Journal of Lie Theory Volume 12 (2002) 265–288 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

On the Structure of Graded Transitive Lie Algebras

Gerhard Post

Communicated by P. Olver

Abstract. We study finite-dimensional Lie algebras 𝔏 of polynomial vector fields in n variables that contain the vector fields \frac{∂}{∂x_{i}} (i = 1, …, n) and

x_{1} \frac{∂}{∂x_{1}} + ⋯ + x_{n} \frac{∂}{∂x_{n}} . We show that the maximal ones always contain a

semi-simple subalgebra 𝔤̄, such that \frac{∂}{∂x_{i}} ∈ 𝔤̄ (i = 1, …, m) for an m with 1 ≤ m ≤ n. Moreover a maximal algebra has no trivial 𝔤̄-modules in the space spanned by \frac{∂}{∂x_{i}} (i = m + 1, …, n). The possible algebras 𝔤̄ are described in detail, as well as all 𝔤̄-modules that constitute such maximal 𝔏. The maximal algebras are described explicitly for n ≤ 3. Keywords: Lie algebras, vector fields, graded Lie algebras. MSC 1991: 17B66, 17B70, 17B05.

1. Introduction

The classification of Lie algebras of vector fields has a history of more than 100 years. Lie [7] was particularly interested in it, since he tried to create catalogues of differential equations with a fixed symmetry group. Apart from this, there are several motivations for considering this problem [1]. The problem in its full generality is (still) too difficult to attack. Our concern will be transitive algebras, that are also graded. Remember that W_{n} carries a natural filtering, determined by the lowest order terms that appear in the Taylor expansion of the coefficients of a vector field. If one considers with this filtering, the associated graded Lie algebra of a transitive Lie algebra, one obtains the graded transitive Lie algebra that we study here.

This paper is a continuation of [10]. There is studied the relation of the graded transitive Lie algebras 𝔏 with its structure as an abstract Lie algebra. Results are derived about the forms of the Levi subalgebras, the radical and the nilradical of 𝔏. Of particular interest is the product 𝔤̄ of those Levi subalgebras that have elements of degree -1. It follows that 𝔤̄ = 𝔤̄_{−1} ⊕ 𝔤̄_{0} ⊕ 𝔤̄_{1}, where the subscript denotes the degree.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

There were two main conclusions in [10]. The first one was that when studying maximal algebras 𝔏, it is unnatural to take maximality in the class of Lie algebras of fixed maximal degree, cf. [4, 5, 6]. The second conclusion was that to make progression, one will have to study how 𝔏 is a representation over 𝔤̄. Exactly these two points we take into account, and end up with a rather detailed description for the possible forms of 𝔏. It seems difficult to proceed any further in the general case. This is due to the possible presence of certain quadratic terms Q_{α} in 𝔤̄_{1}. These terms can exist only if only S^{2}(V^{∗}) ⊗ V contains a 𝔤̄_{0}- module isomorphic to the 𝔤̄_{0}-module 𝔤̄_{1}. Here V is an arbitrary 𝔤̄_{0}-representation (and Q_{α} ∈ S^{2}(V^{∗}) ⊗ V ). The lowest number of variables where this happens is n = 3. Another complication occurs when the space spanned by ∂_{x_{i}} (i = 1, …, n) contains trivial 𝔤̄-modules. Such algebra 𝔏 can not be maximal, but we cannot prove that 𝔏 is contained in a maximal one in case Q_{α} ≠ 0 and n ≥ 5. We give a more detailed overview of the results in section 2., after introducing the notions that play an important role.

Acknowledgment. I like to thank Jan Draisma for several useful remarks, and Loek Helminck for providing me with the L^{A}TEX-form of the diagrams in table 1.

2. Definitions and description of results

Consider the Lie algebra W_{n} of all polynomial vector fields on ℂ^{n}. An element X ∈ W_{n} has the form

X = ∑_{i=1}^{n} P_{i}(x)∂_{x_{i}}

where the P_{i} are polynomials in n variables x = (x_{1}, x_{2}, …, x_{n}), and ∂_{x_{i}} = \frac{∂}{∂x_{i}}. We say that X has degree k − 1 if the polynomials P_{i} are all homogeneous of degree k. We write X ∈ U_{k−1}. If we define the Euler vector field E = ∑_{i=1}^{n} x_{i}∂_{x_{i}}, then we have exactly that

U_{k} = {X ∈ W_{n}|[E, X] = kX}.  (1)

Thus W_{n} becomes a ℤ-graded Lie algebra,

W_{n} = ⨁_{k=0}^{∞} U_{k−1}; [U_{i}, U_{j}] ⊂ U_{i+j}

Our object is to study subalgebras 𝔏 of W_{n} that satisfy the following conditions:

(0) 𝔏 is finite-dimensional.

(1) 𝔏 is graded: if 𝔏_{k} = 𝔏 ∩ U_{k} then 𝔏 = ⊕L_{k}.

(2) 𝔏 is transitive: 𝔏_{−1} = U_{−1}.

We will pay special attention to the case

(3) 𝔏 is maximal: there exists no finite-dimensional Lie algebra 𝔏′ ⊂ W_{n} containing 𝔏 properly.

We call 𝔏 satisfying (0)-(2) a graded transitive Lie algebra, and those satisfying (0)- (3) a maximal graded transitive Lie algebra. Note that linear changes of coordinates map 𝔏 to a Lie algebra that we call equivalent; more precisely the aim is to study all equivalence classes of maximal graded transitive Lie algebras of vector fields. If 𝔏 is maximal then 𝔏 contains E: if E ∉ 𝔏 then 𝔏′ = 𝔏⋊ℂE is a one-dimensional extension of 𝔏. Once a finite-dimensional Lie algebra of vector fields contains E it is automatically ℤ-graded, thanks to equation (1). Throughout we will assume that E ∈ 𝔏.

Our method is based on results from [10]. These results concern the structure of algebras 𝔏, not necessarily maximal, as an abstract Lie algebra. Usually 𝔏 is not semi-simple; 𝔏 can only be semi-simple if it is contained in U_{−1} ⊕ U_{0} ⊕ U_{1}. Let ℜ denote the radical of 𝔏. If ℜ ≠ {0} then also V = ℜ ∩ U_{−1} ≠ {0}. We can perform a linear change of coordinates such that

V = ⟨∂_{x_{m+1}}, ∂_{x_{m+2}}, …, ∂_{x_{n}}⟩

in the new coordinates (which are again denoted by x_{1}, x_{2}, …, x_{n}). Let I_{V} denote the (graded) ideal^{1} in 𝔏 given by

I_{V} = {X ∈ 𝔏|X = ∑_{i=m+1}^{n} P_{i}(x)∂_{x_{i}}}

The quotient 𝔤 = 𝔏/I_{V} is semi-simple, since ℜ is contained in I_{V} , and also graded. Moreover the elements of 𝔤 are of the form (omitting +I_{V} )

X̄ = ∑_{i=1}^{m} P_{i}(x_{1}, x_{2}, …, x_{m})∂_{x_{i}}.

Note that P_{i} is independent of x_{j} for j > m, since otherwise [∂_{x_{j}}, X] ∉ I_{V} , which contradicts the fact that ∂_{x_{j}} ∈ ℜ.

Now we are left with 𝔤, which is semi-simple. We can therefore decompose 𝔤 into, say r, simple ideals:

𝔤 = 𝔤^{(1)} × 𝔤^{(2)} × ⋯ × 𝔤^{(r)}

Each summand 𝔤^{(i)} is graded, and we have

𝔤^{(i)} = 𝔤_{−1}^{(i)} ⊕ 𝔤_{0}^{(i)} ⊕ 𝔤_{1}^{(i)},

with dim(𝔤_{−1}^{(i)}) = dim(𝔤_{1}^{(i)}) = m_{i} ≠ 0 and ∑ m_{i} = m. Since the summands 𝔤^{(i)} mutually commute, each 𝔤^{(i)} itself can be considered to be a graded Lie algebra of vector fields on ℂ^{m_{i}}. The Euler operator on ℂ^{m_{i}} is a derivation of 𝔤^{(i)}: therefore it is contained in 𝔤^{(i)}.

Now we can describe the contents of this paper. First (section 3.) we describe all possible 𝔤^{(i)}. This description can already be found in [8], and is summarized in table 1. Therefore the Lie algebra 𝔤 can be described explicitly.

1 I_{V} is the maximal ideal J such that J ∩ U_{−1} = V ; this statement, noted by Jan Draisma, simplifies some proofs in [10].

Of great technical importance is lemma 3.6, which describes the elements of 𝔤_{1} in terms of those of 𝔤_{0}.

The Levi-Malcev theorem states that there exists a Lie subalgebra 𝔤̄ in 𝔏 such that 𝔤 = 𝔏/I_{V} . We call 𝔤̄ an extension of 𝔤; we can assume that 𝔤̄ is graded, see section 4.. Let us denote the remaining variables (x_{m+1}, …, x_{n}) by (y_{1}, …, y_{n−m}), so that V = ⟨∂_{y_{1}}, …, ∂_{y_{n−m}}⟩. The new terms that appear in 𝔤̄ are completely determined by the action on V . After some calculations one ends up with the form given in proposition 4.1. In particular 𝔤̄_{−1} = ⟨∂_{x_{1}}, …, ∂_{x_{m}}⟩ is unchanged (after maybe a linear change of coordinates), and X ∈ 𝔤̄_{1} takes the form

X = ∑_{i=1}^{m} P_{i}(x)∂_{x_{i}} + ∑_{i=1}^{n−m} L_{i}(x, y)∂_{y_{i}} + ∑_{i=1}^{n−m} Q_{i}(y)∂_{y_{i}},

where the terms of P_{i}, L_{i} and Q_{i} are of the forms cx_{a}x_{b}, cx_{a}y_{b} and cy_{a}y_{b}, respectively.

