On Infinite Dimensional Lie Algebras Having a Cartan Decomposition

A. J. Calderón Martín ^{1} and M. Forero Piulestán Departamento de Matemáticas. Universidad de Cádiz 11510 Puerto Real. Cádiz, Spain. E-mail: ajesus.calderon@uca.es

Version: March 13, 2003

We introduce the concept of Cartan decomposition relative to a Cartan subalgebra H, in the sense of [2], for Lie algebras of arbitrary dimension. The class of simple Lie algebras having such decomposition is a wide class of Lie algebras closely related to the topologically simple L^{∗}-algebras ([8, 9, 5]), the simple cinvolutive Lie algebras ([6]) and the topologically simple compact Banach-Lie algebras ([4]). We give a description theorem for the complex ones, turning out to be the natural extension of the finite dimensional setup.

Key Words: Infinite dimensional Lie algebras, Cartan decompositions, roots and root spaces.

1. INTRODUCTION AND PREVIOUS DEFINITIONS

In order to make sense, the definition of Cartan subalgebra given below requieres a definition of semisimplicity for infinite dimensional Lie algebras. In this paper, a Lie algebra is defined to be semisimple if it has no non trivial abelian ideals. A critic of this definition is given in [6, apendix to chapter II]. But certainly, any other reasonable definition of semisimplicity will imply the absence of non trivial abelian ideals. We define the Annihilator of a Lie algebra L as the ideal given by Ann(L) = {x ∈ L : [x, y] = 0 for all y ∈ L}. It is clear that any semisimple Lie algebra L has zero annihilator, therefore L is also semisimple in the sense given in [8] for L^{∗}-algebras. We shall say that a Lie algebra L is simple if the product is nonzero and has no nonzero proper ideals. Throughout this paper, Lie algebras L are considered of arbitrary dimension, semisimple and over an algebraically closed field 𝕂 of characteristic 0. For subsets M, N of L we denote by [M, N] the subspace spanned by {[m, n] : m ∈ M, n ∈ N}.

Y. Billing and A. Pianzola introduced in [2] the concept of Cartan subalgebra for Lie algebras of arbitrary dimension as follows:

1The first author is supported in part 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.

A subalgebra H of L is called a Cartan subalgebra if

1. The elements of H act locally ad-nilpotently on H.

2. H is its own normalizer in L, i.e., N_{L}(H) = H.

If L is finite dimensional, then H is nilpotent by Engel’s theorem and the classical definition of Cartan subalgebra is recovered. A root of L relative to H is a mapping α : H ⟶ 𝕂 such that there exists v_{α} ∈ L, v_{α} ≠ 0 satisfying [h, v_{α}] = α(h)v_{α} for any h ∈ H. The root space associated to α is the subspace L_{α} = {v_{α} ∈ L : [h, v_{α}] = α(h)v_{α} for any h ∈ H}. Given a set S of nonzero roots of L, we shall denote by SpℤS the set of mappings

SpℤS = {∑_{i=1}^{n} p_{i}α_{i} : p_{i} ∈ ℤ and α_{i} ∈ S}.

We shall call that L has a Cartan decomposition relative to H if

1. L = H ⊕( ⨁_{α∈Λ} L_{α}), where Λ is the set of all nonzero roots of L relative

to H.

2. Each L_{α}, α ∈ Λ, is finite dimensional.

3. For any finite set S ⊂ Λ we have that SpℤS ∩ Λ is also finite.

4. [L_{α}, L_{−α}] ≠ 0.

Let us observe that in case that L is a simple complex Lie algebra with involution having a Cartan decomposition, Corollary 4.2 will imply that our definition of Cartan subalgebra coincides with the one given by Schue for L^{∗}-algebras in [8], and the one given by L’Harpe for c-involutive Lie algebras in [6, definition 4 on page 32], by adding to H the condition of being also selfadjoint. Hence, our definition of Cartan decomposition recovers the ones given for simple finite dimensional Lie algebras and topologically simple L^{∗}-algebras ([8]), c-involutive Lie algebras ([6])and compact Banach-Lie algebras ([4]).

It is easy to prove that the roots are linear mappings, the root space associated to the zero root is contained in the Cartan subalgebra and, by the Jacobi identity, that if α + β is a root then [L_{α}, L_{β}] ⊆ L_{α+β} and if α + β is not a root then [L_{α}, L_{β}] = 0.

