Cartan subalgebras of 𝔤𝔩_{∞} 1

Cartan subalgebras of 𝔤𝔩_{∞}

Karl-Hermann Neeb * and Ivan Penkov **

Abstract. Let V be a vector space over a field 𝕂 of characteristic zero and V_{∗} be a space of linear functionals on V which separate the points of V . We consider V ⊗V_{∗} as a Lie algebra of finite rank operators on V , and set 𝔤𝔩(V,V_{∗}):=V ⊗V_{∗}. We define a Cartan subalgebra of 𝔤𝔩(V,V_{∗}) as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of 𝔤𝔩(V,V_{∗}) under the assumption that 𝕂 is algebraically closed. A subalgebra of 𝔤𝔩(V,V_{∗}) is a Cartan subalgebra if and only if it equals ⊕_{j}(V_{j}⊗(V_{j})_{∗})⊕(V^{0}⊗V_{∗}^{0}) for some one-dimensional subspaces V_{j}⊆V and (V_{j})_{∗}⊆V_{∗} with (V_{i})_{∗}(V_{j})=δ_{ij}𝕂 and such that the spaces V_{∗}^{0}=∩_{j}(V_{j})^{⊥}⊆V_{∗} and V^{0}=∩_{j}((V_{j})_{∗})^{⊥}⊆V satisfy V_{∗}^{0}(V^{0})={0}. We then discuss explicit constructions of subspaces V_{j} and (V_{j})_{∗} as above. Our second main result claims that a Cartan subalgebra of 𝔤𝔩(V,V_{∗}) can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra 𝔥 which coincides with the maximal locally nilpotent 𝔥-submodule of 𝔤𝔩(V,V_{∗}), and such that the adjoint representation of 𝔥 is locally finite.

AMS Subject Classification 2000: Primary 17B65, Secondary 17B20.

Introduction

It is an interesting question which class of subalgebras of an infinite-dimensional Lie algebra, over a field 𝕂 of characteristic zero, play a role similar to Cartan subalgebras of a finite-dimensional Lie algebra. Despite the fact that infinite-dimensional Lie algebras have been studied extensively in the last 30 years, there is no definitive answer to this question. The best understood cases are those of Kac–Moody algebras and extended affine Lie algebras (see [BP95], [PK83], [AABGP97] and the references therein), whose specific is that their Cartan subalgebras are finite-dimensional. The simplest example of an infinite-dimensional Lie algebra whose Cartan subalgebras are no longer finite-dimensional, is the Lie algebra 𝔤𝔩_{∞} of infinite matrices with finitely many non-zero entries in 𝕂, and in the literature there is no systematic investigation of all Cartan subalgebras of 𝔤𝔩_{∞}. The purpose of the present paper is to fill in this gap for 𝔤𝔩_{∞} and to the larger class of Lie algebras 𝔤𝔩(V, V_{∗}) defined below.

The following three definitions of a Cartan subalgebra 𝔥 of a finite-dimensional Lie algebra 𝔤 are equivalent: (C1) 𝔥 is a locally nilpotent self-normalizing subalgebra; (C2) 𝔥 coincides with the maximal locally nilpotent 𝔥-submodule of 𝔤, i.e., 𝔥 = 𝔤^{0}(𝔥), where

𝔤^{0}(𝔥) = {x ∈ 𝔤: (∃n ∈ ℕ) (ad 𝔥)^{n}(x) = {0}};

(C3) 𝔥 is a locally nilpotent subalgebra which coincides with the set of all elements x ∈ 𝔤 for which ad x commutes with the abelian subalgebra (ad 𝔥)_{s} consisting of all semisimple parts (ad h)_{s} in the Jordan decomposition ad h = (ad h)_{s} + (ad h)_{n} of ad h for elements h ∈ 𝔥 ((ad h)_{n} stands for the nilpotent part).

* Work supported in part by the University of California at Riverside and by a grant of the DFG.

** Work supported in part by MSRI, by an NSF grant, and by the Max Planck Institut für Mathematik in Bonn.

Since 𝔤 is assumed finite-dimensional, “locally nilpotent” in the conditions (C1)-(C3) is of course equivalent to “nilpotent”, but we have stated (C1)-(C3) in a form suitable also for the more general class of locally finite Lie algebras we consider in this paper. The fact that (C1) is equivalent to (C2) is well-known ([Bou90, Ch. VII]). The equivalence of (C2) and (C3) follows from the equalities

𝔤^{0}(𝔥) = ⋂_{h∈𝔥} 𝔤^{0}(ad h) = ⋂_{h∈𝔥} ker(ad h)_{s} = 𝔷_{𝔤}((ad 𝔥)_{s}).

If 𝔤 is reductive over an algebraically closed field, every Cartan subalgebra 𝔥 is maximal toral, i.e., for every 0 ≠ h ∈ 𝔥, ad h is diagonalizable, and 𝔥 is maximal with this property. This is the key to one of the most important properties of a Cartan subalgebra 𝔥 of a reductive Lie algebra 𝔤: that (after a possible field extension) 𝔥 yields a root decomposition of 𝔤.

Let V be a fixed (arbitrary) vector space over 𝕂 and V^{∗} be its dual space. In what follows we set 𝔤 = 𝔤𝔩(V, V_{∗}) := V ⊗ V_{∗}, considered as a Lie algebra of finite rank operators on V , where V_{∗} ⊆ V^{∗} is a subspace separating the points of V . If J is a set, we write 𝕂^{(J)} for the free vector space over J and (e_{j})_{j∈J} for the canonical basis elements. The standard pairing 𝕂^{(J)} × 𝕂^{(J)} → 𝕂 induces an injection 𝕂^{(J)} ↪ (𝕂^{(J)})^{∗} ≅ 𝕂^{J}, and for V = V_{∗} = 𝕂^{(J)} , 𝔤 = 𝔤𝔩(J, 𝕂) := 𝔤𝔩(V, V_{∗}) is the Lie algebra of J × J -matrices with finitely many non-zero entries. We set also 𝔤𝔩_{∞} := 𝔤𝔩(ℕ, 𝕂).

All Cartan subalgebras yielding a root decomposition of 𝔤𝔩_{∞}, i.e., the so called splitting Cartan subalgebras, are well understood, see [NS01], [St01] and [PS03]. It is also known that there are maximal toral subalgebras of 𝔤𝔩_{∞} which do not yield a root decomposition, and therefore also no generalized root decomposition ([PS03]). In particular, even if 𝕂 is algebraically closed, none of the conditions (C1)-(C3) implies the existence of a generalized root decomposition related to 𝔥. In this paper we put the condition (C3) in the spotlight, as it relates 𝔥 in a most transparent way with the abelian subalgebra 𝔥_{s}, consisting of the semisimple parts of all h ∈ 𝔥, and in this way carries the most resemblance with the finite-dimensional case. More precisely, we define a Cartan subalgebra of 𝔤𝔩(V, V_{∗}) as a subalgebra satisfying (C3).

Our main result is a description, in terms of linear algebra, of all Cartan subalgebras of 𝔤 = 𝔤𝔩(V, V_{∗}) for which 𝔥_{s} is toral. (The latter condition is automatic if 𝕂 algebraically closed.) In particular we prove that all Cartan subalgebras are abelian. As a corollary we obtain that there are at most three types of Cartan subalgebras for which 𝔥_{s} is toral: the ones for which the inclusion 𝔥_{s} ⊆ 𝔥 is proper, the toral ones, i.e. those for which 𝔥 = 𝔥_{s}, and finally, the splitting ones for which 𝔥 = 𝔥_{s} and 𝔤 has a root decomposition with respect to 𝔥. We consider examples of pairs V , V_{∗} for which not all types of Cartan subalgebras occur, and we show that all three types do occur for 𝔤𝔩(J, 𝕂).

As each Cartan subalgebra of 𝔤𝔩(V, V_{∗}) is abelian, its adjoint module is trivial and in particular locally finite. It is not clear if this latter property holds for any subalgebra of 𝔤𝔩(V, V_{∗}) (in particular of 𝔤𝔩_{∞}) which satisfies (C1) or (C2). However, our second main result claims that if one strengthens (C1) and (C2) by the very natural additional requirement that the adjoint module of 𝔥 be locally finite, then the so obtained new conditions (C1’) and (C2’) are equivalent to (C3) for subalgebras of 𝔤 = 𝔤𝔩(V, V_{∗}).

Acknowledgments. We thank Helmut Strade for sharing an inspiring idea about describing the toral Cartan subalgebras of 𝔤𝔩_{∞}. We thank also Ivan Dimitrov and Gregg Zuckerman for the permission to present in the Appendix a proposition they recently proved.

I. Preliminaries and notation

All vector spaces (and Lie algebras) are defined over 𝕂 and \overline{𝕂} stands for the algebraic closure of 𝕂. The superscript ∗ denotes dual space. The vector spaces V and V_{∗} are fixed as above, and the sign ⊥ always refers to the pairing V × V_{∗} → 𝕂. If 𝔨 is a Lie algebra, U(𝔨) stands for the enveloping algebra of 𝔨. In this paper ℕ := {1, 2, 3, ...}.

