Lobachevskii Journal of Mathematics Vol. 3, 1999, 147–184 \begin{matrix}c \\ ◯ \end{matrix}Olga Kuzmich

Olga Kuzmich

GRADED NILPOTENT LIE ALGEBRAS IN LOW DIMENSIONS

(submitted by B.N.Shapukov)

Abstract. We classify all graded nilpotent Lie algebras up to dimension 7 and compute their universal prolongations.

Contents

1. Introduction 147 2. Fundamental graded Lie algebras 149 2.1. The classification algorithm 149 2.2. Computations and the classification 151 3. Universal prolongation 155 3.1. Algebraic construction 155 3.2. Geometric construction 156 3.3. Realization of 𝔤(𝔪) in polynomial vector fields 156 3.4. Der_{0}(𝔪)-invariant subspaces 160 3.5. Universal prolongation of direct product 160 4. Computation of universal prolongations 163 4.1. Finite type algebras 164 4.2. Infinite type algebras 166 5. The summary of results 171 5.1. Fundamental Lie algebras of dimension ≤ 6 171 5.2. The universal prolongations of the fundamental graded Lie

algebras of dimension ≤ 6 173

5.3. The classification of nondegenerate 7-dimensional fundamental Lie

algebras 175

References 184

1. Introduction

Graded nilpotent Lie algebras are a central object of so-called nilpotent differential geometry and nilpotent analysis, which is a very fast developing branch of differential geometry and analysis today. The key idea in this field is to pass from smooth manifolds to filtered manifolds, i.e., to smooth manifolds with a fixed filtration of the tangent bundle, where the role of the tangent bundle is played by the graded vector bundle associated with the given filtration. The fibers of this bundle can be naturally endowed with the structure of a graded nilpotent Lie algebra. Thus, in contrast to the standard theory of smooth manifolds, in which all manifolds of the same dimension are locally equivalent, in the case of filtered manifolds, graded nilpotent Lie algebras determine a local invariant (“type of the filtered manifold”) at each point.

Key words and phrases. Polynomial vector fields, graded fundamental Lie algebras, universal prolongation, infinitesimal symmetries of distributions.

The theory of filtered manifolds was developed in the works of Tanaka [4, 5] in connection with the theory of CR-structures and the geometry of parabolic homogeneous spaces (quotient spaces of semisimple Lie groups by parabolic subgroups). It turns out that the theory of G-structures can be completely carried over to the case of filtered manifolds in such a way that the new theory contains the old one as a trivial special case in which the graded Lie algebra that determines the type of the filtered manifold is simply a commutative Lie algebra with trivial filtration. The problem of equivalence of geometric structures on filtered manifolds was studied in the paper of Morimoto [1], in which the author, among other things, introduced the notion of a structure of order k on a filtered manifold, described the process of the prolongation and reduction of these structures, and defined the notion of involutiveness for geometric structures.

A major advantage of the theory of filtered manifolds lies in the fact that G-structures involving invariant distributions can be more naturally described as structures on filtered manifolds. In particular, G-structures of higher order turn into structures of the first order on filtered manifolds, while many Gstructures of infinite type become structures of finite type.

Therefore, there is a need for an independent study of nilpotent graded Lie algebras. The purpose of the present paper is to describe all complex and real nilpotent graded Lie algebras up to dimension 6 inclusive, as well as to compute some of their attributes. In Section 2 we classify all fundamental graded nilpotent Lie algebras using induction on the length of the grading.

Then, in Section 3, we study the structure of the universal prolongations of nilpotent graded Lie algebras and the ways of computing them. The intimate relationship between geometric and purely algebraic methods plays a key part in that study.

Finally, in Section 4, we compute the universal prolongations of the Lie algebras obtained. The degenerate case, which reduces to Lie algebras of lower dimension, is naturally singled out, as well as the case of finite type algebras, in which we explicitly construct the algebra of infinitesimal symmetries of the standard distribution whose associated graded algebra is isomorphic to the sought-for universal prolongation.

149 GRADED NILPOTENT LIE ALGEBRAS IN LOW DIMENSIONS

2. Fundamental graded Lie algebras

Recall that a graded Lie algebra is a graded vector

𝔤 = ∑_{p=−∞}^{∞} 𝔤_{p}

endowed with a Lie algebra structure such that [𝔤_{i}, 𝔤_{j}] ⊂ 𝔤_{i+j} for all i, j ∈ ℤ.

Definition 1. A graded Lie algebra 𝔪 = ∑_{p>0} 𝔪_{−p} is said to be fundamental if it satisfies the following conditions:

(i) dim 𝔪 < ∞; (ii) 𝔪_{−p−1} = [𝔪_{−p}, 𝔪_{−1}] for all p > 0.

It is immediate from this definition that a fundamental algebra 𝔪 is a graded nilpotent Lie algebra generated by 𝔪_{−1}.

Example 1. Let M be an arbitrary smooth manifold on which a completely nonintegrable distribution D is given. Suppose that this distribution D is regular, i.e., there exists a chain of distributions

{0} = D^{(0)} ⊂ D = D^{(1)} ⊂ D^{(2)} ⊂ ⋯ ⊂ D^{(μ)} = T M

such that \underline{D^{(k+1)}} = \underline{D^{(k)}} + [\underline{D^{(k)}}, \underline{D}] for all k ≥ 1 (here \underline{D^{(k)}} denotes the set of all vector fields lying in the distribution D^{(k)}).

Thus, the completely integrable distribution D defines a filtered manifold structure on M. For any point x ∈ M, let 𝔪(x) = ∑_{p≥0} D_{x}^{(p+1)}/D_{x}^{(p)}. The operation of commutation of vector fields naturally induces the structure of a nilpotent graded Lie algebra on 𝔪(x), which, as can be easily verified, is fundamental.

2.1. The classification algorithm. Assume that we know the classification of all graded fundamental Lie algebras whose grading has length ≤ μ (for instance, if μ = 1, these are precisely all commutative Lie algebras). Having made this assumption, we shall now describe all fundamental graded Lie 𝔪 = ∑_{p=1}^{μ+1} 𝔪_{−p} whose grading has the length μ + 1.

Let 𝔪 be a fundamental Lie algebra, its grading having the length μ + 1. Using the direct sum decomposition defined by the grading, we construct, for each 1 ≤ k ≤ μ, the fundamental graded algebras \overline{𝔪}_{k} = ∑_{p=1}^{k} 𝔪_{−p} (as vector spaces) with the multiplication [m, n]_{k} = π_{k}[m, n] for all m, n ∈ \overline{𝔪}_{k}, where π_{k}: 𝔪 → \overline{𝔪}_{k} denotes the natural projection associated with the grading of 𝔪. Let also c(m, n) = π[m, n], where π: 𝔪 → 𝔪_{−μ−1} is the natural projection.

Note that (\overline{𝔪}_{μ}, [,]_{μ}) is a Lie algebra isomorphic to 𝔪/𝔪_{−μ−1}, and it follows from the Jacobi identity that c ∈ Z^{2}(\overline{𝔪}_{μ}, 𝔪_{−μ−1}) is a 2-cocycle of the trivial \overline{𝔪}_{μ}- module 𝔪_{−μ−1}. Conversely, any graded Lie algebra \overline{𝔪}_{μ} whose grading has the length μ and any 2-cocycle of the trivial \overline{𝔪}_{μ}-module 𝔪_{−μ−1} allow to introduce on 𝔪 = \overline{𝔪}_{μ} ⊕ 𝔪_{−μ−1} the structure of a graded Lie algebra whose grading has the length μ + 1.

Here 𝔪 is a fundamental graded Lie algebra if and only if \overline{𝔪}_{μ} is a fundamental graded Lie algebra and

c(𝔪_{−p} ∧ 𝔪_{−1}) = \begin{cases}\{0\}, & p < μ; \\ 𝔪_{−μ−1}, & p = μ. \end{cases}

This condition is equivalent to the condition that c is surjective c(\overline{𝔪}_{μ−1}∧𝔪_{−1}) = {0}.

Consider the problem of isomorphism between two Lie algebras corresponding to the same fundamental graded Lie algebra \overline{𝔪}_{μ} but to different cocycles c ∈ Z^{2}(\overline{𝔪}_{μ}, 𝔪_{−μ−1}). Of course, we are interested only in those isomorphisms of algebras 𝔪 that preserve the grading and the quotient algebra 𝔪/𝔪_{−μ−1} = \overline{𝔪}_{μ}. Consequently, we obtain the natural action of the group A = Aut_{0}(\overline{𝔪}_{μ}) × GL(𝔪_{−μ−1}) on the cocycles c:

c ↦ (α, β).c = β ∘ c ∘ α^{−1}, (α, β) ∈ A,

Here and in what follows, Aut_{0}(𝔪) denotes the group of all grading-preserving automorphisms of 𝔪.

Thus we obtain the following result:

Lemma 1. Let (\overline{𝔪}_{μ}, [,]_{μ}) be a fundamental Lie algebra with grading of length μ, let 𝔪_{−μ−1} be the trivial \overline{𝔪}_{μ}-module, and let 𝔪 = \overline{𝔪}_{μ} ⊕ 𝔪_{−μ−1} be the algebra with multiplication

[𝔪, 𝔪_{−μ−1}] = \{0\}; [m, n] = [m, n]_{μ} + c(m, n), ∀m, n ∈ \overline{𝔪}_{μ},

where c: \overline{𝔪}_{μ} ∧\overline{𝔪}_{μ} → 𝔪_{−μ−1}. Then 𝔪 is a fundamental Lie algebra with grading of length μ + 1 if and only if the following conditions are satisfied:

1) c is surjective;

2) c ∈ Z^{2}(\overline{𝔪}_{μ}, 𝔪_{−μ−1});

3) c(\overline{𝔪}_{μ−1} ∧ 𝔪_{−1}) = \{0\};

4) 𝔪_{−μ−1} ≠ {0}. Two algebras 𝔪 of this kind are isomorphic if and only if the corresponding cocycles c are equivalent up to the following action: c ↦ β ∘ c ∘ α^{−1}, where α ∈ Aut_{0}(\overline{𝔪}_{μ}), β ∈ GL(𝔪_{−μ−1}).

The results obtained so far are not yet convenient enough to be used for computations. Therefore, we shall now restate all the conditions in terms of dual spaces. By Lemma 1, the classification of fundamental Lie algebras with grading of length μ + 1 reduces to the classification of cocycles

c: \overline{𝔪}_{μ} ∧ \overline{𝔪}_{μ} → 𝔪_{−μ−1}.

Consider the properties of the dual mapping

c^{∗}: 𝔪_{−μ−1}^{∗} → \overline{𝔪}_{μ}^{∗} ∧ \overline{𝔪}_{μ}^{∗}.

Lemma 2. In the conditions of Lemma 1, 𝔪 is a fundamental Lie algebra with grading of length μ+1 if and only if the following conditions are satisfied:

1) c^{∗} is surjective;

2) im c^{∗} ⊂ Z^{2}(\overline{𝔪}_{μ});

3) im c^{∗} ⊂ (\overline{𝔪}_{μ−1} ∧ 𝔪_{−1})^{⊥}; 4) im c^{∗} ≠ {0}. Two algebras 𝔪 corresponding to two different mappings c^{∗} are isomorphic if and only if the subspaces im c^{∗} ⊂ Z^{2}(\overline{𝔪}_{μ}) are conjugate with respect to the natural action of Aut_{0}(\overline{𝔪}_{μ}) on Z^{2}(\overline{𝔪}_{μ}).

In doing explicit computations, we always assume that a certain basis of \overline{𝔪}_{μ} is fixed which is in agreement with the grading. We use the following identifications: first, we assume that the mappings \overline{𝔪}_{μ}^{∗} ∧ \overline{𝔪}_{μ}^{∗} are just skewsymmetric matrices, while Z^{2}(\overline{𝔪}_{μ}) is a subspace of these matrices. Then the action of the group Aut_{0}(\overline{𝔪}_{μ}) assumes the natural form

A.X = ^{t}A^{−1}XA^{−1}.

(In the sequel, for the convenience of notation, we work with the right action X.A = A^{−1}.X = ^{t}AXA.) Second, using the fact that c^{∗} is injective, we identify 𝔪_{−μ−1}^{∗} with im c^{∗}, and c^{∗} itself with the natural embedding im c^{∗} ↪ \overline{𝔪}_{μ}^{∗} ∧ \overline{𝔪}_{μ}^{∗}.

We now present the final algorithm for classifying fundamental graded Lie algebras with grading of length μ + 1.

1) Describe all fundamental graded Lie algebras \overline{𝔪}_{μ} with grading of length μ.

2) Describe all nonzero subspaces U ⊂ Z^{2}(\overline{𝔪}_{μ}) ⋂(\overline{𝔪}_{μ−1} ∧ 𝔪_{−1})^{⊥} up to the natural action of the group Aut_{0}(\overline{𝔪}_{μ}): A.X = ^{t}A^{−1}XA^{−1} for all X ∈ U.

3) Describe the product of the Lie algebras 𝔪 = \overline{𝔪}_{μ} ⊕ U using the following rule:

[𝔪, U] = \{0\}; [m, n] = [m, n]_{μ} + c(m, n), ∀m, n ∈ \overline{𝔪}_{μ},

where c(m, n) = ∑ u_{k}(m, n) · u_{k} and {u_{k}} is an arbitrary basis of U.

2.2. Computations and the classification. The above algorithm is used in this paper to classify all fundamental Lie algebras of dimension ≤ 7. As an illustration, we shall now go through the key stages of the classification for the case of dimension ≤ 6. The results of the classification of fundamental Lie algebras of dimension ≤ 6 are presented in Table 1, while Table 3 contains the classification results for the 7-dimensional case.

Let μ = 1; then 𝔪 is commutative, and the corresponding algebras are m1, m2 1, m3 1, m4 1, m5 1, m6 1.

If μ = 2, then 𝔪 is metabelian, and conversely, every metabelian algebra is a fundamental graded Lie algebra with grading 𝔪_{−2} = C^{2}𝔪 and 𝔪_{−1} is an arbitrary complement of 𝔪_{−2}. Therefore, we need to classify all metabelian algebras, which is equivalent to the classification of subspaces in the space of skew-symmetric matrices Z^{2}\overline{𝔪}_{1} = \overline{𝔪}_{1} ∧ \overline{𝔪}_{1} up to the natural action of the group GL(\overline{𝔪}_{1}) for each commutative Lie algebra \overline{𝔪}_{1}. We now describe these subspaces for all possible dimensions of \overline{𝔪}_{1}.

If dim \overline{𝔪}_{1} = 1, then \overline{𝔪}_{1} ∧ \overline{𝔪}_{1} = 0, and the case is exhausted.

If dim \overline{𝔪}_{1} = 2, then the space \overline{𝔪}_{1} ∧ \overline{𝔪}_{1} is one-dimensional, and the corresponding Lie algebra is m3 2.

Now let dim \overline{𝔪}_{1} = 3. In this case, the dimension of a subspace (over any field) is an invariant of the subspace, so that we obtain the following three Lie algebras: m4 2, m5 2 1, and m6 2 1.