The paper is organized as follows. In the second section we study the roots and root spaces associated to a Cartan subalgebra of a Lie algebra of arbitrary dimension, our main results in this section is that the root spaces associated to nonzero roots are one-dimensional and that the only integral multiplies kα of a nonzero root α which are roots are 0 and ±α. In the third section we study the direct limit of a direct system of Lie algebras, we prove the existence of the direct limit (unique up to isomorphism), and we study some of its properties. Finally, in the fourth section we obtain a description theorem for infinite dimensional simple complex Lie algebras having a Cartan decomposition L, by approximating L as the direct limit of a suitable family of finite dimensional simple Lie subalgebras, we note that the introduction of the concept of connection of roots to construct an adequate direct system of Lie algebras is perhaps the most interesting novelty in this paper. It is shown that this class is written as simple Lie subalgebras of certain simple associative algebras. The result turn out to be the natural extension of the finite dimensional theory, the L^{∗}-algebras theory ([8, 9, 5]) and the theory of compact Banach-Lie algebras ([4]). Specifically, the following theorem is proved

Theorem 4.3. Let L be an infinite dimensional simple complex Lie algebra having a Cartan decomposition relative to a Cartan subalgebra H. Then L is isomorphic to one of the followings:

1. A^{−}, with A a simple associative algebra.

2. Skw(A, ξ) with A as above and ξ an involutive antiautomorphism of A.

2. ROOTS AND CARTAN DECOMPOSITIONS

Lemma 2.1. Let L be a semisimple Lie algebra having a Cartan decomposition relative to a Cartan subalgebra H, and let α, β be two nonzero roots. Then we have:

1. If β+α and β−α are not roots then β(h_{α}) = 0 for any h_{α} ∈ [L_{α}, L_{−α}].

2. β(h_{α}) = rα(h_{α}) with r ∈ ℚ and h_{α} as above.

Proof. We obtain 1. as an easy consequence of Jacobi identity and the facts [L_{α}, L_{β}] = [L_{α}, L_{−β}] = 0.

2. Let fix h_{α} = [v_{α}, v_{−α}] with v_{α} ∈ L_{α} and v_{−α} ∈ L_{−α}. Let consider V = L(L_{β+jα} : j ∈ ℤ) the linear space generated by {L_{β+jα} : j ∈ ℤ}. It is clear that V is a finite dimensional vector space invariant for ad(v_{α}), ad(v_{−α}) and ad(h_{α}) = ad(v_{α})ad(v_{−α}) − ad(v_{−α})ad(v_{α}). The fact that the trace of ad(h_{α}) on V is 0 gives us the equation mβ(h_{α}) + kα(h_{α}) = 0 with m ≠ 0 and m, k ∈ ℤ. Hence β(h_{α}) = \frac{k}{m}α(h_{α}).

Corollary 2.1. Let L be a semisimple Lie algebra having a Cartan decomposition relative to a Cartan subalgebra H, and let α be a nonzero root. Then α(h_{α}) ≠ 0 for any 0 ≠ h_{α} ∈ [L_{α}, L_{−α}].

Proof. Let suppose that α(h_{α}) = 0. Since h_{α} ∈ L_{0} and, by Lemma 2.1 -2, β(h_{α}) = 0 for all root β, we have [h_{α}, L] = 0. Hence, h_{α} ∈ Ann(L) and then h_{α} = 0, a contradiction.

Theorem 2.1. Let L be a semisimple Lie algebra having a Cartan decomposition relative to a Cartan subalgebra H, and let α be a nonzero root. Then L_{α} is one-dimensional and the only integral multiplies kα which are roots are 0 and ±α.

Proof. By Lemma 2.1-2 we can fix 0 ≠ v_{α} ∈ L_{α} and 0 ≠ v_{−α} ∈ L_{−α} such that 0 ≠ h_{α} = [v_{α}, v_{−α}]. Let consider

L′ = 𝕂[v_{α}, L_{−α}] ⊕ 𝕂v_{α} ⊕ (⨁_{j=1} L_{−jα}).

We have that L′ is a finite dimensional Lie algebra invariant for ad(v_{α}), ad(v_{−α}), and ad(h_{α}) = [ad(v_{α}), ad(v_{−α})]. Since the trace of ad(h_{α}) on L′ is 0 and since [h_{α}, v_{α}] = α(h_{α})v_{α}, [h_{α}, v_{−jα}] = −jα(h_{α})v_{−jα}, and [h_{α}, h′] = 0 for any h′ ∈ 𝕂[v_{α}, L_{−α}], we have

α(h_{α})(1 + ∑(−j)d_{j}) = 0

where d_{j} is the dimension of L_{−jα}. By Corollary 2.1 α(h_{α}) ≠ 0, hence 1 + ∑(−j)d_{j} = 0. Thus −2α, −3α, ⋯ are not roots and d_{1} = 1. Since we can replace α by −α in the argument, we have both conclusions of the theorem.