We call a Lie algebra 𝔨 locally finite (resp., locally nilpotent) if every finite subset of 𝔨 is contained in a finite-dimensional (resp., nilpotent) subalgebra. We call an 𝔨-module M locally finite if each element m ∈ M is contained in a finite-dimensional submodule, and we call M locally nilpotent if for any m ∈ M there exists an i ∈ ℕ with 𝔨^{i} · m = {0}. Furthermore, we say that an 𝔨-module is a generalized weight 𝔨-module if M = ⨁_{λ∈𝔨^{∗}} M^{λ}(𝔨), where

M^{λ}(𝔨) := {m ∈ M: (∃i ∈ ℕ)(∀x ∈ 𝔨) (x − λ(x)1)^{i} · m = 0}.

We define M to be a weight module, if, in addition,

M^{λ}(𝔨) = M_{λ}(𝔨) := {m ∈ M: (∀x ∈ 𝔨) x · m = λ(x)m}

for each λ ∈ 𝔨^{∗}. The support in 𝔨^{∗} of a module M is the set

supp M := {α ∈ 𝔨^{∗}: M^{α}(𝔨) ≠ {0}}.

If 𝔨 is any Lie algebra and 𝔥 ⊆ 𝔨 is a locally nilpotent Lie subalgebra, we say that 𝔨 admits a generalized 𝔥-root decomposition (resp., an 𝔥-root decomposition) if 𝔨 is a generalized 𝔥-weight module (resp., an 𝔥-weight module) with respect to the adjoint action, and, in addition, 𝔥 coincides with the maximal locally nilpotent 𝔥-submodule of 𝔨, i.e., 𝔥 = 𝔨^{0}(𝔥).

In this paper we denote by 𝔤 a fixed Lie algebra of the form 𝔤𝔩(V, V_{∗}), where V_{∗} ⊆ V^{∗} is a subspace separating the points of V . Typical examples of this situation are as follows.

(a) V_{∗} = V^{∗}. Then 𝔤𝔩(V, V_{∗}) is the Lie algebra of finite rank operators on V .

(b) V = 𝕂^{(J)} = V_{∗} for a set J. Then 𝔤𝔩(V, V_{∗}) ≅ 𝔤𝔩(J, 𝕂).

(c) V is a locally convex real or complex vector space and V_{∗} is the space of continuous linear functionals. As a consequence of the Hahn-Banach Extension Theorem, V_{∗} separates the points of V . Here 𝔤𝔩(V, V_{∗}) is the Lie algebra of continuous finite rank operators on V . In general, the structure of the Lie algebra 𝔤𝔩(V, V_{∗}) depends essentially on the choice of

the subspace V_{∗}. As the following proposition shows, this is not the case when V and V_{∗} are of countable dimension.

Proposition I.1. If V and V_{∗} are of countable dimension, then 𝔤𝔩(V, V_{∗}) ≅ 𝔤𝔩_{∞}. Proof. We have to find a basis (f_{n})_{n∈ℕ} of V for which the dual basis (f_{n}^{∗})_{n∈ℕ} ⊆ V^{∗} spans V_{∗}.

Fix a basis (e_{n})_{n∈ℕ} of V and a basis (φ_{n})_{n∈ℕ} of V_{∗}. We first change the enumeration of the basis (φ_{n})_{n∈ℕ} by a permution σ: ℕ → ℕ according to the following rule. Put V_{n} := span{e_{1}, …, e_{n}} for n ∈ ℕ, and let σ_{1} be the minimal number j with φ_{j}(e_{1}) ≠ 0. Inductively we proceed as follows. If σ_{1}, …, σ_{k} are chosen such that the restrictions of φ_{σ_{1}}, …, φ_{σ_{k}} to V_{k} are linearly independent, then we choose σ_{k+1} as the minimal element in ℕ \ {σ_{1}, …, σ_{k}} for which the restriction of φ_{σ_{k+1}} to V_{k+1} is linearly independent from the restrictions of φ_{σ_{1}}, …, φ_{σ_{k}}. As the sequence (φ_{n})_{n∈ℕ} separates the points of V , the above procedure never stops and defines an injection σ: ℕ → ℕ. To see that σ is surjective, hence a permutation, we argue by contradiction. Assume that σ is not surjective and pick the minimal element m ∈ ℕ \ σ(ℕ). Suppose that {1, …, m − 1} ⊆ {σ_{1}, …, σ_{k}}. Then there exist λ_{1}, …, λ_{k} such that the linear functional

φ′_{m} := φ_{m} − ∑_{j=1}^{k} λ_{j}φ_{σ_{j}}

vanishes on e_{1}, …, e_{k}. From the linear independence of the sequence (φ_{n})_{n∈ℕ} we infer that φ′_{m} is non-zero, so that there exists a minimal N ∈ ℕ with φ_{m}(e_{N}) ≠ 0. Then the restrictions of φ_{σ_{1}}, …, φ_{σ_{N−1}}, φ′_{m} to V_{N} are linearly independent, hence the restrictions of φ_{σ_{1}}, …, φ_{σ_{N−1}}, φ_{m} to V_{N} are linearly independent. Thus σ_{N} = m, in contradiction with the choice of m. This proves that σ is a permutation, and hence that the functionals φ_{σ_{k}} form a basis of V_{∗}.

Let (V_{∗})_{n} := span{φ_{σ_{1}}, …, φ_{σ_{n}}}. Then (V_{∗})_{n} |_{V_{n}} = V_{n}^{∗}, and we can choose inductively a basis (f_{n})_{n∈ℕ} for which V_{n} = span{f_{1}, …, f_{n}} for n ∈ ℕ, and φ_{σ_{j}}(f_{i}) = δ_{ij} for i ≥ j. In the next step we alterate the basis (φ_{σ_{n}})_{n∈ℕ} of V_{∗} to a basis (ν_{n})_{n∈ℕ} with (V_{∗})_{n} = span{ν_{1}, …, ν_{n}} and ν_{j}(f_{i}) = δ_{ij} for all i, j ∈ ℕ, i.e., (ν_{n})_{n∈ℕ} is the dual basis to (f_{n})_{n∈ℕ}. This proves that the pair V, V_{∗} is equivalent to the pair 𝕂^{(ℕ)}, 𝕂^{(ℕ)} with the standard pairing.

The next proposition shows that the requirement dim V = dim V_{∗} does not always lead to a pairing equivalent to the standard pairing 𝕂^{(J)} × 𝕂^{(J)} → 𝕂 for some J.

Proposition I.2. Let 𝕂 = ℝ, ℂ, and V be an infinite-dimensional Hilbert space and V_{∗} be the space of continuous linear functionals on V . Then there is no (vector space) basis of V such that the dual basis belongs to V_{∗}.

Proof. We argue by contradiction and assume that (f_{j})_{j∈J} is a (vector space) basis of V for which the dual basis belongs to V_{∗}. Let (e_{s})_{s∈S} be an orthonormal Hilbert basis of V . Each e_{s} is a finite linear combination e_{s} = ∑_{j∈J} a_{js}f_{j}. Let S_{0} ⊆ S be an infinite countable subset. Then the set J_{0} := {j ∈ J: (∃s ∈ S_{0})a_{js} ≠ 0} is also countable. Furthermore,

H := ⋂_{j∈J\J_{0}} ker f_{j}^{∗}

is a closed subspace of V with (f_{j})_{j∈J_{0}} as a (vector space) basis. On the other hand, Baire’s Category Theorem implies that H is not the union of an ascending chain of finite-dimensional subspaces, hence not of countable dimension. Contradiction.

Any element x ∈ 𝔤 = 𝔤𝔩(V, V_{∗}) is a finite rank operator on V , hence has a Jordan decomposition x = x_{s} + x_{n} into a semisimple part x_{s} and a nilpotent part x_{n}. As 𝔤 is locally finite, the operator ad x is locally finite for any x ∈ 𝔤, and has a Jordan decomposition ad x = (ad x)_{s} + (ad x)_{n}. As ad x_{s} is semisimple and ad x_{n} is nilpotent, both Jordan decompositions are compatible, i.e., (ad x)_{s} = ad x_{s}, (ad x)_{n} = ad x_{n}.

We call a subalgebra 𝔱 of a Lie algebra 𝔨 toral if for every element x ∈ 𝔱 the operator ad x : 𝔨 → 𝔨 is diagonalizable. In particular, every non-zero element of a toral subalgebra of 𝔤𝔩(V, V_{∗}) is semisimple (and if 𝕂 is algebraically closed, a subalgebra is toral if and only if all its nonzero elements are diagonalizable).

Lemma I.3. Every toral subalgebra of a Lie algebra is abelian.

Proof. Let x, y ∈ 𝔱. Since ad x |_{𝔱} is diagonalizable, we can write y as y = ∑_{λ} y_{λ} with [x, y_{λ}] = λy_{λ} for λ ∈ 𝕂. Then, for any λ, (ad y_{λ})^{2}(x) = 0, and as ad y_{λ} is also diagonalizable, [y_{λ}, x] = (ad y_{λ})(x) = 0. Therefore [y, x] = ∑_{λ}[y_{λ}, x] = 0.

