Journal of Lie Theory Volume 12 (2002) 423–447 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

Complete Filtered Lie Algebras over a Vector Space of Dimension Two

Thomas W. Judson

Communicated by E. B. Vinberg

Abstract. There may exist many non-isomorphic complete filtered Lie algebras with the same graded algebra. In [6], we found elements in the Spencer cohomology that determined all complete filtered Lie algebras having certain graded algebra provided that obstructions do not exist in the cohomology at higher levels. In this paper we use the Spencer cohomology to classify all graded and filtered algebras over a real vector space of dimension two.

1. Introduction

Closed transitive Lie algebras are subalgebras of the Lie algebra D(𝕂^{n}) of formal vector fields. If 𝕂 a field of characteristic zero, X is a formal vector field in D(𝕂^{n}) if

X = ∑_{i} X_{i}(x_{1}, …, x_{n})\frac{∂}{∂x_{i}} ,

where X_{i} in 𝕂[[x_{1}, …, x_{n}]]. The vector space D(𝕂^{n}) is a Lie algebra under the usual bracket operation

[X, Y] = ∑_{i,j} \{X^{i} \frac{∂Y^{j}}{∂x_{i}} − Y^{i} \frac{∂X^{j}}{∂x_{i}} \} \frac{∂}{∂x_{j}} .

If D^{k}(𝕂^{n}) is the set of X ∈ D(𝕂^{n}) such that each X^{i} has no terms of degree k or less, then D(𝕂^{n}) has a natural filtration

D(𝕂^{n}) ⊃ D^{0}(𝕂^{n}) ⊃ D^{1}(𝕂^{n}) ⊃ D^{2}(𝕂^{n}) ⊃ · · · .

Guillemin and Sternberg studied local geometries by examining Lie algebras of formal vector fields [3]. More specifically, if we choose a coordinate system and replace each infinitesimal automorphism (which is a vector field) with its Taylor series expansion about the origin, we obtain a subalgebra L of D(𝕂^{n}). Letting L_{k} = D^{k}(𝕂^{n}) ∩ L, we have

L ⊃ L_{0} ⊃ L_{1} ⊃ L_{2} ⊃ ⋯

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag with [L_{i}, L_{j}] ⊂ L_{i+j}. Guillemin and Sternberg limited their study to transitive geometries. That is, for any two points there exists a local transformation that takes one point to other. In infinitesimal terms, there exists an X ∈ L such that X(0) = v for each v ∈ 𝕂^{n}. We also demand that L be closed. If X ∈ D(𝕂^{n}) and there exists an X_{i} ∈ L such that X and X_{i} agree on terms of up to order i for i = 1, 2, …, then X ∈ L. A subalgebra L ⊂ D(𝕂^{n}) satisfying these properties is a closed transitive Lie algebra. Two such algebras are isomorphic when they are equivalent by a formal change of coordinates.

A complete filtered Lie algebra over a field 𝕂 of characteristic zero is a Lie algebra with a decreasing sequence of subalgebras L = L_{−1} ⊃ L_{0} ⊃ L_{1} ⊃ ⋯ satisfying the following conditions.

1. ⋂_{i} L_{i} = 0.

2. [L_{i}, L_{j}] ⊂ L_{i+j} (by convention L_{−2} = L).

3. dim L_{i}/L_{i+1} < ∞.

4. If x ∈ L_{i} for i ≥ 0 and [L, x] ⊂ L_{i}, then x ∈ L_{i+1}.

5. Whenever {x_{i}} is a sequence in L such that x_{i} − x_{i+1} ∈ L_{i} for i ≥ 0, then there exists an x ∈ L such that x − x_{i} ∈ L_{i}.

Every complete filtered Lie algebra is isomorphic to a closed transitive subalgebra of D(𝕂^{n}) [3].

A graded Lie algebra is a Lie algebra ∏_{p=−1}^{∞} G_{p} that satisfies the following conditions.

1. [G_{i}, G_{j}] ⊂ G_{i+j} (by convention G_{−2} = 0).

2. dim G_{i} < ∞.

3. If x ∈ G_{i} for i ≥ 0 and [G_{−1}, x] = 0, then x = 0.

Any graded Lie algebra is a complete filtered Lie algebra if we let L_{i} = G_{i} ×G_{i+1} × · · ·. Conversely, if L is a complete filtered Lie algebra, then the bracket operation on L induces a bracket operation on

G_{L} = ∏_{p=−1}^{∞} L_{p}/L_{p+1}.

We refer to G_{L} as the associated graded algebra of L. An isomorphism of two complete filtered Lie algebras is a Lie algebra isomorphism preserving the filtration. Similarly, an isomorphism of two graded Lie algebras is a Lie algebra isomorphism preserving the gradation.

There may exist many non-isomorphic complete filtered Lie algebras with the same graded algebra. Given a graded Lie algebra ∏ G_{p}, it is an interesting problem to try to reconstruct all complete filtered Lie algebras L whose associated graded algebras are isomorphic to ∏ G_{p}. One of the primary tools for analyzing this problem has been the Spencer cohomology. A complete filtered Lie algebra is isomorphic to its graded algebra provided certain cohomology groups vanish [3, 7, 9, 12]. It is more difficult to determine the complete filtered Lie algebras that are not isomorphic to their graded algebras. Many of the known results have hypothesis that are difficult to verify. In [6] we outlined a theory, where certain elements in the Spencer cohomology determine all the complete filtered Lie algebras having a certain graded algebra provided that obstructions do not exist in the cohomology at a higher level. In this paper we use the theory to classify all graded and filtered algebras over a real vector space of dimension two. Cartan first classified these algebras as pseudogroups on ℝ^{2} [1].

2. Graded Algebras with dim G_{−1} = 2

If ∏ G_{p} is a graded Lie algebra, then V = G_{0} is a linear Lie algebra acting faithfully on G_{−1} by [G_{0}, G_{−1}] ⊂ G_{−1}. For p ≥ 0, we may consider G_{p} to be a subspace of V ⊗S^{p+1}(V^{∗}). If X ∈ G_{p} and v_{0}, …, v_{p} ∈ V , define \overline{X} ∈ V ⊗S^{p+1}(V^{∗}) by

\overline{X}(v_{0}, …, v_{p}) = [⋯ [[X, v_{0}], v_{1}], ⋯ v_{p}].

Since [G_{−1}, G_{−1}] = 0, the Jacobi identity implies that \overline{X}(v_{0}, …, v_{p}) is symmetric in v_{0}, …, v_{p}. The bracket operation on ∏ G_{p} then becomes

[\overline{X}, \overline{Y}](v_{0}, …, v_{p+q}) = \frac{1}{p!(q + 1)!} ∑ \overline{X}(\overline{Y} (v_{j_{0}}, …, v_{j_{q}}), v_{j_{q+1}}, …, v_{j_{p+q}})
− \frac{1}{(p + 1)!q!} ∑ \overline{Y} (\overline{X}(v_{k_{0}}, …, v_{k_{p}}), v_{k_{p+1}}, …, v_{k_{p+q}}).