The next corollary is now immediate

Corollary 2.2. Under the hypothesis of Theorem 2.1, if α, β and α + β are nonzero roots of L relative to H then [L_{α}, L_{β}] = L_{α+β}.

3. DIRECT LIMITS OF LIE ALGEBRAS

Let (I, ≤) be a direct set and let {L_{i}}_{i∈I} be a family of Lie algebras such that for any i, j ∈ I with i ≤ j there exists a monomorphism e_{ji} : L_{i} ⟶ L_{j} such that e_{ji} ∘ e_{ik} = e_{jk} and e_{ii} = Id for all i, j, k with k ≤ i ≤ j. Then we shall say that S = ({L_{i}}_{i∈I}, {e_{ji}}_{i≤j}) is a direct system of Lie algebras.

Given S we define a direct limit, lim_{→}S, as a couple (L, {e_{i}}_{i∈I}) where L is a Lie algebra, e_{i} : L_{i} ⟶ L is a monomorphism that satisfies

e_{i} = e_{j} ∘ e_{ji}

for any j ≥ i and (L, {e_{i}}_{i∈I}) is universal for this property in the sense that if (L′, {t_{i}}_{i∈I}) is another such couple, then there exists a unique monomorphism θ : L ⟶ L′ such that t_{i} = θ ∘ e_{i} for all i ∈ I. It is clear that if a direct limit exists, then it is unique up to isomorphism. We define the concepts of direct system and direct limit for associative algebras in a similar way.

If A is an associative algebra then A^{−} will denote the Lie algebra whose product is given by [x, y] = xy − yx.

As in [3], we can prove that any direct system of Lie algebras S has a direct limit, and state the following results:

Theorem 3.1. Let S = ({A_{i}}_{i∈I}, {e_{ji}}_{i≤j}) be a direct system of associative algebras. Then S^{−} = ({A_{i}^{−}}_{i∈I}, {e_{ji}}_{i≤j}) is a direct system of Lie algebras and lim_{→}S^{−} = (lim_{→}S)^{−}.

Theorem 3.2. Let S = ({A_{i}}_{i∈I}, {e_{ji}}_{i≤j}) be a direct system of associative algebras, with {ξ_{i}}_{i∈I} a family of involutive antiautomorphisms, ξ_{i} : A_{i} → A_{i}, such that ξ_{j} ∘ e_{ji} = e_{ji} ∘ ξ_{i} for i ≤ j. Write A = lim_{→}S, then:

1. There exists a unique involutive antiautomorphism ξ : A → A satisfying ξ ∘ e_{i} = e_{i} ∘ ξ_{i} for any i ∈ I.

2. If we consider the Lie subalgebra of A_{i}^{−}, Skw(A_{i}, ξ_{i}) then

Skw(S, ξ) := ({Skw(A_{i}, ξ_{i})}_{i∈I}, {e_{ji}|_{Skw(A_{i},ξ_{i})}}_{i≤j})

is a direct system of Lie algebras and lim_{→}(Skw(S, ξ)) = Skw(lim_{→}S, ξ).

4. THE DESCRIPTION THEOREM

We shall prove in this section that if L is an infinite dimensional simple complex Lie algebra having a Cartan decomposition relative to a Cartan subalgebra H, then L is isomorphic to lim_{→}S, S = ({L_{i}}_{i∈I}, {i_{ji}}_{i≤j}) being a direct system, with the inclusion, of simple finite dimensional Lie subalgebras of a same type A_{l}, B_{l}, C_{l} or D_{l}. Finally, the properties of the direct limits, shown in §3, will give us the description theorem.

In the remainder of this section L will denote an infinite dimensional semisimple complex Lie algebra having a Cartan decomposition relative to a Cartan subalgebra H, and Λ will be the set of nonzero roots of L relative to H.