Then we come to the study of the module: I_{V} is a module over 𝔤. This we study in section 5., for the case that V contains no trivial 𝔤̄_{0}-modules. In this case we prove that there exists a unique maximal algebra (assuming 𝔤̄ ≠ {0} is fixed).

If V contains a trivial 𝔤̄_{0}-module, the situation is more complicated, see section 6.. Though 𝔏 is never maximal, we only can prove that it is contained in a maximal algebra, in case that Q_{i} = 0 for all i = 1, …, m. If some Q_{i} ≠ 0, it is possible that 𝔏 is contained in a maximal algebra by suitably extending 𝔤̄. Essentially this depends on the existence of a suitable ℤ-grading, called the zdegree, in (an extension of) 𝔏.

Finally section 7. discusses the cases n ≤ 3, while section 8. contains the conclusion.

3. Lie algebras of depth 1

We discuss a class of ℤ-graded Lie algebras, which are closely related to Lie algebras of vector fields. For convenience we use

Definition 3.1. Suppose 𝔤 = ⊕𝔤_{k} is a ℤ-graded Lie algebra. We say that 𝔤 is of depth 1 if

• 𝔤_{k} = {0} for all k < −1 and 𝔤_{−1} ≠ {0}.

• 𝔤_{+} = ⊕_{k≥0}𝔤_{k} contains no non-zero ideal of 𝔤.

It follows that for X, Y ∈ 𝔤_{k}: [∂, X] = [∂, Y] for all ∂ ∈ 𝔤_{−1} ⇒ X = Y (k ≥ 0). Inductively, we obtain that any X ∈ 𝔤_{k} is uniquely determined by the products

[∂_{1}, [∂_{2}, …, [∂_{k+1}, X] …]],

where ∂_{1}, ∂_{2}, …, ∂_{k+1} ∈ 𝔤_{−1} and k ≥ 0. This way X ∈ 𝔤_{k} is given by the linear map A = φ_{k}(X):

A : \underbrace{𝔤_{−1} × ⋯ × 𝔤_{−1}} → 𝔤_{−1}
k + 1

given by

A(∂_{1}, …, ∂_{k+1}) = \frac{1}{k!}[∂_{1}, [∂_{2}, …, [∂_{k+1}, X] …]].

Due to Jacobi’s identity and [∂_{i}, ∂_{j}] = 0, A is symmetric, hence A ∈ S^{k+1}(𝔤_{−1}^{∗}) ⊗ 𝔤_{−1}. Choosing a basis in 𝔤_{−1}, we obtain a linear map φ_{k} : 𝔤_{k} → U_{k}, where U_{k} denotes the space of all vector fields on ℂ^{m}, m = dim(𝔤_{−1}), of degree k.

Composing all φ_{k}, we obtain a linear map φ : 𝔤 → W_{m}, with φ|_{𝔤_{k}} = φ_{k}. Clearly φ is injective. One can prove (see [5]) that φ is a Lie algebra morphism, of degree 0. Hence we find that W_{m} contains all Lie algebras of depth 1 with dim(𝔤_{−1}) = m as subalgebras.

This point of view gives us the opportunity to compare two Lie algebras 𝔤 of depth 1 and with dim(𝔤_{−1}) = m. Since all are embedded in W_{m} it can happen that one is contained in another; those finite-dimensional ones that are not contained in any other of finite dimension, we call maximal.

Next we turn to the case that 𝔤 is simple. Since the Killing form is a non-degenerate pairing between 𝔤_{k} and 𝔤_{−k}, we see that necessarily

𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} with dim(𝔤_{−1}) = dim(𝔤_{1}).

Conversely, suppose 𝔤 is a simple Lie algebra with a ℤ-grading such that 𝔤_{k} = 0 if and only if |k| > 1. Since 𝔤 contains no non-trivial ideals at all, 𝔤 is of depth 1. Therefore we will investigate now which simple (abstract) Lie algebras 𝔤 can be given a ℤ-grading such that 𝔤_{k} = 0 if and only if |k| > 1. Let us fix a Cartan subalgebra 𝔥 in 𝔤, of dimension ℓ. Now 𝔤 is ℤ^{ℓ}-graded by the (coefficients of) the roots. Suppose θ = n_{1}α_{1} + ⋯ + n_{ℓ}α_{ℓ} is the highest root. If n_{s} = 1 for a certain s, we can put a ℤ-grading on 𝔤 by deg(e_{α}) = m_{s} if α = ∑_{i=1}^{ℓ} m_{i}α_{i}. Since |m_{s}| ≤ n_{s}, we have indeed that 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}.

The simple Lie algebras of depth 1, obtained in this way, are given in table 1; we depict the Dynkin diagram of 𝔤 = X_{ℓ} (where X = A, B, C, D or E) with the point s colored black and write X_{ℓ;s} for this graded Lie algebra.

𝔤  Diagram  𝔤_{0}  dim(𝔤_{−1})
A_{ℓ;s}  [figure: 1; 2; s; ℓ−1; ℓ] A_{s−1} × ℂ × A_{ℓ−s} s(ℓ + 1 − s)
B_{ℓ}  [figure: 1; 2; ℓ−1; ⟹; ℓ]  B_{ℓ−1} × ℂ  2ℓ − 1
C_{ℓ}  [figure: 1; 2; ℓ−1; ⟸; ℓ]  A_{ℓ−1} × ℂ  \binom{ℓ+1}{2}

D_{ℓ;1} [figure: 1; 2; ℓ−1; ℓ]  D_{ℓ−1} × ℂ  2ℓ − 2

D_{ℓ;ℓ} [figure: 1; 2; ℓ−1; ℓ]  A_{ℓ−1} × ℂ  \binom{ℓ}{2}

E_{6}  [figure: 1; 3; 4; 2; 5; 6]  D_{5} × ℂ  16

E_{7} [figure: 1; 3; 4; 2; 5; 6; 7]  E_{6} × ℂ  27

Table 1. Simple Lie algebras of depth 1

In cases where s is unique, we omitted it in the notation; moreover we have put A_{0} = {0}. Obviously A_{ℓ;s} and A_{ℓ;ℓ−s+1} are isomorphic, as well as D_{ℓ;ℓ} and D_{ℓ;ℓ−1}.

The structure of 𝔤_{0} is described by the Dynkin diagram of 𝔤, with s omitted; the summand ℂ corresponds to the space spanned by the Euler vector field; it is central in 𝔤_{0}. Finally dim(𝔤_{−1}) can be calculated by

dim(𝔤_{−1}) = \frac{1}{2}(dim 𝔤 − dim 𝔤_{0}).

The following theorem tells that the simple Lie algebras of depth one are all (gradedly) isomorphic to one from the table above.

Theorem 3.2. (Morozov, [8],§6) The simple Lie algebras of depth 1 are those given in table 1.

Example 3.3. Let us construct A_{ℓ;s} in terms of vector fields. Generators for A_{ℓ} are {E_{ij}} with i, j = 1, …, ℓ + 1 and there is one linear relation among them: E_{11} + E_{22} + ⋯ + E_{ℓ+1,ℓ+1} = 0. The commutator is defined by

[E_{ij}, E_{ab}] = δ_{ja}E_{ib} − δ_{ib}E_{aj}.

Moreover

𝔤_{−1} = ⟨E_{ij}|i ≥ s + 1 and j ≤ s⟩.

Let us denote

∂_{ij} = ∂_{x_{ij}} = E_{s+i,j} (i ≤ l − s + 1, j ≤ s)

and

X_{ij} = E_{j,s+i} (i ≤ l − s + 1, j ≤ s).

For E_{ab} (a, b ≤ s), we have

[∂_{ij}, E_{ab}] = [E_{s+i,j}, E_{ab}] = δ_{aj}E_{s+i,b} = δ_{aj}∂_{ib}

so that

E_{ab} = ∑_{i=1}^{ℓ−s+1} x_{ia}∂_{ib} (a, b ≤ s).

Similarly, [∂_{ij}, E_{s+a,s+b}] = −δ_{bi}E_{s+a,j} = −δ_{bi}∂_{aj}, and we find

E_{s+a,s+b} = − ∑_{j=1}^{s} x_{bj}∂_{aj} (a, b ≤ ℓ − s + 1).

Finally,

[∂_{ij}, X_{ab}] = [E_{s+i,j}, E_{b,s+a}] = δ_{jb}E_{s+i,s+a} − δ_{ai}E_{bj}.

Hence by integrating we find

X_{ab} = − ∑_{i,j} x_{aj}x_{ib}∂_{ij}.

Note that

E_{11} + E_{22} + ⋯ + E_{ℓ+1,ℓ+1} = ∑_{k=1}^{s} ( ∑_{i=1}^{ℓ−s+1} x_{ik}∂_{ik}) + ∑_{k=1}^{ℓ−s+1} (− ∑_{j=1}^{s} x_{kj}∂_{kj}) = 0.

The procedure above shows how inductively we can obtain the realization of 𝔤 in terms of polynomial vector fields. Once a (graded) basis of 𝔤 is known, as well as the corresponding structure constants, the realization is fixed.

We now describe this construction in terms of roots and root vectors. Let {e_{α}, h_{i}} be a Chevalley basis of 𝔤, where α ∈ R and i = 1, 2, …, ℓ.