Let dim \overline{𝔪}_{1} = 4. Every one-dimensional subspace 𝔪_{−2} ⊂ \overline{𝔪}_{1} ∧ \overline{𝔪}_{1} is equivalent to either ⟨e^{3} ∧ e^{4}⟩ (m5 2 2) or ⟨e^{1} ∧ e^{2} + e^{3} ∧ e^{4}⟩ (m5 2 3), where e^{1}, e^{2}, e^{3}, e^{4} is the dual basis of the space \overline{𝔪}_{1}. Note that this is again true over any field.

Now we describe all 2-dimensional subspaces 𝔪_{−2}. If 𝔪_{−2} is degenerate, then it is equivalent to the space ⟨e^{2} ∧ e^{4}, e^{3} ∧ e^{4}⟩ (m6 2 2).

Let 𝔪_{−2} be nondegenerate. Suppose that 𝔪_{−2} contains a decomposable tensor. Then its basis can be written as x^{1} = e^{1} ∧ e^{2} and x^{2} = a ∧ e^{1} + b ∧ e^{2} + γe^{3} ∧ e^{4}, where e^{1}, e^{2}, e^{3}, e^{4} is the dual basis of \overline{𝔪}_{1}, and a, b ∈ ⟨e^{3}, e^{4}⟩ and γ ∈ k are certain parameters. Let us describe these parameters up to the group GL(4, k). If dim⟨a, b⟩ ≤ 1, we find that 𝔪_{−2} is equivalent to the space ⟨e^{1} ∧ e^{2}, e^{3} ∧ e^{4}⟩. Assume that dim⟨a, b⟩ = 2. Then if γ = 0, then 𝔪_{−2} is equivalent to ⟨e^{1} ∧ e^{2}, e^{1} ∧ e^{3} + e^{2} ∧ e^{4}⟩, and if γ ≠ 0, then 𝔪_{−2} to one of the spaces obtained already.

Assume now that the subspace 𝔪_{−2} contains no nonzero decomposable tensors. We write its basis in the form x^{1} = e^{1} ∧ e^{2} + e^{3} ∧ e^{4}, x^{2} = a∧e^{1} +b∧e^{2} +γe^{3} ∧e^{4}, where a, b ∈ ⟨e^{3}, e^{4}⟩, and γ ∈ k. It is clear that dim⟨a, b⟩ = 2, for otherwise the tensor x^{2} would be decomposable. We can therefore assume that x^{2} = e^{1} ∧ e^{3} + βe^{2} ∧ e^{4} + γe^{3} ∧ e^{4} (β ≠ 0), and by means of a suitable coordinate change of the form e^{1} ↦ e^{1} + te^{4}, e^{3} ↦ e^{3} + te^{2}, we obtain x^{2} = e^{1} ∧ e^{3} + αe^{2} ∧ e^{4}. The matrix of the bilinear form ∧ on 𝔪_{−2} relative to

the basis \{x^{1}, x^{2}\}. has the form (\begin{matrix}2 & 0 \\ 0 & −2α \end{matrix}). Consequently, the subspace 𝔪_{−2}

contains no nonzero decomposable vectors if and only if the basic field is ℝ (this is a first distinction between the classifications over ℂ and ℝ) and the parameter α is < 0. Up to homotheties we can assume that α ∼ −1.

Thus 𝔪_{−2} is equivalent to one of the following subspaces:

1. ⟨e^{1} ∧ e^{2}, e^{3} ∧ e^{4}⟩ (m6 2 3); (1r) ⟨e^{1} ∧ e^{2} + e^{3} ∧ e^{4}, e^{1} ∧ e^{3} − e^{2} ∧ e^{4}⟩ (m6 2 3r);

2. ⟨e^{1} ∧ e^{2}, e^{1} ∧ e^{3} + e^{2} ∧ e^{4}⟩ (m6 2 4).

To see that these subspaces are essentially different from each other, it will suffice to compare the corresponding restrictions of the bilinear form ∧.

Finally, let dim \overline{𝔪}_{1} = 5. We are interested only in one-dimensional subspaces 𝔪_{−2} ⊂ \overline{𝔪}_{1} ∧ \overline{𝔪}_{1}, and we obtain

1. ⟨e^{1} ∧ e^{2}⟩ (m6 2 5);

2. ⟨e^{1} ∧ e^{2} + e^{3} ∧ e^{4}⟩ (m6 2 6).

Now let μ = 3. As \overline{𝔪}_{2} we take each of the Lie algebras obtained above whose dimension is ≤ 5. These are m3 2, m4 2, m5 2 1, m5 2 2, and m5 2 3.

Let \overline{𝔪}_{2} = m3 2. Its automorphism group (of grading 0) can be written as

Aut_{0}(\overline{𝔪}_{2}) = \{(\begin{matrix}det A & 0 \\ 0 & A \end{matrix})| A ∈ GL(2, k)\} ,

and the corresponding subspace in Z^{2}\overline{𝔪}_{2} can be written as

(\overline{𝔪}_{1} ∧ 𝔪_{−1})^{⊥} = \{(\begin{matrix}0 & X \\ −^{t}X & 0 \end{matrix})| X ∈ Mat(1 × 2, k)\} .

The action is as follows: X ↦ det A XA, which is equivalent to X ↦ XA on the subspaces. Hence, two subspaces of the same dimension are equivalent (over any field), and the corresponding algebras are m4 3 and m5 3 1.

Now let \overline{𝔪}_{2} = m4 2. Then

Aut_{0}(\overline{𝔪}_{2}) = \{(\begin{matrix}det A & 0 & 0 \\ 0 & ∗ & ∗ \\ 0 & 0 & A \end{matrix})| A ∈ GL(2, k)\} ,

(\overline{𝔪}_{1} ∧ 𝔪_{−1})^{⊥} = \{(\begin{matrix}0 & 0 & X \\ 0 & 0 & 0 \\ −^{t}X & 0 & 0 \end{matrix})| X ∈ Mat(1 × 2, k)\} .

The action is the same as in the preceding case. The corresponding algebras are m5 3 2 and m6 3 1.

Consider the Lie algebra \overline{𝔪}_{2} = m5 2 1. We have

Aut_{0}(\overline{𝔪}_{2}) = \{(\begin{matrix}At & 0 & 0 \\ 0 & A & B \\ 0 & 0 & t \end{matrix})| A ∈ GL(2, k), B ∈ Mat(2 × 1, k), t ∈ k^{∗}\} ,

(\overline{𝔪}_{1} ∧ 𝔪_{−1})^{⊥} = \{(\begin{matrix}0 & X & Y \\ −X & 0 & 0 \\ −^{t}Y & 0 & 0 \end{matrix} )| X ∈ 𝔤𝔩(2, k), X = ^{t}X, Y ∈ Mat(2 × 1, k)\} .

The equivalence relation is as follows:

\begin{cases}X ↦ t^{t}AXA = X′ \\ Y ↦ t^{2} ^{t}AY + X′B′, \end{cases}

where B′ = A^{−1}B is an arbitrary matrix.

Thus, we obtain the following one-dimensional subspaces and algebras:

(X, Y) ∼ (0, (\begin{matrix}0 \\ 1 \end{matrix})) (m6 3 2), ((\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) , 0) (m6 3 3), ((\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}1 \\ 0 \end{matrix})) (m6 3 4),

((\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) , 0) (m6 3 5), ((\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , 0) (m6 3 5r).

For \overline{𝔪}_{2} = m5 2 2, we have

Aut_{0}(\overline{𝔪}_{2}) = \{(\begin{matrix}det A & 0 & 0 \\ 0 & ∗ & ∗ \\ 0 & 0 & A \end{matrix})| A ∈ GL(2, k)\} ,

(\overline{𝔪}_{1} ∧ 𝔪_{−1})^{⊥} = \{(\begin{matrix}0 & 0 & X \\ 0 & 0 & 0 \\ −^{t}X & 0 & 0 \end{matrix})| X ∈ Mat(1 × 2, k)\} ,

and the action has the form X ↦ det A XA. Thus, we obtain the algebra m6 3 6.

The case \overline{𝔪}_{2} = m5 2 3 is impossible, because then (\overline{𝔪}_{−1} ∧ 𝔪_{−1})^{⊥} = \{0\}. Now let μ = 4. Then \overline{𝔪}_{3} can be one of the following algebras: m4 3, m5 3 1,

and m5 3 2.

Olga Kuzmich 154

For \overline{𝔪}_{3} = m4 3, we have

Aut_{0}(\overline{𝔪}_{3}) = \{(\begin{matrix}at^{2} & 0 & 0 & 0 \\ 0 & at & 0 & ∗ \\ 0 & 0 & a & ∗ \\ 0 & 0 & 0 & t \end{matrix})| a, t ∈ k^{∗}\} ,

(\overline{𝔪}_{2} ∧ 𝔪_{−1})^{⊥} = \{(\begin{matrix}0 & 0 & 0 & x \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ −x & 0 & 0 & 0 \end{matrix})| x ∈ k\} ,

and the action has the form x ↦ at^{3}x. The corresponding algebra is m5 4 1. In the case \overline{𝔪}_{3} = m5 3 1, we have

Aut_{0}(\overline{𝔪}_{3}) = \{(\begin{matrix}det A A & 0 & 0 \\ 0 & det A & 0 \\ 0 & 0 & A \end{matrix})| A ∈ GL(2, k)\} ,

(\overline{𝔪}_{2} ∧ 𝔪_{−1})^{⊥} = \{(\begin{matrix}0 & 0 & X \\ 0 & 0 & 0 \\ −X & 0 & 0 \end{matrix})| X ∈ 𝔤𝔩(2, k), X = ^{t}X\} ,

and the action is as follows:

X ↦ det A ^{t}AXA.

Thus, we obtain the following 1-dimensional subspaces and algebras:

X ∼ (\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) (m6 4 1), (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) (m6 4 2), (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) (m6 4 2r).

Now let \overline{𝔪}_{3} = m5 3 2. Here

Aut_{0}(\overline{𝔪}_{3}) = \{(\begin{matrix}at^{2} & 0 & 0 & 0 & 0 \\ 0 & at & 0 & 0 & 0 \\ 0 & 0 & ∗ & ∗ & ∗ \\ 0 & 0 & 0 & a & ∗ \\ 0 & 0 & 0 & 0 & t \end{matrix})| a, t ∈ k^{∗}\} ,

(\overline{𝔪}_{2} ∧ 𝔪_{−1})^{⊥} = \{(\begin{matrix}0 & 0 & 0 & 0 & x \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ −x & 0 & 0 & 0 & 0 \end{matrix})| x ∈ k\} ,

and the action has the form x ↦ at^{3}x. The corresponding algebra is m6 4 3. Finally, let μ = 5. The only possibility here is \overline{𝔪}_{4} = m5 4 1, in which case

Aut_{0}(\overline{𝔪}_{4}) = \{(\begin{matrix}at^{3} & 0 & 0 & 0 & 0 \\ 0 & at^{2} & 0 & 0 & 0 \\ 0 & 0 & at & 0 & 0 \\ 0 & 0 & 0 & a & b \\ 0 & 0 & 0 & 0 & t \end{matrix})| a, t ∈ k^{∗}, b ∈ k\} ,

(\overline{𝔪}_{3} ∧ 𝔪_{−1})^{⊥} = \{(\begin{matrix}0 & 0 & 0 & x & y \\ 0 & 0 & −x & 0 & 0 \\ 0 & x & 0 & 0 & 0 \\ −x & 0 & 0 & 0 & 0 \\ −y & 0 & 0 & 0 & 0 \end{matrix})| x, y ∈ k\} ,

and the action has the form

\begin{cases}x ↦ a^{2}t^{3}x, \\ y ↦ at^{4}y + abt^{3}x. \end{cases}

The Lie algebras corresponding to the equivalence classes under this action are m6 5 1 and m6 5 2.

Remark 1. The classification of fundamental graded algebras of dimension ≤ 5 is the same over any filed of characteristic zero, whereas in the 6-dimensional case, differences arise even on passing from the real to the complex numbers.

3. Universal prolongation

Let 𝔪 = ∑_{p>0} 𝔪_{−p} be an arbitrary fundamental graded Lie algebra.

Definition 2. A graded Lie algebra 𝔤(𝔪) = ⨁_{p∈ℤ}𝔤_{p}(𝔪) is called the universal

prolongation of 𝔪 if the following conditions are satisfied:

(1) 𝔤_{p}(𝔪) = 𝔪_{p} for p < 0;

(2) if x ∈ 𝔤_{p} for p ≥ 0 and [x, 𝔪_{−1}] = {0}, then x = 0;

(3) 𝔤(𝔪) is maximal among all graded algebras satisfying conditions (1) and (2).

3.1. Algebraic construction. In this subsection we show that the universal prolongation exists and provide an explicit algebraic construction for it. For detailed proofs, see the paper [5], in which the notion of universal prolongation was introduced for the first time.

Let 𝔪 = ∑_{p>0} 𝔪_{−p} be a fundamental graded Lie algebra. We construct a graded Lie algebra

𝔤(𝔪) = ∑_{p∈ℤ}𝔤_{p}(𝔪)

such that 𝔤_{p}(𝔪) = 𝔪_{p} for all p < 0, 𝔤_{0}(𝔪) = Der_{0}(𝔪) and such that

𝔤_{p}(𝔪) = {u ∈ ⨁_{q<0} 𝔤_{p+q}(𝔪)⊗𝔪_{q}^{∗} | u([X, Y]) = [u(X), Y]−[u(Y), X], ∀X, Y ∈ 𝔪}

for all p > 0. The commutation in this algebra is defined in the obvious way: [X, Y] for X, Y ∈ 𝔪 is already defined; [u, X] = u(X) for u ∈ 𝔤_{p}(𝔪) (p ≥ 0), X ∈ 𝔪, and [u, v](X) = [[u, X], v] + [u, [v, X]] for u ∈ 𝔤_{p}(𝔪), v ∈ 𝔤_{q}(𝔪) (p, q ≥ 0) and for all X ∈ 𝔪.

It can be easily verified that the algebra 𝔤(𝔪) so defined is indeed a graded Lie algebra and that it satisfies all the axioms of the universal prolongation of 𝔪.

3.2. Geometric construction. The Lie algebra 𝔤(𝔪) has the following geometric meaning. Let M be a connected Lie group whose Lie algebra is 𝔪, and let D be a left-invariant distribution on M such that D_{e} = 𝔤_{−1}. Denote by A the Lie algebra of germs of infinitesimal symmetries of D at the unity element. Define two subspaces A_{0} and A_{−1} in A as follows:

A_{0} =\{\underline{X} ∈ A | X_{e} = 0\};
A_{−1} =\{\underline{X} ∈ A | X_{e} ∈ D_{e}\}

(here \underline{X} denotes the germ of a vector field X at the point e). Now let

A_{−p−1} =[A_{−p}, A_{−1}] for all p ⩾ 2;
A_{p} ={ξ ∈ A_{p−1} | [ξ, A_{−1}] ⊂ A_{p−1}} for all p ⩾ 1.

