Seminar Sophus Lie 2 (1991) 175–183

Cartan algebras in symmetric Lie algebras

Norbert Dörr

Cartan algebras proved to be useful in the classification of semisimple Lie algebras and in the determination of invariant cones in Lie algebras. In the discussion of infinitesimally ordered symmetric spaces, cones of the form W = 𝔥⊕C with C pointed and generating in the associated symmetric Lie-algebra 𝔤 = 𝔥 ⊕ 𝔮 which satisfy e^{ad 𝔥}C = C play an important role. The idea is to use Cartan algebras also to achieve a classification for those wedges.

In this article the existence of invariant Cartan algebras in a symmetric Lie algebra 𝔤 = 𝔥 ⊕ 𝔮 over a field of characteristic zero is established. In the solvable case it is shown, that these Cartan algebras are conjugate under elementary inner automorphisms coming from a fixed element under the involution. We state a general conjugation theorem for an algebraically closed field. In the end we discuss some relations between symmetric Cartan algebras and Levi complements.

1. Some facts on Cartan algebras

We recall some notation and basic facts concerning Cartan algebras in arbitrary Lie algebras. The reader who is not familiar with Cartan algebras is referred to [1] Ch. VII, which is an excellent source of reference on Cartan algebras in general.

In that follows 𝕂 denotes an arbitrary field of characteristic zero. Let ρ: 𝔫 → gl(V) be a finite dimensional representation of a nilpotent Lie-Algebra 𝔫. The nilspace of an element x ∈ 𝔫 is defined by

V^{0}(x) =^{def} {v ∈ V | (∃n ∈ ℕ) ρ(x)^{n}v = 0}.

For λ ∈ 𝕂 we set

V^{λ}(x) =^{def} {v ∈ V | (∃n ∈ ℕ) (ρ(x) − λ·1)^{n}v = 0}.

Consider a function λ: 𝔫 → 𝕂. The weightspace of 𝔫 with weight λ is given by

V^{λ}(𝔫) =^{def} ⋂_{x∈𝔫} V^{λ(x)}(x).

The space V^{0}(𝔫) is called the nilspace of 𝔫.

In this article, we consider always a nilpotent subalgebra 𝔫 of a fixed Lie algebra 𝔤, and V is a vector subspace of 𝔤, invariant under the restriction of the adjoint representation ρ: 𝔫 → gl(V), ρ(x) = ad x|_{V} . In the case V = 𝔤 we denote by Λ the set of all λ: 𝔫 → 𝕂, λ ≠ 0 satisfying 𝔤^{λ}(𝔫) ≠ {0}. For λ, μ ∈ Λ ∪ {0} we have

[𝔤^{λ}(𝔫), 𝔤^{μ}(𝔫)] ⊆ 𝔤^{λ+μ}(𝔫).

In particular, 𝔤^{0}(𝔫) is a subalgebra of 𝔤 containing 𝔫, and 𝔤^{λ}(𝔫) is invariant under ad 𝔤^{0}(𝔫). Further, 𝔤^{0}(𝔫) equals its own normalizer. If all ad x, x ∈ 𝔫, are triagonizable, we have

𝔤 = 𝔤^{0}(𝔫) ⊕ 𝔤^{+}(𝔫), where 𝔤^{+}(𝔫) =^{def} ∑_{λ∈Λ} 𝔤^{λ}(𝔫).

There exists an x ∈ 𝔫 with 𝔤^{0}(𝔫) = 𝔤^{0}(x). If 𝕂′ is an extension of 𝕂, then we have

(𝔤 ⊗_{𝕂} 𝕂′)^{λ}(𝔫) = 𝔤^{λ}(𝔫) ⊗_{𝕂} 𝕂′.

Let 𝔤 be a finite dimensional Lie algebra. The normalizer of a subalgebra 𝔱 in 𝔤 is given by

N_{𝔤}(𝔱) =^{def} {x ∈ 𝔤 | [x, 𝔱] ⊆ 𝔱}.

Definition 1.1. A subalgebra 𝔱 of 𝔤 is called a Cartan algebra, if 𝔱 is nilpotent, and equals its own normalizer N_{𝔤}(𝔱).

