Journal of Lie Theory Volume 11 (2001) 111–128 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag Berlin

On the structure of transitively differential algebras

Gerhard Post

Communicated by P. Olver

Abstract. We study finite-dimensional Lie algebras 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 derive some general results on the structure of such Lie algebras, and provide the complete classification in the cases n = 2 and n = 3. Finally we describe a certain construction in high dimensions.

1. Introduction

It was Lie [5] who started to study the classification problem for finite-dimensional Lie algebras 𝔏 of vector fields in ℂ^{n} up to local diffeomorphisms. His results concern the cases n = 1, n = 2 and n = 3 (this last case is not treated completely). He split the study of these algebras in the case of primitive algebras (the corresponding local transformation group has no invariant foliation) and imprimitive ones. The lists for the imprimitive algebras in [5] are, in a sense, without structure: it is impossible to predict what (classes of) algebras one will encounter. The mathematical solution for this kind of problems is adding a structure: one can study for instance simple algebras or primitive algebras. For sure it is very difficult to obtain general results for the general classification problem. Also here we add some structure by assuming that

(O) the coefficients of X ∈ 𝔏 are polynomials;

(A) 𝔏 contains the translations \frac{∂}{∂x_{i}} (i = 1, …, n) (“𝔏 acts transitive”);

(B) 𝔏 contains the Euler vector field E = ∑_{i=1}^{n} x_{i} \frac{∂}{∂x_{i}} (“𝔏 is graded”).

In fact, due to the presence of E in 𝔏, we may assume that the coefficients P_{i} of

X = ∑ P_{i}(x)\frac{∂}{∂x_{i}} ∈ 𝔏 are all homogeneous polynomials of the same degree k; this

follows by considering the commutator [E, X].

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

Note that in any finite-dimensional Lie algebra 𝔤 of C^{∞}-vector fields in ℂ^{n} we have a natural filtering by the minimal degree of the Taylor series (around 0) of the coefficients. The associated graded Lie algebra will be of the type we consider in case 𝔤 (or rather G) acts transitively (around 0). This explains a part of the interest in these algebras. For further motivations, we refer to [1].

Our aim in this paper is to present some general results on the structure of Lie algebras 𝔏 with the structure above. As background we have the classification in a special subcase, namely when 𝔏 is multigraded. Such 𝔏 is described (almost completely) by a diagram made from the integers a_{ij} with

a_{ij} = max{α|x_{i}^{α}∂_{x_{j}} ∈ 𝔏}.

We discuss this case in section 2..

It turns out that there are many cases in which 𝔏 is not multigraded. However assuming that 𝔏 is maximal (so that 𝔏 becomes a so called transitively differential algebra of certain order), most examples drop out; it is a non-trivial task to distinguish the remaining ones.

We describe the contents of the paper. In section 2. we discuss multigraded algebras, and in section 3. we explain how the elements from 𝔏 can be seen as multilinear mappings. In section 4. we give some general results on the Lietheoretical structure of 𝔏. Our main result is that the radical is contained in a multigraded Lie algebra. Next we apply these structural results to the cases n = 2 and n = 3 (section 5.); we obtain a complete classification of transitively differential algebras here. In section 6. we describe an elegant construction in higher dimensions. We end with a discussion in section 7..

Acknowledgement. Useful correspondence with Issai Kantor, especially on the matters in the sections 3. and 4., is gratefully acknowledged.

2. Multigraded algebras

In this section we discuss the definition of (essential) multigraded algebras and describe some results obtained in [4].

Throughout this paper 𝔏 will denote a finite-dimensional Lie algebra of polynomial vector fields on ℂ^{n}, satisfying the requirements (A) and (B) in section 1.. For any X ∈ 𝔏, we see that all its homogeneous components are in 𝔏. If

X = ∑_{i=1}^{n} P_{i}(x)\frac{∂}{∂x_{i}}

where all P_{i} are homogeneous polynomials of degree k, then

[E, X] = (k − 1) · X.  (1)

We call k the order of X. The maximal order ν among X ∈ 𝔏 is called the order ν of 𝔏, ord(𝔏); in this case 𝔏 is said to belong to the class D^{ν}(n) (or D^{ν} if it is clear what n is).

It follows easily from (1) that 𝔏 is ℤ-graded:

𝔏_{d} = {X ∈ 𝔏 | ord(X) = d + 1}

with

𝔏 = ⨁_{d=−1}^{ν−1} 𝔏_{d} and [𝔏_{d_{1}}, 𝔏_{d_{2}}] ⊂ 𝔏_{d_{1}+d_{2}}.

In particular we have 𝔏_{−1} = ⟨∂_{1}, …, ∂_{n}⟩. From now on we write ∂_{i} = \frac{∂}{∂x_{i}} for

notational convenience.

A transitively differential algebra of order ν is a maximal Lie algebra in D^{ν}. To be more explicit

Definition 2.1. Suppose 𝔏 belongs to D^{ν} and 𝔏 is maximal in D^{ν}, which means that 𝔏′ ∈ D^{ν} and 𝔏′ ⊃ 𝔏 implies 𝔏′ = 𝔏, then 𝔏 is called a transitively differential algebra of order ν.

All 𝔏 in the class D^{ν} are ℤ-graded; sometimes will also encounter 𝔏 that are even ℤ^{n}-graded. Remember that ℂ[x_{1}, …, x_{n}] has a ℤ^{n}-graded, mdeg, by

mdeg(x^{α}) = α

Correspondingly, the polynomial vector fields, being realized as special elements of End(ℂ[x_{1}, …, x_{n}]), attain a ℤ^{n}-grading by

mdeg(x^{α}∂_{j}) = (α_{1}, …, α_{j−1}, α_{j} − 1, α_{j+1}, …, α_{n}).  (2)

We say that 𝔏 is multigraded if it has a basis of homogeneous elements in the ℤ^{n}-grading (2). Equivalently 𝔏 has a basis of simultaneous eigenvectors for the operators ad x_{1}∂_{1}, ad x_{2}∂_{2}, …, ad x_{n}∂_{n}. Hence we have a decomposition

𝔏 = ⨁_{α∈ℤ^{n}} 𝔏_{α} ; [𝔏_{α}, 𝔏_{β}] ⊂ 𝔏_{α+β}

where for α = (α_{1}, …, α_{n}) we have

{𝔏_{α} = {X ∈ 𝔏 | [x_{i}∂_{i}, X] = α_{i}X for i = 1 … n}

Consequently any multigraded transitively differential algebra contains the elements x_{1}∂_{1}, …, x_{n}∂_{n}.