Then the family of subspaces {A_{p}}_{p∈ℤ} defines a decreasing filtration of A, and the Lie algebra 𝔤(𝔪) can be identified with the associated graded algebra, i.e., 𝔤_{p}(𝔪) ≡ A_{p}/A_{p+1} for all p ∈ ℤ.

3.3. Realization of 𝔤(𝔪) in polynomial vector fields. Throughout this subsection we assume that k is an arbitrary field of characteristic 0.

Let 𝔪 = ∑_{−μ≤p<0} 𝔪_{p} be a fundamental graded Lie algebra of dimension n. Suppose deg x = p ⇔ x ∈ 𝔪_{p}. Fix a homogeneous basis e_{1}, …, e_{n} in 𝔪.

Denote by T (𝔪) the tensor algebra of 𝔪. It is clear that T (𝔪) with a grading induced from 𝔪 is an associative graded algebra with unity. Noting that all elements of T (𝔪) of the form x⊗y −y ⊗x−[x, y] (x, y ∈ 𝔪) are homogeneous, we obtain a grading on the universal enveloping algebra U(𝔪) as well.

Let U^{∗}(𝔪) denote the direct sum

U^{∗}(𝔪) = ∑_{p≥0} U_{p}^{∗}(𝔪),

where U_{p}^{∗}(𝔪) = (U_{−p}(𝔪))^{∗} are graded linear mappings U(𝔪) → k of degree p. On U^{∗}(𝔪) there exists a natural structure of an associative commutative algebra with unity:

(f_{1}f_{2})(u) = (f_{1} ⊗ f_{2})(Δu), ∀f_{1}, f_{2} ∈ U^{∗}(𝔪), u ∈ U(𝔪),

where Δ: U(𝔪) → U(𝔪) ⊗ U(𝔪) is the coproduct in the universal enveloping algebra. Let k[𝔪] = k[x_{1}, …, x_{n}] be the algebra of polynomials, with grading given by deg x_{i} = − deg e_{i}. It is easily seen that the mapping U^{∗}(𝔪) → k[𝔪] that assigns the polynomial ∑_{α} \frac{1}{α!}f(e^{α})x^{α} to each function f ∈ U^{∗}(𝔪) is an isomorphism of associated graded algebras. In what follows, we use this isomorphism to identify U^{∗}(𝔪) and k[𝔪].

The following object that we are interested in is the algebra Der(U^{∗}(𝔪)) of derivations of the algebra U^{∗}(𝔪). Since U^{∗}(𝔪) ≡ k[𝔪], we see that Der(U^{∗}(𝔪)) is precisely the Lie algebra of all polynomial vector fields

V (𝔪) = \{f^{1} \frac{∂}{∂x_{1}} + ⋯ + f^{n} \frac{∂}{∂x_{n}} | f^{i} ∈ k[𝔪]\} .

Note that V (𝔪) can be naturally endowed with the structures of a graded Lie algebra and a graded k[𝔪]-module, where the grading in V (𝔪) is defined by deg (f \frac{∂}{∂x_{i}} ) = deg f − deg x_{i} for all polynomials f ∈ k[𝔪] and i = 1, …, n.

A graded homomorphism ρ: 𝔪 → V (𝔪) is called a faithful representation of 𝔪 in the Lie algebra of polynomial vector fields if ρ(𝔪) forms the system of generators of the k[𝔪]-module V (𝔪). Note that in this case the homomorphism ρ must be injective.

To give an example of a faithful representation, consider the action of the algebra 𝔪 on U^{∗}(𝔪):

(x.f)(u) = f(ux) ∀x ∈ 𝔪, f ∈ U^{∗}(𝔪), u ∈ U(𝔪).

It is easily seen that the mapping τ(x): f ↦ x.f is a derivation. Thus we obtain a mapping τ: 𝔪 → V (𝔪), which, as can be easily verified, is a faithful representation of 𝔪. We shall call this representation the standard (right) representation of the Lie algebra 𝔪.

Any automorphism φ of the graded algebra k[𝔪] with unity uniquely determines an automorphism φ̂ of the graded Lie algebra V (𝔪):

φ̂: d ↦ φ ∘ d ∘ φ^{−1} for all d ∈ V (𝔪).

We say that two faithful representations ρ_{1}, ρ_{2} : 𝔪 → V (𝔪) of the Lie algebra 𝔪 are equivalent if φ̂ ∘ ρ_{1} = ρ_{2} for some automorphism φ of the graded algebra k[𝔪].

Theorem 1. Any two faithful representations of 𝔪 are equivalent.

Proof. To prove this theorem, we shall need the following lemma:

Lemma 3. Let ρ: 𝔪 → V (𝔪) be a faithful representation of 𝔪, and let e_{1}, …, e_{n} be a basis in 𝔪 that agrees with the grading in 𝔪 and such that [e_{i}, e_{j}] = ∑_{k=1}^{n} c_{ij}^{k}e_{k}.

Let f_{1}, …, f_{n} ∈ k[𝔪] be a collection of polynomials such that

ρ(e_{i})f_{j} − ρ(e_{j})f_{i} − ∑_{k} c_{ij}^{k}f_{k} = 0 for all i, j = 1, …, n. (1)

Then there exists a unique (up to a constant factor) polynomial f such that ρ(e_{i})f = f_{i} for all i = 1, … n.

If deg f_{i} = deg e_{i} +k for all i = 1, …, n, then we can assume that deg f = k.

Proof. We first proof the lemma for the case of the standard representation ρ = τ. In terms of U(𝔪), the assertion of the lemma means that if e_{i}.f_{j} − e_{j}.f_{i} − ∑_{k} c_{ij}^{k}f_{k} = 0 for some functions f_{1}, …, f_{n} ∈ U^{∗}(𝔪), there there is a unique (up to a constant factor) function f ∈ U^{∗}(𝔪) such that e_{i}.f = f_{i}. To construct this function, we first define f ∈ T^{∗}(𝔪) by letting

\begin{aligned}f(1) & = c ∈ k \\ f(ue_{i}) & = f_{i}(π(u)), \end{aligned} (2)

where π: T (𝔪) → U(𝔪) is the natural projection and u ∈ T (𝔪) runs through the standard basis for T (𝔪). It is easy to see that the function f so defined indeed belongs to T^{∗}(𝔪), and moreover, all elements of the form u(e_{i}e_{j} − e_{j}e_{i} − ∑_{k} c_{ij}^{k})v with u, v ∈ U(𝔪) belong to the kernel of f. In other words, f ∈ U^{∗}(𝔪). The uniqueness follows from the fact that any function f ∈ U^{∗}(𝔪) satisfying conditions 1 must satisfy conditions 2.

Now, let ρ be an arbitrary faithful representation. Denote by v_{i} = τ(e_{i}) the images of the basis of 𝔪 under the standard representation. note that [v_{i}, v_{j}] = ∑_{k} c_{ij}^{k}v_{k}. Since the collections of vector fields ρ(e_{1}), …, ρ(e_{n}) and v_{1}, …, v_{n} form bases of the k[𝔪]-module V (𝔪), we can write v_{i} = ∑_{j=1}^{n} f_{i}^{j}ρ(e_{j}) and ρ(e_{i}) = ∑_{j=1}^{n} g_{i}^{j}v_{j}, where (f_{i}^{j}) and (g_{i}^{j}) are inverse matrices over the ring k[𝔪]. If we now let g_{i} = ∑_{j=1}^{n} f_{i}^{j}f_{j}, then by means of simple (but quite bulky) computations we find that v_{i}(g_{j}) − v_{j}(g_{i}) − ∑_{k} c_{ij}^{k}g_{k} = 0 for all i, j = 1, …, n. According to what was proved above, there exists a function f ∈ k[𝔪] such that v_{i}(f) = g_{i}. Finally, it is easy to show that e_{i}(f) = f_{i}.

It is obvious that f is homogeneous.

We turn back to the proof of the theorem. We need to construct an isomorphism φ: k[𝔪] → k[𝔪] such that φ̂ · ρ_{1} = ρ_{2}. Let u_{i} = ρ_{1}(e_{i}) and v_{i} = ρ_{2}(e_{i}). It suffices to construct a mapping φ: k[𝔪] → k[𝔪] satisfying the condition v_{i}(φ(f)) = φ(u_{i}(f)) (f ∈ k[𝔪], i = 1, …, n). Since we can choose a homogeneous basis of k[𝔪] over k, we constructed the required mapping φ only on homogeneous polynomials f by induction on deg f. For deg f = 0, let φ(1) = 1. Suppose deg f > 0. Then, since deg u_{i}(f) < deg f for all i = 1, …, n, we can assume by induction that v_{i}(φ(u_{j}(f))) = φ(u_{i}(u_{j}(f))) for all i, j = 1, …, n. It follows that v_{i}φ(u_{j}(f)) − v_{j}φ(u_{i}(f)) − ∑_{k} c_{ij}^{k}φ(u_{k}(f)) = 0. Thus, by Lemma 3, there exists a unique polynomial g such that v_{i}(g) = φ(u_{i}(f)). We let φ(f) = g.

We show that φ is an automorphism of the graded algebra k[𝔪]. Indeed, φ(1) = 1, and it can be shown by induction on deg f_{1}f_{2} that v_{i}(φ(f_{1})φ(f_{2})) = φ(u_{i}(f_{1}f_{2})). Therefore φ(f_{1}f_{2}) = φ(f_{1})φ(f_{2}) for any f_{1}, f_{2} ∈ k[𝔪].

We now find ker φ. Let f ∈ ker φ with the least possible deg f. Then 0 = v_{i}(φ(f)) = φ(u_{i}(f)). Therefore, all u_{i}(f) lie in ker φ, and since deg f has the least possible value, we have u_{i}(f) = 0. By Lemma 3, it follows that f ∈ k. But by construction φ(f) = f = 0, so that ker φ = {0}.

To find im φ, let f ∉ im φ with deg f having the least possible value. Then v_{i}(f) = f_{i} ∈ im φ. Therefore there exist g_{i} such that v_{i}(f) = φ(g_{i}). Since φ(e_{i}g_{j} − e_{j}g_{i} − ∑_{k} c_{ij}^{k}g_{k}) = 0, by Lemma 3 there exists an element g such that e_{i}(g) = g_{i}. Therefore v_{i}(φ(g)) = φ(e_{i}(g)) = φ(g_{i}) = v_{i}(f), so that f = φ(g + const). Thus im φ = k[𝔪].

Since all faithful representations of the Lie algebra 𝔪 are equivalent, we restrict our consideration to the standard representation τ: V (𝔪). In so doing, we identify 𝔪 with its image τ(𝔪) ⊂ V (𝔪), and, by an abuse of language, say that all fields in 𝔪 are left-invariant.

Let

𝔤 = {x ∈ V (𝔪) | [x, 𝔪] = {0}}.

By an abuse of language, we call all vector fields x ∈ 𝔤 right-invariant. Now if e_{1}, …, e_{n} is a homogeneous basis of 𝔪, there exist homogeneous vector fields u_{1}, …, u_{n} such that the correspondence e_{i} ↦ u_{i} defines an anti-isomorphism of 𝔪 and 𝔤. Indeed, in the case of the standard representation, we can choose u_{i} to be the derivation of U^{∗}(𝔪) defined by f ↦ f.e_{i}, where f.x(u) = f(xu) (x ∈ 𝔪, f ∈ U^{∗}(𝔪), u ∈ U(𝔪)) is the right action of 𝔪 on U^{∗}(𝔪).

For any subset S ⊂ V (𝔪), let ⟨S⟩ denote the module over k[x_{1}, …, x_{n}] generated by S. We define the symmetry algebra of S as

Sym(S) = {x ∈ V (𝔪) | [x, S] ⊂ ⟨S⟩}.

Theorem 2. The subalgebra Sym(𝔪_{−1}) is a graded subalgebra of V (𝔪) and is isomorphic to the universal prolongation 𝔤(𝔪) of the fundamental graded Lie algebra 𝔪.

Proof. We split the proof into several lemmas.

Lemma 4. The Lie algebra 𝔤(𝔪) can be embedded into V (𝔪) as follows: 𝔤_{p} = ⟨u_{i} | e_{i} ∈ 𝔪_{p}⟩ (as a k-module), p < 0

𝔤_{p} = {v ∈ V_{p} | [v, 𝔤_{−1}] ⊂ 𝔤_{p−1}}, p ≥ 0

Proof. It suffices to consider the case p ≥ 0. The proof is by induction on p. Suppose that the assertion of the lemma holds for all k < p, and let d ∈ 𝔤_{p}(𝔪). Then d can be identified with a certain mapping d: 𝔪 → ∑_{i<p} 𝔤_{i}. We find a vector field D ∈ V_{p} such that [D, u_{i}] = d(u_{i}) for all i = 1, …, n. We express it in the form D = ∑_{α=1}^{n} f^{α}e_{α}. Then [D, u_{i}] = −u_{i}(f^{α})e_{α}. We let f_{i}^{α} = u_{i}(f^{α}) for all i = 1, …, n. It then follows from

d([u_{i}, u_{j}]) − [u_{i}, d(u_{j})] − [d(u_{i}), u_{j}] = 0 for all i, j = 1, … n (3)

that the functions f_{1}^{α}, …, f_{n}^{α} for any α = 1, …, n satisfy equations (1) and, by Lemma 3, determine the desired function f^{α} uniquely up to a constant factor. Since D ∈ V_{p} (p ≥ 0), we find that the vector field D is uniquely defined.

Conversely, let D be any element of V_{p} such that [D, 𝔤_{−1}] ⊂ 𝔤_{p−1}. Then D defines a nonzero mapping d: 𝔪 → ∑_{i<p} 𝔤_{i}, which obviously satisfies condition (3), and hence D lies in 𝔤(𝔪).

Now let d_{1} ∈ 𝔤_{p} and d_{2} ∈ 𝔤_{q}, and let D_{1} ∈ V_{p} and D_{2} ∈ V_{q} be the corresponding vector fields. It is then easy to see that the vector [d_{1}, d_{2}] ∈ 𝔤_{p+q} corresponds to the vector field [D_{1}, D_{2}] ∈ V_{p+q}.

Thus, we have constructed a required embedding 𝔤(𝔪) → V (𝔪).

Lemma 5. For all p ∈ ℤ, we have 𝔤_{p} = V_{p} ∩ Sym(𝔪_{−1}).

Proof. Let v ∈ V_{p} (p < 0). Then the condition v ∈ Sym(𝔪_{−1}) implies that [v, e_{i}] ∈ V_{p−1} ∩ ⟨𝔪_{−1}⟩ = 0, i.e., v is a right-invariant vector field. For nonnegative p, the proof is by induction on p.