For any subalgebra 𝔞 ⊆ 𝔤, we denote by 𝔷_{𝔤}(𝔞) the centralizer of 𝔞 in 𝔤, by 𝔷(𝔞) - the center of 𝔞, and by 𝔫_{𝔤}(𝔞) - the normalizer of 𝔞 in 𝔤.

Lemma I.4. Let 𝔥 ⊆ 𝔤 = 𝔤𝔩(V, V_{∗}) be a locally nilpotent subalgebra, and 𝔥_{s} = {h_{s}: h ∈ 𝔥} be the set of semisimple Jordan components of elements of 𝔥. Then the following assertions hold:

(1) 𝔥_{s} is an abelian Lie algebra;

(2) 𝔥 ⊆ 𝔷_{𝔤}(𝔥_{s});

(3) 𝔫_{𝔤}(𝔥) ⊆ 𝔷_{𝔤}(𝔥_{s});

(4) 𝔥 := 𝔷_{𝔤}(𝔥_{s}) is a self-normalizing subalgebra of 𝔤.

Proof. (2) For each finite-dimensional nilpotent Lie algebra 𝔨 the set (ad 𝔨)_{s} commutes with ad 𝔨, which implies [𝔥_{s}, 𝔥] ⊆ 𝔷(𝔥). Hence (ad 𝔥_{s})^{2}(𝔥) = {0}, and the semisimplicity of the elements of 𝔥_{s} leads to [𝔥_{s}, 𝔥] = {0}.

(1) For x, y ∈ h assertion (2) implies that y_{s} commutes with x, and hence also with x_{s}. Therefore x_{s} + y_{s} is semisimple and (2) shows that it commutes with x_{n} + y_{n}. From the finite-dimensional case we derive that x_{n} +y_{n} is nilpotent, thus x+y = (x_{s} +y_{s})+(x_{n} +y_{n}) is the additive Jordan decomposition of x + y. This implies that 𝔥_{s} is a subspace, hence, in view of (2), an abelian Lie algebra.

(3) In view of (2), for each x ∈ 𝔫_{𝔤}(𝔥) and each h ∈ 𝔥 we have the relation [h, x] ∈ 𝔥. Since the semisimple semisimple part (ad h)_{s} = ad h_{s} of ad h can be obtained by applying a polynomial without constant term to ad h, we also obtain (ad h_{s})(x) ∈ 𝔥, so (2) leads to (ad h_{s})^{2}(x) = 0. As ad h_{s} is semisimple, we obtain [h_{s}, x] = 0. This proves (3).

(4) follows directly from (3).

II. Maximal toral subalgebras

Lemma II.1. An element x ∈ 𝔤 is ad-diagonalizable if and only if x is diagonalizable as an operator on V .

Proof. Decompose V as a direct sum U ⊕W, where x·W = {0} and U is a finite-dimensional x-invariant subspace. Then V_{∗} = W^{⊥} ⊕ U^{⊥}, W^{⊥} ≅ U^{∗}, and x ∈ 𝔤𝔩(U, W^{⊥}) ≅ U ⊗ U^{∗}.

If x is diagonalizable as an operator on V , then ad x is diagonalizable with eigenvalues λ_{i} − λ_{j}, where λ_{i} are the eigenvalues of x.

Assume now that ad x is diagonalizable and observe that this implies that ad x |_{𝔤𝔩(U,W^{⊥})} is diagonlizable. This implies that x is semisimple. Let λ_{1}, …, λ_{n} denote the eigenvalues of x in \overline{𝕂}. Then λ_{i} − λ_{j} are the eigenvalues of ad x, and λ_{i} − λ_{j} ∈ 𝕂. We may therefore write λ_{i} = λ + μ_{i} with μ_{i} ∈ 𝕂. As the set of all λ_{i} is invariant under the Galois group Aut_{𝕂}(\overline{𝕂}), the affine space generated by all λ_{i} contains a fixed point, i.e., an element of 𝕂. On the other hand, this affine space is contained in λ + 𝕂, which gives λ ∈ 𝕂. Therefore λ_{1}, …, λ_{n} ∈ 𝕂, i.e., x is diagonalizable on U, and therefore on V .

In this section we consider a fixed toral subalgebra 𝔱 ⊆ 𝔤 = 𝔤𝔩(V, V_{∗}). We write V′ (resp., V′_{∗}, 𝔤′) for the maximal locally finite 𝔱-submodule of V (resp., V_{∗}, 𝔤). Since each element x ∈ 𝔱 is diagonalizable (Lemma II.1), the action of 𝔱 on the locally finite modules V′ and V′_{∗} is simultaneously diagonalizable, i.e., V′ and V′_{∗} are weight 𝔱-modules. Let

V′ = ⨁_{α∈supp V} V^{α} and V′_{∗} = ⨁_{β∈supp V_{∗}} V_{∗}^{β}

be the corresponding weight decompositions.

Lemma II.2.

(1) 𝔱 · V ⊆ V′ and 𝔱 · V_{∗} ⊆ V′_{∗}.

(2) 𝔤′ = V′ ⊗ V′_{∗}.

(3) 𝔤′ = V′ ⊗ V′_{∗} is an associative subalgebra of 𝔤 = V ⊗ V_{∗} and a weight 𝔱-module with respect to the adjoint action. w

Proof. (1) For x ∈ 𝔱, x · V is a finite-dimensional subspace of V which is y -invariant for every y ∈ 𝔱 as 𝔱 is abelian. Hence x · V ⊂ V′. Similarly x · V_{∗} ⊂ V′_{∗}.

(2) is a direct consequence of Proposition A in the Appendix.

(3) V′ ⊗ V′_{∗} is obviously an associative subalgebra of V ⊗ V_{∗}. Furthermore, V′ ⊗ V′_{∗} is the tensor product of the weight 𝔱-modules V′ and V′_{∗}, and is thus itself a weight 𝔱-module.

In view of Lemma II.2(3), the weight decompositions of V′ and V′_{∗} yield the root decomposition

𝔤′ = ⨁_{α,β} V^{α} ⊗ V_{∗}^{β} = 𝔷_{𝔤}(𝔱) ⊕ (⨁_{γ≠0} 𝔤^{γ}), where 𝔤^{γ} = ⨁_{α+β=γ} V^{α} ⊗ V_{∗}^{β}.

Furthermore, 𝔷_{𝔤}(𝔱) ⊆ 𝔤′ implies

𝔱 ⊆ 𝔷_{𝔤}(𝔱) = 𝔤^{0} = ⨁_{α∈(supp V)∩(− supp V_{∗})} V^{α} ⊗ V_{∗}^{−α}.

In the sequel we are mainly interested in the centralizers of maximal toral subalgebras. We start by a description of maximal toral subalgebras in terms of their action on V and V_{∗}.

Proposition II.3. The subalgebra 𝔱 is maximal toral if and only if the following conditions are satisfied: (M1) (supp V) \ {0} = −(supp V_{∗}) \ {0} and dim V^{α} = dim V_{∗}^{−α} = 1 for any α ∈ supp V \ {0}; (M2) V_{∗}^{0}(V^{0}) = {0} and (V^{0} ⊗ V_{∗}^{0}) ∩ 𝔱 = {0}; (M3) 𝔱 = ⨁_{0≠α∈supp V} V^{α} ⊗ V_{∗}^{−α};

(M4) V_{∗}^{0} = ⋂_{0≠α∈supp V} (V^{α})^{⊥} and V^{0} = ⋂_{0≠β∈supp V_{∗}}(V_{∗}^{β})^{⊥}. If these conditions are satisfied, then

𝔷_{𝔤}(𝔱) = 𝔱 ⊕ (V^{0} ⊗ V_{∗}^{0}).

Proof. Assume first that 𝔱 is maximal toral. (M1) From 𝔱 ⊆ ⨁_{α∈(supp V)∩(− supp V_{∗})} V^{α} ⊗ V_{∗}^{−α}, V_{∗}^{β}(V^{α}) = {0} for β ≠ −α, and 𝔱 · V^{α} ≠ {0} for α ≠ 0, it follows that V_{∗}^{−α}(V^{α}) ≠ {0}. Pick v ∈ V^{α} and φ ∈ V_{∗}^{−α} with φ(v) = 1. Then v ⊗ φ ∈ 𝔷_{𝔤}(𝔱) is diagonalizable, hence contained in 𝔱 by maximality. Therefore

{0} = [v ⊗ φ, V^{α} ⊗ V_{∗}^{−α}].

This implies in particular

{0} = [v ⊗ φ, (ker φ ∩ V^{α}) ⊗ V_{∗}^{−α}] = (ker φ ∩ V^{α}) ⊗ φ.