Let us discuss the structure of a multigraded Lie algebra 𝔏 in the class D^{ν}. We assume that the elements x_{1}∂_{1}, … x_{n}∂_{n} are in 𝔏; this can be done without restriction. According to [4] the remaining homogeneous elements X ∈ 𝔏 fall in two classes (up to a multiple):

X = x_{1}^{α_{1}} ⋯ x_{n}^{α_{n}}∂_{j} with α_{j} = 0  (3)

or ord(X) = 2 and

X = X_{j} = ∑_{i=1}^{n} a_{ji}x_{i}x_{j}∂_{i} with a_{jj} = 1.  (4)

If the element X_{j} is in 𝔏 then we call j a special point. There is a remarkable relation between the coefficients a_{ij} appearing in (4) and the maximal power α such that x_{i}^{α}∂_{j} ∈ 𝔏. Indeed, investigations in [4] reveal that A = (a_{ij}) is the matrix given by

a_{ij} = max{α|x_{i}^{α}∂_{j} ∈ 𝔏}  (5)

There are two characteristic conditions for A and the special points:

(I) a_{ij}a_{jk} ≤ a_{ik} for all i, j, k = 1..n;

(II) j is special and a_{ij} ≠ 0 for an i implies a_{ij} = a_{ji} = 1 and moreover i is special as well.

Clearly, for any multigraded 𝔏 in the class D^{ν} we can define the matrix A by formula (5), and mark the special points S ⊂ {1, 2, …, n}. A natural question is whether A and the special points S determine 𝔏. This is not the case, but it turns out that there is a unique maximal algebra 𝔏(A, S) with matrix A and special points S, see [4], section V. This provides a rather detailed description of all multigraded algebras.

Example 2.2. Let us take n = 4, A = (\begin{matrix}1 & 3 & 11 & 11 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{matrix}) and S = \{1\}. Then

𝔏(A, S) = ⟨∂_{1}, x_{i}∂_{i} (i = 1..4), x_{3}∂_{4}, x_{4}∂_{3}, x_{1}^{2}∂_{1} + 3x_{1}x_{2}∂_{2} + 11x_{1}x_{3}∂_{4} + 11x_{1}x_{4}∂_{4},

x_{1}^{k}∂_{2} (k ≤ 3), x_{1}^{k}x_{2}^{ℓ}∂_{i} (k + 3ℓ ≤ 11, ℓ ≤ 2, i = 3, 4)⟩  (6)

Now 𝔏(A, S) contains multigraded subalgebras with same matrix A and the same special point 1. An example is

𝔏 = ⟨∂_{1}, x_{i}∂_{i} (i = 1..4), x_{3}∂_{4}, x_{4}∂_{3}, x_{1}^{2}∂_{1} + 3x_{1}x_{2}∂_{2} + 11x_{1}x_{3}∂_{4} + 11x_{1}x_{4}∂_{4},

x_{1}^{k}∂_{2} (k ≤ 3), x_{1}^{k}∂_{i} (k ≤ 11, i = 3, 4), x_{1}^{k}x_{2}∂_{i} (k ≤ 3, i = 3, 4), x_{2}^{2}∂_{i} (i = 3, 4)⟩

In fact 𝔏 is the minimal multigraded Lie algebra with matrix A and special point 1.

If we define A′ = (\begin{matrix}1 & 3 & 11 & 11 \\ 0 & 1 & 3 & 3 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{matrix}) then 𝔏(A, S) ⊂ 𝔏(A′, S). Hence

𝔏(A, S) is not a transitively differential algebra of order 11, but 𝔏(A′, S) is.

One could hope that any multigraded transitively differential algebra 𝔏 is of the form 𝔏(A, S). This is not the case, though 𝔏 is contained in 𝔏(A, S), for certain A and S. Possibly 𝔏(A, S) is of higher order than 𝔏. The simplest example where this happens is n = 3 and

𝔏 = ⟨∂_{i}, x_{j}∂_{j}, x_{2}∂_{3}, x_{2}^{2}∂_{3}, x_{2}^{2}∂_{2} + 2x_{2}x_{3}∂_{3}, x_{1}∂_{3}, x_{1}^{2}∂_{3}, x_{1}x_{2}∂_{3}, x_{1}x_{2}^{2}∂_{3}⟩.

Any 𝔏(A, S) ⊃ 𝔏 contains the element x_{1}^{2}x_{2}^{2}∂_{3}, which is of order 4, while ord(𝔏) = There is a convenient way to construct multigraded Lie algebras. Suppose we give all variables x_{1}, x_{2}, …, x_{n} a degree:

deg(x_{1}) = d_{1}; deg(x_{2}) = d_{2}; …; deg(x_{n}) = d_{n};

with d_{i} > 0 for all i. This induces a degree on the vector field terms by

deg(x_{1}^{α_{1}}x_{2}^{α_{2}} … x_{n}^{α_{n}}∂_{j}) = α_{1}d_{1} + α_{2}d_{2} + ⋯ + α_{n}d_{n} − d_{j}.  (7)

Now suppose that 𝔏 is the Lie algebra with basis consisting of all vector field terms of degree 0 or less. By our assumption that d_{i} > 0 we have that 𝔏 is indeed finitedimensional. Moreover it is clear that ∂_{i} and the Euler vector field E are contained in 𝔏. Hence 𝔏 belong to D^{ν} for ν = ⌊max{d_{1}, …, d_{n}}/ min{d_{1}, …, d_{n}}⌋.

Let us return to Lie algebras 𝔏 in the class D^{ν}, not necessarily multigraded. Let L : ℂ^{n} → ℂ^{n} be a linear, invertible transformation. This transformation induces a transformation on the polynomial vector fields (as represented in some coordinate system), preserving the order. In particular a Lie algebra in the class D^{ν} is mapped to another one in D^{ν}; these two will be called equivalent. Note that the multigrading of an element is not preserved.

We will call 𝔏 essentially multigraded if 𝔏 is equivalent to a multigraded Lie algebra; this means that coordinates can be chosen such that 𝔏 is multigraded with respect to these coordinates.

To end this section, we state a lemma that gives a way to test if 𝔏 is possibly essential multigraded, without performing a linear transformation. For

X = ∑_{i=1}^{n} P_{i}∂_{i} ∈ 𝔏, we can form the Jacobian J(X) with

J_{ij}(X) = \frac{∂P_{i}}{∂x_{j}} .

Lemma 2.3. Let 𝔏 be in class D^{ν}, with ν ≥ 3 and dim(𝔏) = d. If 𝔏 is essentially multigraded, then 𝔏 has a basis {X_{1}, X_{2}, …, X_{d}} such that rank J(X) = 1 for all X ∈ {X_{1}, X_{2}, …, X_{d}} with ord(X) ≥ 3.

Proof. The rank of J(X) is independent of the (linear) coordinates chosen. Hence, if 𝔏 is essentially multigraded, we can choose coordinates such that 𝔏 is multigraded. But for multigraded Lie algebras there is a homogeneous basis {X_{1}, X_{2}, …, X_{d}} such that the elements of order 3 and higher have the form X = x_{1}^{α_{1}} … x_{n}^{α_{n}}∂_{j}, see formula (3). Those X obviously have rank J(X) = 1.