Suppose that 𝔤_{i−1} = V_{i−1} ∩ Sym(𝔪_{−1}) for all i < p. Let v ∈ V_{p}, e_{i} ∈ 𝔪_{−1}, and [v, e_{i}] = ∑_{α=1}^{n} f_{i}^{α}e_{α}. Note that v ∈ Sym(𝔪_{−1}) if and only if f_{i}^{α} = 0 for all deg e_{α} < −1. On the other hand, v ∈ 𝔤_{p} if and only if [v, u_{j}] ⊂ 𝔤_{p−1} for all u_{j} ∈ 𝔤_{−1}. But

[[v, u_{j}], e_{i}] = [[v, e_{i}], u_{j}] = − ∑_{α=1}^{n} u_{j}(f_{i}^{α})e_{α} for all i, j = 1, …, n.

Thus [v, u_{j}] ∈ 𝔤_{p−1} ⊂ Sym(𝔪_{−1}) if and only if u_{j}(f_{i}^{α}) = 0 for all deg e_{α} < −1. But since in this case deg f_{i}^{α} ≥ 1, these two conditions are equivalent. Hence [v, 𝔤_{−1}] ⊂ 𝔤_{p−1} if and only if v ∈ Sym(𝔪_{−1}).

This completes the proof of the theorem.

3.4. Der_{0}(𝔪)-invariant subspaces.

Lemma 6. Let 𝔞 ⊂ 𝔪_{−1} be a subspace. Then Sym(𝔪_{−1}) ⊂ Sym(𝔞) if and only if 𝔞 is invariant under Der_{0}(𝔪).

Proof. Necessity. If d ∈ Der_{0}(𝔪), then there exists a unique vector field v ∈ V_{0} such that [v, e_{i}] = d(e_{i}). It follows that v ∈ Sym(𝔪_{−1}), and hence v ∈ Sym(𝔞), so that [v, 𝔞] ⊂ ⟨𝔞⟩. Taking into account v ∈ V_{0}, we find that [v, 𝔞] ⊂ V_{−1}, so that [v, 𝔞] ⊂ 𝔞. Consequently , d(𝔞) ⊂ 𝔞.

Sufficiency. Let Der_{0}(𝔪)(𝔞) ⊂ 𝔞. We prove by induction on p that 𝔤_{p} ⊂ Sym(𝔞). Let v ∈ 𝔤_{p}. If p < 0, then v is right-invariant, and hence [v, 𝔪] = 0. Thus [v, 𝔞] = 0 and v ∈ Sym(𝔞). If p = 0, we have [v, 𝔪] ⊂ 𝔪, and hence the mapping d: 𝔪 → 𝔪, x ↦ [v, x] is a derivation of 𝔪. Thus [v, 𝔞] ⊂ 𝔞 and v ∈ Sym(𝔞).

Now let p > 0 and [𝔤_{k}, 𝔞] ⊂ ⟨𝔞⟩ for all k < p. Without loss of generality we can assume that the basis of 𝔞 is a subbasis of the basis of 𝔪. Let [v, e_{i}] = ∑_{α=1}^{n} f_{i}^{α}e_{α} for the basis vectors e_{i} ∈ 𝔞. Then

∑_{α=1}^{n} −u_{j}(f_{i}^{α})e_{α} = [[v, e_{i}], u_{j}] = [[v, u_{j}], e_{i}] ∈ ⟨𝔞⟩.

Consequently, f_{i}^{α} = const for all e_{α} ∉ 𝔞. Further, since 𝔞 ∈ 𝔪_{−1}, we obtain f_{i}^{α} > 0, and hence [v, 𝔞] ⊂ ⟨𝔞⟩.

3.5. Universal prolongation of direct product.

3.5.1. Degenerate algebras.

Definition 3. We say that a fundamental algebra 𝔪 is degenerate if Z(𝔪) ∩ 𝔪_{−1} ≠ {0}, where Z(𝔪) denotes the center of 𝔪.

Before stating a result on the universal prolongation of degenerate algebras, we introduce some new notation. Let 𝔪 be an arbitrary fundamental graded Lie algebra, and let T (𝔪) be its tensor algebra. It is clear that T (𝔪) is a graded Lie algebra (the grading is induced from 𝔪). Noticing that the elements of the form x ⊗ y − y ⊗ x − [x, y] ∈ T (𝔪) are homogeneous, we obtain a grading on the universal enveloping algebra:

U(𝔪) = T (𝔪)/⟨x ⊗ y − y ⊗ x − [x, y]⟩.

Then

U(𝔪) = ∑_{p≤0} U_{p}(𝔪),

where

U_{0}(𝔪) = ⟨1⟩,

U_{p}(𝔪) = ⟨m_{i_{1}} … m_{i_{k}} | m_{i_{j}} ∈ 𝔪_{i_{j}}, i_{1} + ⋯ + i_{k} = p⟩ = π⟨m_{i_{1}} ⊗ ⋯ ⊗ m_{i_{k}} | m_{i_{j}} ∈ 𝔪_{i_{j}}, i_{1} + ⋯ + i_{k} = p⟩ (here π: T (𝔪) → U(𝔪) is the canonical projection). Denote by U^{∗}(𝔪) the direct sum

U^{∗}(𝔪) = ∑_{p≥0} U_{p}^{∗}(𝔪),

where U_{p}^{∗}(𝔪) = (U_{−p}(𝔪))^{∗} are graded linear mappings U(𝔪) → U_{0}(𝔪) of degree p.

Proposition 1. Let 𝔪 be a nondegenerate fundamental Lie algebra, and let 𝔷 be a commutative Lie algebra. Then

𝔤(𝔪 × 𝔷) = 𝔤(𝔪) ⋌ (U^{∗}(𝔪 × 𝔷) ⊗ 𝔷)

and

𝔤_{p}(𝔪 × 𝔷) = 𝔤_{p}(𝔪) ⊕ (U_{p+1}^{∗}(𝔪 × 𝔷) ⊗ 𝔷).

Proof. We use the following notation:

𝔪^{p} = 𝔪 ⊕ 𝔤_{0}(𝔪) ⊕ ⋯ ⊕ 𝔤_{p−1}(𝔪) (p > 0),
𝔷_{p} = U_{p+1}^{∗}(𝔪 × 𝔷) ⊗ 𝔷 (p ≥ −1),
𝔷^{p} = 𝔷 ⊕ 𝔷_{0} ⊕ ⋯ ⊕ 𝔷_{p−1} (p > 0).

We prove the following equalities by induction on p:

𝔤_{p}(𝔪 × 𝔷) = 𝔤_{p}(𝔪) ⊕ 𝔷_{p},
[𝔤_{p}(𝔪), 𝔷] = {0},
[𝔤^{p+1}(𝔪), 𝔤^{p+1}(𝔪)] ∩ 𝔤^{p+1}(𝔪 × 𝔷) ⊂ 𝔤^{p+1}(𝔪),
[𝔷^{p+1}, 𝔤^{p+1}(𝔪 × 𝔷)] ∩ 𝔤^{p+1}(𝔪 × 𝔷) ⊂ 𝔷^{p+1}.

For p = 0 we compute 𝔤_{0}(𝔪×𝔷). Denote by π: 𝔪×𝔷 → 𝔪 and φ: 𝔪×𝔷 → 𝔷 the natural projections. We take an arbitrary u ∈ 𝔤_{0}(𝔪 × 𝔷) and let v = π ∘ u, w = φ ∘ u. It is clear that u = v + w (since 1 = π + φ) and that v(𝔷) = 0 (since 𝔷 = Z(𝔪 × 𝔷) ∩ (𝔪 × 𝔷)_{−1}). Verifying the condition u[x, y] = [ux, y] + [x, uy], we get v ∈ 𝔤_{0}(𝔪). Thus 𝔤_{0}(𝔪 × 𝔷) = 𝔤_{0}(𝔪) ⋌ (𝔪_{−1} × 𝔷)^{∗} ⊗ 𝔷.

Now let p > 0. We define two projections π: (𝔪 × 𝔷)^{p} = 𝔤^{p}(𝔪) ⊕ 𝔷^{p} → 𝔤^{p}(𝔪) and φ: (𝔪×𝔷)^{p} = 𝔤^{p}(𝔪)⊕𝔷^{p} → 𝔷^{p} (this can be done according to the induction assumption), and let u ∈ 𝔤_{p}(𝔪 × 𝔷), v = π ∘ u, and w = φ ∘ u, as earlier. Then v: 𝔪 × 𝔷 → 𝔤^{p}(𝔪) and w: 𝔪 × 𝔷 → 𝔷^{p} are graded mappings of degree p and u = v + w. We now check the condition

v[x, y] + w[x, y] = [vx, y] + [wx, y] + [x, vy] + [x, wy] for all x, y ∈ 𝔪 × 𝔷.

If x, y ∈ 𝔪, we find that v|_{𝔪} ∈ 𝔤_{p}(𝔪) and w[x, y] = [wx, y] + [x, wy]; for x, y ∈ 𝔷, we get [wx, y] + [x, wy] = 0; and the condition x ∈ 𝔪, y ∈ 𝔷 gives [𝔪, v(𝔷)] = 0 and [wx, y] + [x, wy] = 0. Thus we obtain the following two conditions on v: v|_{𝔪} ∈ 𝔤_{p}(𝔪) and v(𝔷) = 0, and the following condition on w: w[x, y] = [wx, y] + [x, wy]. We shall rewrite this condition on w in a different form. To do this, we identify the space Hom_{p}(𝔪 × 𝔷, 𝔷^{p}) of graded linear mappings 𝔪 × 𝔷 → 𝔷^{p} of degree p with the space ∑_{q=1} μ𝔪_{−q}^{∗} ⊗ 𝔷_{p−q} + 𝔷^{∗} ⊗ 𝔷_{p−1}. Then for all x ∈ (𝔪×𝔷)_{−q}, y ∈ (𝔪×𝔷)_{−r} we have [wx, y] = w(x)(y) = w(x⊗y) if q + r ≤ p + 1, and [wx, y] = 0 if q + r > p + 1. We have thus obtained the following condition on w:

w(x ⊗ y − y ⊗ x − [x, y]) = 0 for all x ⊗ y ∈ T (𝔪 × 𝔷), deg x ⊗ y ≥ −(p + 1)

From this formula and from the definition of 𝔷_{p} = U_{p+1}^{∗}(𝔪 × 𝔷) ⊗ 𝔷 we conclude that w ∈ 𝔷_{p}.

To complete the proof, it remains to notice that for all u ∈ 𝔤_{q}(𝔪 × 𝔷), v ∈ 𝔷_{r} (q, r ≥ 0 and q + r ≤ p) and for any x ∈ 𝔪 × 𝔷, we have

[u, v](x) = [ux, v] + [u, vx] ⊂ [𝔤^{q}(𝔪 × 𝔷), 𝔷_{r}] + [𝔤_{p}(𝔪 × 𝔷), 𝔷^{r}] ⊂ 𝔷^{q+r},

so that [u, v] ∈ 𝔷_{q+r} ⊂ 𝔷^{p+1}, which can easily be proved by induction.

Remark 2. Using Theorem 2, we shall now find the universal prolongation of a degenerate algebra 𝔪 × 𝔷 as a subalgebra in Sym(𝔪_{−1} × 𝔷). Noting that 𝔷 is a Der_{0}(𝔪 × 𝔷)-invariant subspace, we have by Lemma 6 that Sym(𝔪_{−1} × 𝔷) ⊂ Sym(𝔷).

Describe Sym(𝔷). To this end, we identify V (𝔪) with a subalgebra in V (𝔪× 𝔷) and let A = ⟨𝔷⟩. It is obvious that Sym(𝔷) = V (𝔪) ⋌ A.

Now, checking the condition [Sym(𝔪_{−1} × 𝔷), 𝔪_{−1}] ⊂ ⟨𝔪_{−1}, 𝔷⟩, we obtain Sym(𝔪_{−1} × 𝔷) = Sym_{𝔪}(𝔪_{−1}) ⋌ A, where Sym_{𝔪}(𝔪_{−1}) denotes all symmetries of the subspaces 𝔪_{−1} that lie in the subalgebra V (𝔪) ⊂ V (𝔪 × 𝔷). Thus, 𝔤(𝔪 × 𝔷) = A ⋋ 𝔤(𝔪), where, as one can easily see, A ≅ U^{∗}(𝔪 × 𝔷) ⊗ 𝔷.

3.5.2. Direct product of nondegenerate algebras.

Proposition 2. If 𝔪, 𝔪′ are nondegenerate fundamental algebras, then 𝔤(𝔪× 𝔪′) = 𝔤(𝔪) × 𝔤(𝔪′).

Proof. We show by induction that 𝔤_{p}(𝔪×𝔪′) = 𝔤_{p}(𝔪)×𝔤_{p}(𝔪′). This obviously holds true for p < 0. Let p = 0, and let π: 𝔪 × 𝔪′ → 𝔪 and π′: 𝔪 × 𝔪′ → 𝔪′ denote the natural projections. We choose an arbitrary element D ∈ 𝔤_{0}(𝔪×𝔪′) and let d = π ∘ D, d′ = π′ ∘ D; it is clear that D = d + d′ (since 1 = π + π′). We verify the condition D[x, y] = [Dx, y] + [x, Dy] that determines D. For x, y ∈ 𝔪 we obtain d|_{𝔪} ∈ 𝔤_{0}(𝔪) and d′([𝔪, 𝔪]) = {0}. Similarly, for x, y ∈ 𝔪′ we obtain d′|_{𝔪′} ∈ 𝔤_{0}(𝔪′) and d([𝔪′, 𝔪′]) = {0}, and the condition x ∈ 𝔪, y ∈ 𝔪′ gives d(𝔪′) ⊂ Z(𝔪) and d′(𝔪) ⊂ Z(𝔪′).

Now, since the algebra 𝔪′ is nondegenerate and the degree of d is zero, we get d(𝔪′_{−1}) ⊂ Z(𝔪) ∩ 𝔪_{−1} = {0}. Hence d(𝔪′) = {0}. Similarly, d′(𝔪) = {0}. Thus 𝔤_{0}(𝔪 × 𝔪′) = 𝔤_{0}(𝔪) × 𝔤_{0}(𝔪′). Note that [𝔤_{0}(𝔪), 𝔤_{0}(𝔪′)] = {0}.

Now let p > 0. We assume by induction that (𝔪 × 𝔪′)^{p} = 𝔪^{p} × (𝔪′)^{p} and [𝔪^{p}, (𝔪′)^{p}] = {0}. Let π: (𝔪 × 𝔪′)^{p} → 𝔪^{p} and π′: (𝔪 × 𝔪′)^{p} → (𝔪′)^{p} be the natural projections. Consider the mapping D: 𝔪 × 𝔪′ → (𝔪 × 𝔪′)^{p} ∈ 𝔤_{p}(𝔪 × 𝔪′), and let d = π ∘ D, d′ = π′ ∘ D. The rest of the proof is similar to the proof in the case p = 0, in which the same determining condition D[x, y] = [Dx, y] + [x, Dy] is used, and the axiom (2) of the universal prolongation is employed instead of the nondegeneracy assumption.

Remark 3. In geometric terms, using the fact that the nondegeneracy of the ideals 𝔪 and 𝔪′ implies that they are Der_{0}(𝔪 × 𝔪′)-invariant, we find from Theorem 2 and Lemma 6 that

𝔤(𝔪 × 𝔪′) ≅ Sym(𝔪_{−1} × 𝔪′_{−1}) ⊂ Sym(𝔪_{−1}) ∩ Sym(𝔪′_{−1}).