A subset Λ_{0} of Λ is called a root system (relative to H) if it satisfies the conditions: α ∈ Λ_{0} implies −α ∈ Λ_{0}; and α, β ∈ Λ_{0}, α + β ∈ Λ implies α+β ∈ Λ_{0}. From now on, given α ∈ Λ we denote by h_{α} the unique nonzero element of L_{0} of the form h_{α} = [e_{α}, e_{−α}] with 0 ≠ e_{α} ∈ L_{α}, 0 ≠ −e_{α} ∈ L_{−α} and such that α(h_{α}) = 1, (by Corollary 2.1 there exists such h_{α} for any α ∈ Λ). If we define H_{Λ_{0}} = Sp{h_{α} : α ∈ Λ_{0}} and V_{Λ_{0}} = ⨁_{α∈Λ_{0}} L_{α}, it

is straightforward to verify that L_{Λ_{0}} = H_{Λ_{0}} ⊕ V_{Λ_{0}} is a Lie subalgebra of L, with Cartan subalgebra H_{Λ_{0}} = H ∩ L_{Λ_{0}}, whose roots relative to H_{Λ_{0}} are precisely the roots in Λ_{0}. We shall say that L_{Λ_{0}} is the Lie subalgebra associated to the root system Λ_{0}.

Our next goal is to prove that the Lie subalgebra L_{Λ_{0}} associated to a finite root system Λ_{0} in L is semisimple.

Lemma 4.1. Let L_{Λ_{0}} be the Lie subalgebra associated to a finite root system Λ_{0} in L. Write < ·, · > the Killing form on L. Then we have:

1. < h_{α}, h_{α} >≠ 0 for any α ∈ Λ_{0}.

2. < h, v_{α} >= 0 for any h ∈ H_{Λ_{0}} and v_{α} ∈ L_{α}, α ∈ Λ_{0}.

3. < v_{α}, v_{β} >= 0 for any v_{α} ∈ L_{α}, v_{β} ∈ L_{β}, α, β ∈ Λ_{0} and β ≠ −α.

4. < v_{α}, v_{−α} >≠ 0 for any 0 ≠ v_{iα} ∈ L_{iα}, i ∈ {±1} and α ∈ Λ_{0}.

Proof. 1. We have < h_{α}, h_{α} >= trz(ad(h_{α}) ∘ ad(h_{α})) = α(h_{α})^{2} + ∑_{γ∈Λ_{0}−{α}} γ(h_{α})^{2}. As in the proof of Lemma 2.1-2 we obtain γ(h_{α}) =

r_{γ}α(h_{α}) with r_{γ} ∈ ℚ, and finally we conclude from α(h_{α}) = 1 that

< h_{α}, h_{α} >= 1 + ∑_{γ∈Λ_{0}−{α}} r_{γ}^{2} ≠ 0.

2. Since < ·, · > is invariant in the sense of [7, p. 69], we have

< h, v_{α} >=< h, [h_{α}, v_{α}] >=< [h, h_{α}], v_{α} >= 0.

3. It is clear that < v_{α}, v_{β} >= trz(ad(v_{α}) ∘ ad(v_{β})) = 0.

4. Since

< h_{α}, h_{α} >=< [e_{α}, e_{−α}], h_{α} >=< e_{α}, [h_{α}, e_{−α}] >= − < e_{α}, e_{−α} >,

by applying 1. we have < e_{α}, e_{−α} >≠ 0. Hence, as dimL_{±α} = 1 we conclude < v_{α}, v_{−α} >≠ 0.

Lemma 4.2. Under the hypothesis of Lemma 4.1, if < x, L_{Λ_{0}} >= 0 for some x ∈ L_{Λ_{0}} then x ∈ H_{Λ_{0}}.

Proof. Write x = h + ∑_{α∈Λ_{0}} w_{α} ∈ L_{Λ_{0}}, with h ∈ H_{Λ_{0}} and w_{α} ∈ L_{α}.

Since < x, v_{−α} >= 0 for any v_{−α} ∈ L_{−α}, α ∈ Λ_{0}, Lemma 4.1-2,3 shows that < w_{α}, v_{−α} >= 0, and therefore w_{α} = 0 by Lemma 4.1-4.

Proposition 4.1. The Lie subalgebra L_{Λ_{0}} associated to a finite root system Λ_{0} in L is semisimple.

Proof. Let us firstly observe that [L_{Λ_{0}}, L_{Λ_{0}}] = L_{Λ_{0}} and that

Rad(L_{Λ_{0}}) ⊂ H_{Λ_{0}}.