3. 𝔏 and multilinear mappings

A very useful description of vector fields is in term of multilinear mappings, see for example [2]. Let X ∈ 𝔏, ord(X) = k, and denote U_{−1} = ⟨∂_{1}, …, ∂_{n}⟩. We associate to X the k-linear map A : U_{−1} × U_{−1} × ⋯ × U_{−1} → U_{−1} by

A(v_{1}, v_{2}, …, v_{k}) = \frac{1}{k!}[v_{1}, [v_{2}, …, [v_{k}, X] …]]

for v_{1}, v_{2}, …, v_{k} ∈ U_{−1}. We will write A = φ_{k}(X). Using the Jacobi identity one easily checks that A is symmetric. Therefore A is completely determined by its diagonal A(x, …, x), x ∈ U_{−1}. If A = φ_{k}(X) (k ≥ 1) and B = φ_{ℓ}(Y) (ℓ ≥ 1), then C = φ_{k+ℓ−1}([X, Y]) satisfies

C(x, …, x) = ℓB(A(x, …, x), x, …, x) − kA(B(x, …, x), x, …, x).

If ord(X) = −1, then we put v = X ∈ U_{−1} and we have

C(x, …, x) = kB(v, x, …, x)

Definition 3.1. Let V be a linear subspace of U_{−1}, and 𝔏 in D^{ν}.

(a) We call V a reducing subspace of 𝔏 if for all k ≥ 0 and all X ∈ 𝔏_{k−1},

A = φ_{k}(X) satisfies

A(v, x, …, x) ∈ V

for all v ∈ V and x ∈ U_{−1}.

(b) We call 𝔏 reducible if there exists a non-zero reducing subspace V for 𝔏, with V ≠ U_{−1}. If such V does not exist, 𝔏 is called irreducible.

Lemma 3.2. Let V be a reducing subspace of 𝔏 and define

I_{V,k−1} = {X ∈ 𝔏_{k−1} | for A = φ_{k}(X) : A(x, …, x) ∈ V for all x ∈ U_{−1}} (8)

and I_{V} = ⨁_{k=0}^{ν} I_{V,k−1}. Then : I_{V} is an ideal in 𝔏.

Proof. Let us give a proof of this useful lemma^{1}. Remember A(v, x, …, x) is (up to a factor k) simply the commutator of A of order k and v ∈ U_{−1}. Fix coordinates {∂_{1}, …, ∂_{n}} of U_{−1} such that ⟨∂_{r+1}, …, ∂_{n}⟩ = V . Since V is a reducing subspace, any X ∈ 𝔏 is of the form

X = ∑_{i=1}^{r} P_{i}(x_{1}, …, x_{r})∂_{i} + ∑_{i=r+1}^{n} P_{i}(x_{1}, …, x_{n})∂_{i}.

Define ψ : 𝔏 → 𝔏̄ by

ψ(X) = ∑_{i=1}^{r} P_{i}(x_{1}, …, x_{r})∂_{i}.

One easily checks that ψ is a Lie algebra morphism. Hence I_{V} =ker(ψ) is an ideal.

1 𝔏/I_{V} (“die verkürzte Gruppe”) appears already in [5] as an important computational technique.

Example 3.3. Let us look at 𝔏(A, S) define in (2.2). We see two reducing subspaces, W_{1} = ⟨∂_{3}, ∂_{4}⟩ and W_{2} = ⟨∂_{2}, ∂_{3}, ∂_{4}⟩. Lemma 3.2 tells that I_{W_{2}} is an ideal. Clearly 𝔏(A, S)/I_{W_{2}} ≅ sl_{2}(ℂ). The ideal I_{W_{1}} contains (but is not equal to) ⟨x_{3}∂_{3} − x_{4}∂_{4}, x_{4}∂_{3}, x_{3}∂_{4}⟩ ≅ sl_{2}(ℂ).

Since V ⊂ I_{V} we conclude that any simple 𝔏 is irreducible. The converse is also true for ord(𝔏) ≠ 1; this follows from proposition 4.5 below (if 𝔑 ≠ {0} then one finds 𝔑 = U_{−1}, so ord(𝔏) ≤ 1). A counter example for ord(𝔏) = 1 is the transitively differential algebra 𝔏 of order 1, 𝔏 = U_{−1} ⊕ U_{0}, of all polynomial vector fields of order 0 and 1.

Let us say some words on the case that 𝔏 is simple. This can only happen if ord(𝔏) = 2. Take any X ∈ 𝔏 of order 2. Then A = φ_{2}(X) is a bilinear map A : U_{−1} × U_{−1} → U_{−1}. So A defines a multiplication ∗ on U_{−1}: v ∗ w = A(v, w) for v, w ∈ U_{−1}. This multiplication is commutative:

v ∗ w = w ∗ v ⇔ [v, [w, X]] = [w, [v, X]]

for all v, w ∈ U_{−1}, and moreover

(v ∗ w) ∗ v^{2} = v ∗ (w ∗ v^{2}).

Commutative algebras with this property are called Jordan algebras; so there is a relation between algebras 𝔏 in D^{2} and Jordan algebras. This relation can be exploited to obtain a full classification of all simple Lie algebras; however it turns out that one has to define Jordan triples, a generalization of Jordan algebras. This way one gets a 1-1 correspondence between simple Lie algebras and simple Jordan triples; we refer to [3] for more information.

4. The Lie-theoretic structure of 𝔏

Let 𝔏 be a Lie algebra in D^{ν}. We first study the radical ℜ of 𝔏 (the maximal solvable ideal) and the nilradical 𝔑 of 𝔏 (the maximal nilpotent ideal). Of course, 𝔑 ⊂ ℜ; according to [6], § 5, N^{∘} 3, we have that

𝔑 = {X ∈ ℜ|ad X is nilpotent}.

Consequently if X ∈ ℜ and ord(X) ≠ 1 then X ∈ 𝔑.

Proposition 4.1. Let 𝔏 be in D^{ν} with ν ≠ 2 and 𝔑 the nilradical of 𝔏. Then 𝔑 ≠ 0.

Proof. If ν < 2 then U_{−1} ⊂ 𝔑. Hence assume ν > 2. Let K(X, Y) be the Killing form of X, Y ∈ 𝔏. We show that if ord(X) = k > 2 then K(X, Y) = 0 for all Y ∈ 𝔏, hence X ∈ 𝔑.

If ord(Y) = ℓ, then

(k − 1)K(X, Y) = K([E, X], Y) = −K(X, [E, Y]) = −(ℓ − 1)K(X, Y).

Hence (k + ℓ − 2)K(X, Y) = 0. Since k ≥ 3 and ℓ ≥ 0, we have K(X, Y) = 0.

It follows that any semi-simple 𝔏 in D^{ν} is of order 2.

Example 4.2. Consider the Lie algebra 𝔏(A, S) in (2.2). The elements of the form x_{1}^{k}∂_{2}, x_{1}^{k}x_{2}^{ℓ}∂_{3}, x_{1}^{k}x_{2}^{ℓ}∂_{4} generate the ideal 𝔑. In view of example 3.3 we see that ℜ = ⟨𝔑, x_{2}∂_{2}, x_{3}∂_{3} + x_{4}∂_{4}⟩.

The following notion is useful for us.