Thus ker φ ∩ V^{α} = {0}, which yields V^{α} = 𝕂v for 0 ≠ α ∈ (supp V) ∩ (− supp V_{∗}). We likewise see that V_{∗}^{−α} = 𝕂φ. In particular, V^{α} ⊗ V_{∗}^{−α} ⊆ 𝔱. As 𝔱 · V′ = ⨁_{0≠α∈supp V} V^{α} and 𝔱 · V′_{∗} = ⨁_{0≠β∈supp V_{∗}} V_{∗}^{β}, we further see that supp V_{∗} \ {0} = − supp V \ {0}. (M2) Suppose that there exists φ ∈ V_{∗}^{0} and v ∈ V^{0} with φ(v) = 1. As above, we see that v ⊗ φ ∈ 𝔱, contradicting (v ⊗ φ) · v = v and v ∈ V^{0} . Therefore V_{∗}^{0}(V^{0}) = {0}, which in turn implies that each element in V^{0} ⊗ V_{∗}^{0} is nilpotent. Hence 𝔱 ∩ (V^{0} ⊗ V_{∗}^{0}) = {0}. (M3) Since 𝔱 contains all the spaces V^{α}⊗V_{∗}^{−α} for α ≠ 0 and is contained in ⨁_{α∈supp V} V^{α}⊗V_{∗}^{−α}, we obtain

𝔱 = (𝔱 ∩ (V^{0} ⊗ V_{∗}^{0})) ⊕ ( ⨁_{0≠α∈supp V} V^{α} ⊗ V_{∗}^{−α}).

Now (M3) follows from (M2). (M4) follows from the equality V^{0} = {v ∈ V: 𝔱 · v = {0}} as, in view of (M3), the space {v ∈ V: 𝔱 · v = {0}} coincides with the common annihilator of the spaces V_{∗}^{−α}, α ≠ 0. A similar argument applies to V_{∗}^{0} .

Conversely, assume that (M1)-(M4) are satisfied. Then dim V^{α} ⊗ V_{∗}^{−α} = 1 for 0 ≠ α ∈ supp V , and V_{∗}^{β}(V^{α}) ≠ {0} for β ≠ −α imply that 𝔱 is abelian and that each element of 𝔱 is diagonalizable. Therefore 𝔱 is a toral subalgebra of 𝔤 = V ⊗ V_{∗} (Lemma II.1). The centralizer of 𝔱 in 𝔤 is contained in 𝔤′ and coincides with 𝔱 ⊕ (V^{0} ⊗ V_{∗}^{0}). Now (M2) implies that each element in V^{0} ⊗ V_{∗}^{0} is nilpotent, so 𝔱 is maximal toral.

Finally,

𝔷_{𝔤}(𝔱) = 𝔤^{0}(𝔱) = ⨁_{α+β=0} V^{α} ⊗ V_{∗}^{β} = 𝔱 ⊕ (V^{0} ⊗ V_{∗}^{0}).

Corollary II.4. If 𝔱 is a maximal toral subalgebra, then (supp V) \ {0} ⊂ 𝔱^{∗} is a linearly independent set.

Proof. The statement follows from the equality 𝔱 = ⨁_{0≠α∈supp V} V^{α} ⊗ V_{∗}^{−α} and from the fact that α vanishes on ⨁_{δ≠α∈supp V} V^{δ} ⊗ V_{∗}^{−δ}.

The next proposition shows that for a maximal toral subalgebra 𝔱 the spaces V^{α} for α ≠ 0 determine the space V^{0} (resp., V_{∗}^{β} for β ≠ 0 determine V_{∗}^{0} ).

Proposition II.5. Let 𝔱 ⊆ 𝔤 be a maximal toral subalgebra. Then

V^{0} = ⋂_{0≠α∈supp V} ( ⋂_{α≠δ∈supp V} (V^{δ})^{⊥})^{⊥} and V_{∗}^{0} = ⋂_{0≠β∈supp V_{∗}}( ⋂_{β≠η∈supp V_{∗}}(V_{∗}^{η})^{⊥})^{⊥}.

Proof. By Proposition II.3, V_{∗}^{0} = ⋂_{0≠α∈supp V} (V^{α})^{⊥}. Fix 0 ≠ α ∈ supp V and pick f_{α} ∈ V^{α} and f_{α}^{∗} ∈ V_{∗}^{−α} with f_{α}^{∗}(f_{α}) = 1. Consider an element φ ∈ ⋂_{α≠δ∈supp V} (V^{δ})^{⊥}. Then φ − φ(f_{α})f_{α}^{∗} ∈ V_{∗}^{0} leads to φ ∈ V_{∗}^{0} + V_{∗}^{−α}, and therefore to

V_{∗}^{0} + V_{∗}^{−α} = ⋂_{α≠δ∈supp V} (V^{δ})^{⊥}.

As V_{∗}^{0}(V^{0}) = {0}, we have

V^{0} = ⋂_{0≠α∈supp V} (V_{∗}^{−α})^{⊥} = ⋂_{0≠α∈supp V} (V_{∗}^{0} + V_{∗}^{−α})^{⊥} = ⋂_{0≠α∈supp V} ⋂_{α≠δ∈supp V} (V^{δ})^{⊥}.

The second equality is established in a similar way.

III. The structure of Cartan subalgebras

Definition III.1. (a) We define a Cartan subalgebra 𝔥 ⊆ 𝔤 = 𝔤𝔩(V, V_{∗}) as a locally nilpotent subalgebra 𝔥 with 𝔥 = 𝔷_{𝔤}(𝔥_{s}).

(b) A toral Cartan subalgebra of 𝔤 is a toral subalgebra 𝔱 ⊆ 𝔤 with 𝔷_{𝔤}(𝔱) = 𝔱.

(c) A generalized splitting Cartan subalgebra of 𝔤 is a subalgebra 𝔥 ⊆ 𝔤 for which 𝔤 has a generalized root decomposition 𝔤 = 𝔥 ⊕ (⨁_{α∈Δ} 𝔤^{α}), where Δ := (supp 𝔤) \ {0}. The Cartan subalgebra 𝔥 is splitting if, in addition, 𝔤 is a weight 𝔥-module.

As all toral subalgebras are abelian by Lemma I.3, toral Cartan subalgebras are in particular Cartan subalgebras. For the same reason, toral Cartan subalgebras are maximal abelian, hence in particular maximal toral subalgebras of 𝔤, and are therefore covered by Proposition II.3. Moreover, if 𝔥 is a generalized splitting Cartan subalgebra, then the generalized root spaces are common eigenspaces of (ad 𝔥)_{s} = ad 𝔥_{s} corresponding to non-zero eigenfunctionals. This immediately implies that 𝔷_{𝔤}(𝔥_{s}) = 𝔤^{0}(𝔥) = 𝔤_{0}(𝔥_{s}) = 𝔥. Therefore 𝔥 is a Cartan subalgebra in the sense of Definition III.1(a).

Lemma III.2. For a maximal toral subalgebra 𝔱 ⊆ 𝔤 the following are equivalent:

(1) 𝔱 is a toral Cartan subalgebra.

(2) V^{0} = {0} or V_{∗}^{0} = {0}.

Proof. This follows from the equality 𝔷_{𝔤}(𝔱) = 𝔱 ⊕ (V^{0} ⊗ V_{∗}^{0}) (Proposition II.3).

Proposition III.3. For any maximal toral subalgebra 𝔱 ⊆ 𝔤, 𝔥 := 𝔷_{𝔤}(𝔱) is an abelian selfnormalizing subalgebra of 𝔤 with 𝔥 = 𝔤^{0}(𝔥) = 𝔷_{𝔤}(𝔥_{s}), and in particular a Cartan subalgebra.

Proof. By Proposition II.3,

𝔥 = 𝔱 ⊕ (V^{0} ⊗ V_{∗}^{0}).

Furthermore, the equality V_{∗}^{0}(V^{0}) = {0} implies that V^{0} ⊗ V_{∗}^{0} is an abelian Lie algebra such that (V^{0} ⊗ V_{∗}^{0})^{2} = {0}, and thus 𝔥 is an abelian subalgebra of 𝔤 with 𝔥_{s} = 𝔱 and 𝔥_{n} = V^{0} ⊗ V_{∗}^{0} . Hence 𝔥 = 𝔷_{𝔤}(𝔥_{s}), and Lemma I.4 implies that 𝔥 is self-normalizing. Finally, 𝔥 ⊆ 𝔤^{0}(𝔥) ⊆ 𝔤^{0}(𝔱) = 𝔷_{𝔤}(𝔱) = 𝔥 shows that 𝔥 = 𝔤^{0}(𝔥).

The following theorem is our first main result. It implies that if 𝕂 is algebraically closed, all Cartan subalgebras of 𝔤 are centralizers of maximal toral subalgebras, and hence are as in Proposition III.3.