As usual, we denote h_{α} = ∑ m_{i}h_{i} for α = ∑ m_{i}α_{i}. Let, as above, s be such that deg(e_{α_{s}}) = 1. We define

R^{∘} = {α = ∑ m_{i}α_{i} ∈ R|m_{s} = 0} = {α ∈ R| deg(e_{α}) = 0}

and

R̂ = {α = ∑_{i≠s} m_{i}α_{i}|m_{i} ≥ 0 for all i and α + α_{s} ∈ R}.

Note that 0 ∈ R̂. Clearly,

R = (−α_{s} − R̂) ∪ R^{∘} ∪ (α_{s} + R̂) (disjoint union),

corresponding to the degrees -1,0 and 1, respectively. For α ∈ R̂ we denote

∂_{α} = e_{−α_{s}−α} and X_{α} = e_{α_{s}+α}.

The realization of 𝔤 is by vector fields in the variables x_{α}, α ∈ R̂. Hence |R̂| = m and

𝔤_{−1} = ⟨∂_{α}|α ∈ R̂⟩, 𝔤_{1} = ⟨X_{α}|α ∈ R̂⟩,

while 𝔤_{0} consists of vector fields of the form

∑_{α,β∈R̂} c_{αβ}x_{α}∂_{β}.

Hence 𝔤_{0} acts naturally on the space S^{1} = ⟨x_{α}|α ∈ R̂⟩.

Lemma 3.4. The 𝔤_{0}-modules 𝔤_{1} and S^{1} are isomorphic.

Proof. The space 𝔤_{1} is the module dual to 𝔤_{−1} by its non-degenerate pairing with respect to the Killing form (∂|X), with ∂ ∈ 𝔤_{−1} and X ∈ 𝔤_{1}; the 𝔤_{0}-action on 𝔤_{1} is dual to the 𝔤_{0}-action on 𝔤_{−1}:

(∂|[g, X]) = −([g, ∂]|X).

We show that S^{1} satisfies the same relation, with respect to the pairing (., .) given by

(∂_{α}, x_{β}) = δ_{αβ}.

Indeed, for g = ∑ c_{γϵ}x_{γ}∂_{ϵ} on one hand we have

(∂_{α}, [g, x_{β}]) = (∂_{α}, ∑_{γ} c_{γβ}x_{γ}) = c_{αβ},

while

([g, ∂_{α}], x_{β}) = −c_{αβ}.

Consequently, also S^{1} is the module dual to 𝔤_{−1}.

Remark 3.5. In the ADE cases, the Killing form can be normalized such that the isomorphism of the 𝔤_{0}-module S^{1} and 𝔤_{1} is given by x_{α} ↦ X_{α}. In general an isomorphism is given by x_{α} ↦ n_{α}X_{α}, where n_{α} = 2/(α|α).

For α, β ∈ R, with α ≠ −β we put

[e_{α}, e_{β}] = c(α, β)e_{α+β}.

We have c(α, β) ≠ 0 if and only if α + β ∈ R. Moreover we put c(α, β) = 0 if α + β ∉ R. Now we can describe 𝔤_{0} in terms of vector fields. For e_{α} ∈ 𝔤_{0} we have

[∂_{β}, e_{α}] = c(−α_{s} − β, α)e_{−α_{s}−β+α} = c(−α_{s} − β, α)∂_{β−α}.

Hence

e_{α} = ∑_{β−α∈R̂} c(−α_{s} − β, α)x_{β}∂_{β−α}.

Similarly for h_{i} ∈ 𝔥 we obtain

h_{i} = ∑_{β}⟨β, α_{i}⟩x_{β}∂_{β}

where

⟨β, α_{i}⟩ = \frac{2(β|α_{i})}{(α_{i}|α_{i})}.

Next we consider X_{α} ∈ 𝔤_{1}. The following expression is essential to us.

Lemma 3.6. Suppose for X_{α} ∈ 𝔤_{1} and ∂_{β} ∈ 𝔤_{−1} we have

[∂_{β}, X_{α}] = L_{α}^{β} ∈ 𝔤_{0}.

Then

X_{α} = \frac{1}{2} ∑_{β∈R̂} x_{β}L_{α}^{β}.

Proof. We have

X_{α} = ∑_{β,γ,ϵ} c_{βγϵ}^{α}x_{β}x_{γ}∂_{ϵ};

we can assume that c_{βγϵ}^{α} = c_{γβϵ}^{α}. Now

[∂_{β}, X_{α}] = ∑_{γ,ϵ} c_{βγϵ}^{α}x_{γ}∂_{ϵ} + ∑_{γ,ϵ} c_{γβϵ}^{α}x_{γ}∂_{ϵ} = L_{α}^{β}.

Hence

\frac{1}{2} ∑ x_{β}L_{α}^{β} = \frac{1}{2} ∑ x_{β} (∑_{γ,ϵ} c_{βγϵ}^{α}x_{γ}∂_{ϵ} + ∑_{γ,ϵ} c_{γβϵ}^{α}x_{γ}∂_{ϵ}) = X_{α}

We can express L_{α}^{β} in the Chevalley basis by

L_{α}^{β} = \begin{cases}c(−α_{s} − β, α_{s} + α)e_{α−β} & (α ≠ β) \\ −h_{α_{s}+α} & (α = β). \end{cases}

Moreover in view of remark 3.5 we have for e_{γ} ∈ 𝔤_{0}:

[e_{γ}, X_{α}] = \frac{1}{2} ∑[e_{γ}, x_{β}]L_{α}^{β} + \frac{1}{2} ∑ x_{β}[e_{γ}, L_{α}^{β}]

= \frac{1}{2} ∑ c(γ, α_{s} + β)n_{β}n_{β+γ}^{−1}x_{β+γ}L_{α}^{β} + \frac{1}{2}x_{β}[e_{γ}, L_{α}^{β}]

= \frac{1}{2} ∑ x_{β} (c(γ, α_{s} + β − γ)n_{β−γ}n_{β}^{−1}L_{α}^{β−γ} + [e_{γ}, L_{α}^{β}]) .

On the other hand [e_{γ}, X_{α}] = c(γ, α_{s} + α)X_{α+γ}. Comparing the coefficents of x_{β}

we see that for all α, β ∈ R̂ and γ ∈ R^{∘} holds

c(γ, α_{s} + β − γ)n_{β−γ}n_{β}^{−1}L_{α}^{β−γ} + [e_{γ}, L_{α}^{β}] = c(γ, α_{s} + α)L_{γ}^{β}  (2)

Note that this is a relation between elements in 𝔤_{0}, containing only linear combinations and commutators. Consequently it holds in all representations of 𝔤_{0} as well.

4. Extensions of simple algebras of depth 1

We proceed to investigate how the semi-simple algebra 𝔤 of depth 1, as considered in section 3. above, can be extended to vector fields in more variables. So we assume that 𝔤 is realized in terms of vector fields in m = dim 𝔤_{−1} variables, which we call x_{1}, x_{2}, …, x_{m}, and we try to reconstruct the original algebra 𝔤̄ ⊂ 𝔏 in n variables, such that 𝔤 = 𝔤̄/I_{V} . For convenience we call 𝔤̄ an extension of 𝔤; we stress that 𝔤̄ and 𝔤 are isomorphic Lie algebras. The remaining n − m variables, we will call y_{1}, y_{2}, …, y_{n−m}. As to the existence of 𝔤̄, the Levi-Malcev theorem tells that 𝔏 contains a subalgebra 𝔤̄ such that 𝔤̄/I_{V} and 𝔤 are isomorphic. In case that all ideals in 𝔏 are ℤ-graded, it is possible to choose 𝔤̄ to be ℤ-graded as well^{2}. Hence 𝔤̄ = 𝔤̄_{−1} ⊕ 𝔤̄_{0} ⊕ 𝔤̄_{1}. Let us now study 𝔤̄ step by step.

First g = ∂_{x_{i}} ∈ 𝔤_{−1} corresponds to ḡ ∈ 𝔤̄_{−1} of the form

ḡ = ∂_{x_{i}} + ∑_{j=1}^{n−m} a_{ij}∂_{y_{j}}.

Introducing new coordinates in ℂ^{n} by

\begin{cases}x_{j} = x′_{j} \\ y_{j} = y′_{j} + ∑ a_{ij}x′_{i} \end{cases}

we obtain ḡ = ∂_{x′_{i}}. Hence we can assume that ∂_{x_{i}} ∈ 𝔤̄ (dropping ’).

Next we turn to g ∈ 𝔤_{0}. For g = ∑ c_{ij}x_{i}∂_{x_{j}} we have

ḡ = g = ∑_{i,j≤m} c_{ij}x_{i}∂_{x_{j}} + ∑_{j=1}^{n−m} f_{j}(x, y)∂_{y_{j}}.

Now [∂_{x_{i}}, ḡ] ∈ 𝔤̄, and also

[∂_{x_{i}}, ḡ] = ∑_{j} c_{ij}∂_{x_{j}} + ∑_{j} \frac{∂f_{j}}{∂x_{i}} ∂_{y_{j}}.

This implies that \frac{∂f_{j}}{∂x_{i}} = 0 for all i, j. Hence

ḡ = ∑_{i,j≤m} c_{ij}x_{i}∂_{x_{j}} + ∑_{i,j≤n−m} d_{ij}y_{i}∂_{y_{j}}.

In particular, we have for e_{α} ∈ 𝔤_{0} that

ē_{α} = e_{α} + E_{α} with E_{α} = ∑ d_{ij}y_{i}∂_{y_{j}}

and similarly

h̄_{α} = h_{α} + H_{α}.

