Journal of Lie Theory Volume 11 (2001) 483–489 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

Derivation Algebras of Certain Nilpotent Lie Algebras ^{1}

J. M. Cabezas and J. R. Gómez

Communicated by K. H. Hofmann

Abstract. We compute the dimension of the algebra of derivations of any n-dimensional nilpotent real or complex Lie algebra whose Goze’s invariant (or characteristic sequence) is (n − 3, 1, …, 1).

1. Introduction

Let 𝔤 be a nilpotent Lie algebra of dimension n. The adjoint ad(X) of an element X ∈ 𝔤 is defined by ad(X)(Y) = [X, Y]. For all X ∈ 𝔤 − [𝔤, 𝔤], c(X) = (c_{1}(X), c_{2}(X), …, 1) is the sequence, in decreasing order, of the dimensions of the characteristic subspaces of the nilpotent operator ad(X). The sequence c(𝔤) = sup{c(X) : X ∈ 𝔤−[𝔤, 𝔤]} is called the characteristic sequence or Goze’s invariant of 𝔤. The filiform, quasifiliform, and abelian Lie algebras of dimension n have as their Goze invariant (n − 1, 1), (n − 2, 1, 1) and (1, 1, …, 1), respectively.

Definition 1.1. A nilpotent Lie algebra 𝔤, of dimension n, is called p-filiform if its Goze invariant is (n − p, 1, …, 1).

We will denote by

C^{0}𝔤 = 𝔤 ⊇ C^{1}𝔤 = [𝔤, 𝔤] ⊇ C^{2}𝔤 = [𝔤, [𝔤, 𝔤]] ⊇ ⋯ ⊇ C^{n−p}𝔤 = {0}, 1 ≤ p < n,

the descending central series of 𝔤.

Remark 1.2. If 𝔤 is p-filiform then dim C^{k}𝔤/C^{k+1}𝔤 ≥ 1, 1 ≤ k ≤ n − p − 1.

Obviously, for a p-filiform algebra, the relation

n = dim \frac{𝔤}{[𝔤, 𝔤]} + ∑_{k=1}^{n−p−1} \frac{C^{k}𝔤}{C^{k+1}𝔤} ≥ dim \frac{𝔤}{[𝔤, 𝔤]} + n − p − 1

implies dim 𝔤/[𝔤, 𝔤] ≤ p + 1.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Remark 1.3. If 𝔤 is 1-filiform then dim 𝔤/[𝔤, 𝔤] = 2 and dim C^{k}𝔤/C^{k+1}𝔤 = 1, 1 ≤ k ≤ n − 2. That is, dim C^{k}𝔤 = n − k − 1, 1 ≤ k ≤ n − 1, and 𝔤 is a filiform Lie algebra.

1Partially supported by the PAICYT, FQM143, of the Junta de Andalucía (Spain) and the research project, BFM2000-1047, of the Ministerio de Ciencia y Tecnología of Spain.

Remark 1.4. We note that (n − 1)-filiform algebras are abelian, and that an (n−2)-filiform algebra is a direct sum of a (2k+1)-dimensional Heisenberg algebra, 1 ≤ k ≤ [\frac{n−1}{2}] + 1, and an abelian algebra [9].

The next more general case among p-filiform algebras is that of (n − 3)-filiform algebras, and this is the case we consider. The structure of these algebras was already clarified by the authors together with Jiménez-Merchán in [9]. Here we use this information in order to compute for any (n − 3)-filiform nilpotent Lie algebra 𝔤 the dimension of the derivation algebra Der 𝔤.

If 𝔤 is an arbitrary Lie algebra, then the derivations of 𝔤 are precisely the 1-cocycles on 𝔤 with values in the adjoint module 𝔤, and the inner derivations ad x, x ∈ 𝔤 are the 1-coboundaries. Hence the adjoint morphism ad is embedded into an exact sequence 0 → 𝔷(𝔤) → 𝔤 →^{ad} Der(𝔤) → H^{1}(𝔤, 𝔤) → 0, where 𝔷(𝔤) is the center of 𝔤. Accordingly, as an application of our calculation, we shall compute dim H^{1}(𝔤, 𝔤) for any (n − 3)-filiform nilpotent Lie algebra.