Definition 4.3. Let S ⊂ 𝔏 be a linear subspace. We call S layered if [v, s] ∈ S for all v ∈ U_{−1} and s ∈ S.

In particular 𝔏 itself is layered, as U_{−1} = 𝔏_{−1}. We note that also any ideal I ⊂ 𝔏 is layered. Moreover we have

Lemma 4.4. Let Z(I) be the center of an ideal I in 𝔏. Then Z(I) is layered.

Proof. Choose x ∈ I, z ∈ Z(I) and v ∈ U_{−1} arbitrary. We have to prove that [x, [v, z]] = 0. This follows directly from the Jacobi identity:

[x, [v, z]] = [[x, v], z] + [v, [x, z]] = 0 + 0 = 0.

Of special interest is the case I = 𝔑.

Proposition 4.5. Let W = Z(𝔑) ∩ U_{−1} be as above. Then W is a reducing subspace.

Proof. Let X ∈ 𝔏_{k−1} and A = φ_{k}(X). We have to prove that for w ∈ W, A(w, x, …, x) ∈ W for all x ∈ U_{−1}. Fix an x̄ ∈ U_{−1} and define Ā ∈ φ_{0}(𝔏_{0}) by

Ā(v) = A(v, x̄, …, x̄)

Take Y ∈ 𝔑 ∩ 𝔏_{p−1} and B = φ_{p}(Y). Then

[Ā, B](x, …, x) = pB(Ā(x), x, …, x) − Ā(B(x, …, x))

and by commutation with w we get

0 =[Ā, B](w, x, …, x) = pB(Ā(w), x, …, x)
+ p(p − 1)B(Ā(x), w, x, …, x) − Ā(B(w, …, x))

Now clearly the last two terms of the right-hand side are 0. Hence also the first term:

B(Ā(w), x, …, x) = 0

for all x ∈ U_{−1} and B = φ_{p}(Y). Hence [Ā(w), Y] = 0 for all Y ∈ 𝔑, or Ā(w) ∈ W. So we obtain

A(w, x̄, …, x̄) ∈ W

for all x̄ ∈ U_{−1}.

Let us summarize our results. If ℜ ≠ 0, then also 𝔑 ≠ 0 and V = ℜ ∩ U_{−1} = 𝔑 ∩ U_{−1}. This space V is a reducing subspace, as is easily checked. It is possible that V = U_{−1}, for example with n = 2 one can take

𝔏 = ⟨x_{1}^{k}∂_{2} (k ≤ ν), ∂_{1}⟩

Moreover if ℜ ≠ 0 then also W = Z(𝔑) ∩ U_{−1} ≠ 0. This W is also a reducing subspace. Inductively, using lemma 3.2, we have a flag of subspaces 0 = W_{0} ⊂ W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{s} = V in V such that any X ∈ 𝔑 satisfies

[W_{i+1}, X] ∈ W_{i} for all i = 0, …, s − 1.  (9)

If W = U_{−1} then 𝔑 = U_{−1} and hence 𝔏 ⊂ U_{−1} ⊕ U_{0}, where U_{0} denotes the space of all vector fields of order 0.

Next we study the semi-simple part of 𝔏.

Let 𝔏 be of class D^{ν} and ℜ its radical. Then 𝔏/ℜ is semi-simple; there exists a subalgebra 𝔖 ⊂ 𝔏 such that 𝔖 ≅ 𝔏/ℜ. Such 𝔖 is called a Levi factor. 𝔖 is the direct sum of (say d) simple Lie algebras 𝔖_{1}, …, 𝔖_{d}. Set, as before V = ℜ∩U_{−1}, and I_{V} ⊃ ℜ the corresponding ideal. It is possible that I_{V} contains some of the factors 𝔖_{1}, …, 𝔖_{d}. However we have

Proposition 4.6. Let 𝔖_{i} ⊂ I_{V} , then 𝔖_{i} ⊂ U_{0}.

Proof. Consider the Killing form K on I_{V} . For X ∈ V , K(X, Y) = 0 for all Y ∈ I_{V} , as V ⊂ 𝔑. Using the proof of proposition 4.1, it follows for X ∈ I_{V} and ord(X) ≠ 1, that K(X, Y) = 0 for all Y ∈ I_{V} . So the semi-simple part of I_{V} is contained in the part of order 1.

Using all results till now we are able to formulate a proposition on the structure of 𝔏 in case that V = U_{−1}.

Proposition 4.7. Suppose 𝔏 is a Lie algebra in the class D^{ν} with radical ℜ such that ℜ ∩ U_{−1} = U_{−1}. Then 𝔏 is a subalgebra of a multigraded 𝔏̄ of class Dν̄ with ν̄ ≥ ν.

Proof. By proposition 4.5 we have a flag

U_{−1} = W_{r} ⊃ W_{r−1} ⊃ ⋯ ⊃ W_{1} ⊃ W_{0} = {0}

with W_{r}/W_{r−1} a reducing subspace for 𝔏/I_{W_{r−1}} . In particular we have that all elements of 𝔏 have a common “triangular” form: if x^{(s)} = (x_{1}^{(s)}, x_{2}^{(s)}, …, x_{r_{s}}^{(s)}) are coordinates for W_{s}/W_{s−1} then X ∈ 𝔏 takes the form

X = P_{r}(x^{(r)})∂_{x^{(r)}} + P_{r−1}(x^{(r−1)}, x^{(r)})∂_{x^{(r−1)}} + ⋯ + P_{n}(x^{(1)}, x^{(2)}, …, x^{(r)})∂_{x^{(1)}}

where we used some obvious vector notations. If X ∈ 𝔑 then X even takes the strictly triangular form (ord(X) ≥ 1):

X = P_{r−1}(x^{(r)})∂_{x^{(r−1)}} + ⋯ + P_{n}(x^{(2)}, …, x^{(r)})∂_{x^{(1)}}

We inductively give degrees to the variables in x^{(r)}, x^{(r−1)} down to x^{(1)}, cf. (7). First we put deg(x_{i}^{(r)}) = 1. Then look at all terms in P_{r−1}(x^{(r−1)}, x^{(r)}) that are independent of x^{(r−1)}. Suppose d_{r−1} is the maximal degree (which in this case coincides with the maximal polynomial degree), then we put deg(x_{i}^{(r−1)}) = d_{r−1}. Now look at all terms in P_{r−2} independent of x^{(r−2)}. Let d_{r−2} be the maximal degree. Then we put deg(x_{i}^{(r−2)}) = d_{r−2}. This way we continue, and obtain that all these terms have degree 0 or less. It remains to consider the “diagonal terms” in P_{i}. We know from ℜ ∩ U_{−1} = U_{−1} and proposition 4.6 that the diagonal terms are of order 1, hence of the form x^{(i)}∂_{x^{(i)}} . Consequently these terms have degree 0. Hence 𝔏 is a subalgebra of the multigraded Lie algebra 𝔏′ consisting of all terms of degree 0 or less in the grading constructed above.