Note that [e_{α}, E_{β}] = [e_{α}, H_{γ}] = [h_{γ}, E_{α}] = [h_{γ}, H_{ϵ}] = 0 for all α, β ∈ R^{∘} and all γ, ϵ ∈ R. Because 𝔤_{0} and 𝔤̄_{0} are isomorphic, we obtain that {E_{α}, H_{α}} form a representation of 𝔤_{0}, realized by vector fields in the variables y_{1}, …, y_{n−m}.

Finally we arrive at X_{α} ∈ 𝔤_{1}. Similar to lemma 3.6 we need that

X̄_{α} = \frac{1}{2} ∑ x_{β}L_{α}^{β} + ∑ x_{β}(L̄_{α}^{β} − L_{α}^{β}) + Q_{α},

2One can follow the proof of the Levi-Malcev theorem step by step, and finds that all occurring spaces can be chosen to be graded.

where

L̄_{α}^{β} = \begin{cases}c(−α_{s} − β, α_{s} + α)ē_{α−β} & (α ≠ β) \\ −h̄_{α_{s}+α} & (α = β), \end{cases}

and L_{α}^{β} is as in lemma 3.6, and

Q_{α} = ∑_{i,j,k} c_{ijk}^{α}y_{i}y_{j}∂_{y_{k}}.

We still need to check the commutators [𝔤̄_{0}, 𝔤̄_{1}] and [𝔤̄_{1}, 𝔤̄_{1}]. For ē_{γ} ∈ 𝔤̄_{0} and X̄_{α}, using remark 3.5, we have

[ē_{γ}, X̄_{α}] = [e_{γ} + E_{γ}, \frac{1}{2} ∑ x_{β}L_{α}^{β} + + ∑ x_{β}(L̄_{α}^{β} − L_{α}^{β}) + Q_{α}]

= ∑[e_{γ}, X_{α}] + ∑[e_{γ}, x_{β}](L̄_{α}^{β} − L_{α}^{β}) + ∑ x_{β}[E_{γ}, L̄_{α}^{β} − L_{α}^{β}] + [E_{γ}, Q_{α}]
= c(γ, α_{s} + α)X_{α+γ} + [E_{γ}, Q_{α}]

+ ∑ c(γ, α_{s} + β)n_{β}n_{β+γ}^{−1}x_{β+γ}(L̄_{α}^{β} − L_{α}^{β}) + ∑ x_{β}[E_{γ}, L̄_{α}^{β} − L_{α}^{β}]

= c(γ, α_{s} + α)X_{α+γ} + [E_{γ}, Q_{α}]
+ ∑ x_{β} (c(γ, α_{s} + β − γ)n_{β−γ}n_{β}^{−1}(L̄_{α}^{β−γ} − L_{α}^{β−γ}) + [E_{γ}, L̄_{α}^{β} − L_{α}^{β}])

Using equation (2) the last sum can be rewritten, and we obtain

[ē_{γ}, X̄_{α}] = c(γ, α_{s} + α) (X_{α+γ} + ∑ x_{β}(L̄_{α+γ}^{β} − L_{α+γ}^{β})) + [E_{γ}, Q_{α}].

Hence if (and only if) [E_{γ}, Q_{α}] = c(γ, α_{s} + α)Q_{α+γ}, we obtain

[ē_{γ}, X̄_{α}] = c(γ, α_{s} + α)X̄_{α+γ}.

For [h̄_{γ}, X̄_{α}] we have the same reasoning.

Finally we need to consider [X̄_{α}, X̄_{β}] = 0: clearly we need [Q_{α}, Q_{β}] = 0. All other terms (if present) involve some x_{γ}; hence its suffices to prove that [∂_{γ}, [X̄_{α}, X̄_{β}]] = 0. Now

[∂_{γ}, [X̄_{α}, X̄_{β}]] = [[∂_{γ}, X̄_{α}], X̄_{β}] + [X̄_{α}, [∂_{γ}, X̄_{β}]] = [L̄_{α}^{γ}, X̄_{β}] + [X̄_{α}, L̄_{β}^{γ}].

These commutators are commutators of 𝔤̄_{0} and 𝔤̄_{1}, which we just considered. Hence we obtain

[∂_{γ}, [X̄_{α}, X̄_{β}]] = \overline{[L_{α}^{γ}, X_{β}] + [X_{α}, L_{β}^{γ}]}.

But this is 0, since

[L_{α}^{γ}, X_{β}] + [X_{α}, L_{β}^{γ}] = [∂_{γ}, [X_{α}, X_{β}]] = 0.

Combining all results we obtain

Proposition 4.1. Suppose 𝔤̄ ⊂ W_{n} is a graded extension of a semi-simple 𝔤 ⊂ W_{m} of depth 1, with notations as before. Then, up to a linear change of coordinates, 𝔤̄ is of the form

• for degree -1: ¯∂_{α} = ∂_{α} for all α ∈ R̂; • for degree 0:

\begin{cases}ē_{α} = e_{α} + E_{α} \\ h̄_{α} = h_{α} + H_{α} \end{cases}

with {E_{α}, H_{α}} a representation of 𝔤_{0};

• for degree 1:

X̄_{α} = \frac{1}{2} ∑ x_{β}L_{α}^{β} + ∑ x_{β}(L̄_{α}^{β} − L_{α}^{β}) + Q_{α}

with [Q_{α}, Q_{β}] = 0, and X̄_{α} ↦ Q_{α} is a morphism of 𝔤̄_{0}-modules.

5. 𝔏 as a 𝔤̄-module; no trivial modules in V

Now we know the form of 𝔤̄, we study 𝔏, or rather I_{V} , as a 𝔤̄-module. As before, we write

𝔤̄ = 𝔤̄^{(1)} × 𝔤̄^{(2)} × ⋯ × 𝔤̄^{(r)},

with each 𝔤̄^{(i)} simple of depth 1. The special simple root vector in 𝔤̄^{(i)} of degree 1, we denote by e_{s}^{(i)}. In each 𝔤̄^{(i)} we have a partial Euler operator Ē^{(i)}, satisfying [Ē^{(i)}, 𝔤̄_{0}] = 0 and [Ē^{(i)}, X_{0}] = 1, where X_{0} = e_{s}^{(i)} ∈ 𝔤̄^{(i)}. If α_{1}^{(i)}, …, α_{ℓ_{i}}^{(i)} are the simple roots of 𝔤̄^{(i)} with corresponding Cartan elements h̄_{1}^{(i)}, …, h̄_{ℓ_{i}}^{(i)}, then

Ē^{(i)} = ∑_{j=1}^{ℓ_{i}} c_{j}h̄_{j}^{(i)},

The coefficients c = (c_{1}, …, c_{ℓ_{i}}) are given below for the different simple algebras of depth 1.

Algebra the vector c = (c_{1}, …, c_{ℓ})
A_{ℓ;s}  \frac{1}{ℓ+1}(ℓ + 1 − s, 2(ℓ + 1 − s), …, s(ℓ + 1 − s), s(ℓ − s), …, 2s, s)
B_{ℓ}  (1, 1, …, 1)
C_{ℓ}  (1, 2, …, ℓ − 1, \frac{1}{2}ℓ)
D_{ℓ;1} (1, 1, …, 1, \frac{1}{2}, \frac{1}{2})
D_{ℓ;ℓ}  \frac{1}{2}(1, 2, …, ℓ − 2, \frac{1}{2}ℓ, \frac{1}{2}ℓ)
E_{6}  \frac{1}{3}(4, 3, 5, 6, 4, 2)
E_{7}  \frac{1}{2}(2, 3, 4, 6, 5, 4, 3)

Table 2. The Euler operator expressed in simple Cartan elements

Essential in the sequel is that in all cases c_{j} > 0, for all j = 1, …, ℓ_{i}.

As 𝔤̄ is semi-simple, the representation theory of (finite-dimensional) representations is well-established. In our case the representation to consider is I_{V} . We will study this representation by its lowest weight vectors. We say that v is x-independent if v is independent of all x_{α} (α ∈ R̂).

Lemma 5.1. For v ∈ I_{V} the following statements are equivalent:

• v is a lowest weight vector for 𝔤̄;

• v is an x-independent lowest weight vector for 𝔤̄_{0}.

Proof. First suppose v is lowest weight vector for 𝔤̄. Since ∂_{α} ∈ 𝔤̄_{−1}, we have [∂_{α}, v] = 0, hence v is independent of all x_{α}. Moreover v is a lowest weight vector for 𝔤̄_{0} ⊂ 𝔤̄ as well. The converse is similar.

This lemma is important for us, as it turns the study of lowest weight vectors of 𝔤̄ to the study of x-independent lowest weight vectors of 𝔤̄_{0}. Note that

𝔤̄_{0} = D𝔤̄_{0} × ℂ^{r},

where D𝔤̄_{0} = [𝔤̄_{0}, 𝔤̄_{0}] and ℂ^{r} is the r-dimensional center of 𝔤̄_{0} spanned by the r partial Euler operators in 𝔤̄_{0}. These Euler operators belong to the (chosen) Cartan subalgebra of 𝔤̄, and therefore acts diagonizably on I_{V} .

For the time being we only consider v ∈ W_{n}, independent of x, and study the 𝔤̄_{0}-action on it. We will denote

W[y] = {v ∈ W_{n}|v = ∑_{i=1}^{n−m} P_{i}(y)∂_{y_{i}}}.