In particular, if X ∈ G_{p} with p > 0 and v ∈ G_{−1}, then

[\overline{X}, v](v_{1}, …, v_{p}) = \overline{X}(v, v_{1}, …, v_{p}).

Conversely, given a sequence V = G_{−1}, G_{0}, G_{1}, … in V ⊗ S^{p+1}(V^{∗}), we know that ∏ G_{p} is a graded algebra under the bracket operation described above if [G_{p}, G_{q}] ⊂ G_{p+q}.

Given a finite sequence V = G_{−1}, G_{0}, G_{1}, …, G_{n−1} with G_{p} ⊂ V ⊗S^{p+1}(V^{∗}) and [G_{p}, G_{q}] ⊂ G_{p+q} with p, q, and p + q all less than n, we wish to impose conditions on subspaces G_{i} ⊂ V ⊗S^{i+1}(V^{∗}) with i ≥ n that will allow ∏ G_{p} to be a graded algebra. Define the first prolongation Λ^{1}P of a subspace P ⊂ V ⊗S^{p+1}(V^{∗}) to be the subspace of maps T ∈ V ⊗ S^{p+2}(V^{∗}) such that for all fixed v ∈ V , T (v, v_{1}, …, v_{p}) ∈ P. The k-th prolongation is defined inductively by Λ^{1}Λ^{k−1}P. Thus, G_{n} ⊂ Λ^{1}G_{n−1} and [G_{n}, G_{0}] ⊂ G_{n}. Hence, G_{n} must be an invariant subspace under this representation. Since [G_{p}, G_{q}] ⊂ G_{n} whenever p < n, q < n, and p + q = n, we must not choose G_{n} to be too small. If such a G_{n} can be selected, then we are guaranteed a graded algebra containing G_{n}.

For a given Lie algebra G_{0} ⊂ 𝔤𝔩(V) acting on a vector space V = G_{−1}, it is often possible to compute all graded algebras arising from G_{−1} and G_{0}. Suppose that dim V = 2 and G_{0} is a subalgebra of 𝔤𝔩(V). The prolongation Λ^{1}G_{0} of G_{0} consists of T ∈ V ⊗ S^{2}(V^{∗}) such that for v ∈ G_{−1}, T (v) ∈ G_{0}. We can represent elements T ∈ V ⊗ S^{2}(V^{∗}) using matrices

( \begin{matrix}a_{11}^{1} & a_{12}^{1} & a_{22}^{1} \\ a_{11}^{2} & a_{12}^{2} & a_{22}^{2} \end{matrix} ) ,

where

T (e_{i}, e_{j}) = a_{ij}^{1}e_{1} + a_{ij}^{2}e_{2},

if {e_{1}, e_{2}} is a fixed basis for V . Hence, T is in Λ^{1}G_{0} if and only if the matrix is in G_{0} whenever the first or the last column of the matrix is deleted. In general, we shall write

(\begin{matrix}a_{11···1}^{1} & a_{1···12}^{1} & ⋯ & a_{12···2}^{1} & a_{22···2}^{1} \\ a_{11···1}^{2} & a_{1···12}^{2} & ⋯ & a_{12···2}^{2} & a_{22···2}^{2} \end{matrix})

for an element in Λ^{n}G_{0}.

Proposition 2.1. Let V be a real vector space with dim V = 2. The following subalgebras G_{0} are the only subalgebras of 𝔤𝔩(V) up to conjugation.

1. dim G_{0} = 1 and λ ∈ ℝ,

(\begin{matrix}0 & a \\ 0 & 0 \end{matrix}) , (\begin{matrix}a & 0 \\ 0 & λa \end{matrix}) , (\begin{matrix}a & a \\ 0 & a \end{matrix}) , (\begin{matrix}λa & −a \\ a & λa \end{matrix}) .

2. dim G_{0} = 2 and λ ∈ ℝ,

(\begin{matrix}a & 0 \\ 0 & b \end{matrix}) , (\begin{matrix}a & b \\ 0 & a \end{matrix}) , (\begin{matrix}a & −b \\ b & a \end{matrix}) , (\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) .

3. dim G_{0} = 3,

(\begin{matrix}a & b \\ 0 & c \end{matrix}) , 𝔰𝔩(V).

4. dim G_{0} = 4, 𝔤𝔩(V).

A complete determination of Lie algebras of dimension less than or equal to three can be found in Jacobson [5]. To construct the faithful representations of these algebras in 𝔤𝔩(V) up to conjugation, see [4, 5].

Proposition 2.2. Let V be a real vector space of dimension two. The prolongations of G_{0} ⊂ 𝔤𝔩(V) are the algebras (1), (3), (4), (6)–(8), (11), (14), (16), (21), (25), (35), and (37) in Table 1.

As an example, we will compute the prolongations in (7) and (21). Let e_{1}, e_{2} be a basis for V and recall that we can represent elements T ∈ V ⊗ S^{2}(V^{∗}) using matrices

( \begin{matrix}a_{11}^{1} & a_{12}^{1} & a_{22}^{1} \\ a_{11}^{2} & a_{12}^{2} & a_{22}^{2} \end{matrix} ) ,

where

T (e_{i}, e_{j}) = a_{ij}^{1}e_{1} + a_{ij}^{2}e_{2}.

Since T is in Λ^{1}G_{0} if and only if the matrix is in G_{0} whenever the first or the last column of the matrix is deleted, the first prolongation of (7) must be zero. On the other hand, the first prolongation of (21) is

( \begin{matrix}a_{1} & a_{2} & a_{3} \\ 0 & 0 & 0 \end{matrix} ) .

Continuing, we see that the nth prolongation is

( \begin{matrix}a_{1} & a_{2} & ⋯ & a_{n+1} & a_{n+2} \\ 0 & 0 & ⋯ & 0 & 0 \end{matrix} ) .

For a more in depth treatment of prolongation, see [3, 12].

Judson 427

Theorem 2.3. Table 1 is a complete list of all graded algebras up to isomorphism with V , a real vector space of dimension two, and G_{0} ⊂ 𝔤𝔩(V).

Table 1e. Graded Algebras ∏ G_{p} with G_{0} ⊂ 𝔤𝔩(2, ℝ)

1. (\begin{matrix}0 & a \\ 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & a \\ 0 & 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & 0 & a \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

2. G_{k} = 0 for k > n

(\begin{matrix}0 & a \\ 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & a \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & a \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (0) − ⋯

3. λ ≠ 0

(\begin{matrix}a & 0 \\ 0 & λa \end{matrix}) − (0) − ⋯

4.