Since the inversion of this inclusion is obviously true, we obtain

Sym(𝔪_{−1} × 𝔪′_{−1}) = Sym(𝔪_{−1}) ∩ Sym(𝔪′_{−1}).

Now note that

Sym(𝔪_{−1}) ∩ Sym(𝔪′_{−1}) = Sym_{𝔪}(𝔪_{−1}) × Sym_{𝔪′}(𝔪′_{−1}),

and moreover Sym_{𝔪}(𝔪_{−1}) ≅ 𝔤(𝔪) and Sym_{𝔪′}(𝔪′_{−1}) ≅ 𝔤(𝔪′). Consequently,

𝔤(𝔪 × 𝔪′) = 𝔤(𝔪) × 𝔤(𝔪′).

Example 2. In the case of fundamental graded Lie algebras of dimension ≤ 6, the above proposition can be applied to the Lie algebra m6 2 3 = m3 2×m3 2. The Lie algebra m3 2 is nondegenerate, so that

𝔤(m3 2 × m3 2) = 𝔤(m3 2) × 𝔤(m3 2).

3.5.3. Relationship between the real and complex cases. Below for any complex Lie algebra 𝔤, let 𝔤_{ℝ} denote the real Lie algebra which is obtained by restricting the scalar field. In turn, given a real vector space V, let ℂV denote the complex vector space which is obtained from V by extending the scalar field.

Proposition 3. Let 𝔪 be a nondegenerate fundamental Lie algebra over the complex numbers. Then the universal prolongation of the algebra 𝔪_{ℝ} is precisely the algebra 𝔤(𝔪)_{ℝ}.

Proof. It is clear that the negative part of the Lie algebra 𝔤(𝔪)_{ℝ} is naturally isomorphic to 𝔪_{ℝ}. Hence, the Lie algebra 𝔤(𝔪)_{ℝ} is the universal extension of 𝔪_{ℝ} if and only if H^{p,1}(𝔪_{ℝ}, 𝔤(𝔪)_{ℝ}) = {0} for all p ≥ 0 (see [5, Lemma 1.14]). Consider the complexifications of these cohomology spaces. There exists a natural isomorphism

ℂH^{p,1}(𝔪_{ℝ}, 𝔤(𝔪)_{ℝ}) ≅ H_{ℂ}^{p,1}(ℂ𝔪_{ℝ}, ℂ𝔤(𝔪)_{ℝ}.)

Furthermore, the Lie algebras ℂ𝔪_{ℝ} and ℂ𝔤(𝔪)_{ℝ} are naturally isomorphic to the complex algebras 𝔪 × 𝔪 and 𝔤(𝔪) × 𝔤(𝔪), respectively. It follows from Proposition 2 that the universal proposition of the algebra 𝔪 × 𝔪 is precisely the algebra 𝔤(𝔪) × 𝔤(𝔪). Thus, the above cohomology spaces are all zero, which proves the proposition.

Example 3. In the case of fundamental graded Lie algebras of dimension ≤ 6, the above proposition is applicable to the Lie algebra m6 2 3r. This Lie algebra appears only in the real classification. Over the field of complex numbers, it is isomorphic to m6 2 3. It is easily verified that m6 2 3r is isomorphic to m3 2_{ℝ}, where m3 2 is viewed over the complex numbers. Thus, by Proposition 3, we have

𝔤(m6 2 3r) = 𝔤(m3 2)_{ℝ}.

4. Computation of universal prolongations

In this section, we construct the universal prolongations of all fundamental graded algebras of dimension ≤ 6.

4.1. Finite type algebras.

Definition 4. A fundamental Lie algebra 𝔪 is called a finite type algebra if its universal prolongation 𝔤(𝔪) is finite dimensional.

The paper [2] gives an effective criterion to determine whether a given fundamental algebra is of finite type. Namely, let 𝔤_{0} be the subalgebra of 𝔤_{0}(𝔪) defined by

𝔤_{0} = {d ∈ 𝔤_{0}(𝔪) | d(𝔪_{−p}) = {0} ∀p ≥ 2}.

The algebra 𝔤_{0} can be naturally identified with a subalgebra in 𝔤𝔩(𝔪_{−1}).

Theorem 3 ([2]). The universal prolongation 𝔤(𝔪) is finite dimensional if and only if the algebra 𝔤_{0}, viewed as a subalgebra in 𝔤𝔩(𝔪_{−1}), is of finite type (in the sense of Sternberg [3]).

Using this criterion, we immediately find that any finite type fundamental Lie algebra is isomorphic to one of the following algebras: m5 3 1, m6 2 1, m6 3 5, m6 3 5r, m6 4 1, m6 4 2, m6 4 2r, and m6 5 2.

4.1.1. Algebras with simple universal prolongation. Consider separately those fundamental graded algebras 𝔪 whose universal prolongation 𝔤(𝔪) is a simple Lie algebra and those for which 𝔤(𝔪) is not simple. The first case is studied in detail in [6], where, in particular, all such algebras are described. In the second case, we compute the universal prolongation by explicitly constructing the subspaces 𝔤_{p}(𝔪) for p ≥ 0.

Definition 5. Let 𝔤 = ⨁_{p∈ℤ} 𝔤_{p} be a semisimple graded Lie algebra. We say that 𝔤 is effective if 𝔤′ = ⨁_{p≥0} 𝔤_{p} contains no ideals of 𝔤. The grading of 𝔤 is called contact if 𝔤 is effective and if 𝔪 = ⨁_{p<0} 𝔤_{p} satisfies the following conditions:

(1) 𝔪 = 𝔤_{−2} ⊕ 𝔤_{−1} is such that dim 𝔤_{−2} = 1.

(2) The operation [,]: 𝔤_{−1} × 𝔤_{−1} → 𝔤_{−2} is nondegenerate.

It follows from these definitions that if 𝔤 is effective, then it is simple.

Now let 𝔤 be a simple complex Lie algebra of type X_{l}, let 𝔥 be a Cartan subalgebra of 𝔤, and let Δ = {α_{1}, …, α_{l}} be the system of simple roots of the root system Φ of 𝔤 relative to 𝔥. Choose any nonempty subset Δ_{1} ⊂ Δ and let

Φ_{k}^{+} = {α = ∑_{i=1}^{l} n_{i}(α)α_{i} ∈ Φ^{+} | ∑_{α_{i}∈Δ_{1}} n_{i}(α) = k} for k ≥ 0.

Setting

𝔤_{0} = 𝔥 ⊕ ⨁_{α∈Φ_{0}^{+}}(𝔤_{α} ⊕ 𝔤_{−α}),

𝔤_{k} = ⨁_{α∈Φ_{k}^{+}} 𝔤_{α}, 𝔤_{−k} = ⨁_{α∈Φ_{k}^{+}} 𝔤_{−α} (k > 0),

where 𝔤_{α} denotes the root subspace corresponding to α ∈ Φ, we get a grading {𝔤_{p}}_{p∈ℤ} satisfying 𝔤_{p} = [𝔤_{p+1}, 𝔤_{−1}] for p < −1. We denote this grading by (X_{l}, Δ_{1}). In the real case, Satake diagrams are used instead of Dynkin diagrams.

The following theorems show when a simple graded Lie algebra 𝔤 = ⨁_{p∈ℤ} 𝔤_{p} is the universal prolongation of a fundamental graded algebra 𝔪 = ⨁_{p<0} 𝔤_{p}.

Theorem 4. Let 𝔤 = ⨁_{p∈ℤ} 𝔤_{p} be a simple graded Lie algebra over ℂ such that 𝔤_{p} = [𝔤_{p+1}, 𝔤_{−1}] for p < −1. Then 𝔤 = ⨁_{p∈ℤ} 𝔤_{p} is the prolongation of 𝔪 = ⨁_{p<0} 𝔤_{p} over ℂ, except for the following three cases:

(1) 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} has the depth 1.

(2) 𝔤 = ⨁_{p=−2}^{2} 𝔤_{p} is a (complex) contact grading.

(3) 𝔤 = ⨁_{p∈ℤ} 𝔤_{p} is isomorphic to (A_{l}, {α_{1}, α_{i}}) (1 < i < l) or (C_{l}, {α_{1}, α_{l}}).

Theorem 5. Let 𝔤 = ⨁_{p∈ℤ} 𝔤_{p} be a simple graded Lie algebra over ℝ such that 𝔤_{p} = [𝔤_{p+1}, 𝔤_{−1}] for p < −1. Then 𝔤 = ⨁_{p∈ℤ} 𝔤_{p} is the prolongation of 𝔪 = ⨁_{p<0} 𝔤_{p}, except for the following three cases:

(1) 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} has the depth 1.

(2) 𝔤 = ⨁_{p=−2}^{2} 𝔤_{p} is a real or complex contact grading.

(3) 𝔤 = ⨁_{p∈ℤ} 𝔤_{p} is isomorphic to (A_{l}, {α_{1}, α_{i}}), (C_{l}, {α_{1}, α_{l}}), or to their normal real forms (AI, {α_{1}, α_{i}}), (CI, {α_{1}, α_{l}}); here 1 < i < l.

Using these theorems, we can compute the universal prolongation for the algebras m5 3 1, m6 2 1, m6 3 5, m6 3 5r, and m6 5 2.

4.1.2. Algebras with finite dimensional nonsimple algebra 𝔤(𝔪). Here we consider the algebras m6 4 1, m6 4 2, and m6 4 2r. Their universal prolongations 𝔤(𝔪) are obtained by straightforward computation.

4.1.3. 𝔪 = m6 4 1. We describe 𝔤(𝔪) by its multiplication table:

e_{1} e_{2} e_{3}  e_{4}  e_{5}  e_{6}  e_{7}  e_{8}  e_{9}  e_{10} e_{11}
e_{1}  0  0  0  0  0  0  −e_{1}  0  5e_{1} e_{2} 5e_{3}
e_{2}  0  0  0  0  0  0 −2e_{2} 0  0  0  0
e_{3}  0  0  0  0  0  e_{1} −e_{3} −e_{2} 3e_{3}  0 8e_{4}
e_{4}  0  0  0  0  e_{2}  e_{3} −e_{4}  0  e_{4}  0 9e_{5}
e_{5}  0  0  0  −e_{2}  0  e_{4} −e_{5}  0  −e_{5}  0 8e_{8}
e_{6}  0  0 −e_{1} −e_{3} −e_{4}  0  0  −e_{5} 2e_{6} −e_{8} −e_{9}
e_{7} e_{1} 2e_{2} e_{3}  e_{4}  e_{5}  0  0  e_{8}  0  e_{10} 0
e_{8}  0  0  e_{2}  0  0  e_{5} −e_{8}  0  −3e_{8} 0 5e_{10}
e_{9} −5e_{1} 0 −3e_{3} −e_{4} e_{5} −2e_{6} 0  3e_{8}  0 5e_{10} 2e_{11}
e_{10} −e_{2} 0  0  0  0  e_{8} −e_{10}  0 −5e_{10} 0  0
e_{11} −5e_{3} 0 −8e_{4} −9e_{5} −8e_{8} e_{9}  0 −5e_{10} −2e_{11} 0  0

Here the Lie algebra 𝔤𝔩(2, k) with the basis

e_{6} = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) , e_{7} = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , e_{9} = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) and e_{11} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix})

acts on the graded vector space V = ⟨e_{1}, e_{3}, e_{4}, e_{5}, e_{8}, e_{10}⟩. It is clear that this action is irreducible, and the center ⟨e_{7}⟩ of 𝔤𝔩(2, k) acts by means of scalar matrices:

e_{6} = (\begin{matrix}0 & −1 & 0 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}) , e_{7} = (\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}) ,

e_{9} = (\begin{matrix}−5 & 0 & 0 & 0 & 0 & 0 \\ 0 & −3 & 0 & 0 & 0 & 0 \\ 0 & 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 5 \end{matrix}) , e_{11} = (\begin{matrix}0 & 0 & 0 & 0 & 0 & 0 \\ −5 & 0 & 0 & 0 & 0 & 0 \\ 0 & −8 & 0 & 0 & 0 & 0 \\ 0 & 0 & −9 & 0 & 0 & 0 \\ 0 & 0 & 0 & −8 & 0 & 0 \\ 0 & 0 & 0 & 0 & −5 & 0 \end{matrix}) .

This action preserves the nondegenerate bilinear form β: V ∧ V → k defined by [x, y] = β(x, y)e_{2}. Note that any invariant nondegenerate bilinear form on V coincides with β up to a scalar multiplier.

4.1.4. 𝔪 = m6 4 2. Here 𝔤(𝔪) = 𝔪 ⋋ 𝔤_{0}(𝔪), where 𝔤_{0}(𝔪) is viewed as a subspace in Hom_{0}(𝔪, 𝔪):

(\begin{matrix}2x + 2y & 0 & 0 & 0 & 0 & 0 \\ 0 & 2x + y & 0 & 0 & 0 & 0 \\ 0 & 0 & x + 2y & 0 & 0 & 0 \\ 0 & 0 & 0 & x + y & 0 & 0 \\ 0 & 0 & 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 & 0 & y \end{matrix})

4.1.5. 𝔪 = m6 4 2r. Here (as above) 𝔤(𝔪) = 𝔪 ⋋ 𝔤_{0}(𝔪), where 𝔤_{0}(𝔪) ⊂ Hom_{0}(𝔪, 𝔪):

(\begin{matrix}4x & 0 & 0 & 0 & 0 & 0 \\ 0 & 3x & y & 0 & 0 & 0 \\ 0 & −y & 3x & 0 & 0 & 0 \\ 0 & 0 & 0 & 2x & 0 & 0 \\ 0 & 0 & 0 & 0 & x & y \\ 0 & 0 & 0 & 0 & −y & x \end{matrix})

4.2. Infinite type algebras. The universal prolongations for infinite type algebras can be computed using Theorem 2. These computations are pretty much the same and present no real difficulties. Therefore, we restrict ourselves to two most inter-sting examples. The final results for the rest of algebras can be found in Subsection 5.2.

Example 4. Consider the fundamental algebra 𝔪 = m6 3 3. Let e_{1}, …, e_{6} be a basis of 𝔪 such that

[e_{3}, e_{5}] = e_{1}, [e_{4}, e_{6}] = e_{2}, [e_{5}, e_{6}] = e_{3}.

Consider the following representation of 𝔪 in terms of polynomial vector fields:

e_{i} = \frac{∂}{∂x_{i}} (i = 1, 2, 4, 5),
e_{3} = \frac{∂}{∂x_{3}} − x_{5} \frac{∂}{∂x_{1}} ,
e_{6} = \frac{∂}{∂x_{6}} + x_{4} \frac{∂}{∂x_{2}} + x_{5} \frac{∂}{∂x_{3}} − 1/2x_{5}^{2} \frac{∂}{∂x_{1}} .