Indeed, if x ∈ L_{Λ_{0}} then

x = ∑_{α∈Λ_{0}} λ_{α}h_{α} + ∑_{α∈Λ_{0}} v_{α} = ∑_{α∈Λ_{0}} λ_{α}[e_{α}, e_{−α}] + ∑_{α∈Λ_{0}}[h_{α}, v_{α}] ∈ [L_{Λ_{0}}, L_{Λ_{0}}]

and so [L_{Λ_{0}}, L_{Λ_{0}}] = L_{Λ_{0}}. Since the radical of a finite dimensional Lie algebra L′ is characterized as the ideal Rad(L′) = {x ∈ L′ :< x, [L′, L′] >= 0}, where < ·, · > denotes the Killing form (see [7, p.73]), the fact [L_{Λ_{0}}, L_{Λ_{0}}] = L_{Λ_{0}} and Lemma 4.2 show Rad(L_{Λ_{0}}) ⊂ H_{Λ_{0}}.

Secondly, we assert that

Rad(L_{Λ_{0}}) = Ann(L_{Λ_{0}}).

Indeed, Ann(L_{Λ_{0}}) is a solvable ideal and therefore is included in Rad(L_{Λ_{0}}). If h ∈ Rad(L_{Λ_{0}}) ⊂ H_{Λ_{0}} we have, by the character of ideal of Rad(L_{Λ_{0}}), [h, v_{α}] ∈ Rad(L_{Λ_{0}}) ⊂ H_{Λ_{0}} for any 0 ≠ v_{α} ∈ L_{α} and α ∈ Λ_{0}, therefore α(h) = 0. Hence, we have for any x ∈ L_{Λ_{0}},

[h, x] = [h, ∑_{α∈Λ_{0}} λ_{α}h_{α} + ∑_{α∈Λ_{0}} v_{α}] = [h, ∑_{α∈Λ_{0}} λ_{α}h_{α}] + ∑_{α∈Λ_{0}}[h, v_{α}] = 0

and so h ∈ Ann(L_{Λ_{0}}).

Finally, as by Levi’s theorem, ([7, p.91]),

L_{Λ_{0}} = Rad(L_{Λ_{0}}) ⊕ T_{Λ_{0}},

with T_{Λ_{0}} a semisimple subalgebra of L_{Λ_{0}}, we have

L_{Λ_{0}} = [L_{Λ_{0}}, L_{Λ_{0}}] = [Ann(L_{Λ_{0}}) ⊕ T_{Λ_{0}}, Ann(L_{Λ_{0}}) ⊕ T_{Λ_{0}}] ⊂ [T_{Λ_{0}}, T_{Λ_{0}}] ⊂ T_{Λ_{0}}

and so L_{Λ_{0}} = T_{Λ_{0}}, the proof is complete.

We shall say that a finite set of nonzero roots {α_{i}} is linearly independent if the set {h_{α_{i}}} is linearly independent. J. R. Schue introduced for any non zero root α of a semisimple L^{∗}−algebra L with inner product < ·|· > and with a Cartan decomposition relative to a Cartan subalgebra H, the elements 0 ≠ h′_{α} ∈ H such that α(h) =< h|h′_{α} > for any h ∈ H. As any complex finite dimensional semisimple Lie algebra admits an, essentially unique, structure of L^{∗}−algebra satisfying (L_{α})^{∗} = L_{−α} (see [8]). Following [8, Definition on page 73] we have h′_{α} ∈ [L_{α}, L_{α}^{∗}]. Therefore h′_{α} = kh_{α} with 0 ≠ k ∈ ℂ. Hence, [1, Lemma 1], [1, Corollary 2] and the observations in §1 give us the following two results:

Lemma 4.3. Let L be a finite dimensional semisimple Lie algebra having a Cartan decomposition relative to a Cartan subalgebra H. Write L as

L = ⨁_{j=1}^{m} L_{j} where L_{j} are simple Lie algebras. If α is a nonzero root relative

to H, then L_{α} belongs precisely to one L_{j}. If we denote by

Λ_{j} = {α : L_{α} ⊆ L_{j}},

then Sp{[e_{α}, e_{−α}] : α ∈ Λ_{j}} is a Cartan subalgebra H_{j} of L_{j} and the restrictions to H_{j} of the α ∈ Λ_{j} are precisely the roots of L_{j}.

Corollary 4.1. Under the hipotheses of Lemma 4.3, let us suppose {α_{1,}...., α_{n}} is a linearly independent set of nonzero roots of L. If there

exists a root γ of L such that γ = ∑_{i=1}^{n} c_{i}α_{i} c_{i} ≠ 0, then all α_{i} and γ are

roots of the same simple component L_{j}.