It remains to study the situation in which V ≠ U_{−1}. In this case we can consider 𝔏′ = 𝔏/I_{V} . This is a Lie algebra of vector fields on ℂ^{n′} with n′ = n − dim(V). Note that the Euler field is E′ = E mod I_{V} . Moreover 𝔏′ belongs to D^{2}(n′), as 𝔏′ is semi-simple. Let 𝔖′_{1}, …, 𝔖′_{d′} be the simple Lie algebras that constitute 𝔏′. Thank to the presence of E′ it is immediate that also 𝔖′_{1}, …, 𝔖′_{d′} are graded. In particular if we denote U′_{−1} = U_{−1}/V we have

U′_{−1} = ⨁_{i=1}^{d′} 𝔖′_{i} ∩ U′_{−1}

Clearly for any 𝔖′_{i} we have 𝔖′_{i} ∩ U′_{−1} ≠ {0}. Let us put n_{i} = dim(𝔖′_{i} ∩ U′_{−1}), so that n_{1} + ⋯ + n_{d} + dim(V) = n. With this definition we have

Proposition 4.8. Let 𝔏′ = 𝔖′_{1} ⊕ ⋯ ⊕ 𝔖′_{d′} be as above. Then the simple summand 𝔖′_{i} is in D^{2}(n_{i}). Combining the propositions 4.6 and 4.8, we see that 𝔏 contains two types of simple subalgebras 𝔖; either 𝔖 is in U_{0} or 𝔖 is in U_{−1} ⊕ U_{0} ⊕ U_{1}; see also example 3.3.

5. Transitive Differential algebras in low dimensions

We will discuss the structure of Lie algebras in the class D^{ν}(n) for n = 2 and n = 3. Apart from being interesting in its own right, we think that this is a good demonstration of the theorems from section 4..

We first consider n = 2. In section 4. we introduced the spaces V = 𝔑∩U_{−1} and W = Z(𝔑) ∩ U_{−1}. If dim V = 2, we have that 𝔏 is a subalgebra of a multigraded one (proposition 4.7). Hence we see (cf. [4]) that 𝔏 is subalgebra of

⟨∂_{x}, ∂_{y}, x∂_{y}, …, x^{ν}∂_{y}, x∂_{x}, y∂_{y}⟩ (ν ≥ 2)

or 𝔏 is contained in U_{−1} ⊕ U_{0}.

At the other extreme, we have that V = {0}, which means that 𝔏 is semi-simple. This gives two (multigraded) possibilities, namely

𝔏 = ⟨y^{2}∂_{y}, x^{2}∂_{x}, y∂_{y}, x∂_{x}, ∂_{y}, ∂_{x}⟩ ≅ sl_{2} ⊕ sl_{2}

or

𝔏 = ⟨y^{2}∂_{y} + xy∂_{x}, x^{2}∂_{x} + xy∂_{y}, y∂_{y}, x∂_{y}, y∂_{x}, x∂_{x}, ∂_{x}, ∂_{y}⟩ ≅ sl_{3}

Finally we have the case that dim V = dim W = 1. Now 𝔏 = sl_{2} ⋉ ℜ and X ∈ 𝔑 is of the form

X = x^{k}∂_{y} (k ≤ ν)

while 𝔏 contains an element Y of the form

Y = x^{2}∂_{x} + (αx^{2} + βxy + γy^{2})∂_{y}

Now [∂_{y}, Y] ∈ 𝔑 implies that γ = 0. Moreover [Y, x^{ν}∂_{y}] = 0 implies β = ν. For ν ≤ 1 it follows that one can take α = 0, while for ν ≥ 2 we have x^{2}∂_{y} ∈ 𝔏, hence we can assume that α = 0. All together we find that 𝔏 is a subalgebra of

⟨x^{ν}∂_{y}, x^{ν−1}∂_{y}, …, ∂_{y}, x^{2}∂_{x} + νxy∂_{y}, x∂_{x}, y∂_{y}, ∂_{x}⟩

Now we turn to the case n = 3. Now not all 𝔏 in D^{ν} are multigraded. A simple counterexample is the smallest Lie algebra in D^{ν} containing the element P (x_{1}, x_{2})∂_{x_{3}} for a homogeneous polynomial P of order ν. The construction of a transitively differential algebra that is not (essentially) multigraded is not easy. We discuss this problem by considering different cases for the dimensions of W = Z(𝔑) ∩ U_{−1} and V = 𝔑 ∩ U_{−1}. We will not discuss dim W = 0; in this case 𝔏 is a direct sum of simple Lie algebras in D^{2}, see proposition 4.8.

If dim W = 3, we have that 𝔏 ⊂ U_{−1} ⊕ U_{0}, hence by maximality, 𝔏 = U_{−1} ⊕ U_{0}.

If dim W = 2, we can assume that W = ⟨∂_{y}, ∂_{z}⟩ and 𝔑 contains ⟨x^{k}∂_{z}, x^{ℓ}∂_{y}⟩ for k ≤ κ, ℓ ≤ λ (and possibly ∂_{x} ∈ 𝔑). We can assume that κ ≥ λ and κ ≥ 1. The only possible element of order 2 can be put in the form

X = x^{2}∂_{x} + λxy∂_{y} + (κxz + γxy)∂_{z}.

A straightforward calculation (omitted here) yields that 𝔏 is not maximal in case that dim W = 2.

We end up with the most difficult case, dim W = 1. This case we split in two subcases, namely V = W, and dim V ≥ 2. In the first case, the nilradical has only elements of the form X = P (x, y)∂_{z}. Let κ be the maximal k such that x^{k}∂_{x} ∈ 𝔑 and λ the maximal ℓ such that y^{ℓ}∂_{z} ∈ 𝔑. We know that 𝔏̄ = 𝔏/I_{V} is semi-simple, and according to the beginning of this section, we have only two possibilities: 𝔏̄ = sl_{2} ⊕ sl_{2} or 𝔏̄ = sl_{3}.

If 𝔏̄ = sl_{2} ⊕ sl_{2}, then 𝔏 contains

Y_{1} = x^{2}∂_{x} + Q_{1}(x, y, z)∂_{z} and Y_{2} = y^{2}∂_{y} + Q_{2}(x, y, z)∂_{z}

for some quadratic polynomials Q_{1} and Q_{2}. As [Y_{1}, x^{κ}∂_{z}] = 0 and [Y_{2}, y^{λ}∂_{z}] = 0 and [Y_{1}, Y_{2}] = 0, we find

Q_{1}(x, y, z) = κxz + α_{1}x^{2} + α_{2}xy and Q_{2}(x, y, z) = λyz + β_{2}xy + β_{3}y^{2}.

for some α_{i} and β_{i}. We may assume that these α_{i} and β_{i} are 0 in case κ ≥ 2 and λ ≥ 2. By direct calculations one can show that this holds in the remaining cases κ ≤ 1 or λ ≤ 1 as well. Hence 𝔏 has order ν = κ + λ and basis

{x^{k}y^{ℓ}∂_{z} (k ≤ κ, ℓ ≤ λ), z∂_{z}, x^{2}∂_{x} + κxz∂_{z}, x∂_{x}, ∂_{x}, y^{2}∂_{y} + λyz∂_{z}, y∂_{y}, ∂_{y}}

Next we consider the case that 𝔏̄ = sl_{3}. Now 𝔏 contains elements of the form

Y_{1} = x^{2}∂_{x} + xy∂_{y} + Q_{1}(x, y, z)∂_{z} and Y_{2} = xy∂_{x} + y^{2}∂_{y} + Q_{2}(x, y, z)∂_{z}.