Apart from e_{1}, …, e_{6}, we shall work with the vector fields u_{i} = Ae_{i} (i = 1, …, 6), where the operator A is defined by the formula

A = 1 − x_{4}L_{e_{4}} − x_{5}L_{e_{5}} + 1/2x_{5}^{2}L_{e_{5}}^{2}.

Computing them explicitly, we get

u_{i} = e_{i} = \frac{∂}{∂x_{i}} (i = 1, 2, 4, 5),
u_{3} = e_{3} + x_{5}e_{1} = \frac{∂}{∂x_{3}} ,
u_{6} = e_{6} − x_{4}e_{2} − x_{5}e_{3} − 1/2x_{5}^{2}e_{1} = \frac{∂}{∂x_{6}} ,

so that u_{1}, …, u_{6} form a commutative Lie algebra.

We shall find an arbitrary symmetry S ∈ Sym(𝔪_{−1}) in the form S = ∑ f^{i}u_{i}. Using the fact that the subspace 𝔞 = ⟨e_{4}, e_{5}⟩ is Der_{0}(𝔪)-invariant and that 𝔪 is nondegenerate, we can identify every symmetry S with its projection to ⟨u_{1}, u_{2}, u_{3}, u_{6}⟩ parallel to ⟨u_{4}, u_{5}⟩. Therefore we are interested only in the functions f^{1}, f^{2}, f^{3}, and f^{6}. Now, using [e_{6}, u_{4}] = −e_{2} and [e_{6}, u_{5}] = −e_{3}, we get

[e_{4}, S] = e_{4}(f^{i})u_{i},
[e_{5}, S] = e_{5}(f^{i})u_{i},
[e_{6}, S] = e_{6}(f^{i})u_{i} − f^{4}e_{2} − f^{5}e_{3}.

Hence, we can write the condition S ∈ Sym(𝔪_{−1}) in the form of the following system:

e_{4}(f^{i}) = 0 (i ≠ 4),
e_{5}(f^{i}) = 0 (i ≠ 4, 5),
e_{6}(f^{1}) + x_{5}e_{6}(f^{3}) − 1/2x_{5}^{2}e_{6}(f^{6}) = 0,
e_{6}(f^{2}) − x_{4}e_{6}(f^{6}) = f^{4},
e_{6}(f^{3}) − x_{5}e_{6}(f^{6}) = f^{5},

which is equivalent to the equation

(u_{6} + x_{4}u_{2} + x_{5}u_{3} − 1/2x_{5}^{2}u_{1})(f^{1} + x_{5}f^{3} − 1/2x_{5}^{2}f^{6}) = 0.

Now, since all vector fields and functions that appear in this equation are independent of x_{4}, x_{5}, we can let the coefficients of the various powers of x_{4} and x_{5} be equal to zero:

\begin{matrix}1 : & u_{6}(f^{1}) = 0, & x_{5} : & u_{3}(f^{1}) + u_{6}(f^{3}) = 0, \\ x_{4} : & u_{2}(f^{1}) = 0, & x_{5}^{2} : & −1/2u_{1}(f^{1}) + u_{3}(f^{3}) − 1/2u_{6}(f^{6}) = 0, \\ x_{4}x_{5} : & u_{2}(f^{3}) = 0, & x_{5}^{3} : & −1/2u_{1}(f^{3}) − 1/2u_{3}(f^{6}) = 0, \\ x_{4}x_{5}^{2} : & −1/2u_{2}(f^{6}) = 0, & x_{5}^{4} : & 1/4u_{1}(f^{6}) = 0. \end{matrix}

This system is in its turn equivalent to the conditions

\begin{matrix}f^{1} ∈ k[x_{1}, x_{3}] \\ f^{2} ∈ k[x_{1}, x_{2}, x_{3}, x_{6}] \\ f^{3} ∈ k[x_{1}, x_{3}, x_{6}] \\ f^{6} ∈ k[x_{3}, x_{6}] \end{matrix} \begin{matrix}u_{1}(f^{3}) = −u_{3}(f^{6}) \\ u_{3}(f^{3}) = 1/2(u_{1}(f^{1}) + u_{6}(f^{6})) \\ u_{6}(f^{3}) = −u_{3}(f^{1}) \end{matrix}

It is clear that f^{3} exists if and only if the following relations hold:

[u_{1}, u_{3}](f^{3}) = 1/2u_{1}^{2}(f^{1}) + u_{3}^{2}(f^{6}) = 0
[u_{1}, u_{6}](f^{3}) = −u_{1}u_{3}(f^{1}) + u_{3}u_{6}(f^{6}) = 0
[u_{3}, u_{6}](f^{3}) = −u_{3}^{2}(f^{1}) − 1/2u_{6}^{2}(f^{6}) = 0

This system is equivalent to the following one:

\begin{matrix}u_{1}^{2}(f^{1}) = 2α, u_{1}u_{3}(f^{1}) = −γ, u_{3}^{2}(f^{1}) = −β, \\ u_{3}^{2}(f^{6}) = −α, u_{3}u_{6}(f^{6}) = −γ, u_{6}^{2}(f^{6}) = 2β, \end{matrix}

where α, β, γ ∈ k are constants.

Thus we obtain

f^{1} = αx_{1}^{2} − γx_{1}x_{3} − 1/2βx_{3}^{2} + λx_{1} − μx_{3} + ν,
f^{6} = βx_{6}^{2} − γx_{6}x_{3} − 1/2αx_{3}^{2} + ρx_{6} − σx_{3} + τ,

where λ, μ, ν, ρ, σ, τ ∈ k are arbitrary constants. From

u_{1}(f^{3}) = γx_{6} + αx_{3} + σ,
u_{3}(f^{3}) = αx_{1} + βx_{6} − γx_{3} + 1/2(λ + ρ),
u_{6}(f^{3}) = γx_{1} + βx_{3} + μ

it follows that

f^{3} = αx_{1}x_{3} + βx_{6}x_{3} + γx_{1}x_{6} − 1/2γx_{3}^{2} + σx_{1} + μx_{6} + 1/2(λ + ρ)x_{3} + δ, where δ ∈ k is a constant.

Thus we get

𝔤(𝔪) = ⟨\frac{∂}{∂x_{1}} , \frac{∂}{∂x_{3}} , −x_{3} \frac{∂}{∂x_{1}} + x_{6} \frac{∂}{∂x_{3}} , \frac{∂}{∂x_{6}} ,

x_{1} \frac{∂}{∂x_{1}} + 1/2x_{3} \frac{∂}{∂x_{3}} , 1/2x_{3} \frac{∂}{∂x_{3}} + x_{6} \frac{∂}{∂x_{6}} ,

− 1/2x_{3}^{2} \frac{∂}{∂x_{1}} + x_{3}x_{6} \frac{∂}{∂x_{3}} + x_{6}^{2} \frac{∂}{∂x_{6}} , x_{1} \frac{∂}{∂x_{3}} − x_{3} \frac{∂}{∂x_{6}} ,

− x_{1}x_{3} \frac{∂}{∂x_{1}} + (x_{1}x_{6} − 1/2x_{3}^{2})\frac{∂}{∂x_{3}} − x_{3}x_{6} \frac{∂}{∂x_{6}} ,

x_{1}^{2} \frac{∂}{∂x_{1}} + x_{1}x_{3} \frac{∂}{∂x_{3}} − 1/2x_{3}^{2} \frac{∂}{∂x_{6}} ⟩ ⋌ \{f \frac{∂}{∂x_{2}} | f ∈ k[x_{1}, x_{2}, x_{3}, x_{6}]\}.

Or 𝔤(𝔪) = B_{2} ⋌ S^{∗}(k^{4}), where the multiplication is given by

[x, f] = x(f), x ∈ B_{2}, f ∈ k[x_{1}, x_{2}, x_{3}, x_{6}],

[f_{1}, f_{2}] = f_{1} \frac{∂f_{2}}{∂x_{2}} − f_{2} \frac{∂f_{1}}{∂x_{2}} , f_{1}, f_{2} ∈ k[x_{1}, x_{2}, x_{3}, x_{6}].

Example 5. Consider the algebra 𝔪 = m6 3 4, and assume that the basis vectors e_{1}, …, e_{6} of 𝔪 are such that

[e_{2}, e_{6}] = e_{1}, [e_{3}, e_{5}] = e_{1}, [e_{4}, e_{6}] = e_{2}, [e_{5}, e_{6}] = e_{3}.

Consider the following representation of 𝔪 in terms of polynomial vector fields:

e_{i} = \frac{∂}{∂x_{i}} (i = 1, 2, 4, 5),
e_{3} = \frac{∂}{∂x_{3}} − x_{5} \frac{∂}{∂x_{1}} ,
e_{6} = \frac{∂}{∂x_{6}} + x_{2} \frac{∂}{∂x_{1}} + x_{4} \frac{∂}{∂x_{2}} + x_{5} \frac{∂}{∂x_{3}} − 1/2x_{5}^{2} \frac{∂}{∂x_{1}} .

We apply the operator

A = 1 − x_{2}L_{e_{2}} − x_{4}L_{e_{4}} − x_{5}L_{e_{5}} + 1/2x_{5}^{2}L_{e_{5}}^{2}

to these fields and let u_{i} = Ae_{i} (i = 1, …, 6). We obtain

u_{i} = e_{i} = \frac{∂}{∂x_{i}} (i = 1, 2, 4, 5),
u_{3} = e_{3} + x_{5}e_{1} = \frac{∂}{∂x_{3}} ,
u_{6} = e_{6} − x_{2}e_{1} − x_{4}e_{2} − x_{5}e_{3} − 1/2x_{5}^{2}e_{1} = \frac{∂}{∂x_{6}} ,

so that u_{1}, …, u_{6} form a commutative Lie algebra.

An arbitrary symmetry S ∈ Sym(𝔪_{−1}) can be found in the form S = ∑ f^{i}u_{i}. Note that the subspace 𝔞 = ⟨e_{4}, e_{5}⟩ is Der_{0}(𝔪)-invariant, and since 𝔪 is nondegenerate, we can identify each symmetry S with its projection onto ⟨u_{1}, u_{2}, u_{3}, u_{6}⟩ parallel to ⟨u_{4}, u_{5}⟩. We are therefore interested only in the functions f^{1}, f^{2}, f^{3}, and f^{6}. If we now note that

[e_{4}, S] = e_{4}(f^{i})u_{i},
[e_{5}, S] = e_{5}(f^{i})u_{i},
[e_{6}, S] = e_{6}(f^{i})u_{i} − f^{2}e_{1} − f^{4}e_{2} − f^{5}e_{3},

we can write the condition S ∈ Sym(𝔪_{−1}) as

e_{4}(f^{i}) = 0 (i ≠ 4),
e_{5}(f^{i}) = 0 (i ≠ 4, 5),
e_{6}(f^{1}) + x_{5}e_{6}(f^{3}) − x_{2}e_{6}(f^{6}) − 1/2x_{5}^{2}e_{6}(f^{6}) = f^{2},
e_{6}(f^{2}) − x_{4}e_{6}(f^{6}) = f^{4},
e_{6}(f^{3}) − x_{5}e_{6}(f^{6}) = f^{5}.

Thus, we have obtained the following equation:

(u_{6} + x_{2}u_{1} + x_{4}u_{2} + x_{5}u_{3} − 1/2x_{5}^{2}u_{1})(f^{1} − x_{2}f^{6} + x_{5}f^{3} − 1/2x_{5}^{2}f^{6}) + x_{4}f^{6} = f^{2}.

Since all vector fields and functions appearing in this equation are independent of x_{4} and x_{5}, we can set the coefficients of the various powers of these variables to be equal to zero:

1 : u_{6}(f^{1}) + x_{2}(u_{1}(f^{1}) − u_{6}(f^{6})) − x_{2}^{2}u_{1}(f^{6}) = f^{2},
x_{5} : u_{3}(f^{1}) + u_{6}(f^{3}) + x_{2}(u_{1}(f^{3}) − u_{3}(f^{6})) = 0,
x_{5}^{2} : −1/2u_{1}(f^{1}) + u_{3}(f^{3}) − 1/2u_{6}(f^{6}) = 0,
x_{5}^{3} : −1/2u_{1}(f^{3}) − 1/2u_{3}(f^{6}) = 0,
x_{5}^{4} : 1/4u_{1}(f^{6}) = 0,
x_{4} : u_{2}(f^{1}) − x_{2}u_{2}(f^{6}) = 0,
x_{4}x_{5} : u_{2}(f^{3}) = 0,
x_{4}x_{5}^{2} : −1/2u_{2}(f^{6}) = 0.

From the first and the last three equations we find that f^{2} = f^{2}(f^{1}, f^{6}) and u_{2}(f^{i}) = 0 (i = 1, 3, 6), and it follows that the algebra Sym(𝔪_{−1}) can be projected onto ⟨u_{1}, u_{3}, u_{6}⟩ parallel to ⟨u_{2}⟩. Hence, we need only to find the functions f^{1}, f^{3}, and f^{6}. The remaining equations are equivalent to the system

u_{3}(f^{1}) + u_{6}(f^{3}) = 0,
u_{1}(f^{3}) − u_{3}(f^{6}) = 0,
u_{1}(f^{1}) − 2u_{3}(f^{3}) + u_{6}(f^{6}) = 0,
u_{1}(f^{3}) + u_{3}(f^{6}) = 0,
u_{1}(f^{6}) = 0,

which in turn is equivalent to the conditions

f^{1} ∈ k[x_{1}, x_{3}, x_{6}], f^{3} ∈ k[x_{3}, x_{6}], f^{6} ∈ k[x_{6}],
u_{1}(f^{1}) = 2u_{3}(f^{3}) − u_{6}(f^{6}),
u_{3}(f^{1}) = −u_{6}(f^{3}).

Find the conditions for the existence of f^{1}. We obtain [u_{1}, u_{3}](f^{1}) = −2u_{3}^{2}(f^{3}) = 0. This implies that

f^{3} = 2αx_{3} + β, f^{6} = γ,

where α, β, γ ∈ k[x_{6}] are polynomials in x_{6}. Solving the equations u_{1}(f^{1}) = 4α − γ_{6} and u_{3}(f^{1}) = −2α_{6}x_{3} − β_{6}, we get

f^{1} = −α_{6}x_{3}^{2} − β_{6}x_{3} + (4α − γ_{6})x_{1} + δ,

where δ ∈ k[x_{6}], and the subscript 6 denotes the derivative with respect to x_{6}, i.e., α_{6} = u_{6}(α), β_{6} = u_{6}(β), and γ_{6} = u_{6}(γ).

Thus,

𝔤(𝔪) = \{(−α_{6}x_{3}^{2} − β_{6}x_{3} + (4α − γ_{6})x_{1} + δ)\frac{∂}{∂x_{1}} +

+ (2αx_{3} + β)\frac{∂}{∂x_{3}} + γ \frac{∂}{∂x_{6}} | α, β, γ, δ ∈ k[x_{6}]\}.