2. Some background and history

Nilpotent algebras over a field with characteristic 0 were characterized up to dimension 6 by Morozov in 1958 [17] and in dimension 7 by Ancochea Bermúdez and Goze [3]. For dimensions beyond 7, classifications are known only for filiform algebras through the work of Ancochea Bermúdez and Goze for dimension 8 [2], and of the second author and Echarte for dimension 9 [10]. Descriptions of the variety of nilpotent Lie algebras are available for dimension 7 [4] and dimension 8 [5]; however, beyond dimension 8, again one has information only for filiform algebras [1], [2], [4], [11], and [15]. It appears to be a general phenomenon that structural information on nilpotent Lie algebras without dimensional limitations is readily accessible only for the class of filiform algebras. For finite groups, the concept of “filiformity” was introduced by Norman Blackburn in [6] and discussed in the textbook literature [16], pp. 361–377.

The concept of a p-filiform nilpotent Lie algebra is more general than that of a filiform algebra; the role of filiform algebras in the structure theory of nilpotent Lie algebras motivates us to contribute information on this more general class of nilpotent Lie algebras—at least for low nilpotent class, i.e. large p, namely, that of p = n − 3 [9], p = n − 4 [8], and p = n − 5 [7]. In this article we deal with p = n − 3 exclusively.

3. Algebras of derivations: The results

In [9] it is shown that any n-dimensional nilpotent (n − 3)-filiform Lie algebra 𝔤 over a ground field 𝕂 = ℝ or 𝕂 = ℂ with n > 4 has a basis {X_{0}, X_{1}, X_{2}, X_{3}} ∪ {Y_{1}, …, Y_{n−4}} such that 𝔤 is isomorphic to one of the following n − 2 algebras, where [m] is the largest integer not exceeding m:

𝔤^{2q−1} for q ∈ \{1, …, [\frac{n − 2}{2}]\}, 𝔤^{2q} for q ∈ \{1, …, [\frac{n − 3}{2}]\}, and 𝔤^{n−2},

which are defined by the following equations:

𝔤^{2q−1} : {[X_{0}, X_{1}] = X_{2}, \atop [X_{0}, X_{2}] = X_{3},}  [Y_{2k−1}, Y_{2k}] = X_{3}, 1 ≤ k ≤ q − 1,
𝔤^{2q} : {[X_{0}, X_{1}] = X_{2}, \atop [X_{0}, X_{2}] = X_{3},} [X_{1}, Y_{n−4}] = X_{3}, [Y_{2k−1}, Y_{2k}] = X_{3}, 1 ≤ k ≤ q − 1,
𝔤^{n−2} : {[X_{0}, X_{1}] = X_{2}, \atop [X_{0}, X_{2}] = X_{3}}, [X_{1}, X_{2}] = Y_{n−4}.

With this notation we can formulate our main result.

Main Theorem. For an n-dimensional nilpotent (n − 3)-filiform Lie algebra 𝔤 over the field 𝕂 = ℝ or 𝕂 = ℂ, the dimension of the algebra of derivations is

dim ( Der(𝔤^{2q−1})) = n^{2} − (3 + 2q)n + 2q^{2} + 3q + 6, q ∈ \{1, …, [\frac{n−2}{2}]\},
dim ( Der(𝔤^{2q})  = n^{2} − (4 + 2q)n + 2q^{2} + 5q + 8, q ∈ \{1, …, [\frac{n−3}{2}]\}, and
dim ( Der(𝔤^{n−2})) = n^{2} − 6n + 15.

The proof will be given in the following section.

4. Algebras of derivations: The proofs

In this section, we will determine the dimension of the algebra of derivations of each (n − 3)-filiform Lie algebra of dimension n. However, we shall deal only with the algebra of derivations Der(𝔤^{2q−1}); the corresponding results for the other algebras are proved analogously.