Lemma 4.4. Let L be a finite dimensional semisimple Lie algebra having a Cartan decomposition relative to a Cartan subalgebra H. If α and β are two nonzero roots such that α ≠ ±β then α and β are linearly independent.

Proof. Suppose α and β are not linearly independent, then h_{α} = ch_{β} with 0 ≠ c ∈ ℂ. Let consider L as an L^{∗}-algebra with inner product < ·|· >. By the above observation, there exist non zero elements h′_{α}, h′_{β} ∈ H such that α(h) =< h|h′_{α} > and β(h) =< h|h′_{β} > for any h ∈ H and h′_{α} = k_{α}h_{α}, h′_{β} = k_{β}h_{β} with k_{α}, k_{β} ∈ ℂ − {0}. Hence,

α(h) =< h|k_{α}h_{α} >=< h|k_{α}ch_{β} >=< h|k_{α}ck_{β}^{−1}h′_{β} >= \overline{k_{α}ck_{β}^{−1}}β(h)

for any h ∈ H. From the theory of finite dimensional split semisimple Lie algebras, this is only possible if α = ±β.

Let α and β be two nonzero roots of a semisimple Lie algebra, we shall say that α and β are connected if there exist α_{1}, ...α_{n} ∈ Λ such that

{α_{1}, α_{1} + α_{2}, α_{1} + α_{2} + α_{3}, ....., α_{1} + ... + α_{n−1} + α_{n}}

is a family of nonzero roots, α_{1} is a fixed element of {α, −α} and α_{1} + ... + α_{n−1} + α_{n} = β. We shall also say that {α_{1}, ...α_{n}} is a connection from α

to β. It is clear that α_{p} ≠ ± ∑_{i=1}^{p−1} α_{i}, p = 2, .., n. We denote by

Λ_{α} := {β ∈ Λ : α and β are connected}

Let us observe that {α} is a connection from α to itself and therefore α ∈ Λ_{α}.

Lemma 4.5. Under the hipotheses of Lemma 4.3, if furthermore α and β are two connected nonzero roots, then L_{α} and L_{β} belong to the same simple Lie algebra L_{j}.

Proof. We have α_{1}, ..., α_{n} ∈ Λ such that

{α_{1}, α_{1} + α_{2}, ....., α_{1} + ... + α_{n−1} + α_{n}}

are nonzero roots, α_{1} ∈ {α, −α} and α_{1} + ... + α_{n−1} + α_{n} = β. If we consider α_{1}, α_{2}, and α_{1}+α_{2}, by Lemma 4.4 and Corollary 4.1, we have that L_{α_{1}}, L_{α_{2}}and L_{α_{1}+α_{2}} belong to the same simple Lie algebra L_{j}. The same argument with α_{1}+α_{2}, α_{3} and α_{1}+α_{2}+α_{3} gives us that L_{α_{3}}, L_{α_{1}+α_{2}+α_{3}} ⊂ L_{j}. Following this process we finally obtain that L_{α}, L_{β} ⊂ L_{j}.

Proposition 4.2. Let L be an infinite dimensional simple Lie algebra having a Cartan decomposition relative to a Cartan subalgebra H, and let α be a nonzero root. Then the following assertion hold:

1. Λ_{α} is a root system.

2. There exists β ∈ Λ_{α} such that β ≠ ±α.

3. If γ is a nonzero root such that γ ∉ Λ_{α}, then [L_{β}, L_{γ}] = 0 and γ(h_{β}) = 0 for any β ∈ Λ_{α}.

Proof. 1. If β ∈ Λ_{α} then there exists a connection {α_{1}, ....., α_{n}} from α to β. It is easy to check that {−α_{1}, ....., −α_{n}} is a connection from α to −β and therefore −β ∈ Λ_{α}. If β, γ ∈ Λ_{α} and β + γ ∈ Λ, then there exists a connection {α_{1}, ....., α_{n}} from α to β. Hence, {α_{1}, ....., α_{n}, γ} is a connection from α to β + γ and so β + γ ∈ Λ_{α}.