(\begin{matrix}a & 0 \\ 0 & 0 \end{matrix}) − (\begin{matrix}a & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) − (\begin{matrix}a & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

5. G_{k} = 0 for k > n

(\begin{matrix}a & 0 \\ 0 & 0 \end{matrix}) − (\begin{matrix}a & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}a & 0 & ⋯ & 0 \\ 0 & 0 & ⋯ & 0 \end{matrix}) − (0) − ⋯

6.

(\begin{matrix}a & a \\ 0 & a \end{matrix}) − (0) − ⋯

7. λ ∈ ℝ

(\begin{matrix}λa & −a \\ a & λa \end{matrix}) − (0) − ⋯

8.

(\begin{matrix}a & 0 \\ 0 & b \end{matrix}) − (\begin{matrix}a & 0 & 0 \\ 0 & 0 & b \end{matrix}) − (\begin{matrix}a & 0 & 0 & 0 \\ 0 & 0 & 0 & b \end{matrix}) − ⋯

9.

(\begin{matrix}a & 0 \\ 0 & b \end{matrix}) − (\begin{matrix}a & 0 & 0 \\ 0 & 0 & b \end{matrix}) − (\begin{matrix}a & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

or

(\begin{matrix}a & 0 \\ 0 & b \end{matrix}) − (\begin{matrix}a & 0 & 0 \\ 0 & 0 & b \end{matrix}) − (0) − ⋯

10. G_{k} = 0 for k > n

(\begin{matrix}a & 0 \\ 0 & b \end{matrix}) − (\begin{matrix}a & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}a & 0 & ⋯ & 0 \\ 0 & 0 & ⋯ & 0 \end{matrix}) − (0) − ⋯

11.

(\begin{matrix}a & b \\ 0 & a \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & a \end{matrix}) − (\begin{matrix}0 & 0 & a & b \\ 0 & 0 & 0 & a \end{matrix}) − ⋯

12.

(\begin{matrix}a & b \\ 0 & a \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & a \end{matrix}) − (\begin{matrix}0 & 0 & 0 & b \\ 0 & 0 & 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & 0 & 0 & b \\ 0 & 0 & 0 & 0 & 0 \end{matrix}) − ⋯

or

(\begin{matrix}a & b \\ 0 & a \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & a \end{matrix}) − (0) − ⋯

428 Judson

13. G_{k} = 0 for k > n

(\begin{matrix}a & b \\ 0 & a \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (0) − ⋯

14.

(\begin{matrix}a & −b \\ b & a \end{matrix}) − (\begin{matrix}a & −b & −a \\ b & a & −b \end{matrix}) − (\begin{matrix}a & −b & −a & b \\ b & a & −b & −a \end{matrix}) − ⋯

15. G_{k} = 0 for k > n

(\begin{matrix}a & −b \\ b & a \end{matrix}) − (\begin{matrix}a & −b & −a \\ b & a & −b \end{matrix}) − ⋯ − (\begin{matrix}a & −b & −a & ⋯ \\ b & a & −b & ⋯ \end{matrix}) − (0) − ⋯

16. λ ≠ −1

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & λa & b \\ 0 & 0 & (λ + 1)a \end{matrix}) − ⋯

− (\begin{matrix}0 & ⋯ & 0 & λa & b \\ 0 & ⋯ & 0 & 0 & (λ + 1)a \end{matrix}) − ⋯

17. λ ≠ −1 and G_{k} = 0 for k > n

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (0) − ⋯

or

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − ⋯

18. λ ≠ −1

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & λa & b \\ 0 & 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & 0 & 0 & b \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

19. λ = 1

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & λa & b \\ 0 & 0 & (λ + 1)a \end{matrix}) − (0) − ⋯

20. λ = −(n + 1)/(n − 1), n = 2, 3, … and G_{k} = 0 for k > n

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & λa & b \\ 0 & 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & 0 & 0 & b \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

− (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (0) − ⋯

21.

(\begin{matrix}a_{1} & a_{2} \\ 0 & 0 \end{matrix}) − (\begin{matrix}a_{1} & a_{2} & a_{3} \\ 0 & 0 & 0 \end{matrix}) − (\begin{matrix}a_{1} & a_{2} & a_{3} & a_{4} \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

22.

(\begin{matrix}a & b \\ 0 & 0 \end{matrix}) − (\begin{matrix}a & b & c \\ 0 & 0 & 0 \end{matrix}) − (\begin{matrix}0 & a & b & c \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & 0 & a & b & c \\ 0 & 0 & 0 & 0 & 0 \end{matrix}) − ⋯

23. dim G_{k} = 1 for k > n

(\begin{matrix}a & b \\ 0 & 0 \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & a & b \\ 0 & ⋯ & 0 & 0 & 0 \end{matrix}) − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − ⋯

or

(\begin{matrix}a & b \\ 0 & 0 \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & a & b \\ 0 & ⋯ & 0 & 0 & 0 \end{matrix}) − ⋯

24. G_{k} = 0 for k > n

(\begin{matrix}a & b \\ 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (0) − ⋯

25.

(\begin{matrix}a_{1} & a_{2} \\ 0 & b \end{matrix}) − (\begin{matrix}a_{1} & a_{2} & a_{3} \\ 0 & 0 & b \end{matrix}) − (\begin{matrix}a_{1} & a_{2} & a_{3} & a_{4} \\ 0 & 0 & 0 & b \end{matrix}) − ⋯

26. G_{k} = 0 for k > n

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (0) − ⋯

27. dim G_{k} = 1 for k > n

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & a & b \\ 0 & ⋯ & 0 & 0 & 0 \end{matrix}) − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − ⋯

or

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & a & b \\ 0 & ⋯ & 0 & 0 & 0 \end{matrix}) − ⋯

28.

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & c \end{matrix}) − (\begin{matrix}0 & 0 & 0 & b \\ 0 & 0 & 0 & c \end{matrix}) − ⋯

or

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & c \end{matrix}) − (\begin{matrix}0 & 0 & 0 & b \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

29.

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & c \end{matrix}) − (\begin{matrix}0 & 0 & a & b \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

or

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & c \end{matrix}) − (\begin{matrix}0 & 0 & a & b \\ 0 & 0 & 0 & c \end{matrix}) − ⋯

30.