For each q, the algebra 𝔤^{2q−1} is the direct sum 𝔥_{1}^{2q−1} ⊕ 𝔥_{2}^{2q−1} of the subalgebra 𝔥_{1}^{2q−1} = ⟨X_{0}, X_{1}, X_{2}, X_{3}, Y_{1}, Y_{2}, …, Y_{2q−3}, Y_{2q−2}⟩ and the central ideal 𝔥_{2}^{2q−1} = ⟨Y_{2q−1}, Y_{2q}, …, Y_{n−4}⟩. Any linear map d ∈ Hom(𝔤^{2q−1}, 𝔤^{2q−1}) decomposes uniquely into a sum d = d̄_{1} + d̄_{2} + d̄_{12} + d̄_{21} with d̄_{i} =∈ Hom(𝔥_{i}^{2q−1}, 𝔥_{i}^{2q−1}), i = 1, 2, and d̄_{ij} ∈ Hom(𝔥_{i}^{2q−1}, 𝔥_{j}^{2q−1}), (i, j) = (1, 2), (2, 1). It is verified straightforwardly that each derivation d ∈ Der(𝔤^{2q−1}) maps 𝔥_{i}^{2q−1} into itself for = 1, 2, whence d̄_{i} ∈ Der 𝔥_{i}^{2q−1}, i = 1, 2. Therefore, if D(𝔥_{1}^{2q−1}, 𝔥_{2}^{2q−1}) denotes that subspace of Hom(𝔥_{i}^{2q−1}, 𝔥_{j}^{2q−1}) which is induced by Der(𝔤^{2q−1}), we have a direct sum decomposition

(∗) Der(𝔤^{2q−1}) = Der(𝔥_{1}^{2q−1}) ⊕ Der(𝔥_{2}^{2q−1}) ⊕ D(𝔥_{1}^{2q−1}, 𝔥_{2}^{2q−1}) ⊕ D(𝔥_{2}^{2q−1}, 𝔥_{1}^{2q−1}),

such that each d ∈ Der(𝔤^{2q−1}) decomposes uniquely in the form d = d̄_{1} + d̄_{2} + d̄_{12}+d̄_{21} with d̄_{i} ∈ Der(𝔥_{i}^{2q−1}), i = 1, 2, d̄_{ij} ∈ D(𝔥_{i}^{2q−1}, 𝔥_{i}^{2q−1}), (i, j) = (1, 2), (2, 1). Accordingly,

(∗∗) dim Der(𝔤^{2q−1}) = dim Der(𝔥_{1}^{2q−1}) + dim Der(𝔥_{2}^{2q−1}) + dim D(𝔥_{1}^{2q−1}, 𝔥_{2}^{2q−1}) + dim D(𝔥_{2}^{2q−1}, 𝔥_{1}^{2q−1})

and we have to determine the dimensions of the individual summands. It will be helpful to record here that d̄_{ij} ∈ D(𝔥_{i}^{2q−1}, 𝔥_{j}^{2q−1}) implies

d̄_{ij}(𝔥_{i}^{2q−1}) ⊆ Z(𝔥_{j}^{2q−1}), d̄_{ij}([𝔥_{i}^{2q−1}, 𝔥_{i}^{2q−1}]) = {0} and d̄_{ij}(𝔥_{j}^{2q−1}) = {0}

for (i, j) ∈ {(1, 2), (2, 1)}.

Calculation of Der(𝔥_{2}^{2q−1}). Since 𝔥_{2}^{2q−1} = ⟨Y_{2q−1}, Y_{2q}, …, Y_{n−4}⟩ is an abelian Lie algebra of dimension n − 2q − 2, it follows trivially that Der(𝔥_{2}^{2q−1}) = 𝔤𝔩(𝔥_{2}^{2q−1}). Thus dim(Der(𝔥_{2}^{2q−1})) = (n − 2q − 2)^{2}.