If v ∈ I_{V} ∩ W[y] then, due to the complete reducibility of 𝔤̄_{0}-modules and lemma 5.1, we see that v is in the 𝔤̄_{0}-module, generated by all lowest weight vectors of 𝔤̄. Note, that since 𝔤̄_{0} does not increase the (polynomial) degree of the vector field that it acts upon, any v ∈ W[y] generates a finite-dimensional 𝔤̄_{0}-module. However, a priori, the 𝔤̄-module generated by v can be infinite-dimensional. Let us investigate this situation in more detail.

First consider the space V = ⟨∂_{y_{1}}, ∂_{y_{2}}, …, ∂_{y_{n−m}}⟩, which is a 𝔤̄_{0}-submodule of 𝔏. In general 𝔤̄_{0} does not act irreducibly on V . Correspondingly, we split V ,

V = ⨁_{i} V_{(i)},

where each V_{(i)} is an irreducible module. We choose coordinates in V_{(i)} such that the Cartan elements h̄_{α} act diagonally, i.e. in these coordinates,

h̄_{α} = ∑_{i=1}^{m} a_{i}x_{i}∂_{x_{i}} + ∑_{j=1}^{n−m} b_{j}y_{j}∂_{y_{j}}.

From [10] we have some more knowledge on the modules in V . We denote by 𝔑 the largest nilpotent ideal in 𝔏. Remember that

𝔑 = {X ∈ ℜ| ad X is nilpotent}

In [10] it is proved that there exists a flag 0 = W_{t+1} ⊂ W_{t} ⊂ ⋯ ⊂ W_{1} = V of subspaces in V such that [𝔑, I_{W_{i+1}}] ⊂ I_{W_{i}} and moreover [𝔏, I_{W_{i}}] ⊂ I_{W_{i}}. For a subspace W ⊂ V , we define

I_{W} = {X ∈ 𝔏|X = ∑_{i=1}^{k} P_{i}(x, y, z)∂_{y_{i}}}.

Note that I_{W_{i}} is an ideal in 𝔏, not merely a linear subspace. In particular W_{i} is a 𝔤̄_{0}-submodule of V . Hence any W_{i} is a direct sum of the irreducible 𝔤̄_{0}-modules V_{(j)}. We choose coordinates ∂_{y_{j}^{(i)}} (i = 1, …, t; j = 1, …, t_{i} = dim W_{i} − dim W_{i−1} in V such that W_{(i)} = ⟨∂_{y_{1}^{(i)}}, …, ∂_{y_{t_{i}}^{(i)}}⟩ is a 𝔤̄_{0}-submodule of V , and W_{k} =

⊕_{i≥k}W_{(i)}. In these coordinates, the relation [𝔑, I_{W_{i+1}}] ⊂ I_{W_{i}} turns into:

Any X ∈ 𝔑 is of the form

X = ∑_{i,j} P_{ij}(x, y^{(1)}, …, y^{(i−1)})∂_{y_{j}^{(i)}},  (3)

while [𝔏, I_{W_{i}}] ⊂ I_{W_{i}} tells that in any term P (x, y)∂_{y_{j}^{(i)}} , occurring in X ∈ 𝔏 we have

P (x, y) = P (x, y^{(1)}, …, y^{(i)})  (4)

In fact for terms in Q_{α} we can do slightly better.

Lemma 5.2. Let Q_{α} be as in proposition 4.1, and suppose y_{i}y_{j}∂_{y_{k}} is a term in Q_{α}. Then if y_{i} ∈ W_{(i_{1})}^{∗} and y_{j} ∈ W_{(i_{2})}^{∗}, then ∂_{y_{k}} ∈ W_{(i_{3})} with i_{3} > i_{1} and i_{3} > i_{2}.

Proof. Since ∂_{y_{i}} ∈ 𝔑 for all i = 1, …, n − m, we obtain [∂_{y_{i}}, Q_{α}] ∈ 𝔑. The lemma now follows from (3).

Corollary 5.3. Suppose V is irreducible as a 𝔤̄_{0}-module. Then Q_{α} = 0 for all α ∈ R̂. The situation above holds in particular if dim V = 1.

Throughout the remainder of this section, we assume that V contains no trivial 𝔤̄_{0}-modules. The case that V contains some trivial 𝔤̄_{0} modules is postponed to section 6.. If V contains no trivial submodules, then any y_{i} occurs in some h̄_{j}, because [h_{j}, v] = 0 for all j implies [X, v] = 0 for all X ∈ 𝔤̄_{0}: v ∈ V generates a finite-dimensional 𝔤̄_{0}-module inside V , with all weights 0.

Lemma 5.4. Suppose V contains no trivial 𝔤̄_{0}-modules. Then W[y] contains only a finite number of linearly independent 𝔤̄-lowest weight vectors, that generate a finite-dimensional 𝔤̄-module.

Proof. Remember that a 𝔤̄-lowest weight vector v is automatically in W[y], and if it generates a finite-dimensional module, then h_{i}(v) = −λ_{i}v with all λ_{i} non-negative integers. Since the Euler operator Ē^{(j)} ∈ 𝔤̄^{(j)} is a positive sum of the h_{k}^{(j)}, this implies that Ē^{(j)}(v) = −λv with λ ≥ 0 for all lowest weight vectors v that generate a finite-dimensional 𝔤̄-module. Since Ē^{(j)} is central in 𝔤̄_{0}, it is a scalar on the irreducible 𝔤̄_{0}-submodules; in particular Ē^{(j)} in constant on each V_{(i)}. Hence [Ē^{(j)}, ∂_{y_{i}}] = −b_{i}∂_{y_{i}}, with b_{i} ≥ 0 for all i = 1, …, k. So if we define

Ē = ∑_{j} Ē^{(j)} = ∑_{i=1}^{m} x_{i}∂_{x_{i}} + ∑_{i=1}^{n−m} b_{i}y_{i}∂_{y_{i}},

we have b_{i} ≥ 0 for all i. Since y_{i} appears in at least one h_{k}^{(j)} (and hence in at least one Ē^{(j)}), we even have that b_{i} > 0 for all i = 1, …, n − m. We put b = max_{i} b_{i}. Then for a monomial y_{1}^{α_{1}}y_{2}^{α_{2}} … y_{k}^{α_{k}}∂_{y_{a}} we obtain

[Ē, y_{1}^{α_{1}}y_{2}^{α_{2}} … y_{k}^{α_{k}}∂_{y_{a}}] = βy_{1}^{α_{1}}y_{2}^{α_{2}} … y_{k}^{α_{k}}∂_{y_{a}}

with β ≥ α_{i}b_{i} − b for all i = 1, …, n − m. If the lowest weight vector v ∈ W[y] generates a finite-dimensional 𝔤̄-module, and v contains the term y^{α}∂_{y_{a}} then necessarily α_{i} ≤ b/b_{i}, for all i = 1, …, n − m. This restrict the possibilities for v generating a finite-dimensional 𝔤̄-module to a finite-dimensional space.

As a consequence to the proof of the lemma above, we state

Corollary 5.5. Consider the 𝔤̄-module S, generated by 1 and the monomials x_{i}, for i = 1, …, m. The only finite-dimensional submodule in S is ⟨1⟩.

Proof. The method of proof is the same as above; we have [Ē, x_{1}^{a_{1}} … x_{m}^{a_{m}}] = (a_{1} + a_{2} + ⋯ + a_{m})x_{1}^{a_{1}} … x_{m}^{a_{m}}. Hence the lowest weight vectors, except those in ⟨1⟩ all have positive weights; so all generated modules are infinite-dimensional.

Lemma 5.4 allows is to prove the following proposition.

Proposition 5.6. Suppose 𝔏 is such that V contains no trivial 𝔤̄_{0}-modules. Then 𝔏 is contained in a maximal graded transitive Lie algebra 𝔏′.

Proof. There are three cases to consider:

(a) V = {0}, so 𝔏 is semi-simple. Suppose 𝔏′ ⊃ 𝔏. Then we can decompose 𝔏′ into irreducible 𝔏-modules. Let M be a submodule. If X ∈ M, then by suitable differentiations, hence action with elements ∂_{x_{i}} ∈ 𝔏 we obtain that ∂_{x_{j}} ∈ M. Hence M ⊂ 𝔏, and 𝔏′ = 𝔏. So 𝔏 itself is already maximal.

(b) V ≠ {0} and V ≠ U_{−1}. We construct 𝔏′ by adding to 𝔏 all finitedimensional submodules in W_{n}. According to lemma 5.4 𝔏′ is finitedimensional, and taking commutators among elements in 𝔏′ does not take us outside 𝔏′. Clearly, 𝔏′ is maximal.

(c) V = U_{−1}. Then 𝔏 is contained in a multi-graded transitive Lie algebra 𝔏′, which can be chosen maximal (see [10, 9]). ■

A special situation occurs when Q_{α} = 0 for all α ∈ R̂. In this case we can characterize the finite-dimensional 𝔤̄-modules by the weight.

Lemma 5.7. Let 𝔏 be such that [V, [V, 𝔤̄]] = 0, and suppose that v ∈ W[y] is a lowest weight vector for 𝔤̄, satisfying [h_{i}, v] = −λ_{i}v with all λ_{i} non-negative integers. Then v generates a finite-dimensional 𝔤̄-module.

Proof. It is enough to prove that (ad ē_{i})^{λ_{i}+1}v = 0, as these last relations are the defining relations for the irreducible module of lowest weight (λ_{1}, …, λ_{m}) (see [3], §21). Due to the structure of sl_{2}(ℂ)-modules this is equivalent to proving that (ad ē_{i})^{a}v = 0 for a natural number a. For ē_{i} ∈ 𝔤̄_{0} this is clear, since the 𝔤̄_{0}-module generated by v is finite-dimensional. So now consider ē_{s}^{(i)}, with deg(ē_{s}) = 1. (We use the notations from section 4., but write X̄_{0} = ē_{s}^{(i)} omitting ^{(i)} everywhere.) This element has the form

X̄_{0} = \frac{1}{2} ∑ x_{β}L_{0}^{β} + ∑ x_{β}(L̄_{0}^{β} − L_{0}^{β}).