(\begin{matrix}a_{1} & a_{2} \\ 0 & b \end{matrix}) − (\begin{matrix}a_{1} & a_{2} & a_{3} \\ 0 & 0 & 0 \end{matrix}) − (\begin{matrix}a_{1} & a_{2} & a_{3} & a_{4} \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

or

(\begin{matrix}a_{1} & a_{2} \\ 0 & b \end{matrix}) − (\begin{matrix}a_{1} & a_{2} & a_{3} \\ 0 & 0 & 0 \end{matrix}) − (\begin{matrix}0 & a_{1} & a_{2} & a_{3} \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

31.

(\begin{matrix}a_{1} & a_{2} \\ 0 & b \end{matrix}) − (\begin{matrix}a_{1} & a_{2} & a_{3} \\ 0 & 0 & b \end{matrix}) − (\begin{matrix}0 & a_{1} & a_{2} & a_{3} \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

or

(\begin{matrix}a_{1} & a_{2} \\ 0 & b \end{matrix}) − (\begin{matrix}a_{1} & a_{2} & a_{3} \\ 0 & 0 & b \end{matrix}) − (\begin{matrix}a_{1} & a_{2} & a_{3} & a_{4} \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

or

(\begin{matrix}a_{1} & a_{2} \\ 0 & b \end{matrix}) − (\begin{matrix}a_{1} & a_{2} & a_{3} \\ 0 & 0 & b \end{matrix}) − (\begin{matrix}0 & a_{1} & a_{2} & a_{3} \\ 0 & 0 & 0 & b \end{matrix}) − ⋯

32. λ ≠ 0

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & λa & b \\ 0 & 0 & a \end{matrix}) − (\begin{matrix}0 & 0 & λa & b \\ 0 & 0 & 0 & a \end{matrix}) − ⋯

or

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & λa & b \\ 0 & 0 & a \end{matrix}) − (\begin{matrix}0 & 0 & 0 & b \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

33. λ = 1/2

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & λa & 0 \\ 0 & 0 & a \end{matrix}) − (0) − ⋯

34. λ = (n − 1)/2, n = 3, 4, …, and G_{k} = 0 for k > n

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & λa & b \\ 0 & 0 & a \end{matrix}) − (\begin{matrix}0 & 0 & 0 & b \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (0) − ⋯

35. 𝔰𝔩(2, ℝ) − Λ^{1}𝔰𝔩(2, ℝ) − Λ^{2}𝔰𝔩(2, ℝ) − ⋯

36. 𝔰𝔩(2, ℝ) − (0) − ⋯

37. 𝔤𝔩(2, ℝ) − Λ^{1}𝔤𝔩(2, ℝ) − Λ^{2}𝔤𝔩(2, ℝ) − ⋯

38. 𝔤𝔩(2, ℝ) − (0) − ⋯ or 𝔤𝔩(2, ℝ) − Λ^{1}𝔰𝔩(2, ℝ) − Λ^{2}𝔰𝔩(2, ℝ) − ⋯ or

𝔤𝔩(2, ℝ) − (\begin{matrix}2a & b & 0 \\ 0 & a & 2b \end{matrix}) − (0) − ⋯

For the proof of (35)–(38) refer to Singer and Sternberg [12]. Koch proved (25)–(34) in [10]. It remains to show that (1)–(24) are the only possible graded algebras with dim G_{0} = 1 or 2. We will calculate the Lie brackets on a basis for each graded algebra obtained from the prolongation of G_{0} with dim G_{0} ≤ 2 in the following lemmas. The proof of the theorem follows directly from the following lemmas and the fact that [G_{p}, G_{q}] ⊂ G_{p+q}. For the remainder of the paper we shall let {e_{1}, e_{2}} be a canonical basis for V = G_{−1}.

Lemma 2.4. Let {e_{1}, e_{2}, A_{0}, A_{1}, … } be a basis for (1), where

A_{0} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) , A_{1} = (\begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) , ….

Then the only nonzero bracket relations are [A_{0}, e_{2}] = e_{1} and [A_{i}, e_{2}] = A_{i−1}, where i ≥ 1.

Proof. Clearly, these relations hold as well as the relations [e_{1}, e_{2}] = 0 and [A_{i}, e_{1}] = 0 for i ≥ 0. It remains to show that [A_{i}, A_{j}] = 0. For j > 0 and k = 1 or 2, we have

[[A_{0}, A_{j}], e_{k}] = [A_{0}, [A_{j}, e_{k}]] + [A_{j}, [A_{0}, e_{k}]].

If k = 1, the righthand expression is zero. If k = 2, then

[[A_{0}, A_{j}], e_{2}] = [A_{0}, A_{j−1}] = 0

by induction on j; hence, [A_{0}, A_{j}] = 0. Similarly, if we fix j and induct on i, then [A_{i}, A_{j}] = 0.

The proofs of the following lemmas are similar.

Lemma 2.5. Let {e_{1}, e_{2}, A_{0}, A_{1}, … } be a basis for (4), where

A_{0} = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) , A_{1} = (\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) , ….

The only nonzero bracket operations are [A_{0}, e_{1}] = e_{1} and [A_{i}, e_{1}] = A_{i−1} for i ≥ 1.

Lemma 2.6. Let {e_{1}, e_{2}, A_{0}, A_{1}, …, B_{0}, B_{1}, … } be a basis for (8), where

A_{0} = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) , A_{1} = (\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) , …,

and

B_{0} = (\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) , B_{1} = (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) , ….

The nonzero bracket operations are

\begin{aligned}[A_{0}, e_{1}] & = & e_{1}, & [A_{i}, e_{1}] & = & A_{i−1}, \\ [B_{0}, e_{2}] & = & e_{2}, & [B_{i}, e_{2}] & = & B_{i−1}, \end{aligned}

for i ≥ 1, and

[A_{i}, A_{j}] = \frac{(i − j)(i + j + 1)!}{(i + 1)!(j + 1)!}A_{i+j},
[B_{i}, B_{j}] = \frac{(i − j)(i + j + 1)!}{(i + 1)!(j + 1)!}B_{i+j}.

Lemma 2.7. Let {e_{1}, e_{2}, A_{0}, A_{1}, …, B_{0}, B_{1}, … } be a basis for (11) where

A_{0} = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , A_{1} = (\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) , …

and

B_{0} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) , B_{1} = (\begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) , ….

The nonzero bracket operations are

\begin{aligned}[A_{0}, e_{1}] & = & e_{1}, & [A_{i}, e_{1}] & = & B_{i−1}, \\ [A_{0}, e_{2}] & = & e_{2}, & [A_{i}, e_{2}] & = & A_{i−1}, \\ [B_{0}, e_{2}] & = & e_{1}, & [B_{i}, e_{2}] & = & B_{i−1}, \end{aligned}

for i ≥ 1, and

[A_{i}, A_{j}] = \frac{(i − j)(i + j + 1)!}{(i + 1)!(j + 1)!}A_{i+j},
[A_{i}, B_{j}] = \frac{(i − j)(i + j + 1)!}{(i + 1)!(j + 1)!}B_{i+j}.