By the same method as in the previous case, we obtain κ = λ and that we can put α_{i} = 0 and β_{i} = 0 in all cases. Now 𝔏 has order ν = κ and a basis for 𝔏 is

{x^{k}y^{ℓ}∂_{z} (k + ℓ ≤ κ), z∂_{z}, x^{2}∂_{x} + xy∂_{y} + κxz∂_{z},
x∂_{x}, y∂_{x}, ∂_{x}, xy∂_{x} + y^{2}∂_{y} + κyz∂_{z}, x∂_{y}, y∂_{y}, ∂_{y}}

Finally we arrive in the case that dim W = 1 and dim V ≥ 2. By choosing good coordinates in U_{−1} we can assume that 𝔑 contains elements of the form

X = P (x, y)∂_{z} and Y = x^{ℓ}∂_{y} + Q(x, y)∂_{z}

Let λ be the maximal ℓ such that 𝔑 contains an element Y = x^{ℓ}∂_{y} + Q(x, y)∂_{z} for some polynomial Q. For λ = 0 or for λ = 1 and κ ≠ 0, there is no transitively differential algebra; one always can add terms such that we end up in the case with dim V = 1 and the semi-simple part 𝔏̄ being sl_{2} ⊕ sl_{2} or sl_{3}, respectively (this again needs calculations that are omitted here). The case λ = 1 and κ = 0 is a special case in the series below. So from now on we assume λ ≥ 2 and try to construct an 𝔏 that is not (essentially) multigraded. Let κ be the maximal k such that x^{k}∂_{z} ∈ 𝔏. Now consider all terms that x^{a}y^{b}∂_{z} occurring in some P (x, y)∂_{z} ∈ 𝔏, and take the P for which a + λb is maximal for some a, b. As P is homogeneous, b is maximal among the terms x^{a}y^{b} occurring in P. By applying ad Y exactly b times see that x^{a+λb}∂_{z} ∈ 𝔏. By definition of κ, we have:

a + λb ≤ κ  (10)

Now we can give the variable x, y, z degrees, according the scheme in the proof of proposition 4.7. We put

deg(x) = 1, deg(y) = λ and deg(z) = κ.

Thanks to (10) all terms in 𝔑 have degree 0 or less except, possibly, terms occurring in Q. We can assume that this happens in case that ℓ = λ. If all terms in Q have degree 0 or less, one can prove to end up with 𝔏 being

⟨x^{a}y^{b}∂_{z} (a + λb ≤ κ), x^{ℓ}∂_{y} (ℓ ≤ λ), x^{2}∂_{x} + λxy∂_{y} + κxz∂_{z}, x∂_{x}, ∂_{x}, y∂_{y}, z∂_{z}⟩

So assume Y = x^{λ}∂_{y}+Q(x, y)∂_{z} and assume that x^{c}y^{d} occurs in Q with c+dλ > κ. We take d maximal. If d = 0 then Q does not depend on y, and we can eliminate Q(x) = βx^{λ} by the change of variables y′ = y − βz; so the case d = 0 belongs to the multigraded case above. When Q depends on y, we consider

[Y, [∂_{x}, Y]] = x^{λ−1}(deg_{x} Q − λQ)\frac{∂Q}{∂y}∂_{z}

Hence [Y, [∂_{x}, Y]] contains the term x^{λ−1+c}y^{d−1}∂_{z}. Using (10) for a = λ − 1 + c and b = d − 1 gives that λ − 1 + c + λ(d − 1) ≤ κ, or c + λb ≤ κ + 1. Together with c + λd > κ and c + d = λ we obtain that d = \frac{κ}{λ−1} − 1. Hence necessarily κ = α(λ−1) for an integer α. Consequently d = α−1 (so α ≥ 2) and c = λ+1−α. Hence also α ≤ λ + 1. By rescaling z we can assume that the coefficient of x^{c}y^{d} in Q is 1. We now derived more or less the structure of 𝔑. However, it could be that dim V = 2, so that 𝔏̄ = 𝔏/I_{V} = sl_{2}. Direct calculations show that this is only possible if α = 2. Hence for α ≥ 3 we have

𝔏 = ⟨x^{a}y^{b}∂_{z} (a + λb ≤ κ), (ad ∂_{x})^{k} (x^{λ}∂_{y} + x^{λ−α+1}y^{α−1}∂_{z}) (k ≤ λ),
x∂_{x} + y∂_{y} + z∂_{z}, ∂_{x}, y∂_{y} + αz∂_{z}⟩,  (11)

In the case α ≥ 3 we see that 𝔏 is a subalgebra of

⟨x^{a}y^{b}∂_{z} (a + λb ≤ κ + 1), x^{ℓ}∂_{y} (ℓ ≤ λ), x∂_{x}, y∂_{y}, z∂_{z}, ∂_{x}⟩,

which is multigraded, but of order κ + 1.

For α = 2 we can add to 𝔏 in (11)

Z = x^{2}∂_{x} + λxy∂_{y} + (κxz + \frac{1}{2}y^{2})∂_{z}.  (12)

In this case 𝔏 is maximal in the sense that adding any vector field will generate an infinite-dimensional Lie algebra.

To summarize this section we formulate the following proposition.

Proposition 5.1. Let 𝔏 be a transitively differential algebra in 3 dimensions that is not semi-simple. Then either 𝔏 is (essentially) multigraded or 𝔏 belongs to the series of transitively differential algebras given in (11) and (12) with integers λ ≥ 1, 2 ≤ α ≤ λ + 1 and κ = α(λ − 1). Moreover ord 𝔏 = κ for λ ≥ 2 and ord 𝔏 = 2 if λ = 1.

6. A construction in high dimensions

We describe a construction for transitively differential algebras 𝔏 of order 3, which generalizes the construction in [7]. Before giving the detailed construction, let us the global structure of 𝔏. As ν = 3, the nilradical of 𝔏 is non-zero; hence we have an invariant subspace W = ⟨z_{1}, z_{2}, …, z_{m}⟩; W = U_{−1} ∩ Z(𝔑). Next we consider 𝔏̄ = 𝔏/I_{W}. Now 𝔏̄ has also order 3, and we obtain by a similar procedure W̄ = ⟨y_{1}, y_{2}, …, y_{ℓ}⟩. The remaining algebra ¯ 𝔏̄ = 𝔏̄/IW̄ is the affine algebra U_{−1} ⊕ U_{0} in the variables ¯ W̄ = ⟨x_{1}, x_{2}, …, x_{k}⟩. It will turn out that ℓ is related to k by ℓ = \binom{k+1}{2}. So 𝔏 only depends on the two parameters k and m, and n = k + ℓ + m.