Since v is x-independent, we have

[X̄_{0}, v] = ∑ x_{β}[L̄_{0}^{β}, v].

Now L̄_{0}^{β} = [∂_{β}, X̄_{0}] ∈ 𝔤̄_{−} for β ≠ 0. Hence in [L̄_{0}^{β}, v] only the term [L̄_{0}^{0}, v] = [−h̄_{s}, v] contributes. Hence

[X̄_{0}, v] = x_{0}[−h̄_{s}, v] = x_{0}λ_{s}v.

The next step gives, by using [X̄_{0}, x_{0}] = [\frac{1}{2}x_{0}h̄_{s}, x_{0}] = −x_{0}^{2} (lemma 3.4)

[X̄_{0}, [X̄_{0}, x]] = [X̄_{0}, x_{0}]λ_{s}v + x_{0}λ_{s}[X̄_{0}, v] = −x_{0}^{2}λ_{s}v + x_{0}^{2}λ_{s}^{2}v = (λ_{s})(λ_{s} − 1)x_{0}^{2}v.

Inductively we find

(ad X̄_{0})^{a}v = ∏_{i=0}^{a−1}(λ_{s} − i)x_{0}^{a}v.

Hence (ad X̄_{0})^{a}v = 0 for a = λ_{s} + 1.

Remark 5.8. The characterization in lemma 5.7 does not hold when Q_{α} ≠ 0. As an example, consider in three variables 𝔏 with

𝔤̄ = ⟨∂_{x}, 2x∂_{x} + 2y_{1}∂_{y_{1}} + 2y_{2}∂_{y_{2}}, x^{2}∂_{x} + 2xy_{1}∂_{y_{1}} + 2xy_{2}∂_{y_{2}} + y_{1}^{2}∂_{y_{2}}⟩.

In this case all v ∈ W[y] are 𝔤̄-lowest weight vectors. The space of lowest weight vectors v for which [h_{1}, v] = −λv with λ ≥ 0 is

⟨∂_{y_{1}}, ∂_{y_{2}}, y_{1}∂_{y_{1}}, y_{1}∂_{y_{2}}, y_{2}∂_{y_{1}}, y_{2}∂_{y_{2}}⟩,

while the space of lowest weight vectors generating a finite-dimensional 𝔤̄-module is

⟨∂_{y_{1}}, ∂_{y_{2}}, y_{1}∂_{y_{2}}, y_{1}∂_{y_{1}} + 2y_{2}∂_{y_{2}}⟩.

The graded transitive Lie algebra 𝔏 containing 𝔤̄ above and ⟨∂_{y_{1}}, ∂_{y_{2}}, y_{1}∂_{y_{2}}, y_{1} + 2y_{2}∂_{y_{2}}⟩ is maximal (we added all 𝔤̄-modules that are possible). This example appears in section 5 of [10] for the case λ = 2.

6. 𝔏 as a 𝔤̄_{0}-module; trivial modules in V

Now we consider the case that V contains some trivial 𝔤̄_{0}-modules, say V_{(1)}, …, V_{(n−m−k)} with coordinates z_{1}, …, z_{n−m−k}. Let y_{1}, …, y_{k} be the remaining coordinates. From now on W[y] will refer to the variables y_{1}, …, y_{k} only. Since [Ē, y_{i}∂_{z_{j}}] = b_{i}y_{i}∂_{z_{j}} with b_{i} > 0 we see that terms of the form y_{i}∂_{z_{j}} are not present in I_{V} . Also 𝔤̄ contains no z_{i} or ∂_{z_{j}}:

Lemma 6.1. Let V contain n − m − k trivial 𝔤̄-modules, with coordinates z_{1}, …, z_{n−m−k}. Then z_{i} and ∂_{z_{j}} do not occur in 𝔤̄ (i, j = 1, …, n − m − k).

Proof. Since, by assumption [∂_{z_{i}}, X] = 0 for all X ∈ 𝔤̄_{0}, the only place where z_{i} or ∂_{z_{j}} can appear is in the terms of Q_{α} in X_{α} ∈ 𝔤̄_{1}. Since all terms T in Q_{α} satisfy [Ē, T] = T, we see that T contains a y_{k}, hence, like above, ∂_{z_{j}} is not possible. Hence the only possibility is T = y_{k}z_{i}∂_{y_{j}}. But now [∂_{z_{i}}, T] is a term in 𝔑 with Ē eigenvalue +1. This is impossible.

This lemma implies that 𝔏 with trivial modules is an extension of one without trivial modules, namely the algebra in x and y variables only. We denote this algebra by 𝔏(x, y):

𝔏(x, y) = 𝔤̄ ⋉ (I_{V} ∩ ℂ[x_{1}, …, x_{m}]W[y]).

We will study 𝔏 as an extension of 𝔏(x, y). If we split 𝔏(x, y) into irreducible 𝔤̄-modules, we see that the z-dependence within such a module M is “constant”: if P (z)v ∈ 𝔏, for v ∈ M, then also P (z)w ∈ 𝔏 for all w ∈ M. It follows that if M contains an element that doesn’t act nilpotently in 𝔏, then P (z) must be constant. This happens in particular for simple Levi subalgebras in 𝔏, as well as for ℜ/𝔑. Hence only M ⊂ 𝔑 ∩ 𝔏(x, y) can get z-dependent coefficients.

Important is the following lemma.

Lemma 6.2. Suppose V ⊂ 𝔏 contain the maximal trivial 𝔤̄_{0}-module W̃ = ⟨∂_{z_{1}}, …, ∂_{z_{n−m−k}}⟩ with complementary module W = ⟨∂_{y_{1}}, …, ∂_{y_{k}}⟩. Then I_{W} is an ideal in 𝔏.

Proof. Suppose Y ∈ I_{W}. We need to prove that [X, Y] ∈ I_{W} for all X ∈ 𝔏. There are 4 cases to consider:

(1) ∂_{x_{i}} ∈ 𝔤̄_{−1}; [∂_{x_{i}}, Y] ∈ I_{W} is clear.

(2) X ∈ 𝔤̄_{0}; since W is a 𝔤̄_{0}-submodule, [X, Y] ∈ I_{W}.

(3) X_{α} ∈ 𝔤̄_{1}; since Q_{α} contains no terms of the form y_{i}y_{j}∂_{z_{a}} or y_{i}z_{j}∂_{z_{a}}, [X_{α}, Y] ∈ I_{W}.

(4) X ∈ I_{V} ; the argument is the same is in (3).

Hence I_{W} is an ideal.

From this lemma we can obtain a sequence of spaces W_{i} (and hence W_{(j)}) compatible with the z and y-variables. Let us describe this construction. First we consider 𝔏(x, y) := 𝔤̄×(I_{W} ∩W[y]). If Z(𝔑′) denotes the center of the nilradical of 𝔏(x, y), then we define W_{t} = Z(𝔑′) ∩ W, say W_{t} = ⟨y_{1}^{(t)}, …, y_{i_{t}}^{(t)}⟩. Here t is the number of steps that has to be taken in the process below. By construction [∂_{y_{i}^{(t)}}, X] = 0 for i = 1, …, i_{t} and X ∈ 𝔑′; from this using lemma 6.2 one easily obtains that the variables y_{1}^{(t)}, …, y_{i_{t}}^{(t)} appear only in the coefficients of ∂_{y_{1}^{(t)}}, …, ∂_{y_{i_{t}}^{(t)}} .

After this first step we continue with 𝔏(x, y)/I_{W_{s}}, and we obtain in a similar fashion W′_{t−1}, which in 𝔏(x, y) lifts to W_{t−1} = W′_{t−1} + W_{t}. This process we repeat till W is exhausted; then we proceed with 𝔏/I_{W}. Hence we can choose the 𝔤̄_{0}-modules W_{(1)}, …, W_{(t)} such that the first ones are subspaces of W̃ = ⟨∂_{z_{1}}, …, ∂_{z_{n−m−k}}⟩, while the last ones are subspaces of W = ⟨∂_{y_{1}}, …, ∂_{y_{k}}⟩. We can choose the y and z coordinates such that W_{(1)}, …, W_{(t)} are spanned by monomials.

Proposition 6.3. Let 𝔏 be graded transitive Lie algebra, such that V contains trivial 𝔤̄_{0}-modules. Then: 𝔏 is not maximal.

Proof. We construct the spaces W_{(1)}, …, W_{(t)} as described above. Consequently, any X ∈ 𝔑 is of the form

X = ∑_{i,j} P_{ij}(z^{(1)}, …, z^{(i−1)}))∂_{z_{j}^{(i)}} + ∑_{i,j} Q_{ij}(z, y^{(1)}, …, y^{(i−1)})∂_{y_{j}^{(i)}}.  (5)

We give to terms of the form occurring in (5) a ℤ-valued degree, which we call the zdeg, such that all terms in 𝔑 have non-positive degree. (Remember that z-dependent terms only occur in 𝔑.) To start we put zdeg(z_{j}^{(1)}) = 1 =: d_{1} for all j. Next for X ∈ 𝔑 we write

X = ∑_{j} P_{2j}(z^{(1)})∂_{z_{j}^{(2)}} + other terms

Let d′_{2}(X) be the maximum over the zdeg of the monomials occurring in P_{2,j} and d′′_{2} = max_{X} d_{2}(X). We now put zdeg(z_{j}^{(2)}) = d_{2} ≥ d′′_{2}. This way we continue: for d_{3} we take a (natural) number at least as big as the maximum over the zdeg of the monomials occurring in P_{3j}, and we put zdeg(z_{j}^{(3)}) = d_{3}. In the end we gave all z and y variables a degree, which is the same (namely d_{i}) for variables belonging to the same module W_{(i)}.