Lemma 2.8. Let {e_{1}, e_{2}, A_{0}, A_{1}, …, B_{0}, B_{1}, … } be a basis for (14), where

A_{0} = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , A_{1} = (\begin{matrix}−1 & 0 & −1 \\ 0 & 1 & 0 \end{matrix}) , …

and

B_{0} = (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) , B_{1} = (\begin{matrix}0 & −1 & 0 \\ 1 & 0 & −1 \end{matrix}) , ….

Then there exist nonzero bracket operations

[A_{1}, A_{i}] = αA_{i+1},
[A_{1}, B_{i}] = βB_{i+1},
[B_{1}, B_{i}] = γA_{i+1},

where i = 2, 3, … and α, β, γ ≠ 0.

Lemma 2.9. Suppose λ ≠ −1, and let {e_{1}, e_{2}, A_{0}, A_{1}, …, B_{0}, B_{1}, … } be a basis for (16), where

A_{0} = (\begin{matrix}λ & 0 \\ 0 & λ + 1 \end{matrix}) , A_{1} = (\begin{matrix}0 & λ & 0 \\ 0 & 0 & λ + 1 \end{matrix}) , …

and

B_{0} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) , B_{1} = (\begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) , ….

The nonzero bracket operations are

\begin{aligned}[A_{0}, e_{1}] & = & λe_{1}, & [A_{i}, e_{1}] & = & λB_{i−1}, \\ [A_{0}, e_{2}] & = & (λ + 1)e_{2}, & [A_{i}, e_{2}] & = & A_{i−1}, \\ [B_{0}, e_{2}] & = & e_{1}, & [B_{i}, e_{2}] & = & B_{i−1}, \end{aligned}

for i ≥ 1, and

[A_{i}, A_{j}] = \frac{(λ + 1)(i − j)(i + j + 1)!}{(i + 1)!(j + 1)!}A_{i+j},
[A_{i}, B_{j}] = \frac{(λ(i − j) − (j + 1))(i + j + 1)!}{(i + 1)!(j + 1)!}B_{i+j}.

The proofs of (1) through (18) follow directly from the lemmas. The proof of (19) and (20) are special cases of Lemma 2.9. To prove (21) through (24), the following lemma is required.

Lemma 2.10. Consider the basis {e_{1}, e_{2}, A_{k}^{j}, B_{k}} for (21), where

B_{0} = (\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix}) , B_{1} = (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) , …,

A_{1}^{0} = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) , A_{2}^{0} = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ,

A_{1}^{1} = (\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}) , A_{2}^{1} = (\begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}) , … A_{3}^{1} = (\begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}) , ….

Then G_{k} has basis {A_{1}^{k}, …, A_{k+2}^{k}, B_{k}}. The nonzero relations for this algebra are

[A_{1}^{0}, e_{1}] = [A_{2}^{0}, e_{2}] = e_{1},

and for k ≥ 1

[A_{i}^{k}, e_{1}] = A_{i}^{k−1}, 1 ≤ i ≤ k + 1,
[A_{i}^{k}, e_{2}] = A_{i−1}^{k−1}, 2 ≤ i ≤ k + 2,
[A_{i+1}^{i}, A_{j+2}^{j}] = \frac{(i + 1)(i + j + 1)!}{(i + 1)!(j + 1)!}A_{i+j+2}^{i+j},
[A_{1}^{1}, A_{j}^{i−1}] = αA_{j}^{i}, for some α ≠ 0.

3. The Spencer Cohomology

For any graded algebra ∏ G_{p}, define C^{i,j} to be the space of skew-symmetric multilinear maps c : ⋀^{j} G_{−1} → G_{i−1}. If we define the coboundary operator

∂ : C^{i,j} → C^{i−1,j+1}

by

(∂c)(v_{1}, …, v_{j+1}) = ∑_{k}(−1)^{k}[c(v_{1}, …, v̂_{k}, …, v_{j+1}), v_{k}],

then ∂^{2} = 0. The resulting cohomology groups are known as the Spencer cohomology groups, which we will denote by H^{i,j} for i, j ≥ 0. For A ∈ G_{0} define a map c ↦ c^{A} from C^{i,j} to itself by

c^{A}(v_{1}, …, v_{j}) = [A, c(v_{1}, …, v_{j})] − ∑_{k} c(v_{1}, …, [A, v_{k}], …, v_{j}).

Then (∂c)^{A} = ∂(c^{A}). Consequently, G_{0} acts on H^{i,j}, which we shall denote by ξ ↦ ξ^{A}. An element ξ ∈ H^{i,j} is invariant if ξ^{A} = 0 for all A ∈ G_{0}. The set of invariant elements of a cohomology group H^{i,j} is denoted by (H^{i,j})^{I}. If η ∈ Hom(G_{i}, C^{j,l}) and ξ ∈ C^{i+1,k}, define ξ · η ∈ C^{j,k+l} by

ξ · η(v_{1}, …, v_{l+l})

= \frac{1}{k!l!} ∑_{σ∈S_{k+l}}(sgn σ)η(ξ(v_{σ(1)}, …, v_{σ(k)}))(v_{σ(k+1)}, …, v_{σ(k+l)}).

In [6] it was shown that ξ · η ∈ H^{j,k+l}.

The following proposition is due to Kobayashi and Nagano [7].

Proposition 3.1. Let G = ∏ G_{p} be a graded Lie algebra. Then the following statements are true.

1. H^{0,0} = G_{−1}.

2. H^{i,0} = 0 for i ≥ 1.

434 Judson

3. H^{0,1} = 𝔤𝔩(G_{−1})/G_{0}.

4. H^{i,1} = Λ^{1}G_{i−1}/G_{i} for i ≥ 1. In particular, H^{i,1} = 0 if and only if Λ^{1}G_{i−1} = G_{i}. Let L_{p} = G_{p} × G_{p+1} × …, and [ , ] be the usual Lie bracket on a graded

algebra ∏ G_{p}. An n-bracket on ∏ G_{p} is a skew-bilinear map

[ , ]′_{n} : ∏ G_{p} × ∏ G_{p} → ∏ G_{p}

satisfying the following conditions.

1. For X ∈ L_{i}, Y ∈ L_{j}, [X, Y]′_{n} − [X, Y] ∈ L_{i+j+1}.

2. If X, Y , Z ∈ ∏ G_{p}, then

[X, [Y, Z]′_{n}]′_{n} + [Y, [Z, X]′_{n}]′_{n} + [Z, [X, Y]′_{n}]′_{n} ∈ L_{n−1}.

