′′_{0}, t_{α} := \sqrt{\frac{β}{λ}}t′_{α} and e_{α}^{0} := f_{α}^{0} complete the proof.□

Proposition 3.7 Let T be a split Lie triple system respect to H^{0} and let L = L^{0} ⊕ L^{1} be its standard embedding. Then H^{0} ⊕ T_{0} is a MASA of L.

Proof. It is easy to check, taking into account the definition of split Lie triple system and Proposition 3.1.□

Proposition 3.8 Let T be a split Lie triple system respect to H^{0}, let L = L^{0}⊕L^{1} be its standard embedding and let α ∈ Λ_{s,i}^{1}. Then there exist two nonzero linear functionals α_{1}, α_{2} : H^{0} ⊕ T_{0} → 𝕂 such that L_{α}^{0} ⊕ T_{α} = L_{α_{1}} ⊕ L_{α_{2}}.

Proof. By Lemma 3.4 we can choose nonzero elements t_{α} ∈ T_{α}, t_{0} ∈ T_{0}, e_{α}^{0} ∈ L_{α}^{0} such that [e_{α}^{0}, t_{0}] = t_{α} and [t_{α}, t_{0}] = e_{α}^{0}. Since dimT_{α} = dimL_{α}^{0} = 1, we can determine, for any t′_{0} ∈ T_{0}, the unique λ_{t′_{0}}, β_{t′_{0}} ∈ 𝕂 such that [t_{α}, t′_{0}] = λ_{t′_{0}}e_{α}^{0} and [e_{α}^{0}, t′_{0}] = β_{t′_{0}}t_{α}. As [T_{0}, T_{0}] = 0 we have

0 = [[t′_{0}, t_{0}], t_{α}] = −[[t_{0}, t_{α}], t′_{0}]−[[t_{α}, t′_{0}]], t_{0}] = [e_{α}^{0}, t′_{0}]−λ_{t′_{0}}[e_{α}^{0}, t_{0}] = β_{t′_{0}}t_{α}−λ_{t′_{0}}t_{α},

and therefore β_{t′_{0}} = λ_{t′_{0}}.

Thus we can define the mapping

δ_{α} : T_{0} → 𝕂 by δ_{α}(t′_{0}) := λ_{t′_{0}}.

Now, if we also define

α_{1}, α_{2} : H^{0} ⊕ T_{0} → 𝕂

by

α_{1}(h^{0} + t′_{0}) := α(h^{0}) − δ_{α}(t′_{0}) and α_{2}(h^{0} + t′_{0}) := α(h^{0}) + δ_{α}(t′_{0}),

we assert that L_{α_{1}} ⊕ L_{α_{2}} = L_{α}^{0} ⊕ T_{α}. Indeed,

[h^{0} + t′_{0}, e_{α}^{0} + t_{α}] = α(h^{0})(e_{α}^{0} + t_{α}) − λ_{t′_{0}}t_{α} − λ_{t′_{0}}e_{α}^{0} =
= (α(h^{0}) − δ_{α}(t′_{0}))(e_{α}^{0} + t_{α}) = α_{1}(h^{0} + t′_{0})(e_{α}^{0} + t_{α}),

therefore 𝕂(e_{α}^{0} + t_{α}) ⊂ L_{α_{1}}.

In a similar way we can verify that 𝕂(e_{α}^{0} − t_{α}) ⊂ L_{α_{2}}. Then we have L_{α}^{0} ⊕ T_{α} = 𝕂e_{α}^{0} + 𝕂t_{α} = 𝕂(e_{α}^{0} + t_{α}) + 𝕂(e_{α}^{0} − t_{α}) ⊂ L_{α_{1}} ⊕ L_{α_{2}}.

If e_{α_{1}} + e_{α_{2}} ∈ L_{α_{1}} ⊕ L_{α_{2}}, we obtain that

[h^{0}, e_{α_{1}} + e_{α_{2}}] = α_{1}(h^{0})e_{α_{1}} + α_{2}(h^{0})e_{α_{2}} = α(h^{0})(e_{α_{1}} + e_{α_{2}}),

and therefore e_{α_{1}} + e_{α_{2}} ∈ L_{α}^{0} ⊕ T_{α}. The proof is complete.□

Theorem 3.1 Let T be a split Lie triple system respect to H^{0} such that Λ^{1} = Λ_{a,i}^{1} ∪ Λ_{s,i}^{1}. Then its standard embedding L is split respect to H^{0} ⊕ T_{0}.