If 𝔤′ is a subalgebra of 𝔤 and 𝔱 is a Cartan algebra of 𝔤 with 𝔱 ⊆ 𝔤′, then 𝔱 is also a Cartan algebra of 𝔤′. Let 𝕂′ be an extension of the field 𝕂. Then 𝔱 is a Cartan algebra of 𝔤 if and only if 𝔱 ⊗_{𝕂} 𝕂′ is a Cartan algebra of 𝔤 ⊗_{𝕂} 𝕂′.

We consider a Lie algebra 𝔤 with dim 𝔤 = n. For x ∈ 𝔤,

det(λ·1 − ad x) = ∑_{i=0}^{n} a_{i}(x)λ^{i}

with a_{i}(x) ∈ 𝕂 is the characteristic polynomial of the endomorphim ad x. Since a_{i}(x) = (−1)^{n−i} tr(⋀^{n−i} ad x), the assignment x ↦ a_{i}(x) is a polynomial map into 𝕂.

Definition 1.2. The rank rg(𝔤) of 𝔤 is defined as the minimal number l with a_{l} ≠ 0. An element x is called regular, if a_{l}(x) ≠ 0. We denote by

Reg(𝔤) =^{def} {x ∈ 𝔤 | x is regular}

the set of all regular elements.

For x ∈ 𝔤 we have rg(𝔤) ≤ dim 𝔤^{0}(x). Equality holds if and only if x ∈ Reg(𝔤). The set Reg(𝔤) is open and dense with respect to the natural topology and with respect to the Zariski topology. The interplay between Cartan algebras and regular elements is given in the following theorem.

Theorem 1.3. Let 𝔤 be a finite dimensional Lie algebra. Then the following holds:

(i) All Cartan algebras have the same dimensions rg(𝔤) and the same nilpotent length.

(ii) An element x is regular if and only if 𝔱 =^{def} 𝔤^{0}(x) is a Cartan algebra 𝔤. All Cartan algebras of 𝔤 are of this form.

(iii) If 𝔱 is a maximal nilpotent subalgebra of 𝔤 and x ∈ 𝔱 is regular, we have 𝔱 = 𝔤^{0}(x).

(iv) 𝔤 is the sum of its Cartan algebras.

Finally, we state some results concerning the conjugation of Cartan algebras in the Lie algebra 𝔤. First, we consider a solvable Lie algebra 𝔤. We denote by C^{∞}(𝔤) the intersection of the central descending series, i. e.,

C^{∞}(𝔤) =^{def} ⋂_{n∈ℕ} 𝔤^{[n]}.

Since C^{∞}(𝔤) ⊆ [𝔤, 𝔤], this is a nilpotent ideal. In particular, the automorphism e^{ad x} with x ∈ C^{∞}(𝔤) is a polynomial in ad x.

Theorem 1.4. Let 𝔤 be a solvable finite dimensional Lie algebra. Further, let 𝔱 and 𝔱′ be Cartan algebras of 𝔤. There exists an x ∈ C^{∞}(𝔤) with e^{ad x}𝔱 = 𝔱′.

We consider now an arbitrary finite dimensional Lie algebra 𝔤 over an algebraically closed field 𝕂. We have

𝔤 = 𝔤^{0}(𝔱) ⊕ ∑_{λ∈Λ} 𝔤^{λ}(𝔱).

For x ∈ 𝔤^{λ}(𝔱), λ ≠ 0 the endomorphism ad x is nilpotent, and therefore e^{ad x} is a polynomial in ad x. We set

E(𝔱) =^{def} ⟨{e^{ad x} | (∃λ ∈ Λ)x ∈ 𝔤^{λ}(𝔱)}⟩.

If φ ∈ Aut(𝔤), then obviously φE(𝔱)φ^{−1} = E(φ(𝔱)) holds.

Theorem 1.5. Let 𝔤 be a finite dimensional Lie algebra over an algebraically closed field 𝕂. Let 𝔱 and 𝔱′ be Cartan algebras of 𝔤. Then we have E(𝔱) = E(𝔱′). Defining E =^{def} E(𝔱) the following holds: there exists an γ ∈ E, such that 𝔱 = γ(𝔱′), i. e., E acts transitively on the set of all Cartan algebras of 𝔤.

To conclude this course on Cartan algebras in general we cite some properties which we need in the following sections.

Proposition 1.6. Let 𝔤 be a finite dimensional Lie algebra and 𝔤′ a subalgebra of 𝔤. Then the following conditions are equivalent.

(i) 𝔤′ ∩ Reg(𝔤) ≠ Ø and rg(𝔤′) = rg(𝔤).