2. Let us firstly observe that there exists a nonzero root γ ∈ Λ, γ ≠ ±α such that either [L_{α}, L_{γ}] ≠ 0 or [L_{−α}, L_{γ}] ≠ 0. Indeed, if we suppose [L_{α}, L_{γ}] = [L_{−α}, L_{γ}] = 0 for any γ ∈ Λ, γ ≠ ±α, by Lemma 2.1-1 we have γ(h_{α}) = 0 for any γ ∈ Λ, γ ≠ ±α. Let us consider

I := ℂh_{α} ⊕ L_{α} ⊕ L_{−α}.

By the above, it is easy to prove that [I, L] ⊂ I, therefore I is a nonzero finite dimensional ideal of an infinite dimensional simple Lie algebra L, a contradiction. Hence, there exists a nonzero root γ ≠ ±α such that either [L_{α}, L_{γ}] ≠ 0 or [L_{−α}, L_{γ}] ≠ 0. In the first case, {α, γ} is a connection from α to β := α + γ, therefore β ∈ Λ_{α} and β ≠ ±α. In the second case we argue similarly.

3. Let us suppose there exists β ∈ Λ_{α} such that [L_{β}, L_{γ}] ≠ 0. If {α_{1}, ....., α_{n}} is a connection from α to β, we have {α_{1}, ....., α_{n}, γ} is a connection from α to β + γ. Since Λ_{α} is a root system then γ ∈ Λ_{α}, a contradiction. Therefore [L_{β}, L_{γ}] = 0 for any β ∈ Λ_{α} and γ ∉ Λ_{α}. As −β ∈ Λ_{α} for any β ∈ Λ_{α}, we also have [L_{−β}, L_{γ}] = 0. Finally, by Lemma 2.1-1 we conclude γ(h_{β}) = 0 for any β ∈ Λ_{α}.

Theorem 4.1. Let L be an infinite dimensional simple Lie algebra having a Cartan decomposition L = H ⊕ ( ⨁_{α∈Λ} L_{α}) relative to a Cartan sub-

algebra H. Then for any α, β ∈ Λ there exists a connection from α to β.

Proof. Let consider the root system Λ_{α} and the Lie subalgebra

L_{Λ_{α}} = H_{Λ_{α}} ⊕ V_{Λ_{α}}

of L associated to Λ_{α}. We assert that L_{Λ_{α}} is a nonzero ideal of L. Indeed, we have by Proposition 4.2-3 that [L_{β}, L_{γ}] = 0 and that [h_{β}, L_{γ}] = 0 for any β ∈ Λ_{α} and γ ∉ Λ_{α}. Hence,

[L_{Λ_{α}}, L] = [ ∑_{β∈Λ_{α}} ℂh_{β} + ∑_{β∈Λ_{α}} L_{β}, H + ( ∑_{γ∈Λ_{α}} L_{γ}) + ( ∑_{γ∉Λ_{α}} L_{γ})] ⊂ L_{Λ_{α}}.

The simplicity of L implies L_{Λ_{α}} = L and therefore Λ_{α} = Λ.

Corollary 4.2. Under the hipotheses of Theorem 4.1,

H = Sp{h_{β} : β ∈ Λ} = L_{0}.

Proof. By the proof of Theorem 4.1, H = H_{Λ_{α}} = Sp{h_{β} : β ∈ Λ}, α being any nonzero root of Λ. As a consequence H ⊂ L_{0} and the proof is complete.

Let S be a non empty finite subset of Λ. As SpℤS ∩ Λ is a finite root system we have, by Proposition 4.1, that the finite dimensional Lie subalgebra L_{(SpℤS∩Λ)} of L associated to SpℤS ∩ Λ is semisimple. For simplicity of notation we shall denote it by L_{S} := L_{(SpℤS∩Λ)}. It is well known from the theory of semisimple finite dimensional Lie algebras that L_{S} can be written

L_{S} = ⨁_{i=1}^{n_{S}} L_{S_{i}},

with L_{S_{i}}, i = 1, ..., n_{S}, simple finite dimensional Lie algebras. Hence, we can consider the family of simple finite dimensional Lie subalgebras of L, {L_{S_{i}}}_{S∈F,i∈{1,...,n_{S}}}, where F denotes the family of all non empty finite subset of Λ.