Proof. First, if T_{0} = 0 then H^{0} is a MASA of L. We assert that in this case Λ^{0} ∩ Λ^{1} = ∅. Indeed, if α ∈ Λ^{0} ∩ Λ^{1}, α must be strongly integrable and by Corollary 3.1-1, T_{0} ≠ 0, a contradiction. Hence Λ^{0} ∪ Λ^{1} is the root system of L respect to H^{0}, and we have the splitting decomposition

L = H^{0} ⊕ ( ⨁_{β∈Λ^{0}} L_{β}^{0}) ⊕ ( ⨁_{α∈Λ^{1}} T_{α}).

Second, if T_{0} ≠ 0 then H^{0} ⊕ T_{0} is a MASA of L, and: If either α ∉ Λ^{1} and α ∈ Λ^{0}, (then [T_{0}, L_{α}^{0}] = 0), or α is an abelian integrable root of T, (then [T_{0}, T_{α}] = 0); we can define the root α′ : H^{0} ⊕ T_{0} → 𝕂 as α′(h_{0}, t_{0}) := α(h_{0}) for any (h_{0}, t_{0}) ∈ H^{0} ⊕ T_{0}, the associated root space is L_{α′} = L_{α}^{0} in the first case and L_{α′} = T_{α} in the second one.

If α is a strongly integrable root of T then, by Proposition 3.8, there exist two nonzero roots of L respect to H^{0}⊕T_{0}, α′_{1} and α′_{2}, such that L_{α}^{0}⊕T_{α} = L_{α′_{1}} ⊕L_{α′_{2}}. As by Proposition 3.1, L = H^{0} ⊕ ( ⨁_{α∈Λ_{0}} L_{α}^{0}) ⊕ T_{0} ⊕ ( ⨁_{β∈Λ_{1}} T_{β}), we conclude that

L admits the splitting decomposition respect to H^{0} ⊕ T_{0} given by

L = (H^{0} ⊕ T_{0}) ⊕ (⨁_{α′} L_{α′}).

□

4 LOCALLY FINITE SPLIT LIE TRIPLE SYS-TEMS

Definition 4.1 A Lie triple subsystem of a Lie triple system T is a submodule U such that [U, U, U] ⊆ U. We say that a Lie triple system T is locally finite if every finite subset of T is contained in a finite dimensional subtriple of T.

We recall that a similar concept is defined for Lie algebras (c.f. [7] or [10]).

Proposition 4.1 The standard embedding L = L^{0} ⊕ L^{1} of a locally finite Lie triple system T is a locally finite Lie algebra.

Proof. Let V^{0} be a finite subset of L^{0}. By the construction of L^{0}, the subset V^{0} can be described from a suitable finite subset V of T. As T is locally finite, there exists a finite dimensional subtriple of T, U, such that V is contained in U. The Lie subalgebra of L generated by U is finite dimensional and V^{0} is contained in it. Therefore L^{0} is locally finite. It is easy to conclude that L is also locally finite.□

Let us note that in a locally finite split Lie triple system all abelian (resp. strongly) simple roots of Λ^{1} are abelian (resp. strongly) integrable.

Theorem 4.1 Let T be a locally finite split simple Lie triple system such that Λ^{1} = Λ_{a,i}^{1} ∪ Λ_{s,i}^{1}. Then T is isomorphic to one of the followings:

1. T = L where L is a locally finite split simple Lie algebra.

2. T = Skw(L, ξ) with L as in the previous case and ξ an involutive automorphism of L.

Proof. By Propositions 2.1 and 2.2, either T = L or T = Skw(L′, ξ), where L, L′ are simple Lie algebras and ξ is an involutive automorphism of L′. By Proposition 4.1, L and L′ are locally finite. By Proposition 3.1 L is split and finally by Theorem 3.1 L′ is split. The proof is complete.□

Remark 4.1 If T is of infinite dimension, following [8], the algebra L of Theorem 4.1, must be one of the following sl(J, 𝕂), d(J, 𝕂) or sp(J, 𝕂) where J is an infinite set whose cardinality equals the dimension of the Lie algebra.

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