5. The summary of results

5.1. Fundamental Lie algebras of dimension ≤ 6. Below we give the classification of all fundamental graded Lie algebras up to dimension 6 inclusive.

m1, m2, m3 1, m4 1, m5 1, m6 1 are the commutative Lie algebras of dimensions 1, … 6 respectively;

m3 2 \begin{aligned}& e_{1} & e_{2} & e_{3} \\ e_{1} & 0 & 0 & 0 \\ e_{2} & 0 & 0 & e_{1} \\ e_{3} & 0 & −e_{1} & 0 \end{aligned} m4 2 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} \\ e_{1} & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & −e_{1} & 0 \end{aligned}

m4 3 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} \\ e_{1} & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & e_{2} \\ e_{4} & 0 & −e_{1} & −e_{2} & 0 \end{aligned} m5 2 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & −e_{1} & −e_{2} & 0 \end{aligned}

m5 2 2 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 \end{aligned} m5 2 3 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & e_{1} & 0 & 0 \\ e_{3} & 0 & −e_{1} & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 \end{aligned}

m5 3 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{4} & 0 & 0 & −e_{1} & 0 & e_{3} \\ e_{5} & 0 & 0 & −e_{2} & −e_{3} & 0 \end{aligned} m5 3 2 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & −e_{1} & 0 & −e_{2} & 0 \end{aligned}

m5 4 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} \\ e_{1} & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & e_{3} \\ e_{5} & 0 & −e_{1} & −e_{2} & −e_{3} & 0 \end{aligned} m6 2 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{3} & −e_{2} \\ e_{5} & 0 & 0 & 0 & −e_{3} & 0 & e_{1} \\ e_{6} & 0 & 0 & 0 & e_{2} & −e_{1} & 0 \end{aligned}

m6 2 2 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{6} & 0 & 0 & 0 & −e_{1} & −e_{2} & 0 \end{aligned} m6 2 3 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{4} & 0 & 0 & −e_{1} & 0 & 0 & 0 \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{6} & 0 & 0 & 0 & 0 & −e_{2} & 0 \end{aligned}

m6 2 3r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & e_{2} & −e_{1} \\ e_{5} & 0 & 0 & −e_{1} & −e_{2} & 0 & 0 \\ e_{6} & 0 & 0 & −e_{2} & e_{1} & 0 & 0 \end{aligned} m6 2 4 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & −e_{1} & 0 \\ e_{5} & 0 & 0 & 0 & e_{1} & 0 & e_{2} \\ e_{6} & 0 & 0 & −e_{1} & 0 & −e_{2} & 0 \end{aligned}

m6 2 5 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{6} & 0 & 0 & 0 & 0 & −e_{1} & 0 \end{aligned} m6 2 6 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{4} & 0 & 0 & −e_{1} & 0 & 0 & 0 \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{6} & 0 & 0 & 0 & 0 & −e_{1} & 0 \end{aligned}

m6 3 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & −e_{2} & 0 & −e_{3} & 0 \end{aligned} m6 3 2 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & −e_{1} & −e_{2} & −e_{3} & 0 \end{aligned}

173 GRADED NILPOTENT LIE ALGEBRAS IN LOW DIMENSIONS

m6 3 3 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & −e_{2} & −e_{3} & 0 \end{aligned} m6 3 4 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & e_{3} \\ e_{6} & 0 & −e_{1} & 0 & −e_{2} & −e_{3} & 0 \end{aligned}

m6 3 5 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{3} & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{4} & 0 & 0 & −e_{1} & 0 & 0 & e_{2} \\ e_{5} & 0 & −e_{1} & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & −e_{2} & −e_{3} & 0 \end{aligned} m6 3 5r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & −e_{1} & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & −e_{2} & −e_{3} & 0 \end{aligned}

m6 3 6 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{6} & 0 & −e_{1} & 0 & 0 & −e_{2} & 0 \end{aligned} m6 4 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & e_{2} & e_{3} \\ e_{5} & 0 & 0 & 0 & −e_{2} & 0 & e_{4} \\ e_{6} & 0 & 0 & −e_{1} & −e_{3} & −e_{4} & 0 \end{aligned}

m6 4 2 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{2} & e_{3} \\ e_{5} & 0 & 0 & −e_{1} & −e_{2} & 0 & e_{4} \\ e_{6} & 0 & −e_{1} & 0 & −e_{3} & −e_{4} & 0 \end{aligned} m6 4 2r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & e_{2} & e_{3} \\ e_{5} & 0 & −e_{1} & 0 & −e_{2} & 0 & e_{4} \\ e_{6} & 0 & 0 & −e_{1} & −e_{3} & −e_{4} & 0 \end{aligned}

m6 4 3 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & −e_{1} & −e_{2} & 0 & −e_{3} & 0 \end{aligned} m6 5 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{4} \\ e_{6} & 0 & −e_{1} & −e_{2} & −e_{3} & −e_{4} & 0 \end{aligned}

m6 5 2 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{3} & 0 & 0 & 0 & −e_{1} & 0 & e_{2} \\ e_{4} & 0 & 0 & e_{1} & 0 & 0 & e_{3} \\ e_{5} & 0 & −e_{1} & 0 & 0 & 0 & e_{4} \\ e_{6} & 0 & 0 & −e_{2} & −e_{3} & −e_{4} & 0 \end{aligned}

5.2. The universal prolongations of the fundamental graded Lie algebras of dimension ≤ 6. Below, we list the universal prolongations for all nondegenerate fundamental Lie algebras of dimension ≤ 6.

m3 2  𝔤(𝔪) = \{Y (f)X − X(f)Y + fZ | f ∈ k[x, y, z]\}, deg x = deg y = 1, deg z = 2, X = \frac{∂}{∂x} − y \frac{∂}{∂z} , Y = \frac{∂}{∂y} , Z = \frac{∂}{∂z};

m4 3  𝔤(𝔪) = \{Y (f)X − X(f)Y + fZ | f ∈ k[x, y, z]\}, deg x = 1, deg y = 2, deg z = 3, X = \frac{∂}{∂x} − y \frac{∂}{∂z} , Y = \frac{∂}{∂y} , Z = \frac{∂}{∂z};

m5 2 1 𝔤(𝔪) = \{fX + f^{1}Y_{1} + f^{2}Y_{2} | f, f^{1}, f^{2} ∈ k[x, y_{1}, y_{2}]\}, deg x = 1, deg y_{1} = deg y_{2} = 2, X = \frac{∂}{∂x}, Y_{1} = \frac{∂}{∂y_{1}} , Y_{2} = \frac{∂}{∂y_{2}} ;

m5 2 3 𝔤(𝔪) = \{Y_{1}(f)X_{1} + Y_{2}(f)X_{2} − X_{1}(f)Y_{1} − X_{2}(f)Y_{2} + fZ | f ∈ k[x_{1}, x_{2}, y_{1}, y_{2}, z]\}, deg x_{1} = deg x_{2} = deg y_{1} = deg y_{2} = 1, deg z = 2, X_{1} = \frac{∂}{∂x_{1}} − y_{1} \frac{∂}{∂z} , X_{2} = \frac{∂}{∂x_{2}} − y_{2} \frac{∂}{∂z} , Y_{1} = \frac{∂}{∂y_{1}} , Y_{2} = \frac{∂}{∂y_{2}} , Z = \frac{∂}{∂z};

m5 3 1 𝔤(𝔪) = (G_{2}, {α_{1}});

m5 4 1 𝔤(𝔪) = \{Y (f)X − X(f)Y + fZ | f ∈ k[x, y, z]\}, deg x = 1, deg y = 3, deg z = 4, X = \frac{∂}{∂x} − y \frac{∂}{∂z} , Y = \frac{∂}{∂y} , Z = \frac{∂}{∂z};

m6 2 1 𝔤(𝔪) = (B_{3}, {α_{3}}); m6 2 3 𝔤(𝔪) = 𝔤(m3 2) × 𝔤(m3 2); m6 2 3r 𝔤(𝔪) = 𝔤(m3 2)_{ℝ}, where m3 2 is viewed over ℂ;

m6 2 4 𝔤(𝔪) = \{Y (f)X − X(f)Y + fZ + gU | f ∈ k[x, y, z], g ∈ k[x, y, z, u]\}, deg x = deg y = 1, deg z = deg u = 2, X = \frac{∂}{∂x} − y \frac{∂}{∂z} , Y = \frac{∂}{∂y} , Z = \frac{∂}{∂z} , U = \frac{∂}{∂u};

m6 3 2 𝔤(𝔪) = \{Y (f)X − X(f)Y + fZ + gU | f ∈ k[x, y, z], g ∈ k[x, y, z, u]\}, deg x = 1, deg y = deg u = 2, deg z = 3, X = \frac{∂}{∂x} − y \frac{∂}{∂z} , Y = \frac{∂}{∂y} , Z = \frac{∂}{∂z} , U = \frac{∂}{∂u};

m6 3 3 𝔤(𝔪) = (B_{2}, \{α_{1}, α_{2}\}) ⋌ S^{∗}(k^{4}) = ∑_{p=−3}^{3} 𝔤_{p} ⋌ \{f \frac{∂}{∂x_{2}} | f ∈ k[x_{1}, x_{2}, x_{3}, x_{6}]\}, 𝔤_{−3} = ⟨\frac{∂}{∂x_{1}} ⟩, 𝔤_{−2} = ⟨\frac{∂}{∂x_{3}} ⟩, 𝔤_{−1} = ⟨−x_{3} \frac{∂}{∂x_{1}} + x_{6} \frac{∂}{∂x_{3}} , \frac{∂}{∂x_{6}} ⟩, 𝔤_{0} = ⟨x_{1} \frac{∂}{∂x_{1}} + \frac{1}{2}x_{3} \frac{∂}{∂x_{3}} , \frac{1}{2}x_{3} \frac{∂}{∂x_{3}} + x_{6} \frac{∂}{∂x_{6}} ⟩, 𝔤_{1} = ⟨−\frac{1}{2}x_{3}^{2} \frac{∂}{∂x_{1}} + x_{3}x_{6} \frac{∂}{∂x_{3}} + x_{6}^{2} \frac{∂}{∂x_{6}} , x_{1} \frac{∂}{∂x_{3}} − x_{3} \frac{∂}{∂x_{6}} ⟩, 𝔤_{2} = ⟨−x_{1}x_{3} \frac{∂}{∂x_{1}} + (x_{1}x_{6} − \frac{1}{2}x_{3}^{2})\frac{∂}{∂x_{3}} − x_{3}x_{6} \frac{∂}{∂x_{6}} ⟩, 𝔤_{3} = ⟨x_{1}^{2} \frac{∂}{∂x_{1}} + x_{1}x_{3} \frac{∂}{∂x_{3}} − \frac{1}{2}x_{3}^{2} \frac{∂}{∂x_{6}} ⟩, deg x_{6} = 1, deg x_{2} = deg x_{3} = 2, deg x_{1} = 3;

m6 3 4 𝔤(𝔪) = \{(−α_{6}x_{3}^{2} − β_{6}x_{3} + (4α − γ_{6})x_{1} + δ)\frac{∂}{∂x_{1}} + +(2αx_{3} + β)\frac{∂}{∂x_{3}} + γ \frac{∂}{∂x_{6}} | α, β, γ, δ ∈ k[x_{6}]\}; deg x_{6} = 1, deg x_{3} = 2, deg x_{1} = 3;

m6 3 5 𝔤(𝔪) = (A_{3}, {α_{1}, α_{2}, α_{3}}); m6 3 5r 𝔤(𝔪) = 𝔰𝔲(2, 2) (real form of the previous case);

175 GRADED NILPOTENT LIE ALGEBRAS IN LOW DIMENSIONS

m6 4 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} & e_{8} & e_{9} & e_{10} & e_{11} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & −e_{1} & 0 & 5e_{1} & e_{2} & 5e_{3} \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & −2e_{2} & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & −e_{3} & −e_{2} & 3e_{3} & 0 & 8e_{4} \\ e_{4} & 0 & 0 & 0 & 0 & e_{2} & e_{3} & −e_{4} & 0 & e_{4} & 0 & 9e_{5} \\ e_{5} & 0 & 0 & 0 & −e_{2} & 0 & e_{4} & −e_{5} & 0 & −e_{5} & 0 & 8e_{8} \\ e_{6} & 0 & 0 & −e_{1} & −e_{3} & −e_{4} & 0 & 0 & −e_{5} & 2e_{6} & −e_{8} & −e_{9} \\ e_{7} & e_{1} & 2e_{2} & e_{3} & e_{4} & e_{5} & 0 & 0 & e_{8} & 0 & e_{10} & 0 \\ e_{8} & 0 & 0 & e_{2} & 0 & 0 & e_{5} & −e_{8} & 0 & −3e_{8} & 0 & 5e_{10} \\ e_{9} & −5e_{1} & 0 & −3e_{3} & −e_{4} & e_{5} & −2e_{6} & 0 & 3e_{8} & 0 & 5e_{10} & 2e_{11} \\ e_{10} & −e_{2} & 0 & 0 & 0 & 0 & e_{8} & −e_{10} & 0 & −5e_{10} & 0 & 0 \\ e_{11} & −5e_{3} & 0 & −8e_{4} & −9e_{5} & −8e_{8} & e_{9} & 0 & −5e_{10} & −2e_{11} & 0 & 0 \end{aligned}

m6 4 2 \begin{matrix}𝔤(𝔪) = 𝔪 ⋋ 𝔤_{0}(𝔪), \\ 𝔤_{0}(𝔪) = (\begin{matrix}2x + 2y & 0 & 0 & 0 & 0 & 0 \\ 0 & 2x + y & 0 & 0 & 0 & 0 \\ 0 & 0 & x + 2y & 0 & 0 & 0 \\ 0 & 0 & 0 & x + y & 0 & 0 \\ 0 & 0 & 0 & 0 & x & 0 \\ 0 & 0 & 0 & 0 & 0 & y \end{matrix}); \end{matrix}

m6 4 2r \begin{matrix}𝔤(𝔪) = 𝔪 ⋋ 𝔤_{0}(𝔪), \\ 𝔤_{0}(𝔪) = (\begin{matrix}4x & 0 & 0 & 0 & 0 & 0 \\ 0 & 3x & y & 0 & 0 & 0 \\ 0 & −y & 3x & 0 & 0 & 0 \\ 0 & 0 & 0 & 2x & 0 & 0 \\ 0 & 0 & 0 & 0 & x & y \\ 0 & 0 & 0 & 0 & −y & x \end{matrix}); \end{matrix}

m6 5 1 𝔤(𝔪) = \{Y (f)X − X(f)Y + fZ | f ∈ k[x, y, z]\}, deg x = 1, deg y = 4, deg z = 5, X = \frac{∂}{∂x} − y \frac{∂}{∂z} , Y = \frac{∂}{∂y} , Z = \frac{∂}{∂z}; m6 5 2 𝔤(𝔪) = (G_{2}, {α_{1}, α_{2}}). 5.3. The classification of nondegenerate 7-dimensional fundamental Lie algebras.