(ii) 𝔤′ contains a Cartan algebra of 𝔤.

(iii) Every Cartan algebra of 𝔤′ is a Cartan algebra of 𝔤.

Corollary 1.7. Let 𝔫 be a nilpotent subalgebra of 𝔤. Then 𝔤^{0}(𝔫) satisfies the conditions (i), (ii), (iii) of Proposition 1.6.

2. Existence of symmetric Cartan algebras

It is recently proved by Spindler, that every symmetric Lie algebra 𝔤 = 𝔥 ⊕ 𝔮 possesses a Cartan Algebra 𝔱 which is symmetric, i. e., satisfies 𝔱 = 𝔱_{𝔥} ⊕ 𝔱_{𝔮} where 𝔱_{𝔥} = 𝔱 ∩ 𝔥 and 𝔱_{𝔮} = 𝔱 ∩ 𝔮. For semisimple Lie algebras this is well known. We give here another, more direct, proof in the general case, which does not use the result for semisimple Lie algebras. First, we discuss some preliminaries.

Proposition 2.1. Let 𝔤 = 𝔥 ⊕ 𝔮 be a symmetric Lie algebra and x ∈ 𝔥 or x ∈ 𝔮. Then 𝔤_{1} =^{def} 𝔤^{0}(x) is a symmetric subalgebra of 𝔤. Every Cartan subalgebra 𝔱 of 𝔤_{1} is a Cartan subalgebra of 𝔤.

We denote the descending central series of a Lie algebra 𝔤 by 𝔤^{[0]} = 𝔤,…,𝔤^{[m+1]} = [𝔤, 𝔤^{[m]}]. It is well known, that a Lie algebra 𝔤 is nilpotent of class ≤ m if and only if 𝔤 satisfies

[m] [x_{1}, [⋯, [x_{m}, x_{m+1}] ⋯]] = 0 for all x_{1}, …, x_{m+1} ∈ 𝔤.

Lemma 2.2. Let 𝔫 ⊆ 𝔤 be a nilpotent subalgebra and y_{0} ∈ N_{𝔤}(𝔫) such that 𝔫 ⊆ 𝔤^{0}(y_{0}). Then 𝔫̃ =^{def} 𝔫 ⊕ 𝕂·y_{0} is a nilpotent subalgebra.

Proof. Since y_{0} ⊆ N_{𝔤}(𝔫), induction immediately shows that (ad y_{0})𝔫^{[m]} ⊆ 𝔫^{[m]}. Since 𝔫 ⊆ 𝔤^{0}(y_{0}), there exists m_{1} ∈ ℕ such that (ad y_{0})^{m_{1}}𝔫 = {0}. Assume 𝔫 is nilpotent of class m_{2} and set m_{0} =^{def} m_{1}·m_{2}. It is enough to show, that 𝔫̃ satisfies condition [m_{0}] above. Therefore let x_{1}, …, x_{m_{0}+1} ∈ ñ and consider x = [x_{1}, [⋯ [x_{m_{0}}, x_{m_{0}+1}] ⋯]]. Since the Lie bracket is bilinear, we may assume that for i = 1, …, m_{0} + 1 we have x_{i} ∈ 𝔫 or x_{i} = y_{0}. If there are at least m_{2} elements of 𝔫 in the sequence x_{1}, …, x_{m_{0}+1}, we have x = 0 since (ad y_{0})𝔫^{[m]} ⊆ 𝔫^{[m]} and 𝔫 is of class m_{2}. If this is not the case, then by definition of m_{0}, the element y_{0} occurs at least m_{1} times in a row. But then x = 0 by definition of m_{1}. This proves the assertion.

Theorem 2.3. (Existence of symmetric Cartan algebras) Let 𝔤 = 𝔥 ⊕ 𝔮 be a symmetric Lie algebra. Then there exists a symmetric Cartan algebra 𝔱.

Proof. We prove the assertion by induction on the dimension of 𝔤. If dim 𝔤 = 1 nothing is to show. Let 𝔤 = 𝔥 ⊕ 𝔮 be an arbitrary symmetric Lie algebra. If there exists an element x ∈ 𝔥 or x ∈ 𝔮 such that 𝔤^{0}(x) ≠ 𝔤, then 𝔤_{1} =^{def} 𝔤^{0}(x) is a symmetric subalgebra of lower dimension, and the induction hypothesis shows the existence of an symmetric Cartan algebra 𝔱. By Proposition 2.1, 𝔱 is an symmetric Cartan algebra of 𝔤.