Theorem III.4. (Structure Theorem for Cartan Subalgebras) Let 𝔥 ⊆ 𝔤 = 𝔤𝔩(V, V_{∗}) be a Cartan subalgebra for which the abelian subalgebra 𝔥_{s} ⊆ 𝔤 is toral. (The latter is automatic when 𝕂 = \overline{𝕂}.) Then

(1) 𝔥_{s} is a maximal toral subalgebra of 𝔤 with 𝔥 = 𝔷_{𝔤}(𝔥_{s});

(2) 𝔥 = 𝔥_{s} ⊕ (V^{0} ⊗ V_{∗}^{0}) = (⨁_{0≠α∈supp V} V^{α} ⊗ V_{∗}^{−α}) ⊕ (V^{0} ⊗ V_{∗}^{0}), where V′ = ⨁_{α∈supp V} V^{α} and V_{∗} = ⨁_{β∈supp V_{∗}} V_{∗}^{β} are the 𝔥_{s} -module weight decompositions of V′ and V′_{∗};

(3) 𝔥 is abelian;

(4) if 𝔥 is a generalized splitting Cartan subalgebra, then 𝔥 is splitting.

Proof. (1) Let 𝔱 ⊇ 𝔥_{s} be a toral subalgebra. Then 𝔱 is abelian (Lemma I.3) and therefore 𝔱 ⊆ 𝔷_{𝔤}(𝔥_{s}) = 𝔥, i.e., 𝔱 ⊆ 𝔥_{s}.

(2) follows from (1) and Proposition II.3.

(3) is a direct consequence of (1), (2), and the equality V_{∗}^{0}(V^{0}) = {0} (Proposition II.3(M2)).

(4) If 𝔤 has a generalized 𝔥-root decomposition, then 𝔤 is a locally finite 𝔥_{s} -module. Therefore

𝔤 = ⨁_{α,β} V^{α} ⊗ V_{∗}^{β} = V′ ⊗ V′_{∗}

by Lemma II.2, and V′ = V and V′_{∗} = V_{∗}. Furthermore, V_{∗}^{0}(V^{0}) = {0} and V_{∗}^{0}(V^{α}) = {0} for α ≠ 0, i.e., V_{∗}^{0}(V) = {0}. Hence V_{∗}^{0} = {0} (and similarly V^{0} = {0}). This implies that 𝔥 = 𝔥_{s}, i.e., that 𝔥 is splitting.

In [PS03] a statement similar to Theorem III.4(4) is established. Namely the main result of [PS03] claims that, if 𝕂 = \overline{𝕂} any subalgebra 𝔥 which yields a generalized root decomposition of 𝔤 is a splitting Cartan subalgebra.

Corollary III.5. Any Cartan subalgebra 𝔥 ⊆ 𝔤 is abelian.

Proof. If 𝕂 = \overline{𝕂} the statement is proved Theorem III.4. Let

\overline{𝔤} := 𝔤 ⊗_{𝕂} \overline{𝕂} = 𝔤𝔩(V ⊗_{𝕂} \overline{𝕂}, V_{∗} ⊗_{𝕂} \overline{𝕂}).

Then \overline{𝔥} := 𝔥⊗_{𝕂} \overline{𝕂} is a locally nilpotent subalgebra of \overline{𝔤} and \overline{𝔥}_{s} = 𝔥_{s} ⊗_{𝕂} \overline{𝕂}, which directly implies that

𝔷\overline{𝔤}(\overline{𝔥}_{s}) = 𝔷\overline{𝔤}(𝔥_{s}) = 𝔷_{𝔤}(𝔥_{s}) ⊗_{𝕂} \overline{𝕂} = \overline{𝔥}.

Therefore \overline{𝔥} is a Cartan subalgebra of \overline{𝔤}. Hence \overline{𝔥} is abelian by Theorem III.4, and consequently 𝔥 is abelian.

If 𝕂 is algebraically closed, Theorem III.4 enables us to give a description of all Cartan subalgebras of 𝔤𝔩(V, V_{∗}) in terms of pure linear algebra.

We define a dual system of one dimensional subspaces to be a family (V_{j})_{j∈J} of onedimensional subspaces of V , together with a family of one-dimensional subspaces ((V_{j})_{∗})_{j∈J} of V_{∗} such that (V_{i})_{∗}(V_{j}) = δ_{ij}𝕂.

Lemma III.6. Let (V_{j})_{j∈J}, ((V_{j})_{∗})_{j∈J} be a dual system of one-dimensional subspaces. Then it is maximal if and only if the spaces V_{∗}^{0} := ∩_{j}(V_{j})^{⊥} ⊆ V_{∗} and V^{0} := ∩_{j}((V_{j})_{∗})^{⊥} ⊆ V satisfy V_{∗}^{0}(V^{0}) = {0}.

Proof. If V_{∗}^{0}(V^{0}) ≠ {0}, there is an element e ∈ V^{0} and an element e_{∗} ∈ V_{∗}^{0} with e_{∗}(e) = 1, therefore the dual system (V_{j})_{j∈J}, ((V_{j})_{∗})_{j∈J} can be extended by the pair of one-dimensional spaces 𝕂e, 𝕂e_{∗}. Thus the maximality of the family (V_{j})_{j∈J} implies V_{∗}^{0}(V^{0}) = {0}. Conversely, it is clear that V_{∗}^{0}(V^{0}) ≠ {0} if the dual system is not maximal.

The existence of maximal dual families of one-dimensional subspaces follows easily from Zorn’s Lemma. Proposition II.3, Theorem III.4, and Lemma III.6 imply immediately the following proposition.

Proposition III.7. Let (V_{j})_{j∈J}, ((V_{j})_{∗})_{j∈J} be a dual system of one-dimensional subspaces satisfying the equivalent conditions of Lemma III.6. Then 𝔱 := ⨁_{j∈J} V_{j} ⊗(V_{j})_{∗} ⊆ 𝔤 is a maximal toral subalgebra and 𝔥 := 𝔷_{𝔤}(𝔱) is a Cartan subalgebra of 𝔤𝔩(V, V_{∗}). If 𝕂 = \overline{𝕂}, every Cartan subalgebra of 𝔤 is obtained by this construction.

Theorem III.4 and Proposition III.7 imply that, there are the following (mutually exclusive) alternatives for a Cartan subalgebra 𝔥 ⊆ 𝔤 with toral 𝔥_{s}: (I) 𝔥 ≠ 𝔥_{s}; (II) 𝔥 = 𝔥_{s} is toral but not splitting; (III) 𝔥 = 𝔥_{s} is splitting.

Clearly not all cases will always occur, as for instance case (III) implies that the dimensions of V and V_{∗} coincide, while for an infinite-dimensional V they a priori need not coincide. Moreover, Proposition I.2 shows that, when dim V is uncountable, equality of the dimensions of V and V_{∗} is not sufficient for the occurence of case (III). The next three propositions describe precisely which cases among (I)-(III) occur in the following situations: when V = V_{∗} = 𝕂^{(J)} for an infinite set J, when V is a Hilbert space (here 𝕂 = ℝ, ℂ) and V_{∗} is the space of continuous linear functionals on V , and when V is an arbitrary infinite-dimensional vector space and V_{∗} = V^{∗}.

Proposition III.8. Let J be an infinite set. Then in 𝔤 = 𝔤𝔩(J, 𝕂) all three types (I)–(III) of Cartan subalgebras, and, moreover, all dimensions of the spaces V^{0} ⊆ V and V_{∗}^{0} ⊆ V_{∗} do occur.

Proof. Here V = 𝕂^{(J)} = V_{∗}. Write J as a disjoint union J = J_{0} ⊔ J_{1} and assume that η: J_{0} → J_{1} is a surjection such that the inverse image of every element in J_{1} is infinite. Fix a decomposition into two disjoint subsets J_{1} = J_{1}^{+} ⊔ J_{1}^{−} and put J_{0}^{±} := η^{−1}(J_{1}^{±}). If (e_{j})_{j∈J} denotes the canonical basis of V , set

V_{j} := \{ \begin{matrix}𝕂(e_{j} + e_{η(j)}), & j ∈ J_{0}^{+} \\ 𝕂e_{j}, & j ∈ J_{0}^{−} \end{matrix}

and

(V_{j})_{∗} := \{ \begin{matrix}𝕂e_{j}, & j ∈ J_{0}^{+} \\ 𝕂(e_{j} + e_{η(j)}), & j ∈ J_{0}^{−}. \end{matrix}

The families (V_{j})_{j∈J_{0}}, and ((V_{∗})_{j})_{j∈J_{0}}, satisfy (V_{i})_{∗}(V_{j}) = δ_{ij}𝕂 for i, j ∈ J_{0} and thus form a dual system of one-dimensional subspaces. Furthermore, if α ∈ V_{∗} vanishes on all V_{j}, then α(e_{j}) = −α(e_{η(j)}) holds for each j ∈ J_{0}^{+} . For i := η(j) we then have α(e_{i}) = α(e_{j}) for infinitely many indices j with η(j) = i. This implies that α(e_{i}) = 0 for i ∈ J_{1}^{+} , and likewise α(e_{j}) = 0 for j ∈ J_{0}^{+} . We also have α(e_{j}) = 0 for j ∈ J_{0}^{−}, and therefore

V_{∗}^{0} = ∩_{j}(V_{j})^{⊥} = span{e_{j}: j ∈ J_{1}^{−}} ≅ 𝕂^{J_{1}^{−}}.

In a similar way we obtain

V^{0} = ∩_{j}((V_{j})_{∗})^{⊥} = span{e_{j}: j ∈ J_{1}^{+}} ≅ 𝕂^{J_{1}^{+}}.