Calculation of D(𝔥_{1}^{2q−1}, 𝔥_{2}^{2q−1}). Let d̄_{12} ∈ D(𝔥_{1}^{2q−1}, 𝔥_{2}^{2q−1}). From d̄_{12}([𝔥_{1}^{2q−1}, 𝔥_{1}^{2q−1}]) = {0} we conclude d̄_{12}(X_{2}) = 0 and d̄_{12}(X_{3}) = 0, while d̄_{12}(𝔥_{1}^{2q−1}) ⊂ Z(𝔥_{2}^{2q−1}) = 𝔥_{2}^{2q−1} implies d̄_{12}(X_{i}) =

∑_{k=2q−1}^{n−4} a_{ik}Y_{k} for i = 0, 1, and d̄_{12}(Y_{j}) = ∑_{k=2q−1}^{n−4} b_{jk}Y_{k} for 1 ≤ j ≤ 2q − 2. Finally,

d̄_{12}(𝔥_{2}^{2q−1}) = {0} gives us d̄_{12}(Y_{k}) = 0, 2q − 1 ≤ k ≤ n − 4. Thus we obtain a basis for D(𝔥_{1}^{2q−1}, 𝔥_{2}^{2q−1}) in the form of {f̄_{ik}, ḡ_{jk}} where f̄_{ik}(X_{i}) = Y_{k}, ḡ_{jk}(Y_{j}) = Y_{k} for 0 ≤ i ≤ 1, 1 ≤ j ≤ 2q − 2, and 2q − 1 ≤ k ≤ n − 4. Accordingly, we find dim(D(𝔥_{1}^{2q−1}, 𝔥_{2}^{2q−1})) = 2q.(n − 2q − 2).

Calculation of D(𝔥_{2}^{2q−1}, 𝔥_{1}^{2q−1}) Let d̄_{21} ∈ D(𝔥_{2}^{2q−1}, 𝔥_{1}^{2q−1}). From d̄_{21}(𝔥_{2}^{2q−1}) ⊂ Z(𝔥_{1}^{2q−1}) = ⟨X_{3}⟩ it follows that d̄_{21}(Y_{k}) = c_{k}X_{3}, q −1 ≤ k ≤ n−4. Further, d̄_{21}(𝔥_{1}^{2q−1}) = {0} implies d̄_{21}(X_{i}) = 0, 0 ≤ i ≤ 3 and d̄_{21}(Y_{j}) = 0, 1 ≤ j ≤ 2q − 2. Thus we get a basis of D(𝔥_{2}^{2q−1}, 𝔥_{1}^{2q−1}) of the form {h̄_{k}}, where h̄_{k}(Y_{k}) = X_{3}, 2q − 1 ≤ k ≤ n − 4. Hence we obtain dim(D(𝔥_{2}^{2q−1}, 𝔥_{1}^{2q−1})) = n − 2q − 2.

Calculation of Der(𝔥_{1}^{2q−1}) While the previous steps of the calculations were comparatively easy, the computation of the derivations of the non-abelian part of an (n − 3)-filiform Lie algebra is difficult. However, the fact that these algebras can be graded helps in this task. Moreover, the computation will be easiest if the number of subspaces of the gradation is maximal. Therefore we shall now procede to construct a maximal gradation as follows:

𝔥_{1}^{2q−1} = ⨁_{k∈ℤ} 𝔤_{k},  ℤ = {0, ±1, ±2, … }, where
𝔤_{−k} = {0},  q − 2 < k,
𝔤_{−k} = ⟨Y_{−2k+2q−3}⟩, 0 ≤ k ≤ q − 2,
𝔤_{k} = ⟨X_{k−1}⟩, 1 ≤ k ≤ 3,
𝔤_{4}  = ⟨X_{3}, Y_{2q−2}⟩,
𝔤_{k}  = ⟨Y_{−2k+2q+6}⟩, 5 ≤ k ≤ q + 2,
𝔤_{k}  = {0},  q + 2 < k.