Otherwise, for every x ∈ 𝔥 and y ∈ 𝔮 the relation 𝔤^{0}(x) = 𝔤^{0}(y) = 𝔤 holds. In particular, 𝔥 is nilpotent by Engel’s theorem. Let 𝔱 be any maximal nilpotent subalgebra with 𝔥 ⊆ 𝔱. We claim that 𝔱 equals its own normalizer, hence 𝔱 is a Cartan algebra which obviously is symmetric. As 𝔥 ⊆ 𝔱 ⊆ N_{𝔤}(𝔱), the normalizer N_{𝔤}(𝔱) is symmetric. Thus we have to show that N_{𝔤}(𝔱) ∩ 𝔮 ⊆ 𝔱. Suppose y_{0} ∈ N_{𝔤}(𝔱)∩𝔮. By assumption, 𝔤^{0}(y_{0}) = 𝔤, hence 𝔱 ⊆ 𝔤^{0}(y_{0}). Thus the assumptions of Lemma 2.2 are satisfied, showing that 𝔱 ⊕ 𝕂·y_{0} is nilpotent. Therefore y_{0} ∈ 𝔱 by maximality of 𝔱. This proves the theorem.

Theorem 1.3 tells us, that a Lie algebra in general is full of Cartan algebras. Every Cartan algebra is of the form 𝔤^{0}(x) with a regular element x.

Definition 2.4. Let 𝔤 = 𝔥 ⊕ 𝔮 be a symmetric Lie algebra. We set

Sym(𝔤) =^{def} {x ∈ Reg(𝔤) | 𝔤^{0}(x) is a symmetric Cartan algebra}.

We call x ∈ Reg(𝔤) symmetric regular if x ∈ Sym(𝔤).

As usual we denote by τ: 𝔤 → 𝔤 the involutive Lie algebra automorphism associated to the decomposition 𝔤 = 𝔥 ⊕ 𝔮 of the symmetric Lie algebra. Using the relation τ ∘ ad x ∘ τ = ad τ(x) and the definition of the nilspace of an element x ∈ 𝔤, we can easily give another characterization of the set of symmetric regular elements.

Lemma 2.5. Let 𝔤 = 𝔥 ⊕ 𝔮 be a symmetric Lie algebra. Then

Sym(𝔤) = {x ∈ Reg(𝔤) | 𝔤^{0}(x) = 𝔤^{0}(τ(x))}
= {x ∈ Reg(𝔤) | τ(x) ∈ 𝔤^{0}(x)}.

The set Sym(𝔤) is very thin in the topological point of view. We denote by Σ(𝔤) the set of all subalgebras of 𝔤. We endow Σ(𝔤) with the topology of the Grassmann manifold of all vector subspaces of 𝔤. On Reg(𝔤) we have the natural topology and the Zariski topology. One can show, that the map Reg(𝔤) → Σ(𝔤): x ↦ 𝔤^{0}(x) is continuous with respect to the natural topology on Reg(𝔤).

Proposition 2.6. The set of symmetric regular elements Sym(𝔤) is closed in Reg(𝔤) with respect to the natural topology and the Zariski topology. In particular, Sym(𝔤) is nowhere dense or equals Reg(𝔤).

We denote by 𝔠 the span of all symmetric Cartan algebras and define

𝔞 =^{def} 𝔥 + 𝔠.

Lemma 2.7. If φ ∈ Aut(𝔤) satisfies φ ∘ τ = τ ∘ φ, then φ(Sym(𝔤)) = Sym(𝔤). In particular, the set of symmetric regular elements Sym(𝔤) is invariant under e^{ad 𝔥}. Further, we have:

(i) 𝔠 = 𝔠_{𝔥} ⊕ 𝔠_{𝔮} is a symmetric 𝔥-module.

(ii) 𝔞 = 𝔥 ⊕ 𝔠_{q} is a symmetric subalgebra with N_{𝔤}(𝔞) = 𝔞.