In particular, V_{∗}^{0}(V^{0}) = 0, i.e. the dual system (V_{j})_{j∈J_{0}} , ((V_{∗})_{j})_{j∈J_{0}} is maximal. Consequently, for any infinite countable set J the spaces V^{0} and V_{∗}^{0} can have arbitrary prescribed dimensions less or equal |J|.

Proposition III.9. Let 𝕂 = ℝ, ℂ, and V be an infinite-dimensional Hilbert space and V_{∗} be the space of continuous linear functionals on V . Then any Cartan subalgebra 𝔥 ⊆ 𝔤 = 𝔤𝔩(V, V_{∗}) with toral 𝔥_{s} has type (I) or (II), and both cases are possible.

Proof. Proposition I.2 implies that case (III) does not occur. To construct a Cartan subalgebra of type (II), fix an orthonormal Hilbert basis (e_{j})_{j∈J} of V . Set V_{j} := 𝕂e_{j}, and (V_{j})_{∗} = 𝕂e_{j}^{∗}, where v^{∗}(x) := ⟨x, v⟩ is the linear functional corresponding to v ∈ V . Then (V_{j})_{j∈J}, ((V_{j})_{∗})_{j∈J} is a dual system of one-dimensional subspaces with V^{0} = {0} = V_{∗}^{0} , and it defines a Cartan subalgebra of 𝔤𝔩(V, V_{∗}) of type (II).

A Cartan subalgebra of type (I) can be constructed as follows. Here we assume that J = ℕ and set f_{n} := e_{1}^{∗}+e_{n+1}^{∗}−2e_{n+2}^{∗} for n ∈ ℕ. Let V_{n} := 𝕂e_{n+1} and (V_{n})_{∗} := 𝕂f_{n}. Then V_{∗}^{0} = 𝕂e_{1}^{∗} and

V^{0} = ⋂_{n∈ℕ} ker f_{n} = 𝕂 ∑_{n≥2} 2^{−n}e_{n} ⊆ (V_{∗}^{0})^{⊥}.

Hence the maximal dual system of one dimensional spaces (V_{n})_{n∈ℕ}, ((V_{n})_{∗})_{n∈ℕ} defines a Cartan subalgebra of 𝔤𝔩(V, V_{∗}) of type (I).

Proposition III.10. Let V be an infinite-dimensional vector space and V_{∗} = V^{∗}. Then any Cartan subalgebra 𝔥 ⊆ 𝔤 = 𝔤𝔩(V, V_{∗}) with toral 𝔥_{s} has type (II).

Proof. The occurence of type (III) implies equality of the dimensions of V and V_{∗}, which is not the case when V_{∗} = V^{∗}. It remains to show that a Cartan subalgebra cannot have type (I). Assume to the contrary that, for some J, (V_{j})_{j∈J}, ((V_{j})_{∗})_{j∈J} form a maximal dual system of one-dimensional spaces for which V^{0} ≠ {0}, V_{∗}^{0} ≠ {0}. Fix 0 ≠ e ∈ V^{0} . Then, since V^{∗} = V_{∗}, there is a linear functional e_{∗} ∈ V_{∗}^{0} with e_{∗}(e) = 0. Consequently V_{∗}^{0}(V^{0}) ≠ {0}, which is a contradiction.

We complete this section by addressing the problem of conjugacy for maximal toral subalgebras and thus also for Cartan subalgebras.

Proposition III.11. All splitting Cartan subalgebras of 𝔤 = 𝔤𝔩(V, V_{∗}) are conjugate under the group GL(V, V_{∗}) := {g ∈ GL(V): g^{∗} · V_{∗} = V_{∗}}. In general GL(V, V_{∗})-conjugate maximal toral subalgebras have equal respective dimensions of the subspaces V^{0} and V_{∗}^{0} , but equality of those dimensions is not sufficient for GL(V, V_{∗})-cojugacy.

Proof. If 𝔥 is a splitting Cartan subalgebra, Proposition II.3 implies that there is a basis (v_{α})_{α∈A} of V and a dual basis (v_{α}^{∗})_{α∈A} in V_{∗} such that

𝔥 = ⨁_{α∈A}(𝕂v_{α} ⊗ 𝕂v_{α}^{∗}).

In other words, 𝔥 is the set of all elements of 𝔤 which are represented by diagonal matrices with respect to the basis (v_{α})_{α∈A}. This implies immediately the conjugacy of all splitting Cartan subalgebras of V ⊗ V_{∗} under GL(V, V_{∗}) (cf. [NS01] for the case 𝔤𝔩(J, 𝕂)).

It is clear that, if two maximal toral subalgebras are GL(V, V_{∗})-conjugate, their respective dimensions of the spaces V^{0} and V_{∗}^{0} coincide. The following example shows that this is not sufficient for GL(V, V_{∗})-conjugacy.

Set V = V_{∗} := 𝕂^{(ℕ)} . Fix an injective map η: ℕ → ℕ with η(n) > n for each n ∈ ℕ, and let S: V → V, e_{n} ↦ e_{η(n)} be the corresponding shift operator (where (e_{n})_{n∈ℕ} is the standard basis of 𝕂^{(ℕ)} ). Then the endomorphism A := 1 − S ∈ End(V) is injective because S has no eigenvectors in V , and is obviously not surjective as e_{n} ∉ A(V) for any n. Furthermore, the matrix A^{⊤} := 1 − S^{⊤}, considered as an operator on V_{∗}, is locally unipotent with inverse given by ∑_{n∈ℕ_{0}}(S^{⊤})^{n}, and consequently A^{∗} |_{V_{∗}} is an automorphism of V_{∗}.

As A is injective, the one-dimensional subspaces V_{n} := A(𝕂e_{n}) satisfy the conditions of Proposition III.7 with (V_{n})_{∗} := A^{∗} |_{V^{∗}}(𝕂e_{n}) and

V^{0} = ⋂_{n∈ℕ}(V_{n}^{∗})^{⊥} = V_{∗}^{⊥} = {0} and V_{∗}^{0} = ⋂ V_{n}^{⊥} = ker A^{∗} |_{V_{∗}} = {0}.

Since A is not surjective, V′ = span{V_{n} : n ∈ ℕ} ≠ V . Therefore the toral subalgebra 𝔥 is not splitting and is not conjugate to any splitting Cartan subalgebra.

IV. Alternative characterizations of Cartan subalgebras

In this and in the section we consider 𝔤 = 𝔤𝔩(V, V_{∗}) for an arbitrary subspace V_{∗} ⊆ V^{∗} separating the points of V .

The following theorem is our second main result.

Theorem IV.1. A subalgebra 𝔥 ⊆ 𝔤 is a Cartan subalgebra if and only if it satisfies one of the following two equivalent conditions: (C1’) 𝔥 is a locally nilpotent self-normalizing subalgebra whose adjoint module is locally finite; (C2’) 𝔥 coincides with the maximal locally nilpotent 𝔥-submodule of 𝔤 and the adjoint module of

𝔥 is locally finite.

First we observe that conditions (C1’), (C2’) and (C3) are satisfied for a subalgebra 𝔥 ⊆ 𝔤 if and only if they are satisfied for the subalgebra \overline{𝔥} ⊆ \overline{𝔤}, where \overline{𝔨} = 𝔨 ⊗_{𝕂} \overline{𝕂}. Therefore, without restricting generality, we will assume throughout this section that 𝕂 is algebraically closed.

The fact that (C1’) and (C2’) are satisfied for a Cartan subalgebra follows immediately from Proposition III.3 and Theorem III.4. The following Lemma IV.2 and Proposition IV.4 imply that any subalgebra 𝔥 satisfying (C1’) or (C2’) is a Cartan subalgebra. Note that, since 𝕂 = \overline{𝕂}, the subalgebra 𝔥_{s} ⊆ 𝔥 of any subalgebra 𝔥 satisfying (C1’) or (C2’) is toral.

Lemma IV.2. The conditions (C1’) and (C2’) are equivalent.

Proof. Assume that (C1’) is satisfied. Consider the maximal locally nilpotent 𝔥-submodule 𝔤^{0}(𝔥) ⊆ 𝔤. Since the adjoint representation of 𝔥 is locally finite, we have 𝔥 ⊆ 𝔤^{0}(𝔥). Indeed, otherwise for some h ∈ 𝔥 the finite-dimensional submodule U(𝔥)·h would have an 𝔥-eigenvector of non-zero eigenvalue, which would contradict the local nilpotence of 𝔥. To prove that 𝔥 = 𝔤^{0}(𝔥), assume to the contrary that h ∈ 𝔤^{0}(𝔥) is such that 𝔥^{n} · h ∈ 𝔥 for a minimal n > 0. Then 𝔥^{n−1} · h ∈ 𝔫_{𝔤}(𝔥) = 𝔥, contradicting the minimality of n. This shows that 𝔥 = 𝔤^{0}(𝔥), i.e., that (C1’) implies (C2’).