Now let d̄_{1} ∈ Der(𝔥_{1}^{2q−1}). Then d̄_{1} = ∑_{i∈Z} d_{i}, where d_{i} ∈ Der(𝔥_{1}^{2q−1}) and d_{i}(𝔤_{j}) ⊂ 𝔤_{i+j}. From d_{2q}(𝔤_{−q+2}) ⊂ 𝔤_{q+2} and d_{−2q}(𝔤_{q+2}) ⊂ 𝔤_{−q+2} we conclude

that d_{i} = 0 for i > 2q and i < −2q and thus d̄_{1} = ∑_{i=−2q}^{2q} d_{i}. Now we will express

each d_{i}, −2q ≤ i ≤ 2q, as a linear combination of some set B_{i}, −2q ≤ i ≤ 2q, of linearly independent derivations of 𝔥_{1}^{2q−1} such that ⋃_{i=−2q}^{2q} B_{i} is a basis of Der(𝔥_{1}^{2q−1}). Consequently we shall have dim(Der(𝔥_{1}^{2q−1})) = ∑_{i=−2q}^{2q} dim⟨B_{i}⟩. In the calculations we have to keep in mind that the ideals of central descending sequence:

\begin{aligned}C^{0}(𝔥_{1}^{2q−1}) & = & 𝔥_{1}^{2q−1} \\ C^{1}(𝔥_{1}^{2q−1}) & = & [𝔥_{1}^{2q−1}, C^{0}(𝔥_{1}^{2q−1})] = ⟨X_{2}, X_{3}⟩ \\ C^{2}(𝔥_{1}^{2q−1}) & = & [𝔥_{1}^{2q−1}, C^{1}(𝔥_{1}^{2q−1})] = ⟨X_{3}⟩ \end{aligned}

are characteristic ideals, that is, they are stable under derivation. As an example, we will show the computation of d_{−j}, q + 3 ≤ j ≤ 2q. Since d_{−j}(𝔤_{t}) ⊂ 𝔤_{t−j} for all t, we obtain

d_{−j}(Y_{2k}) = β_{2k}^{j}.Y_{4q−2j−2k+3}, 1 ≤ k ≤ 2q − j + 1.

Using that d_{−j} is a derivation, we get

β_{2k}^{j} = β_{4q−2j−2k+4}^{j}, 1 ≤ k ≤ 2q − j + 1.

Thus we have [\frac{2q−j+2}{2}] free parameters. We define derivations δ_{k}^{−j}, 1 ≤ k ≤ [\frac{2q−j+2}{2}], by δ_{k}^{−j}(Y_{2k}) = Y_{4q−2j−2k+3}, δ_{k}^{−j}(Y_{4q−2j−2k+4}) = Y_{2k−1}, 1 ≤ k ≤ [\frac{2q−j+2}{2}] and set B_{−j} = \{δ_{k}^{−j}\}, 1 ≤ k ≤ [\frac{2q−j+2}{2}]. We then verify that d_{−j} ∈ ⟨B_{−j}⟩ and thus dim⟨B_{−j}⟩ = [\frac{2q−j+2}{2}], q + 3 ≤ j ≤ 2q, q ≥ 3.

This was already rather technical; however, the other calculations are even more difficult. Their results are summarized in the table in Section 6 at the end of the paper. As a consequence of this table, we obtain

dim(Der(𝔥_{1}^{2q−1})) = ∑_{i=−2q}^{i=2q} dim⟨B_{i}⟩ = 2q^{2} + q + 4, q ≥ 1,

concluding the calculation of the summands in (∗∗). Indeed, the summary of our findings is

dim(Der(𝔥_{1}^{2q−1}))  = 2q^{2} + q + 4
dim(Der(𝔥_{2}^{2q−1}))  = (n − 2q − 2)^{2}
dim(D(𝔥_{1}^{2q−1}, 𝔥_{2}^{2q−1})) = 2q.(n − 2q − 2)
dim(D(𝔥_{2}^{2q−1}, 𝔥_{1}^{2q−1})) = (n − 2q − 2).

Thus from (∗∗) we finally obtain

dim(Der(𝔤^{2q−1})) = n^{2} − (3 + 2q)n + 2q^{2} + 3q + 6 for 1 ≤ q ≤ [\frac{n − 2}{2}].

as asserted in the Main Theorem.

5. Applications to Cohomology

As we noted in the introduction, information on the algebra of derivations of a Lie algebra may be interpreted in terms of cohomology. Retaining the notation of the Main Theorem in Section 3 and the paragraph that precedes it we obtain the following result.