If [X, Y]′_{n} − [X, Y] ∈ L_{n−1} for X, Y ∈ ∏ G_{p}, then [ , ]′_{n} is a flat n-bracket.

If [ , ]′ is 0-bracket, we can define an element \overline{c} in C^{0,2} by

\overline{c}(u, v) = [u, v]′ mod L_{0},

for u, v ∈ G_{−1}. By definition C^{−1,3} = 0; therefore, ∂\overline{c} = 0. We will let c ∈ H^{0,2} be the element in cohomology represented by \overline{c}. Similarly, if we are given a flat n-bracket with n ≥ 1, we can define elements c ∈ H^{n,2} and η_{i} ∈ Hom(G_{i}, H^{n,1}) for i = 0, …, n − 1. We now state several theorems from [6]. Theorem 3.2. Let [ , ]′ be a 0-bracket on ∏ G_{p}, and suppose that

1. c · c = c^{2} = 0;

2. c ∈ (H^{0,2})^{I}. If H^{k,1} = H^{k,2} = H^{k,3} = 0 for k ≥ 0, then there exists a complete filtered Lie algebra L with Lie algebra bracket [ , ]_{L} on ∏ G_{p} extending [ , ]′ such that ∏ G_{p} under the usual graded bracket is the associated graded algebra of L. Theorem 3.3. Let [ , ] be a n-bracket on ∏ G_{p} with n ≥ 1, and suppose that the following equations are satisfied.

1. η_{0}[A, B] = η_{0}(B)^{A} − η_{0}(A)^{B} for A, B ∈ G_{0}.

2. η_{i}[A, B] = η_{i}(B)^{A} for A ∈ G_{0}, B ∈ G_{i} with i = 1, …, n − 1.

3. η_{i}[A, B] = 0 for A ∈ G_{p}, B ∈ G_{q} with p + q = i, p, q ≥ 1.

4. c^{A} = η_{0}(A) · η_{n−1} for A ∈ G_{0}.

5. c · η_{n−1} = 0.

6. ∂A · η_{n−1} = 0 for A ∈ G_{n}.

7. ∂A · η_{n−1} = −η_{i}(A) · η_{n−1} for A ∈ G_{i}, where i = 1, …, n − 1.

If H^{k,1} = H^{k,2} = H^{k,3} = 0 for k > n, then there exists a complete filtered Lie algebra L with Lie algebra bracket [ , ]_{L} on ∏ G_{p} extending [ , ]′ such that ∏ G_{p} under the usual graded bracket is the associated graded algebra of L.

Let L and M be complete filtered Lie algebras with associated graded algebras isomorphic to ∏ G_{p} and denote the bracket operations on L and M by [ , ]_{L} and [ , ]_{M}, respectively. An n-isomorphism or n-map is a linear map ψ : L → M such that

1. ψ(L_{p}) ⊂ M_{p};

2. L_{p} →^{ψ} M_{p} → M_{p}/M_{p+1} = G_{p} is the map L_{p} → L_{p}/L_{p+1} = G_{p};

3. [ψ(X), ψ(Y)]_{M} − ψ([X, Y]_{L}) ∈ M_{n−1} for X, Y ∈ L.

If an n-map ∏ G_{p} → L exists, we can define c^{L} ∈ H^{n,2} and η_{i}^{L} ∈ Hom(G_{i}, H^{n,1}). These elements satisfy the structure equations in either Theorem 3.2 or Theorem 3.3 depending on whether n = 0 or n ≥ 1.

If α ∈ GL(G_{−1}), then α acts on G_{p} via

A^{α}(v_{1}, …, v_{p}) = αA(α^{−1}v_{1}, …, α^{−1}v_{p}) for A ∈ G_{p} and v_{i} ∈ G_{−1}

which results in an automorphism of ∏ G_{p}. Hence, there is a natural action of Aut(∏ G_{p}) on the cohomology groups H^{i,j} that sends invariant elements to invariant elements. We denote this action by α_{∗} for α ∈ Aut(∏ G_{p}). Furthermore, if η ∈ Hom(G_{p}, H^{i,j}), then the induced action α^{∗} on η is α^{∗}(η)(A) = α_{∗}η(α^{−1}A) for A ∈ G_{p}.

Theorem 3.4. Let L and M be complete filtered Lie algebra with graded algebra ∏ G_{p} and let ψ : L → M be an n-map satisfying the following conditions.

1. (H^{k,2})^{I} = 0 for k > n.

2. For k > n,

\frac{\{η : G_{0} → H^{k,1} : η[A, B] = η(B)^{A} − η(A)^{B}\}}{\{η : G_{0} → H^{k,1} : η(A) = ξ^{A} for some ξ ∈ H^{k,1}\}} = 0.

3. Hom_{G_{0}}(G_{i}, H^{k,1}) = 0 for n < k and 1 ≤ i < k.

If n = 0 and there exists an α ∈ Aut(∏ G_{p}) such that α_{∗}c^{L} = c^{M}, then L ≅ M. If n ≥ 1 and there exist n-maps ϕ_{L} : ∏ G_{p} → L and ϕ_{M} : ∏ G_{p} → M, and for some α ∈ Aut(∏ G_{p}), α_{∗}c^{L} = c^{M} and α^{∗}η_{i}^{L} = η_{i}^{M} for i = 0, …, n − 1, then L ≅ M.

4. The Group of n-maps

The n-maps from ∏ G_{p} to itself act on the cohomological elements c and η_{i}. These n-maps form a group H. There exists a series of subgroups of H

H = H_{0} ⊃ H_{1} ⊃ ⋯ ⊃ H_{n},

where τ ∈ H_{i} whenever τ(v) = v + τ_{i}(v) + τ_{i+1}(v) + ⋯ and τ_{p} ∈ Hom(G_{−1}, G_{p}). In addition, H_{i+1} is normal in H_{i}. Let c^{L}, η_{0}^{L}, …, η_{n−1}^{L} be the elements in cohomology defined by the n-map ϕ : ∏ G_{p} → L. The group H acts on c^{L} and η_{i}^{L} via the n-map ϕσ and gives elements (c^{L})^{σ} and (η_{i}^{L})^{σ}, where σ ∈ H.

Proposition 4.1. Let ϕ : ∏ G_{p} → L be an n-map that defines cohomological elements c^{L} and η_{i}^{L} for i = 0, …, n−1. If σ ∈ H_{n}, then the following statements are true.

1. If σ(v) = v + σ_{0}(v) + σ_{1}(v) + ⋯, v ∈ G_{−1} and σ_{i} ∈ Hom(G_{−1}, G_{i}), then

(c^{L})^{σ} = c^{σ} + c^{L} + ∑_{k=0}^{n−1} σ_{k} · η_{k}^{L}.