3. Conjugation in Lie algebras over an algebraically closed field

In this section 𝕂 is supposed to be an algebraically closed field. Let 𝔤 = 𝔥⊕𝔮 be a symmetric Lie algebra over 𝕂, and 𝔱 a symmetric Cartan algebra of 𝔤. Hence the action of the nilpotent subalgebra 𝔱 gives a weight space decomposition

𝔤 = 𝔱 ⊕ ∑_{λ∈Λ} 𝔤^{λ}(𝔱),

where Λ ⊆ 𝔱̂ as usual is the set of weights. One easily verifies that, if λ ∈ Λ, then λ ∘ τ ∈ Λ. More precisely,

τ(𝔤^{λ}(𝔱)) = 𝔤^{λ∘τ}(𝔱).

Lemma 3.1. Let 𝔤 = 𝔥 ⊕ 𝔮 be a symmetric Lie algebra, 𝔱 a symmetric Cartan algebra and λ ∈ Λ. Then the following conditions are equivalent.

(1) λ|_{𝔱∩𝔥} ≡ 0.

(2) λ ∘ τ = −λ.

This motivates the following definition.

Definition 3.2. Let 𝔱 be a symmetric Cartan algebra in the symmetric Lie algebra 𝔤 = 𝔥 ⊕ 𝔮. Then we set

Λ_{0} =^{def} {λ ∈ Λ | λ ∘ τ = −λ} and Λ_{+} =^{def} Λ \ Λ_{0}.

Let λ ∈ Λ_{+} and x ∈ 𝔤^{λ}(𝔱). Obviously, ad(x + τ(x)) is a nilpotent linear operator. In particular, e^{ad(x+τ(x))} is a polynomial in ad(x + τ(x)).

Definition 3.3. If 𝔱 is a symmetric Cartan algebra, we write

𝔤 = 𝔤_{0} ⊕ 𝔤_{+}

where 𝔤_{0} =^{def} 𝔱 ⊕ ∑_{λ∈Λ_{0}} 𝔤^{λ}(𝔱) and 𝔤_{+} =^{def} ∑_{λ∈Λ_{+}} 𝔤^{λ}(𝔱).

We remark that both 𝔤_{0} and 𝔤_{+} depend on the Cartan algebra 𝔱.

Proposition 3.4. Let 𝔤 = 𝔥⊕𝔮 be a symmetric Lie algebra and 𝔱 a symmetric Cartan algebra. Then the following holds:

(i) 𝔤_{0} is a symmetric subalgebra of 𝔤 with N_{𝔤}(𝔤_{0}) = 𝔤_{0}.

(ii) 𝔤_{+} is a 𝔤_{0}-module, i. e., we have [𝔤_{0}, 𝔤_{+}] ⊆ 𝔤_{+}.

The following proposition describes the structure of the symmetric subalgebra 𝔤_{0}.

Proposition 3.5. Let 𝔱 be a symmetric Cartan algebra of 𝔤 = 𝔥 ⊕ 𝔮, and 𝔤 = 𝔤_{0} ⊕ 𝔤_{+} according Definition 3.3. Denote by 𝔤 = 𝔤^{0}(𝔱_{𝔥}) ⊕ ∑_{μ∈M} 𝔤^{μ}(𝔱_{𝔥}) the weight decomposition of 𝔤 given by the action of the nilpotent subalgebra 𝔱_{𝔥} = 𝔱∩𝔥. Then the following holds:

(i) 𝔤_{0} = 𝔤^{0}(𝔱_{𝔥}), and 𝔤^{0}(𝔱_{𝔥}) = 𝔥^{0}(𝔱_{𝔥}) ⊕ 𝔮^{0}(𝔱_{𝔥}).

(ii) Set Λ_{μ} =^{def} {λ ∈ Λ_{+} | λ|_{𝔱_{𝔥}} ≡ μ}. Then 𝔤_{+} = ∑_{μ∈M} 𝔤^{μ}(𝔱_{𝔥}), where 𝔤^{μ}(𝔱_{𝔥}) = ∑_{λ∈Λ_{μ}} 𝔤^{λ}(𝔱).

(iii) There exists x ∈ 𝔱_{𝔥} such that 𝔤_{0} = 𝔤^{0}(x). Further, 𝔥^{0}(𝔱_{𝔥}) = 𝔥^{0}(x) and 𝔮^{0}(𝔱_{𝔥}) = 𝔮^{0}(x).