Theorem 5.1. For an n-dimensional real or complex nilpotent n − 3-filiform Lie algebra 𝔤 the dimension of the first cohomology group of 𝔤 over the adjoint module 𝔤 is

dim (H^{1}(𝔤^{2q−1}, 𝔤^{2q−1})) = n^{2} − (3 + 2q)n + 2q^{2} + q + 5, q ∈ \{1, …, [\frac{n−2}{2}]\},
dim (H^{1}(𝔤^{2q}, 𝔤^{2q}))  = n^{2} − (4 + 2q)n + 2q^{2} + 3q + 6, q ∈ \{1, …, [\frac{n−3}{2}]\},
dim (H^{1}(𝔤^{n−2}, 𝔤^{n−2})) = n^{2} − 6n + 12.

6. Table

dim⟨B_{−j}⟩ = [\frac{2q−j+2}{2}]  q + 3 ≤ j ≤ 2q  q ≥ 3
dim⟨B_{−j}⟩ = q − j + 1 + [\frac{j−2}{2}]  4 ≤ j ≤ q − 3  q ≥ 7
dim⟨B_{−q+2}⟩ = 3 + [\frac{q−4}{2}]  q ≥ 6
dim⟨B_{−q+1}⟩ = 1 + [\frac{q−3}{2}]  q ≥ 5
dim⟨B_{−q}⟩ = [\frac{q−2}{2}]  q ≥ 4
dim⟨B_{−q−1}⟩ = [\frac{q−1}{2}]  q ≥ 3
dim⟨B_{−q−2}⟩ = [\frac{q}{2}]  q ≥ 2
dim⟨B_{−3}⟩ = \{ \begin{matrix}q − 2 & if & q ≥ 5 \\ 1 & if & q = 4 \\ 0 & if & q ≤ 3 \end{matrix}

dim⟨B_{−2}⟩ = \{ \begin{matrix}q − 1 & if & q ≥ 4 \\ 1 & if & q = 3 \\ 0 & if & q ≤ 2 \end{matrix}

dim⟨B_{−1}⟩ = \{ \begin{matrix}q & if & q ≥ 3 \\ 1 & if & q = 2 \\ 0 & if & q = 1 \end{matrix}
dim⟨B_{0}⟩  = \{ \begin{matrix}q + 2 & if & q ≥ 2 \\ 2 & if & q = 1 \end{matrix}
dim⟨B_{1}⟩  = \{ \begin{matrix}q & if & q ≥ 2 \\ 2 & if & q = 1 \end{matrix}
dim⟨B_{2}⟩  = \{ \begin{matrix}q − 1 & if & q ≥ 3 \\ 2 & if & 1 ≤ q ≤ 2 \end{matrix}
dim⟨B_{3}⟩  = \{ \begin{matrix}q − 2 & if & q ≥ 4 \\ 2 & if & 2 ≤ q ≤ 3 \\ 1 & if & q = 1 \end{matrix}
dim⟨B_{j}⟩  = q − j + 2 + [\frac{j−4}{2}]  4 ≤ j ≤ q − 2 q ≥ 6
dim⟨B_{q−1}⟩ = 3 + [\frac{q−5}{2}]  q ≥ 5
dim⟨B_{q}⟩ = 3 + [\frac{q−4}{2}] q ≥ 4
dim⟨B_{q+1}⟩ = 3 + [\frac{q−3}{2}]  q ≥ 3
dim⟨B_{q+2}⟩ = 2 + [\frac{q−2}{2}]  q ≥ 2
dim⟨B_{j}⟩  = [\frac{2q−j+2}{2}] q + 3 ≤ j ≤ 2q q ≥ 3

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J. M. Cabezas Dpto. de Matemática Aplicada Universidad del País Vasco Nieves Cano 12 E-01006 Vitoria (Spain) mapcamaj@vc.ehu.es J. R. Gómez Dpto. de Matemática Aplicada I Universidad de Sevilla Avda. Reina Mercedes s. n. E-41012 Sevilla (Spain) jrgomez@cica.es

Received February 8, 2000 and in final form December 14, 2000