2. If 0 ≤ p ≤ n − 1 and A ∈ G_{p}, then

(η_{p}^{L})^{σ}(A) = η_{p}^{σ}(A) + η_{p}^{L}(A) + ∑_{k=p+1}^{n−1} η_{k}^{L}(σ_{k}(A)),

where σ(A) = A + σ_{p+1}(A) + σ_{p+2}(A) + ⋯, σ_{i}(A) ∈ G_{i}.

The action of H_{n} on the elements c^{L} and η_{i}^{L} is trivial. Let

σ(v) = v + σ_{n−1}(v) + σ_{n}(v) + ⋯

be a representative for \overline{σ} ∈ H_{n−1}/H_{n} where σ_{i} ∈ Hom(G_{−1}, G_{i}). Since ∂σ_{n−1} = 0, there is a well-defined natural map θ : H_{n−1}/H_{n} → H^{n,1}. Furthermore, θ is surjective.

Proposition 4.2. Let \overline{σ}, \overline{τ} in H_{n−1}/H_{n} have representatives σ, τ ∈ H_{n−1}, respectively. If θ(\overline{σ}) = θ(\overline{τ}), then σ and τ act the same on the elements c^{L} and η_{i}^{L}, 0 ≤ i < n − 1. In addition, if \overline{σ} induces σ_{n−1} ∈ H_{n−1}, then

1. (c^{L})^{σ} = \begin{cases}c^{L} + [σ_{0}, σ_{0}] + σ · η_{0}^{L}, & n = 1 \\ c^{L} + σ_{n−1} · η_{n−1}^{L}, & n ≥ 2; \end{cases}

2. (η_{0}^{L})^{σ}(A) = σ_{n−1}^{A} + η_{0}^{L}(A);

3. (η_{i}^{L})^{σ}(A) = η_{i}^{L}(A), i = 1, …, n − 1.

The action of the groups H_{p−1}/H_{p} on c^{L}, η_{0}^{L}, …, η_{n−1}^{L} for 1 ≤ p < n is partially determined by the adjoint map Ad_{X} : ∏ G_{p} → ∏ G_{p} defined by

Ad_{X} Y = Y + [X, Y] + \frac{1}{2!}[X, [X, Y]] + \frac{1}{3!}[X, [X, [X, Y]]] + ⋯,

where X ∈ G_{p}, p ≥ 1. The map Ad_{X} is both an n-map and an automorphism of ∏ G_{p}. The set Ad_{G_{p}} of all Ad_{X} where X ∈ G_{p} is a subgroup of H_{p−1}, and the subgroup ⟨H_{p} ∪ Ad_{G_{p}}⟩ of H_{p−1} generated by H_{p} and Ad_{G_{p}} is normal in H_{p−1}.

Proposition 4.3. Let X ∈ G_{p}, p = 1, …, n − 1. Then

1. (η_{i}^{L})^{Ad^{X}}(A) = η_{i}^{L}(A) for i = 1, …, n − 1;

2. (η_{0}^{L})^{Ad^{X}}(A) = η_{0}^{L}(A) + η_{p}^{L}(X)^{A};

3. (c^{L})^{Ad^{X}} = c^{L} + ∂X · η_{p−1}^{L}.

Define a map θ : H_{p−1}/H_{p} → (H^{p,1})^{I} for p = 1, …, n − 1 as follows. Let σ(v) = v + σ_{p−1} + ⋯ be a representative for \overline{σ} ∈ H_{p−1}/H_{p}, then ∂σ_{p−1} = 0. Let \overline{σ}_{p−1} ∈ (H^{p,1})^{I} be the element in cohomology represented by σ_{p}.

Proposition 4.4. For p = 1, …, n − 1

θ : H_{p−1}/⟨H_{p} ∪ Ad_{X}⟩ → (H^{p,1})^{I}

is an injection.

The map θ : H_{p−1}/⟨H_{p} ∪ Ad_{X}⟩ → (H^{p,1})^{I} is generally not surjective; however, an element ξ ∈ (H^{p,1})^{I} is the image of some element in H_{p−1}/⟨H_{p} ∪Ad_{X}⟩ under the map θ exactly when ξ is given by an n-derivation on ∏ G_{p}. An n-derivation is a linear map D : ∏ G_{p} → ∏ G_{p} such that

1. D(G_{i}) ⊂ G_{i+1} × G_{i+2} × ⋯;

2. D[X, Y] − [DX, Y] − [X, DY] ∈ G_{n−1} × G_{n} × ⋯, for X, Y ∈ ∏ G_{p}.

Suppose σ ∈ H_{i} (0 ≤ i < n − 1) and σ(v) = v + σ_{i}(v) + σ_{i+1}(v) + · · ·. Then there exists an n-derivation D such that D(v) = σ_{i}(v). Conversely, the map exp D is an n-map. The following theorem gives a method of calculating the action of n-maps on ∏ G_{p} [6].

Theorem 4.5. An element \overline{D} ∈ (H^{p,1})^{I} is the image of some element in

H_{p−1}/⟨H_{p} ∪ Ad_{X}⟩

under the map

θ : H_{p−1}/⟨H_{p} ∪ Ad_{X}⟩ → (H^{p,1})^{I}

exactly when D induces an n-derivation on ∏ G_{p}.

5. Algebras with dim G_{−1} = 2

We are now ready to classify all complete filtered Lie algebras L with graded algebra ∏ G_{p} and dim G_{−1} = 2. We first decide the cases where L is flat; i.e, L ≅ ∏ G_{p}. The following propositions shall prove useful. The proofs of the propositions can be found in Koch’s paper [9].

Proposition 5.1. Koch Let L be a complete filtered Lie algebra with graded algebra ∏ G_{p} such that the following conditions are satisfied.

1. (H^{i,2})^{I} = 0 for i ≥ 0.

2. For j > 0,

\frac{\{η : G_{0} → H^{j,1} : η[A, B] = η(B)^{A} − η(A)^{B}\}}{\{η : G_{0} → H^{j,1} : η(A) = ξ^{A} for some ξ ∈ H^{j,1}\}} = 0.

3. Hom_{G_{0}}(G_{i}, H^{j,1}) = 0 for 1 ≤ i < j.

Then L ≅ ∏ G_{p}.

Proposition 5.2. Gunning Let L be a complete filtered Lie algebra with graded algebra ∏ G_{p} where G_{0} contains the identity map, then L ≅ ∏ G_{p}.

Proposition 5.3. If (H^{i,2})^{I} = 0 and H^{i,1} = 0 for i ≥ 1, then L ≅ ∏ G_{p}.