(iv)

𝔥^{0}(𝔱_{𝔥}) = 𝔱_{𝔥} ⊕ ∑_{λ∈Λ_{0}}(𝔤^{λ}(𝔱) ⊕ 𝔤^{−λ}(𝔱)) ∩ 𝔥

and

𝔮^{0}(𝔱_{𝔥}) = 𝔱_{𝔮} ⊕ ∑_{λ∈Λ_{0}}(𝔤^{λ}(𝔱) ⊕ 𝔤^{−λ}(𝔱)) ∩ 𝔮.

(v) For μ ∈ M, we have 𝔤^{μ}(𝔱_{𝔥}) = 𝔥^{μ}(𝔱_{𝔥}) ⊕ 𝔮^{μ}(𝔱_{𝔥}) with

𝔥^{μ}(𝔱_{𝔥}) = ∑_{λ∈Λ_{μ}}(𝔤^{λ}(𝔱) ⊕ 𝔤^{λ∘τ}(𝔱)) ∩ 𝔥

and

𝔮^{μ}(𝔱_{𝔥}) = ∑_{λ∈Λ_{μ}}(𝔤^{λ}(𝔱) ⊕ 𝔤^{λ∘τ}(𝔱)) ∩ 𝔮.

(vi) 𝔥 = 𝔥^{0}(𝔱_{𝔥}) ⊕ ∑_{μ∈M} 𝔥^{μ}(𝔱_{𝔥}) and 𝔮 = 𝔮^{0}(𝔱_{𝔥}) ⊕ ∑_{μ∈M} 𝔮^{μ}(𝔱_{𝔥}).

Remark 3.6. We see in Lemma 3.5 that 𝔤_{0} = 𝔤^{0}(𝔱_{𝔥}) is also defined over an arbitrary field 𝕂 of characteristic zero. Also, 𝔤_{+} is the uniquely determined 𝔱_{𝔥}- module complement of 𝔤_{0} in 𝔤.

Suppose 𝔱 and 𝔱′ are different symmetric Cartan algebras of 𝔤. What is the relation between 𝔤_{0} = 𝔤^{0}(𝔱_{𝔥}) and 𝔤′_{0} = 𝔤^{0}(𝔱′_{𝔥})?

Lemma 3.7. Let 𝔱 be a symmetric Cartan algebra and φ ∈ Aut(𝔤) such that φ ∘ τ = τ ∘ φ. Then the following holds:

(i) 𝔱′ =^{def} φ(𝔱) is a symmetric Cartan algebra of 𝔤.

(ii) φ induces a bijection Λ → Λ′: λ ↦ λ ∘ φ^{−1}, and φ(𝔤^{λ}(𝔱)) = 𝔤^{λ∘φ^{−1}}(𝔱′).

(iii) λ ∈ Λ_{0} if and only if λ ∘ φ^{−1} ∈ Λ′_{0}. Further, φ induces isomorphisms φ|_{𝔤_{0}}: 𝔤_{0} → 𝔤′_{0} and φ|_{𝔤_{+}}: 𝔤_{+} → 𝔤′_{+}.

The group of of automorphisms

E_{τ} =^{def} ⟨{e^{ad h} | h ∈ 𝔥, ad h nilpotent }⟩

is uniquely determined by the involution τ. Also the group of automorphisms

E_{0} =^{def} ⟨{e^{ad x} | (∃λ ∈ Λ_{0}) x ∈ 𝔤^{λ}(𝔱)}⟩

depends only on the associated subalgebra 𝔤_{0}. With these preparations we can state our main conjugation theorem.

Theorem 3.8. If 𝔱 and 𝔱′ are symmetric Cartan algebras of a symmetric Lie algebra 𝔤 = 𝔥 ⊕ 𝔮, then there exists an element γ ∈ E_{τ} such that 𝔤′_{0} = γ(𝔤_{0}) and 𝔤′_{+} = γ(𝔤_{+}). Further, there exists an α ∈ E_{0}, such that with φ =^{def} γ ∘ α the relations 𝔱′ = φ(𝔱) and φ(𝔤^{λ}(𝔱)) = 𝔤^{λ∘φ^{−1}}(𝔱′) hold.

Corollary 3.9. Let 𝔤 = 𝔥⊕𝔮 be a symmetric Lie algebra, 𝔱 a symmetric Cartan algebra and 𝔤_{0} = 𝔤^{0}(𝔱_{𝔥}). Then the following holds:

(i) If 𝔱 = 𝔤_{0}, then there exists an x ∈ Reg(𝔤) ∩ 𝔥 such that 𝔱 = 𝔤^{0}(x). All symmetric Cartan algebras are conjugate under E_{τ}.