Conversely, let (C2’) be satisfied. Since 𝔥·𝔫_{𝔤}(𝔥) ⊆ 𝔥 and 𝔥 is a locally nilpotent 𝔥-module, 𝔫_{𝔤}(𝔥) is also a locally nilpotent 𝔥-module. Therefore 𝔫_{𝔤}(𝔥) ⊆ 𝔤^{0}(𝔥). Since 𝔤^{0}(𝔥) = 𝔥, this gives 𝔫_{𝔤}(𝔥) ⊆ 𝔥, i.e., 𝔫_{𝔤}(𝔥) = 𝔥.

Lemma IV.3. Condition (C2)’ implies V_{∗}^{0}(V^{0}) = {0}, where V^{0} := V^{0}(𝔥_{s}), V_{∗}^{0} := V_{∗}^{0}(𝔥_{s}).

Proof. By Proposition A in the Appendix, 𝔤′ = V′ ⊗ V′_{∗}, where the superscript ′ indicates maximal locally finite 𝔥_{s} -submodule. Furthermore, the assumption that the adjoint action of 𝔥 on 𝔥 is locally finite implies 𝔥 ⊆ 𝔤′. Hence the generalized weight 𝔥_{s} -module decomposition of V′ and V′_{∗} and the equality 𝔥 = 𝔤^{0}(𝔥) yield

𝔥 = ⨁_{α∈supp V} (V^{α} ⊗ V^{−α}) = 𝔥_{s} ⊕ (V^{0} ⊗ V_{∗}^{0}).

The local nilpotence of 𝔥 implies now V_{∗}^{0}(V^{0}) = {0}.

Proposition IV.4. Condition (C2’) implies that 𝔥 is a Cartan subalgebra.

Proof. The equality 𝔥 = 𝔤^{0}(𝔥) implies that 𝔥 is locally nilpotent. Therefore Lemma I.4 yields the inclusion 𝔥 ⊆ 𝔷_{𝔤}(𝔥_{s}). It remains to establish the opposite inclusion 𝔷_{𝔤}(𝔥_{s}) ⊆ 𝔥.

For h ∈ 𝔥_{s} put U := h · V and W := {v ∈ V : h · v = 0}. Then U is a finite-dimensional space and V = U ⊕ W, V^{∗} = W^{⊥} ⊕ V^{⊥}, and W^{⊥} ≃ U^{∗}. As h ∈ 𝔷(𝔥), 𝔥 preserves the four spaces U, W, U^{⊥}, and W^{⊥}. Therefore the projection

p_{U}: 𝔤 = V ⊗ V_{∗} → U ⊗ W^{⊥} ≅ 𝔤𝔩(U, W^{⊥})

with kernel V ⊗ U^{⊥} + W ⊗ V_{∗} is 𝔥-equivariant. This gives

p_{U}(𝔥) = p_{U}(𝔤^{0}(𝔥)) = 𝔤𝔩(U, W^{⊥})^{0}(𝔥) = 𝔥 ∩ 𝔤𝔩(U, W^{⊥}) =: 𝔥_{U}.

The centralizer of h in 𝔤 = V ⊗ V_{∗}, and therefore the subalgebra 𝔥, is contained in U ⊗ W^{⊥} + W ⊗ U^{⊥}, thus

𝔥 = 𝔥_{U} + (𝔥 ∩ ker p_{U}) = 𝔥_{U} + 𝔥 ∩ 𝔤𝔩(W, U^{⊥}).

As 𝔤𝔩(W, U^{⊥}) commutes with 𝔤𝔩(U, W^{⊥}), we conclude that

𝔤𝔩(U, W^{⊥})^{0}(𝔥_{U}) = 𝔤𝔩(U, W^{⊥})^{0}(𝔥) = 𝔥_{U},

and hence that 𝔥_{U} is a Cartan subalgebra of 𝔤𝔩(U, W^{⊥}) ≅ 𝔤𝔩(U). As 𝔤𝔩(U) is reductive, we have 𝔥_{U} ⊆ 𝔥_{s} and 𝔥_{n} · U = {0}.

By considering 𝔥_{U} ⊆ 𝔤𝔩(U, W^{⊥}), we now see that the weight spaces of 𝔥_{s} in V and V_{∗} with non-zero weights are one-dimensional. From the preceding argument we further conclude that 𝔥_{n} ∘ 𝔥_{s} = {0} in End V , which, in view of [𝔥_{s}, 𝔥_{n}] = {0}, implies 𝔥_{n} · V ⊆ V^{0} . By Proposition A in the Appendix,

𝔤′ = (V ⊗ V_{∗})′ = V′ ⊗ V′_{∗} = ⨁_{α,β} V^{α} ⊗ V_{∗}^{β}.

In particular 𝔥 ⊆ 𝔷_{𝔤}(𝔥_{s}) ⊆ 𝔤′, and therefore

𝔥 ⊆ ⨁_{α} V^{α} ⊗ V_{∗}^{−α}.

For 0 ≠ α we have seen above that dim V^{α} = dim V_{∗}^{−α} = 1, so that

⨁_{α≠0} V^{α} ⊗ V_{∗}^{−α} ⊆ 𝔥_{s},

because 𝔥_{s} · V^{α} ≠ {0} implies V_{∗}^{−α}(V^{α}) ≠ {0}. If A ∈ 𝔥_{s} ∩ (V^{0} ⊗ V_{∗}^{0}), then A annihilates all weight spaces V^{α} with α ≠ 0, and it also annihilates V^{0} . Therefore A · V ⊆ V′ leads to A^{2} = {0},k and hence to A = 0 as 𝔥_{s} consists of semisimple elements (Lemma II.1). This proves

𝔥_{s} = ⨁_{α≠0} V^{α} ⊗ V_{∗}^{−α},

which in turn yields

V^{0} = ⋂_{0≠α}(V_{∗}^{−α})^{⊥} and V_{∗}^{0} = ⋂_{0≠α}(V^{α})^{⊥}.

Since V_{∗}^{0}(V^{0}) = {0} by Lemma IV.3, Proposition II.3 implies that 𝔥_{s} is maximal toral and 𝔷_{𝔤}(𝔥_{s}) = 𝔥_{s} ⊗ (V^{0} ⊗ V_{∗}^{0}) is abelian. Therefore

V^{0} ⊗ V_{∗}^{0} ⊆ 𝔷_{𝔤}(𝔥) ⊆ 𝔤^{0}(𝔥) = 𝔥,

in particular 𝔷_{𝔤}(𝔥_{s}) ⊆ 𝔥.

It is an interesting open problem whether conditions (C1), (C2) and (C3) are equivalent for 𝔤𝔩(V, V_{∗}). Our results reduce the problem to the question of existence for subalgebras of 𝔤𝔩(V, V_{∗}) satisfying (C1) or (C2) and such that their adjoint representation is not locally finite.

V. The structure of 𝔤′

In this section 𝔱 is a fixed maximal toral subalgebra of 𝔤 = 𝔤𝔩(V, V_{∗}), 𝔥 := 𝔷_{𝔤}(𝔱) = 𝔱 ⊕ (V^{0} ⊗ V_{∗}^{0}), and the superscript ’ indicates maximal locally finite 𝔱-submodule.

Lemma V.1. The subalgebra 𝔥 ⊆ 𝔤′ is a maximal toral Cartan subalgebra of 𝔤′ with 𝔷(𝔤′) = V^{0} ⊗V_{∗}^{0} . Furthermore, the decomposition 𝔤′ = 𝔥⊕(⨁_{α∈Δ} 𝔤_{α}) is a root decomposition of 𝔤′ with respect to 𝔥.

Proof. The fact that the decomposition 𝔤′ = 𝔥⊕(⨁_{α∈Δ} 𝔤_{α}) is an 𝔥-root decomposition is clear from the very definition of this decomposition and from the equality 𝔥 = 𝔤^{0}(𝔥) (Proposition III.3).

The equality

[V^{0} ⊗ V_{∗}^{0}, V^{α} ⊗ V_{∗}^{β}] = {0}

for all α, β (Proposition II.3) implies that V^{0} ⊗ V_{∗}^{0} ⊆ 𝔷(𝔤′). Conversely, the inclusion 𝔷(𝔤′) ⊆ 𝔷_{𝔤}(𝔱) = 𝔥 implies easily that 𝔷(𝔤′) is not larger than V^{0} ⊗ V_{∗}^{0} . Furthermore, the equality 𝔥 = 𝔱 + 𝔷(𝔤′) shows that the adjoint action on 𝔤′ of every element x ∈ 𝔥 is diagonalizable, i.e., that 𝔥 is a toral Cartan subalgebra of 𝔤′. The maximality of 𝔤 is an immediate corollary of the equallity 𝔥 = 𝔤^{0}(𝔥).

As a consequence of Lemma V.1, the following theorem ([St99, Th. I.4]) applies to the pair (𝔤′, 𝔥). If 𝔨 is a locally finite Lie algebra which admits a root decomposition with respect to some subalgebra 𝔥_{𝔨}, we call a root α integrable if the subalgebra of 𝔨 generated by the root spaces 𝔨_{±α} is isomorphic to 𝔰𝔩_{2}(𝕂).