m7 2 2 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{6} & 0 & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{7} & 0 & 0 & 0 & −e_{1} & −e_{2} & −e_{3} & 0 \end{aligned}

m7 2 3 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{6} & 0 & 0 & 0 & 0 & −e_{1} & 0 & e_{3} \\ e_{7} & 0 & 0 & 0 & −e_{1} & −e_{2} & −e_{3} & 0 \end{aligned}

m7 2 4 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{2} & 0 \\ e_{6} & 0 & 0 & 0 & 0 & −e_{2} & 0 & e_{3} \\ e_{7} & 0 & 0 & 0 & −e_{1} & 0 & −e_{3} & 0 \end{aligned}

m7 2 4r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{2} & −e_{1} \\ e_{6} & 0 & 0 & 0 & −e_{1} & −e_{2} & 0 & e_{3} \\ e_{7} & 0 & 0 & 0 & −e_{2} & e_{1} & −e_{3} & 0 \end{aligned}

m7 2 5 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} & 0 & e_{2} \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 & e_{3} & 0 \\ e_{6} & 0 & 0 & 0 & 0 & −e_{3} & 0 & e_{1} \\ e_{7} & 0 & 0 & 0 & −e_{2} & 0 & −e_{1} & 0 \end{aligned}

m7 2 5r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} & e_{2} & e_{3} \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 & e_{3} & −e_{2} \\ e_{6} & 0 & 0 & 0 & −e_{2} & −e_{3} & 0 & e_{1} \\ e_{7} & 0 & 0 & 0 & −e_{3} & e_{2} & −e_{1} & 0 \end{aligned}

m7 2 9 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{6} & 0 & 0 & 0 & 0 & −e_{1} & 0 & 0 \\ e_{7} & 0 & 0 & −e_{1} & −e_{2} & 0 & 0 & 0 \end{aligned}

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m7 2 10 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{2} & 0 \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{6} & 0 & 0 & 0 & −e_{2} & −e_{1} & 0 & 0 \\ e_{7} & 0 & 0 & −e_{1} & 0 & −e_{2} & 0 & 0 \end{aligned}

m7 2 13 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & e_{1} & 0 & 0 & 0 & 0 \\ e_{3} & 0 & −e_{1} & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 & 0 & 0 \\ e_{6} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{7} & 0 & 0 & 0 & 0 & 0 & −e_{1} & 0 \end{aligned}

m7 3 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} & e_{2} & 0 \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & −e_{1} & −e_{2} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & 0 & 0 & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 2 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} & e_{2} & e_{2} \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & −e_{1} & −e_{2} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & 0 & −e_{2} & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 3 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} & e_{2} & 0 \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & −e_{1} & −e_{2} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & −e_{2} & 0 & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 4 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{2} & 0 \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & −e_{2} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & 0 & 0 & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 4r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & e_{2} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{2} & −e_{1} & 0 \\ e_{5} & 0 & 0 & −e_{1} & −e_{2} & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & −e_{2} & e_{1} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & 0 & 0 & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 5 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{2} & e_{1} \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & −e_{2} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & 0 & −e_{1} & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 6 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & 0 & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{2} & e_{1} \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & −e_{2} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & −e_{2} & −e_{1} & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 6r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & e_{2} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{2} & −e_{1} & e_{2} \\ e_{5} & 0 & 0 & −e_{1} & −e_{2} & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & −e_{2} & e_{1} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & 0 & −e_{2} & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 7 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & −e_{1} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & 0 & −e_{2} & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 8 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & −e_{1} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & −e_{1} & −e_{2} & −e_{3} & −e_{4} & 0 \end{aligned}

179 GRADED NILPOTENT LIE ALGEBRAS IN LOW DIMENSIONS

m7 3 9 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} & 0 & e_{2} \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & −e_{1} & 0 & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & 0 & −e_{2} & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 10 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{2} & 0 \\ e_{5} & 0 & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & −e_{2} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & −e_{1} & 0 & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 11 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & −e_{1} & e_{2} \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & e_{1} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & 0 & −e_{2} & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 11r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & −e_{1} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & 0 & −e_{2} & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 12 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & 0 & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & −e_{1} & −e_{2} & −e_{3} & −e_{4} & 0 \end{aligned}

m7 3 13(α) \begin{matrix}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −(1 + α)e_{1} & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & αe_{1} \\ e_{5} & 0 & (1 + α)e_{1} & 0 & 0 & 0 & e_{4} & −e_{3} \\ e_{6} & 0 & 0 & −e_{1} & 0 & −e_{4} & 0 & e_{2} \\ e_{7} & 0 & 0 & 0 & −αe_{1} & e_{3} & −e_{2} & 0 \end{matrix}

α ∼ 1/α ∼ −1 − α, α ≠ ±1, −2, −1/2, 0

m7 3 13r(α) \begin{matrix}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −2αe_{1} & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & αe_{1} & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & −e_{1} & αe_{1} \\ e_{5} & 0 & 2αe_{1} & 0 & 0 & 0 & e_{4} & −e_{3} \\ e_{6} & 0 & 0 & −αe_{1} & e_{1} & −e_{4} & 0 & e_{2} \\ e_{7} & 0 & 0 & −e_{1} & −αe_{1} & e_{3} & −e_{2} & 0 \end{matrix}

α ∼ −α, α ≠ 0

m7 3 14 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & −e_{1} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{4} & −e_{3} \\ e_{6} & 0 & 0 & −e_{1} & 0 & −e_{4} & 0 & e_{2} \\ e_{7} & 0 & 0 & 0 & e_{1} & e_{3} & −e_{2} & 0 \end{aligned}

m7 3 14r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & −e_{1} & 0 \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{4} & −e_{3} \\ e_{6} & 0 & 0 & 0 & e_{1} & −e_{4} & 0 & e_{2} \\ e_{7} & 0 & 0 & −e_{1} & 0 & e_{3} & −e_{2} & 0 \end{aligned}

m7 3 15 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −2e_{1} & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{5} & 0 & 2e_{1} & 0 & 0 & 0 & e_{4} & −e_{3} \\ e_{6} & 0 & 0 & −e_{1} & 0 & −e_{4} & 0 & e_{2} \\ e_{7} & 0 & 0 & 0 & −e_{1} & e_{3} & −e_{2} & 0 \end{aligned}

m7 3 16 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & −2e_{1} & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{1} \\ e_{5} & 0 & 2e_{1} & 0 & 0 & 0 & e_{4} & −e_{3} \\ e_{6} & 0 & 0 & −e_{1} & −e_{1} & −e_{4} & 0 & e_{2} \\ e_{7} & 0 & 0 & 0 & −e_{1} & e_{3} & −e_{2} & 0 \end{aligned}

181 GRADED NILPOTENT LIE ALGEBRAS IN LOW DIMENSIONS

m7 3 17 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 & e_{4} & −e_{3} \\ e_{6} & 0 & 0 & 0 & 0 & −e_{4} & 0 & e_{2} \\ e_{7} & 0 & 0 & 0 & 0 & e_{3} & −e_{2} & 0 \end{aligned}

m7 3 18 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & e_{4} & −e_{3} \\ e_{6} & 0 & 0 & 0 & −e_{1} & −e_{4} & 0 & e_{2} \\ e_{7} & 0 & 0 & 0 & 0 & e_{3} & −e_{2} & 0 \end{aligned}

m7 3 24 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & e_{2} & 0 & 0 \\ e_{5} & 0 & 0 & 0 & −e_{2} & 0 & 0 & 0 \\ e_{6} & 0 & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{7} & 0 & 0 & −e_{1} & 0 & 0 & −e_{3} & 0 \end{aligned}

m7 3 25 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & e_{2} & 0 & 0 \\ e_{5} & 0 & −e_{1} & 0 & −e_{2} & 0 & 0 & 0 \\ e_{6} & 0 & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{7} & 0 & 0 & −e_{1} & 0 & 0 & −e_{3} & 0 \end{aligned}

m7 3 25r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{2} & e_{3} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{3} & −e_{2} \\ e_{6} & 0 & 0 & −e_{1} & −e_{2} & −e_{3} & 0 & 0 \\ e_{7} & 0 & −e_{1} & 0 & −e_{3} & e_{2} & 0 & 0 \end{aligned}

m7 3 26 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & −e_{2} & 0 \\ e_{6} & 0 & 0 & 0 & 0 & e_{2} & 0 & e_{3} \\ e_{7} & 0 & 0 & −e_{1} & −e_{2} & 0 & −e_{3} & 0 \end{aligned}

m7 3 27 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & −e_{2} & 0 \\ e_{6} & 0 & 0 & 0 & 0 & e_{2} & 0 & e_{3} \\ e_{7} & 0 & −e_{1} & 0 & −e_{2} & 0 & −e_{3} & 0 \end{aligned}

m7 4 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{3} & e_{4} \\ e_{6} & 0 & 0 & 0 & −e_{1} & −e_{3} & 0 & e_{5} \\ e_{7} & 0 & 0 & −e_{1} & −e_{2} & −e_{4} & −e_{5} & 0 \end{aligned}

m7 4 2 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{3} & e_{4} \\ e_{6} & 0 & 0 & −e_{1} & 0 & −e_{3} & 0 & e_{5} \\ e_{7} & 0 & 0 & 0 & −e_{2} & −e_{4} & −e_{5} & 0 \end{aligned}

m7 4 2r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{2} & −e_{1} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{3} & e_{4} \\ e_{6} & 0 & 0 & −e_{1} & −e_{2} & −e_{3} & 0 & e_{5} \\ e_{7} & 0 & 0 & −e_{2} & e_{1} & −e_{4} & −e_{5} & 0 \end{aligned}

m7 4 5 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & 0 & 0 & 0 & 0 & 0 & e_{4} \\ e_{7} & 0 & −e_{1} & 0 & −e_{2} & −e_{3} & −e_{4} & 0 \end{aligned}

m7 4 6 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & e_{2} & 0 \\ e_{5} & 0 & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{6} & 0 & −e_{1} & 0 & −e_{2} & 0 & 0 & e_{4} \\ e_{7} & 0 & 0 & 0 & 0 & −e_{3} & −e_{4} & 0 \end{aligned}

183 GRADED NILPOTENT LIE ALGEBRAS IN LOW DIMENSIONS

m7 5 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & −e_{1} & 0 & e_{3} \\ e_{5} & 0 & 0 & 0 & e_{1} & 0 & 0 & e_{4} \\ e_{6} & 0 & 0 & −e_{1} & 0 & 0 & 0 & e_{5} \\ e_{7} & 0 & 0 & −e_{2} & −e_{3} & −e_{4} & −e_{5} & 0 \end{aligned}

m7 5 2 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & e_{3} & e_{4} \\ e_{6} & 0 & 0 & 0 & 0 & −e_{3} & 0 & e_{5} \\ e_{7} & 0 & −e_{1} & 0 & −e_{2} & −e_{4} & −e_{5} & 0 \end{aligned}

m7 5 3 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & e_{1} & 0 \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{4} & 0 & 0 & 0 & 0 & −e_{1} & 0 & e_{2} \\ e_{5} & 0 & 0 & 0 & e_{1} & 0 & e_{3} & e_{4} \\ e_{6} & 0 & −e_{1} & 0 & 0 & −e_{3} & 0 & e_{5} \\ e_{7} & 0 & 0 & 0 & −e_{2} & −e_{4} & −e_{5} & 0 \end{aligned}

m7 5 4 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} & e_{2} & 0 \\ e_{5} & 0 & 0 & 0 & −e_{1} & 0 & e_{3} & e_{4} \\ e_{6} & 0 & 0 & 0 & −e_{2} & −e_{3} & 0 & e_{5} \\ e_{7} & 0 & −e_{1} & −e_{2} & 0 & −e_{4} & −e_{5} & 0 \end{aligned}

m7 5 5 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & e_{1} & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & −e_{1} & 0 & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & e_{1} & e_{2} & 0 \\ e_{5} & 0 & 0 & e_{1} & −e_{1} & 0 & e_{3} & e_{4} \\ e_{6} & 0 & −e_{1} & 0 & −e_{2} & −e_{3} & 0 & e_{5} \\ e_{7} & 0 & −e_{1} & −e_{2} & 0 & −e_{4} & −e_{5} & 0 \end{aligned}

m7 5 5r \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & e_{1} & e_{2} & 0 \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{5} & 0 & 0 & −e_{1} & 0 & 0 & e_{3} & e_{4} \\ e_{6} & 0 & 0 & −e_{2} & 0 & −e_{3} & 0 & e_{5} \\ e_{7} & 0 & −e_{1} & 0 & −e_{2} & −e_{4} & −e_{5} & 0 \end{aligned}

m7 6 1 \begin{aligned}& e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ e_{2} & 0 & 0 & 0 & 0 & 0 & 0 & e_{1} \\ e_{3} & 0 & 0 & 0 & 0 & 0 & 0 & e_{2} \\ e_{4} & 0 & 0 & 0 & 0 & 0 & 0 & e_{3} \\ e_{5} & 0 & 0 & 0 & 0 & 0 & 0 & e_{4} \\ e_{6} & 0 & 0 & 0 & 0 & 0 & 0 & e_{5} \\ e_{7} & 0 & −e_{1} & −e_{2} & −e_{3} & −e_{4} & −e_{5} & 0 \end{aligned}

Remark 4. To complete the classification, we provide the list of all degenerate 7-dimensional fundamental Lie algebras.

m7 1 = k^{7};  m7 2 12 = k^{2} × m5 2 3;  m7 3 28 = k^{3} × m4 3;
m7 2 1 = k × m6 2 1;  m7 3 19 = k^{2} × m5 3 1;  m7 4 3 = k × m6 4 1;
m7 2 6 = k^{2} × m5 2 1;  m7 3 20 = k × m6 3 2;  m7 4 4 = k × m6 4 2;
m7 2 7 = k × m6 2 3;  m7 3 21 = k × m6 3 3;  m7 4 4r = ℝ × m6 4 2r;
m7 2 7r = ℝ × m6 2 3r;  m7 3 22 = k × m6 3 4;  m7 4 7 = k^{2} × m5 4 1;
m7 2 8 = k × m6 2 4;  m7 3 23 = k × m6 3 5;  m7 5 6 = k × m6 5 1;
m7 2 11 = k^{4} × m3 2;  m7 3 23r = ℝ × m6 3 5r;  m7 5 7 = k × m6 5 2.

References

[1] T. Morimoto, Geometric structures on filtered manifolds, Hokkaido Math. J., 22 (1993), pp. 263–347.

[2] T. Morimoto, N. Tanaka, The classification of the real primitive infinite Lie algebras, J. Math. Kyoto Univ., 10 (1970), pp. 207–243.

[3] V. Guillemin, S. Sternberg, An algebraic model of transitive differential geometry, Bull. Amer. Math. Soc., 70 (1964), pp. 16–47.

[4] N. Tanaka, On differentail systems, graded Lie algebras and pseudo-groups, J. Math. Kyoto Univ., 10 (1970), pp. 1–82.

[5] N. Tanaka, On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J., 8 (1979), pp. 23–84.

[6] K. Yamaguchi, Differential systems associated with simple graded Lie algebras, Adv. Studies in Pure Math., 22 (1993), pp. 413–494.

Received July 27, 1999