By construction, all terms T occurring in X ∈ 𝔑 have zdeg(T) ≤ 0. Moreover there are only finitely many terms T with zdeg(T) ≤ 0. Thanks to (4) and recalling that I_{V} ∩ 𝔏_{i} ⊂ 𝔑 for i ≠ 0, we see that all terms occurring in I_{V} have zdeg(T) ≤ 0.

Now we make an extension of 𝔏, which we denote by (𝔏(x, y), d), where d = (d_{1}, …, d_{t}). We add to 𝔏(x, y) the following z-dependent elements

⟨z_{1}^{a_{1}} … z_{n−m−k}^{a_{n−m−k}}X|X ∈ W[y] ∩ 𝔑⟩ ⊕ ⟨z_{1}^{b_{1}} … z_{n−m−k}^{b_{n−m−k}}∂_{z_{b}}⟩  (6)

with ∑_{i} a_{i} zdeg(z_{i}) ≤ − zdeg(T) for all terms T in X, and ∑_{i} b_{i} zdeg(z_{i}) ≤ zdeg(z_{b}). By construction, (𝔏(x, y), d) ⊃ 𝔏, and (𝔏(x, y), d) is a graded transitive Lie algebra. However, increasing d_{t} to d′_{t} = d_{t} + 1, while fixing d_{1}, …, d_{t−1}, we obtain 𝔏′, which properly contains 𝔏: z_{1}^{d′_{t}}∂_{y_{k}} is in 𝔏′, but not in 𝔏. Hence 𝔏 is not maximal.

Remark 6.4. When we write (𝔏(x, y), d), we assume throughout that we have a sequence of space W_{(i)} like above, i = 1, …, t, such that the first, say t̃ spaces consists of (all) trivial 𝔤̄_{0}-modules, and d_{i} = zdeg(z_{j}^{(i)}) for all z_{j}^{(i)} ∈ W_{(i)}^{∗}. We can (and will) assume that d_{i} is a non-negative integer, but moreover that d_{i} ≥ 1 for i = 1, …, t̃. Note that the zdegree is constant on all irreducible submodules V_{(i)} in V .

The situation in lemma 6.3 is not completely satisfactory. Take an 𝔏 for which V contains some trivial 𝔤̄-modules. Sometimes it is possible to construct a maximal 𝔏′ properly containing 𝔏. Let us denote V , 𝔤̄ etc. corresponding to 𝔏′ by V′, 𝔤̄′ etc. From proposition 6.3 we know that V′ contains no trivial 𝔤̄′- modules: hence some variables among z_{1}, …, z_{n−m−k} (say the first a) are turned into x-variables. Hence

𝔤̄′ = 𝔤̄^{(1)} × ⋯ × 𝔤̄^{(r)} × ⋯ × 𝔤̄^{(r′)},

with r′ > r. Corresponding to 𝔤̄^{(r+1)}×⋯×𝔤̄^{(r′)} we have the partial Euler operator, denoted by Z:

Z = ∑_{i=1}^{a} z_{i}∂_{z_{i}} + ∑_{i=a+1}^{n−m−k} c_{i}z_{i}∂_{z_{i}} + ∑_{j=1}^{k} b_{j}y_{j}∂_{y_{j}}.

Like in lemma 5.4 we have that c_{i} > 0 and b_{j} ≥ 0 for all i and j. It seems difficult to investigate whether a maximal 𝔏′ containing 𝔏 always exists. A criterion is given in the following lemma.

Lemma 6.5. Suppose V ⊂ 𝔏 contains trivial 𝔤̄_{0}-modules. If there exists an 𝔏′ = (𝔏′(x, y), d) such that

(1) 𝔏 is contained in (𝔏′(x, y), d);

(2) The vector field Y ,

Y = ∑_{i,j} d_{i}y_{j}^{(i)}∂_{y_{j}^{(i)}}

is in 𝔏′(x, y),

then 𝔏 is contained in a maximal graded transitive Lie algebra.

Proof. We define the vector field Z,

Z = ∑_{i,j} d_{i}z_{j}^{(i)}∂_{z_{j}^{(i)}} + ∑_{i,j} d_{i}y_{j}^{(i)}∂_{y_{j}^{(i)}}.

By assumption (2) above, we have that Z ∈ (𝔏′(x, y), d). We will say that a term T in W_{n} has zdegree d if [Z, T] = T. Since Z ∈ (𝔏′(x, y), d), we can find a basis in (𝔏′(x, y), d) of homogeneous elements with respect to the zdegree. If {X_{i}} is a zdegree-homogeneous basis in 𝔏′, then zdeg(X_{i}) < 0 implies X_{i} ∈ 𝔑′. We now extend (𝔏′(x, y), d) to a maximal graded transitive Lie algebra. Let z_{1}, … z_{a} be the variables of zdegree 1. We consider the vector fields Z_{i} defined by

Z_{i} = z_{i}Z for i = 1, …, a.

We show that {X_{j}}∪{Z_{i}} (j = 1, …, dim 𝔏′ and i = 1, …, a) form a basis of a Lie algebra, which we denote by 𝔏′′. First direct calculation shows that [Z, Z_{i}] = Z_{i} and [Z_{i}, Z_{j}] = 0. It remains to check [Z_{i}, X_{j}]. We distinguish several cases:

• X_{j} = ∂_{z_{c}}. Then

[Z_{i}, X_{j}] = [z_{i}Z, ∂_{z_{c}}] = z_{i}[Z, ∂_{z_{c}}] + [z_{i}, ∂_{z_{c}}]Z = −z_{i}∂_{z_{c}} − Z

• X_{j} = z_{b}∂_{z_{i}}. Then

[Z_{i}, X_{j}] = [z_{i}Z, z_{b}∂_{z_{i}}] = [z_{i}, z_{b}∂_{z_{i}}]Z = −Z_{b}

• X_{j} ∉ 𝔑′ and X_{j} does not contain ∂_{z_{i}}. Hence zdeg(X_{j}) = 0 and

[Z_{i}, X_{j}] = [z_{i}Z, X_{j}] = z_{i}[Z, X_{j}] + [z_{i}, X_{j}]Z = 0.

• The remaining case: X_{j} ∈ 𝔑′, zdeg(X_{j}) = d and X_{j} does not contain ∂_{z_{i}}:

[Z_{i}, X_{j}] = z_{i}[Z, X_{j}] + [z_{i}, X_{j}]Z = dz_{i}X_{j}.

If d = 0, then [Z_{i}, X_{j}] = 0, while if d < 0, then z_{i}X_{j} ∈ 𝔏′, as desired.

Finally we see that 𝔏′′ contains an extra simple Levi subalgebra, isomorphic to sl(a + 1, ℂ), with basis {∂_{z_{i}}, z_{i}∂_{z_{j}} (i ≠ j), Z + z_{i}∂_{z_{i}}, Z_{i}} where i, j = 1, …, a. The corresponding Euler operator is exactly Z. Since [Z, ∂_{z_{j}^{(i)}}] = d_{i}∂_{z_{j}^{(i)}} for all i, j and

by our assumption d_{i} ≠ 0 for all i ≤ t̃ (remark 6.4), we see that in V′′ = U_{−1} ∩ ℜ′′ there are no trivial 𝔤̄′′_{0}-modules. By proposition 5.6, 𝔏′′ (and hence 𝔏) is contained in a maximal graded transitive Lie algebra.

Let us demonstrate how this lemma can be used. We say that a term T,

T = x_{1}^{a_{1}} … x_{m}^{a_{m}}y_{j_{1}}^{(i_{1})} … y_{j_{b}}^{(i_{b})}∂_{y_{j}^{(i)}}

has type (i_{1}, …, i_{b}; i). Here we can assume that i_{1} ≤ i_{2} ⋯ ≤ i_{b}. We say that a vector field v has type (i_{1}, …, i_{b}; i) if all terms in v are of this type. Note that all y-dependent term that occur in 𝔤̄_{0} are of type (i; i). Consequently, all terms in a lowest weight vector of a certain type, form by themselves also a lowest weight vector for 𝔤̄_{0}. Unfortunately, the terms Q_{α} that appear in X_{α} ∈ 𝔤̄_{1} break the decomposition in different types. But if all Q_{α} = 0 then we obtain the following result.

Proposition 6.6. Let 𝔏 be such that [V, [V, 𝔤̄]] = 0. Then 𝔏 is contained in a maximal graded transitive Lie algebra.

Proof. We can extend 𝔏(x, y) to a maximal 𝔏′(x, y) (in W_{m+k}) by proposition 5.6. Since all terms in 𝔤̄ preserve the type of a term (Q_{α} = 0 !), we can assume that the 𝔤̄-modules in I_{V} are generated be lowest weight vectors of a certain type. It is possible to choose d = (d_{1}, …, d_{t}) such that (𝔏′(x, y), d) contains 𝔏. Consider

Y = ∑_{i,j} d_{i}y_{j}^{(i)}∂_{y_{j}^{(i)}}.

Clearly [Y, X] = 0 for all vector fields X of type (i; i); hence [Y, 𝔤̄] = 0, which implies Y ∈ 𝔏′(x, y). Applying lemma 6.5, we find that 𝔏 is contained in a maximal graded transitive Lie algebra.