We now give the detailed construction of 𝔏, starting at order 3. First, 𝔏 will contain the elements

x_{a}x_{b}x_{c}∂_{z_{d}}  (a ≤ b ≤ c ≤ k; d ≤ m).  (13)

Secondly we construct the vector

v = (x_{1}^{2}, x_{1}x_{2}, …, x_{1}x_{k}, x_{2}^{2}, …, x_{k}^{2})

of all quadratic monomials in x_{1}, …, x_{k}. Hence v is vector with ℓ = \binom{k+1}{2} components. Let ∂_{y} = (∂_{y_{1}}, …, ∂_{y_{ℓ}}) and v · ∂_{y} = x_{1}^{2}∂_{y_{1}} + ⋯ + x_{k}^{2}∂_{y_{ℓ}}. Then 𝔏 will contain

x_{a}(v · ∂_{y})  (a ≤ k).  (14)

Now we fixed v, we look for all vector w, quadratic in x_{1}, …, x_{k} satisfying

\frac{∂v}{∂x_{a}} · w = 0 (for all a ≤ k)

Note that this automatically implies that also v · w = 0, by using

\frac{∂v}{∂x_{a}} · w = 0 ⇒ ∑ x_{a} \frac{∂v}{∂x_{a}} · w = 0 ⇒ 2v · w = 0.  (15)

The space of quadratic vectors has dimension \binom{k+1}{2} · \binom{k+1}{2}. The linear constraints \frac{∂v}{∂x_{a}} · w = 0 are independent and yield each \binom{k+2}{3} equations on the coefficients. So the space of solutions w has dimension

r = \frac{1}{4}k^{2}(k + 1)^{2} − k\binom{k + 2}{3} = \frac{1}{2}k\binom{k + 1}{3}.

Below we will describe a construction for the space of solutions. For now let w_{1}, …, w_{r} be a basis for this space and y = (y_{1}, …, y_{ℓ}). Then 𝔏 contains the elements

(w_{a} · y)∂_{z_{b}}  (a ≤ r; b ≤ m).  (16)

Next we consider the quadratic elements. These are simply the derivatives of the cubic ones. So the elements in (13) give

x_{a}x_{b}∂_{z_{c}}  (a ≤ b ≤ k; c ≤ m).  (17)

The elements in (14) yield

x_{a} \frac{∂v}{∂x_{b}} · ∂_{y}  (a, b ≤ k)  (18)

By (15) this is exactly the linear span of the elements [∂_{x_{b}}, x_{a}v · ∂_{y}].

Next we have the elements

(\frac{∂w_{a}}{∂x_{b}} · y)∂_{z_{c}}  (a, b ≤ k; c ≤ m).  (19)

These elements are not linearly independent; this will be discussed at the end of the section.

Finally we arrive at the elements of order 1. First we have

x_{a}∂_{z_{c}} x_{a}∂_{y_{b}} y_{b}∂_{z_{c}} and z_{d}∂_{z_{c}}  (a ≤ k; b ≤ ℓ; c, d ≤ m). (20)

Secondly we have the element

y_{1}∂_{y_{1}} + y_{2}∂_{y_{2}} + ⋯ + y_{ℓ}∂_{y_{ℓ}}  (21)

Finally we have elements of the form x_{a}∂_{x_{b}} + . . .. The vector fields x_{a}∂_{x_{b}} act on v · ∂_{y}. This action can be view as a linear action on the space W̄ = ⟨∂_{y_{1}}, … ∂_{y_{ℓ}}⟩. As such, modulo the vector field (21), it can be uniquely represented by a vector field of the form

∑_{c,d} α_{c,d}^{(a,b)}y_{c}∂_{y_{d}}.

If we write v = (v_{1}, …, v_{ℓ}) we have explicitly

x_{a}∂_{x_{b}}(v_{1}∂_{y_{1}} + v_{2}∂_{y_{2}} + ⋯ + v_{ℓ}∂_{y_{ℓ}}) =

−v_{1}(∑_{d} α_{1,d}^{(a,b)}∂_{y_{d}}) − v_{2}(∑_{d} α_{2,d}^{(a,b)}∂_{y_{d}}) − ⋯ − v_{ℓ}(∑_{d} α_{ℓ,d}^{(a,b)}∂_{y_{d}})

By this construction we automatically have that

[x_{a}∂_{x_{b}} − ∑_{c,d} α_{c,d}^{(a,b)}y_{c}∂_{y_{d}}, v · ∂_{y}] = 0.

To 𝔏 we add the vector fields

x_{a}∂_{x_{b}} − ∑_{c,d} α_{c,d}^{(a,b)}y_{c}∂_{y_{d}}  (a, b ≤ k).  (22)

This completes the description of 𝔏.

A direct calculation shows that 𝔏 is indeed a Lie algebra. We will discuss some of the most difficult commutators. Let X be of type (14) and Y of type (16). They commute by

[X, Y] = [x_{a}(v · ∂_{y}), (w_{c} · y)∂_{z_{b}}] = x_{a}(v · w_{c})∂_{z_{b}} = 0

Another commutator to check is type (16) and type (22). For this we consider

0 = [x_{a}∂_{x_{b}} − ∑_{c,d} α_{c,d}^{(a,b)}y_{c}∂_{y_{d}}, [(w_{c} · y)∂_{z_{b}}, v · ∂_{y}]]

By the Jacobi identity we find that

[[x_{a}∂_{x_{b}} − ∑_{c,d} α_{c,d}^{(a,b)}y_{c}∂_{y_{d}}, (w_{c} · y)∂_{z_{b}}], v · ∂_{y}] = 0

Now the inner commutator is of the form (f · y)∂_{z_{b}} with f quadratic in x, and moreover f · v = 0. Similarly we find f · \frac{∂v}{∂x_{i}} = 0 for all i. Hence we conclude that f is a linear combination of w_{1}, …, w_{r}.

Finally, we want to prove that 𝔏 is maximal. Therefore suppose that 𝔏′ is in D^{3} and contains 𝔏. Then 𝔏′ has a reducing subspace W, which must be a subspace of ⟨z_{1}, …, z_{m}⟩. As 𝔏 acts irreducibly on ⟨z_{1}, …, z_{m}⟩, we see that this space coincides with W. So no elements of 𝔏′ contain terms of the form z_{a}∂_{y_{b}} or z_{a}∂_{x_{b}}.

Next we see that y does not appear quadratically by commuting such elements with type (14). After this it is not difficult to check maximality.

That 𝔏 is not essentially multigraded can be seen from the fact that the Jacobian of x_{1}(v · ∂_{y}) has rank greater than 1 (see lemma 2.3). Moreover by giving the variables degrees according to

deg x_{i} = 1;  deg y_{i} = 2;  deg z_{i} = 4

we see that 𝔏 is contained in a multigraded Lie algebra of order 4.