Theorem V.2. (Levi decomposition of locally finite split Lie algebras) Let 𝔨 be a locally finite Lie algebra with root decomposition 𝔨 = 𝔥_{𝔨} ⊕ (⨁_{α∈Δ(𝔨)} 𝔨_{α}) with respect to a toral Cartan subalgebra 𝔥_{𝔨}. Denote the set of integrable root by Δ_{i}(𝔨) ⊆ Δ(𝔨).

(1) The subspace 𝔰 = span Δ̌_{i}(𝔨) + ⨁_{α∈Δ_{i}(𝔨)} 𝔨_{α} is a semisimple subalgebra of 𝔨.

(2) Let Δ_{n}(𝔨) := Δ(𝔨) \ Δ_{i}(𝔨). The subspace 𝔯 := 𝔷_{𝔥_{𝔨}}(𝔰) + ⨁_{α∈Δ_{n}(𝔨)} 𝔨_{α} is the unique maximal locally solvable ideal of 𝔨, and 𝔲 := 𝔷(𝔨) + ⨁_{α∈Δ_{n}(𝔨)} 𝔨_{α} is the unique maximal locally nilpotent ideal of 𝔨.

(3) If 𝔞 is a vector space complement to the subspace 𝔷(𝔨) + span Δ̌_{i}(𝔨) in 𝔥_{𝔨}, we have 𝔨 ≅ 𝔲⋊(𝔰⋊𝔞), where the Lie algebra 𝔩 := 𝔰⋊𝔞 is almost reductive, i.e., has a semisimple commutator algebra.

Lemma V.3. The subspaces V′ ⊗ V_{∗}^{0} and V^{0} ⊗ V′_{∗} are abelian ideals of 𝔤′ with

[V′ ⊗ V_{∗}^{0}, V^{0} ⊗ V′_{∗}] = V^{0} ⊗ V_{∗}^{0} = 𝔷(𝔤′).

Proof. The statement follows from the equalities

[V′ ⊗ V′_{∗}, V^{0} ⊗ V′_{∗}] = (V^{0} ⊗ V′_{∗}) · (V′ ⊗ V′_{∗}) = V^{0} ⊗ V′_{∗},
[V′ ⊗ V′_{∗}, V′ ⊗ V_{∗}^{0}] = (V′ ⊗ V′_{∗}) · (V′ ⊗ V_{∗}^{0}) = V′ ⊗ V_{∗}^{0}

and

[V^{0} ⊗ V′_{∗}, V^{0} ⊗ V′_{∗}] = {0} = [V′ ⊗ V_{∗}^{0}, V′ ⊗ V_{∗}^{0}],

which in turn follow from the equality V_{∗}^{0}(V^{0}) = {0}.

Proposition V.4. If 0 ≠ α, δ ∈ supp V , then the functional α − δ ∈ 𝔥^{∗} is an integrable root of 𝔤′ with

𝔤′_{α−δ} = V^{α} ⊗ V_{∗}^{−δ}.

If V_{∗}^{0} ≠ {0}, then a functional 0 ≠ α ∈ supp V is a non-integrable root with

𝔤′_{α} = V^{α} ⊗ V′_{∗},

and if V^{0} ≠ {0}, a functional 0 ≠ −α ∈ supp V_{∗} is a non-integrable root with

𝔤′_{−α} = V^{0} ⊗ V_{∗}^{−α}.

The space

𝔲 := V′ ⊗ V_{∗}^{0} + V^{0} ⊗ V_{∗} = ⨁_{0≠α∈supp V} (V^{0} ⊗ V_{∗}^{−α} + V^{α} ⊗ V_{∗}^{0})

is the maximal locally nilpotent ideal of 𝔤′. Proof. For 0 ≠ α, δ ∈ supp V we have a root α − δ ∈ Δ with 𝔤′_{α−δ} ⊇ V^{α} ⊗ V_{∗}^{−δ}. Let 0 ≠ f_{α} ∈ V^{α}, α ≠ 0. Define f_{α}^{∗} ∈ V_{∗}^{−α} by f_{α}^{∗}(f_{α}) = 1. Then

h_{α,δ} := [f_{α} ⊗ f_{δ}^{∗}, f_{δ} ⊗ f_{α}^{∗}] = f_{α} ⊗ f_{α}^{∗} − f_{δ} ⊗ f_{δ}^{∗}

satisfies α(h_{α,δ}) = 1 and δ(h_{α,δ}) = −1. Therefore the roots α − δ are integrable, which implies in particular that 𝔤′_{α−δ} is one-dimensional, so that 𝔤′_{α−δ} = V^{α} ⊗ V^{δ}.

Furthermore, Lemma V.3 implies that the root spaces 𝔤′_{α} = V^{α} ⊗ V_{∗}^{0} + V^{0} ⊗ V_{∗}^{−α}, for 0 ≠ α ∈ supp V , are contained in the maximal locally nilpotent ideal of 𝔤′.

The remaining assertions are direct consequences of Theorem V.2.

The following theorem is a direct corollary of Theorem V.2 via the information provided by Lemma V.3 and Proposition V.4.

Theorem V.5. (Structure Theorem for 𝔤′) The Lie algebra 𝔤′ is isomorphic to the semidirect product 𝔲 ⋊ 𝔩, where

𝔲 := V^{0} ⊗ V′_{∗} + V′ ⊗ V_{∗}^{0}

is the Lie algebra with bracket

[(v ⊗ φ), (y ⊗ ψ)] = φ(y) · v ⊗ v^{∗},

and 𝔩 ≅ W′ ⊗ W_{∗} = 𝔤𝔩(W, W_{∗}) for W := 𝔱 · V and W_{∗} := 𝔱 · V_{∗}.

Appendix. A useful general proposition

The following proposition was communicated to us by I. Dimitrov and G. Zuckerman and is a generalization version of a lemma which we had proved for the specific purposes of the present paper.

Let U and W be vector spaces. To any element x ∈ U ⊗ W we assign a subspace U_{x} ⊂ U in the following way. Write x as ∑_{j} u_{j} ⊗ w_{j} with linearly independent w_{j} ∈ W, and set U_{x} := span{u_{i}}. In a similar way we assign to x a subspace W_{x} ⊂ W. To check that U_{x} (and similarly W_{x}) does not depend on the presentation of x as ∑_{j} u_{j} ⊗ w_{j}, it suffices to identify U_{x} with the image of the linear operator ψ(x) ∈ Hom(W^{∗}, U), where ψ is the canonical inclusion

U ⊗ W ↪ Hom(W^{∗}, U), ψ(u ⊗ w)(α) := α(w)u.

This is a straightforward checking which we omit.

Clearly, dim U_{x} < ∞ as ψ(x) is an operator of finite rank. Note also that for any subspace Y ⊂ U we have

Y ⊗ W = {x ∈ U ⊗ W : U_{x} ⊂ Y },

and similarly for any z ∈ W,

U ⊗ Z = {x ∈ U ⊗ W : W_{x} ⊂ Z}.

Let now 𝔨 be a Lie algebra. For any 𝔨-module Q, we denote by Q′ the maximal locally finite 𝔨-submodule of Q.

Proposition A. For any 𝔨-modules M and N, we have

M′ ⊗ N′ = (M ⊗ N)′.

Proof. The inclusion M′ ⊗ N′ ⊂ (M ⊗ N)′ is obvious.

Fix 0 ≠ x ∈ (M ⊗ N)′ and a basis x_{1}, …, x_{n} of U(𝔨) · x with x_{1} = x. Set Y := M_{x_{1}} +…+M_{x_{n}} and Z := N_{x_{1}} +…+N_{x_{n}}. Since x ∈ M_{x} ⊗N_{x} ⊂ Y ⊗Z, it suffices to prove that Y ⊂ M′ and Z ⊂ N′. We will show that Z ⊂ N′ (the argument for Y is completely similar), which will follows from U(𝔨) · Z ⊂ Z. For this it is enough to verify that k · N_{x_{i}} ⊂ Z for any i and any k ∈ 𝔨.

Fix k and i and write x_{i} as ∑_{j} m_{j} ⊗ n_{j} with linearly independent m_{j} ∈ M. Then k · x_{i} = ∑_{j} k · m_{j} ⊗ n_{j} + ∑_{j} m_{j} ⊗ k · n_{j}. Since k · x_{i} ∈ Y ⊗ Z and ∑_{j} k · m_{j} ⊗ n_{j} ∈ M ⊗ Z, we have ∑_{j} m_{j} ⊗ k · n_{j} ∈ M ⊗ Z. Therefore N_{∑_{j} m_{j}⊗k·n_{j}} ⊂ Z, i.e., k · n_{j} ∈ Z. As the n_{j} generate

N_{x_{i}}, this implies k · N_{x_{i}} ∈ Z.

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K.-H. N.:

Fachbereich Mathematik AG5

Schlossgartenstr. 7

D-64289 Darmstadt, Germany

e-mail: neeb@mathematik.tu-darmstadt.de

I.P.:

Department of Mathematics,

University of Califronia at Riverside

Riverside, CA 92521, USA e-mail: penkov@math.ucr.edu