7. The low dimensional cases

We now apply the results of the previous sections to describe all maximal graded transitive Lie algebras 𝔏 in the cases n ≤ 3.

Let us start with n = 1. Since m ≥ 1, we see that n = m, and hence 𝔏 = 𝔤. So 𝔏 is simple, and looking at table 1. we obtain that 𝔏 = A_{1;1}, i.e. 𝔏 = ⟨∂_{x}, x∂_{x}, x^{2}∂_{x}⟩.

For n = 2 we have the possibilities m = 1 and m = 2. If m = 1, then 𝔤 = A_{1;1} as above, and dim V = 1. Using lemma 5.2, this implies that

𝔤̄ = ⟨∂_{x}, x∂_{x} + b_{1}y∂_{y}, x^{2}∂_{x} + \frac{1}{2}b_{1}xy∂_{y}⟩.

with b_{1} ≥ 0. If b_{1} ≠ 0, we can add the lowest weight vectors ∂_{y} and y∂_{y}, and obtain the algebra

𝔏 = ⟨∂_{x}, x∂_{x} + b_{1}y∂_{y}, x^{2}∂_{x} + \frac{1}{2}b_{1}xy∂_{y}, y∂_{y}, x^{k}∂_{y} (k ≤ b_{1})⟩.

If b_{1} = 0 then 𝔏 is contained in A_{1;1} × A_{1;1}, hence 𝔏 is not maximal. If m = 2 there are two possibilities: either 𝔏 is simple, 𝔏 = A_{2;1} or 𝔏 is semi-simple, 𝔏 = A_{1,1} × A_{1;1}. Note that up to now all algebras are multi-graded.

We continue with n = 3, hence m = 1 or m = 2 or m = 3. If m = 1 then 𝔤 = A_{1;1} as above. Now 𝔤_{0} = ℂ, hence all irreducible representations are of dimension 1. Again using lemma 5.2 we obtain

𝔤̄ = ⟨∂_{x}, x∂_{x} + b_{1}y_{1}∂_{y_{1}} + b_{2}y_{2}∂_{y_{2}}, x^{2}∂_{x} + \frac{1}{2}b_{1}xy_{1}∂_{y_{1}} + \frac{1}{2}b_{1}xy_{1}∂_{y_{1}} + cy_{1}^{2}∂_{y_{2}}⟩. (7)

If b_{1} = 0 or b_{2} = 0, then c = 0 (lemma 6.1), and 𝔏 is not maximal (proposition 6.3), but is contained in a maximal one (proposition 6.6). So consider c = 0 and we can assume that b_{2} ≥ b_{1} > 0. Moreover b_{1} and b_{2} are integers. If b is the biggest integer β with β ≤ b_{2}/b_{1} then we obtain that 𝔏 is generated by 𝔤̄ and the 𝔤̄-lowest weight vectors y_{1}∂_{y_{1}}, y_{2}∂_{y_{2}}, ∂_{y_{1}} and y_{1}^{β}∂_{y_{2}} with β ≤ b.

If c ≠ 0, then for X = x^{2}∂_{x} + \frac{1}{2}b_{1}xy_{1}∂_{y_{1}} + \frac{1}{2}b_{1}xy_{1}∂_{y_{1}} + cy_{1}^{2}∂_{y_{2}} we need

[x∂_{x} + b_{1}y_{1}∂_{y_{1}} + b_{2}y_{2}∂_{y_{2}}, X] = X,

which implies that b_{1} + \frac{1}{2}b_{2} = 1, so b_{2} = 2(b_{1} − 1). The lowest weight vectors that generate a finite-dimensional 𝔤̄-module are given by

∂_{y_{1}}, ∂_{y_{2}}, y_{1}∂_{y_{2}}, y_{1}∂_{y_{1}} + 2y_{2}∂_{y_{2}},

compare to remark 5.8. This finishes the case m = 1. If m = 2, then either 𝔤 = A_{2;1} or 𝔤 = A_{1;1} × A_{1;1}. In both cases the lowest weight vectors in I_{V} are given by ∂_{y_{1}} and y_{1}∂_{y_{1}} . If m = 3 then by table 1 there are 4 possibilities: A_{3;1}, A_{2;1} × A_{1;1}, A_{1;1} × A_{1;1} × A_{1;1} and B_{2} (or the isomorphic C_{2}).

We will not discuss the case n = 4 completely. However, there is one interesting case for 𝔏, namely m = 1 and k = 2, i.e. V contains one trivial 𝔤̄_{0}-module. Only in this case, we have a trivial 𝔤̄_{0}-module in V , while Q ≠ 0 is possible; consequently we have no proposition guaranteeing that 𝔏 is contained in a maximal algebra. We show that it is, nevertheless. We have that 𝔤̄ is as given in (7) with b_{2} = 2(b_{1} − 1) > 0. Further that 𝔏(x, y) is contained in the maximal algebra 𝔏′(x, y) with lowest weight vectors (see above) ∂_{y_{1}}, ∂_{y_{2}}, y_{1}∂_{y_{2}}, y_{1}∂_{y_{1}}+2y_{2}∂_{y_{2}} . We try to apply lemma 6.5, with d_{1} = 1 (as always), d_{2} = δ and d_{3} = 2δ, hence δy_{1}∂_{y_{1}} + 2δy_{2}∂_{y_{2}} ∈ 𝔏′(x, y), and δ is chosen big enough to ensure that 𝔏 is contained in (𝔏′(x, y), d). That this is possible is not obvious, since d_{2} and d_{3} are connected by a relation (otherwise we would increase first d_{2} and then d_{3} sufficiently). Let a and b be the maximal numbers such that the terms z^{a}∂_{y_{1}} and z^{b}∂_{y_{2}} appear in 𝔏. Now choose δ = max{a, b}. Then zdeg(z^{a}∂_{y_{1}}) = a−δ ≤ 0 and zdeg(z^{b}∂_{y_{2}}) ≤ b − 2δ ≤ 0. It remains to check the term z^{c}∂_{y_{1}}∂_{y_{2}} appearing in 𝔏. However taking the commutator of this term with ∂_{y_{1}} we see that c ≤ b. Hence zdeg(z^{c}∂_{y_{1}}∂_{y_{2}}) ≤ b + δ − 2δ ≤ 0. Applying lemma 6.5, we see that 𝔏 is contained in a maximal 𝔏′′ with 𝔤̄′′ = A_{1;1} × A_{1;1}.

8. Conclusion

We were able to give a fairly detailed description of the structure of graded transitive Lie algebras 𝔏. In particular the maximal ones are described. It turns out that these maximal ones have so much structure that they can be described by general (n independent) theorems. Two questions are not answered completely satisfactory. First the problem of Q_{α}; can we describe in more detail when Q_{α} ≠ 0 is possible? Second, if Q_{α} ≠ 0 and V contains trivial 𝔤̄_{0}-modules, we do not know whether 𝔏 is contained in a maximal one. Do there exist 𝔏 not contained in a maximal one? For such 𝔏, according to section 7., n ≥ 5.

Going back to Lie’s original problem, we can hope that these results will help in two directions. First the graded, but non-transitive case, and second the transitive, but not necessarily graded case. The last problem is particularly interesting, but also very difficult. The realization theorem of Guillemin and Sternberg (see [2]) states that any pair (𝔏; 𝔏_{+}) of abstract Lie algebras, where 𝔏_{+} is a subalgebra in 𝔏 of codimension n containing no ideals of 𝔏, can be realized as a transitive Lie algebra of formal vector fields in n variables. Moreover this realization is unique up to a (formal) change of coordinates. In particular, if 𝔏 is simple, 𝔏_{+} can be chosen arbitrary. Hence the classification of finitedimensional transitive Lie algebras of formal vector fields entails the classification of all subalgebras of a simple Lie algebra.

Post 287 Appendix – Notations

For the benefit of the reader, we provide a list with notations that are used in different sections.

Name Description
n  dimension of the space
W_{n}  Lie algebra of polynomial vector fields on ℂ^{n}
U_{k}  vector fields in W_{n} of degree k
𝔏  graded transitive Lie algebra (subalgebra of W_{n})
E  Euler operator
ℜ  radical of 𝔏 (largest solvable ideal in 𝔏)
𝔑  nilradical of 𝔏 (largest nilpotent ideal in 𝔏)
V  ℜ ∩ U_{−1} (constant vector fields in ℜ)
I_{V}  {X ∈ 𝔏|X = ∑_{i=m+1}^{n} P_{i}(x)∂_{x_{i}}}
𝔤  𝔏/I_{V} (semi-simple summands of 𝔏 modulo I_{V})
𝔤̄  graded Levi subalgebra of 𝔏, such that 𝔤̄/I_{V} is isomorphic to 𝔤
m  n − dim V = dim(U_{−1} ∩ 𝔤̄)
R  roots of 𝔤 or 𝔤̄
α_{s}  simple root in R such that deg e_{s} = 1
R̂  all α ≥ 0 such that α + α_{s} ∈ R
Ē^{(j)} Euler operator of j^{th} simple summand in 𝔤̄
r  number of simple summands in 𝔤̄
Ē  sum of all Ē^{(j)}
x  the variables x_{1}, …, x_{m} or x_{α} occurring in 𝔤
y  the variables in V^{∗}, not in trivial modules
z  the variables in V^{∗}, but not occurring in 𝔤̄

References

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Gerhard Post Faculty of Mathematical Sciences University Twente P.O. Box 217 7500 AE Enschede, The Netherlands post@math.utwente.nl

Received January 9, 2001 and in final form July 1, 2001