Let us now describe the space of w-functions. For this we consider the condition v·w = 0. This condition splits in \binom{k+3}{4} constraints on the coefficient of monomials of the form

x_{a}^{4}; x_{a}^{3}x_{b}; x_{a}^{2}x_{b}^{2}; x_{a}^{2}x_{b}x_{c}; x_{a}x_{b}x_{c}x_{d}.

Correspondingly, the solutions w naturally split up. We describe the 5 cases. Without loss of generality, we assume (a, b, c, d) = (1, 2, 3, 4).

• (x_{1}^{4}). Then w = (α_{1}x_{1}^{2}, 0, …, 0), and v · w = 0 yields α_{1} = 0.

• (x_{1}^{3}x_{2}). Having v = (x_{1}^{2}, x_{1}x_{2}, …) we get w = (α_{1}x_{1}x_{2}, α_{2}x_{1}^{2}, 0, …, 0). Again v · w = 0 and \frac{∂v}{∂x_{1}} · w = 0 yield w = 0.

• (x_{1}^{2}x_{2}^{2}). Permute the components of v, so that v = (x_{1}^{2}, x_{1}x_{2}, x_{2}^{2}, …). Then w = (α_{1}x_{2}^{2}, α_{2}x_{1}x_{2}, α_{3}x_{1}^{2}, 0, …, 0). It gives one solution

w = (x_{2}^{2}, −2x_{1}x_{2}, x_{1}^{2}, 0, …, 0).  (23)

• (x_{1}^{2}x_{2}x_{3}). If v = (x_{1}^{2}, x_{1}x_{2}, x_{1}x_{3}, x_{2}x_{3}, …) then

w = (x_{2}x_{3}, −x_{1}x_{3}, −x_{1}x_{2}, x_{1}^{2}, 0, …, 0).  (24)

• (x_{1}x_{2}x_{3}x_{4}). Finally the most involved case. If

v = (x_{1}x_{2}, x_{1}x_{3}, x_{1}x_{4}, x_{2}x_{3}, x_{2}x_{4}, x_{3}x_{4}, …)

then we find two linearly independent solutions:

w_{1} = (x_{3}x_{4}, 0, −x_{2}x_{3}, −x_{1}x_{4}, 0, x_{1}x_{2}, 0, …, 0)
and w_{2} = (0, x_{2}x_{4}, −x_{2}x_{3}, −x_{1}x_{4}, x_{1}x_{3}, 0, 0, …, 0)  (25)

Now we can count the dimension of the w-space. For type (23) we have \binom{k}{2}, for type (24) we have k\binom{k−1}{2} and for type (25) we have \binom{k}{4} combinations, and the last one doubled. Hence we have totally

r = \binom{k}{2} + k\binom{k − 1}{2} + 2\binom{k}{4} = \frac{1}{2}k\binom{k + 1}{3}

independent w-functions.

Our w-space coincides with the space T_{n} in the paper [7]. However the dimensions do not agree. This is due to the fact that, starting from n = 4, the space T_{n} does not agree with the space of all symmetric operators on the set of skew-symmetric matrices.

From the explicit form of the w-functions, it is clear that the derivatives \frac{∂w_{a}}{∂x_{b}} appearing in (19) are not linearly independent. These derivatives span exactly the space of all vectors f = (f_{1}, …, f_{ℓ}), with f_{i} linear in x, such that f · v = 0. Counting the dimension s of this space in the same way as we determined the w-space itself, we obtain^{2}

s = 2\binom{k + 1}{3}.

Now we can compute the dimension of 𝔏 depending on k and m. We present this in the following table. We divide the elements in certain classes, namely vertically by order and horizontally by appearing variables.

x∂_{z} x∂_{y} x∂_{x} (x)y∂_{z} y∂_{y} z∂_{z}
order 3 \binom{k+2}{3}m k  –  rm  – –
order 2 \binom{k+1}{2}m k^{2} –  sm  – –
order 1 km  kℓ k^{2}  ℓm  1 m^{2}
order 0  m  ℓ  k  –  – –

Hence for the dimension of 𝔏 we obtain

dim 𝔏 = \frac{1}{12} (k^{4} + 6k^{3} + 17k^{2} + 24k + 12m + 12) m + \frac{1}{2} (k^{3} + 6k^{2} + 5k + 2)

In particular we have for k = 2 and m = 1, that dim 𝔏 = 39, while for k = 3 and m = 6 (this is the case n = 3 in [7]), we have dim 𝔏 = 325.

7. Conclusion

We discussed Lie algebras 𝔏 in the class D^{ν} in Lie-theoretic sense (section 4.) and more concretely for some specific situations in the section 5. and 6.. Our point of view is to be complementary to the cases that 𝔏 is semi-simple or 𝔏 is multigraded. Such algebras are easy to construct, but when restricting to transitively differential algebras it is harder; for n = 3 we find only one series of transitively differential algebras that are not multigraded.

In view of the discussions of sections 2., 4., 5. and 6. one can wonder how useful the common definition of transitively differential algebra of order ν is. For our classification problem it seems better to consider only those algebras that are maximal in the class of all finite-dimensional algebras. Let call such 𝔏 a “transitively differential algebra of order ∞”. In 2. it becomes clear that a multigraded transitively differential algebra of order ∞ is of the form 𝔏(A, S). In section 4. we saw that the radical of any 𝔏 is contained in a multigraded transitively differential algebra of order ∞. Exactly the same holds for the examples of section 5. given in (11) and also for the transitively differential algebras constructed in section 6.: if considered in the class D^{ν+1} instead of D^{ν} all these examples are contained in a multigraded transitively differential algebra 𝔏(A, S) of order ∞.

2This is different in [7] as well.

For future research it would be interesting to concentrate on transitively differential algebras of order ∞: from the results of this paper it becomes clear that we should study a semi-direct product of a (semi-)simple algebra in class D^{2} and the radical. The representation of the first on the second should be analyzed in detail; the abstract representation theory should be connected to the concrete situation of polynomial vector fields.

References

[1] González-López, A., N. Kamran, and P. J. Olver, Lie algebras of vector fields in the real plane, Proc. London Math. Soc. (3) 64 (1992), 339–368.

[2] Kantor, I. L., Classification of irreducible transitive differential groups, Dokl. Akad. Nauk SSSR 158 (1964), 1271–1274.

[3] —, Transitive differential groups and invariant connections in homogeneous spaces, Trudy Sem. Vektor. Tenzor. Anal. 13 (1966), 310–398.

[4] Post, G., A class of graded Lie algebras of vector fields and first order differential operators, J. Math. Phys. 35 (1994), 6838–6856.

[5] Lie, S., ,,Theorie der Transformationsgruppen, Vol. 3, “ B. G. Teubner, Leipzig, 1893.

[6] Bourbaki, N., “Lie Groups and Lie Algebras, Ch. 1-3, “ Springer-Verlag, Berlin, 1989.

[7] Kantor, I. L., and J. Patera, A construction of transitively differential groups of degree 3, J. Math. Phys. 35 (1994), 443–458.

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

Received October 15, 1999 and in final form May 19, 2000