(ii) If 𝔤 = 𝔤_{0}, then there exists an x ∈ Reg(𝔤) ∩ 𝔮 such that 𝔱 = 𝔤^{0}(x).

4. Conjugation in solvable symmetric Lie algebras

In this section let 𝔤 = 𝔥 ⊕ 𝔮 be a solvable symmetric Lie algebra. Suppose 𝔫 is a symmetric ideal. Then the quotient space 𝔤/𝔫 becomes a symmetric Lie algebra in a canonical way. We have 𝔤/𝔫 = (𝔥 + 𝔫/𝔫) ⊕ (𝔮 + 𝔫/𝔫). We denote by C^{∞}(𝔤) the intersection of the descending central series. That is, C^{∞}(𝔤) =^{def} ⋂_{n∈ℕ} 𝔤^{[n]}. Since all 𝔤^{[n]} are symmetric, and C^{∞}(𝔤) ⊆ [𝔤, 𝔤], the set C^{∞}(𝔤) is a nilpotent symmetric ideal. The set {e^{ad x} | x ∈ C^{∞}(𝔤)} is a subgroup of Aut(𝔤), such that every e^{ad x} is a polynomial in ad x.

Theorem 4.1. Let 𝔤 = 𝔥 ⊕ 𝔮 be a solvable symmetric Lie algebra und 𝔱, 𝔱′ symmetric Cartan algebras of 𝔤. Then there exists an x ∈ C^{∞}(𝔤) ∩ 𝔥 such that e^{ad x}𝔱 = 𝔱′.

The proof of this theorem preceeds in an analogous way as the proof of Theorem 1.4. It remains to ensure that the element x, under which the Cartan algebras are conjugated, is contained in 𝔥.

5. Symmetric Cartan algebras and Levi complements

Let 𝔯 be the radical of a symmetric Lie algebra 𝔤 = 𝔥 ⊕ 𝔮. Obviously, 𝔯 is symmetric. In [3], Appendix 9, it is shown, that there exists a symmetric Levi subalgebra 𝔰. A Levi decomposition 𝔤 = 𝔯 ⊕ 𝔰 and a Cartan algebra 𝔱 are called adapted, if 𝔱 = (𝔱∩𝔯)⊕(𝔱∩𝔰). If this is the case, we write 𝔱_{𝔯} =^{def} 𝔱∩𝔯 and 𝔱_{𝔰} =^{def} 𝔱∩𝔰. In [4] the following result is proved. There is also a proof of (i) in [2], which is much shorter and more algebraic than the original one.

Theorem 5.1. Let 𝔤 = 𝔥 ⊕ 𝔮 be a symmetric Lie algebra with radical 𝔯. Then the following holds:

(i) Suppose 𝔰 is a symmetric Levi complement and 𝔱_{𝔰} a symmetric Cartan algebra of 𝔰. Then there exists a symmetric subalgebra 𝔱_{𝔯} of 𝔯 such that 𝔱 =^{def} 𝔱_{𝔯} ⊕ 𝔱_{𝔰} is a symmetric Cartan algebra of 𝔤 adapted to the decomposition 𝔤 = 𝔯 ⊕ 𝔰.

(ii) Conversely, suppose 𝔱 is a symmetric Cartan algebra of 𝔤. Then there exists a symmetric Levi complement 𝔰, a symmetric Cartan algebra 𝔱_{𝔰} of 𝔰, and a symmetric subalgebra 𝔱_{𝔯} of 𝔯 such that 𝔱 = 𝔱_{𝔯} ⊕ 𝔱_{𝔰}.

References

[1] Bourbaki, N., “Groupes et algèbres de Lie”, Chapitres 7 et 8, Hermann, Paris 1975.

[2] Dörr, N., Cartan-Algebren und Ol’shanskiĭ-Keile in symmetrischen Lie-Algebren, Dissertation, TH Darmstadt, 1991.

[3] Kobayashi, S., K. Nomizu, “Foundations of Differential Geometry II”, Wiley, New York, 1969.

[4]  Spindler, K. H., A note on Cartan algebras, 1991, submitted.

Fachbereich Mathematik Technische Hochschule Darmstadt SchloŒgartenstraŒe 7 D-6100 Darmstadt Germany