Theorem 4.2. Let L be an infinite dimensional simple Lie algebra having a Cartan decomposition relative to a Cartan subalgebra H. Then there exists a direct system, with the inclusion, R = ({L_{j}}_{j∈J}, {i_{kj}}_{j≤k}) of simple finite dimensional Lie subalgebras of L such that:

1. L = lim_{→}R.

2. The direct system may be chosen so that all the L_{j} are of a same type A_{l}, B_{l}, C_{l} or D_{l}.

Proof. Let consider the family of simple finite dimensional Lie subalgebras of L, {L_{S_{i}}}_{S∈F,i∈{1,...,n_{S}}}, given above and let {i_{S_{i},T_{j}}} be the inclusion mappings. In order to prove that

R := ({L_{S_{i}}}_{S∈F,i∈{1,...,n_{S}}}, {i_{S_{i},T_{j}}})

is the required direct system we assert that given

L_{S_{i}}, L_{T_{j}} ∈ {L_{S_{i}}}_{S∈F,i∈{1,...,n_{S}}},

there exists

L_{Q_{i_{0}}} ∈ {L_{S_{i}}}_{S∈F,i∈{1,...,n_{S}}}

such that L_{S_{i}}, L_{T_{j}} ⊂ L_{Q_{i_{0}}} . Indeed, let us fix α ∈ S_{i}. By Theorem 4.1, for any β ∈ S_{i} ∪ T_{j} there exists a connection from α to β, which we denote by C_{α,β}. We have that Q := ⋃_{β∈S_{i}∪T_{j}} C_{α,β} is a finite set of Λ and

therefore we can consider the finite dimensional semisimple Lie subalgebra

L_{Q}. Write L_{Q} = ⨁_{i=1}^{n_{Q}} L_{Q_{i}}, L_{Q_{i}} being simple Lie subalgebras of L_{Q}. By Lemma 4.3, there exists L_{Q_{i_{0}}} such that L_{α} ⊂ L_{Q_{i_{0}}} . Finally, by Lemma 4.5, L_{S_{i}}, L_{T_{j}} ⊂ L_{Q_{i_{0}}} . Therefore, R is a direct system with the inclusion.

1. Let us denote lim_{→}R = (L′, {e_{j}}_{j}). As (L, {i_{j}}_{j}), where i_{j} denotes the inclusion mapping, satisfies the conditions of the direct limit for R, the universal property of the direct limits shows the existence of a unique monomorphism Φ : L′ → L such that Φ ∘ e_{j} = i_{j}. Since L′ = ⋃_{j} e_{j}(L_{j}),

(see §3 and [3]), we have Φ(L′) = Φ(⋃_{j} e_{j}(L_{j})) = ⋃_{j} L_{j}, and therefore Φ

is an isomorphism from L′ onto ⋃_{j} L_{j}. Finally, we assert that L = ⋃_{j} L_{j}.

Indeed, if x ∈ L, by the Cartan decomposition of L and Corollary 4.2

x = ∑_{i=1}^{n} λ_{α_{i}}h_{α_{i}} + ∑_{j=1}^{m} v_{β_{j}} with α_{i} ∈ Λ, v_{β_{j}} ∈ L_{β_{j}} and λ_{α_{i}} ∈ ℂ. Consider

T = {α_{i} : i = 1, ..., n} ∪ {β_{j} : j = 1, ..., m} ⊂ Λ and, following the above notation, T′ = ⋃_{β∈T} C_{γ,β}, γ being a fixed element of T. We have T′ is a

finite set of Λ that gives us the semisimple finite dimensional Lie algebra

L_{T} . Write L_{T} = ⨁_{i=1}^{r} L_{T_{i}}, where L_{T_{i}}, i = 1, ..., r are simple finite dimensional

Lie algebras. As R is a direct system for the inclusion then there exists a

finite dimensional simple Lie subalgebra L_{P} such that ⋃_{i=1}^{r} L_{T_{i}} ⊆ L_{P} and

therefore x ∈ L_{P} . The proof of 1, is complete.

To see 2., firstly notice that the infinite dimensional character of L let us remove the exceptional Lie algebras of R, and secondly that (i) each L_{i} is contained in one of type A_{l} or else (ii) there exists L_{i_{0}} so that L_{i} ⊃ L_{i_{0}} implies that L_{i} is of type B_{l}, C_{l} or D_{l}. In each of the two cases is possible to define a subnet satisfying assertion 1.

As consequence of Theorem 4.2 and the properties of the direct limits given in §3 we have the following theorem:

Theorem 4.3 (Main Theorem). Let L be an infinite dimensional simple complex Lie algebra having a Cartan decomposition relative to a Cartan subalgebra H. Then L is isomorphic to one of the followings:

1. A^{−}, with A a simple associative algebra.

2. Skw(A, ξ) with A as above and ξ an involutive antiautomorphism of A.

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