The algebras (3) (λ = 1), (8)–(15), (25)–(34), (37), and (38) have no complete filtered Lie algebras that are not isomorphic to their associated graded algebras since in each algebra G_{0} contains the identity. Singer and Sternberg [12] proved that (35) and (36) are flat. To analyze the remaining cases, it is necessary to compute the cohomology groups of each graded algebra in question. We remark here that H^{i,3} = 0 for i ≥ 0 since dim G_{−1} = 2.

Proposition 5.4. Table 2 is a complete list of all nonzero cohomology groups H^{i,1}, (i ≥ 1) and H^{i,2}, (i ≥ 0) together with the generators for each of the cohomology groups for the graded algebras (1)–(7) and (16)–(24) of Table 1.

We will compute the cohomology for (5) as an example. Using Lemma 2.5, we may take {e_{1}, e_{2}, A_{0}, …, a_{n}} as a basis for this algebra. The nonzero bracket operations are [A_{0}, e_{1}] = e_{1} and [A_{i}, e_{1}] = A_{i−1}, where 1 ≤ i ≤ n. Since dim V = 2, H^{i,j} = 0 for j ≥ 3, and H^{i,1} = 0 for i ≠ n + 1 by Proposition 3.1. To compute H^{n+1,1}, consider the sequence

C^{n+2,0} → C^{n+1,1} → C^{n,2}.

If ξ ∈ C^{n+1,1} is the linear map from V to G_{n} defined by ξ(e_{1}) = aA_{n} and ξ(e_{2}) = bA_{n}, then

∂ξ(e_{1}, e_{2}) = [ξ(e_{1}), e_{2}] − [ξ(e_{2}), e_{1}] = −bA_{n−1}.

Hence, the kernel of ∂ξ consists of linear maps of the form ξ(e_{1}) = aA_{n} and ξ(e_{2}) = 0. Since C^{n+2,0} = 0, H^{n+1,1} = ℝ. To see that there are no invariant elements in H^{n+1,1}, observe that

ξ^{A_{0}}(e_{1}) = [A_{0}, ξ(e_{1})] − ξ([A_{0}, e_{1}]) = −aA_{n}.

To compute H^{0,2}, consider the sequence

C^{1,1} → C^{0,2} → 0.

We may take ξ ∈ C^{1,1} to be the linear map defined by ξ(e_{1}) = aA_{0} and ξ(e_{2}) = bA_{0}. Then

∂ξ(e_{1}, e_{2}) = [ξ(e_{1}), e_{2}] − [ξ(e_{2}), e_{1}] = −be_{1}.

Thus, H^{0,2} = ℝ with representative (e_{1}, e_{2}) ↦ ae_{2}. Since

ξ^{A_{0}}(e_{1}, e_{2}) = [A_{0}, ξ(e_{1}, e_{2})] − ξ([A_{0}, e_{1}], e_{2}) − ξ(e_{1}, [A_{0}, e_{2}]) = −ξ(e_{1}, e_{2}),

there are no invariant elements in H^{0,2}. The computation of H^{n+1,2} and (H^{n+1,2})^{I} follows in a similar manner.

Table 2. Cohomology Groups of ∏ G_{0} with G_{0} ⊂ 𝔤𝔩(2, ℝ). Cohomology Group Generators

(1)  H^{0,2} = ℝ  (e_{1}, e_{2}) ↦ ae_{1}

(H^{0,2})^{I} = H^{0,2}

(2) n ≥ 0  H^{0,2} = ℝ  (e_{1}, e_{2}) ↦ ae_{1}

(H^{0,2})^{I} = H^{0,2} H^{n+1,1} = ℝ e_{1} ↦ 0, e_{2} ↦ aA_{n} (H^{n+1,1})^{I} = H^{n+1,1} H^{n+1,2} = ℝ (e_{1}, e_{2}) ↦ aA_{n} (H^{n+1,2})^{I} = H^{n+1,2}

(3) λ ≠ 0  H^{1,2} = ℝ  (e_{1}, e_{2}) ↦ aA_{0}

(H^{1,2})^{I} = 0 where λ ≠ −1 (H^{1,2})^{I} = H^{1,2} where λ = −1

(4)  H^{0,2} = ℝ  (e_{1}, e_{2}) ↦ ae_{2}

(H^{0,2})^{I} = 0

(5) n ≥ 0  H^{0,2} = ℝ  (e_{1}, e_{2}) ↦ ae_{2}

(H^{0,2})^{I} = 0 H^{n+1,1} = ℝ e_{1} ↦ aA_{n}, e_{2} ↦ 0 (H^{n+1,1})^{I} = 0 H^{n+1,2} = ℝ (e_{1}, e_{2}) ↦ aA_{n} (H^{n+1,2})^{I} = 0

(6)  H^{1,2} = ℝ  (e_{1}, e_{2}) ↦ aA_{0}

(H^{1,2})^{I} = 0

(7)  H^{1,2} = ℝ  (e_{1}, e_{2}) ↦ aA_{0}

(H^{1,2})^{I} = 0 where λ ≠ 0 (H^{1,2})^{I} = H^{1,2} where λ = 0

(16) All cohomology groups vanish.

(17) λ ≠ −1 and dim G_{k} = 1 for k ≥ 1

H^{1,1} = ℝ e_{1} ↦ λaB_{0}, e_{2} ↦ aA_{0} (H^{1,1})^{I} = 0 H^{1,2} = ℝ (e_{1}, e_{2}) ↦ aA_{0} (H^{1,2})^{I} = 0 where λ ≠ −1/2 (H^{1,2})^{I} = H^{1,2} where λ = −1/2 G_{k} = 0 for k ≥ 1

H^{1,1} = ℝ^{2} e_{1} ↦ λaB_{0},

e_{2} ↦ aA_{0} + bB_{0}

(H^{1,1})^{I} = 0 where λ ≠ −2 (H^{1,1})^{I} = ℝ where λ = −2 e_{1} ↦ 0, e_{2} ↦ bB_{0} H^{1,2} = ℝ^{2} (e_{1}, e_{2}) ↦ aA_{0} + bB_{0} (H^{1,2})^{I} = 0

n ≥ 1 and G_{k} = 0 for k > n H^{1,1} = ℝ e_{1} ↦ aλB_{0}, e_{2} ↦ aA_{0} (H^{1,1})^{I} = 0 H^{1,2} = ℝ (e_{1}, e_{2}) ↦ aA_{0} (H^{1,2})^{I} = 0 where λ ≠ −1/2 (H^{1,2})^{I} = H^{1,2} where λ = −1/2 H^{n+1,1} = ℝ e_{1} ↦ 0, e_{2} ↦ aB_{n} (H^{n+1,1})^{I} = 0

where λ ≠ −(n + 2)/(n + 1) (H^{n+1,1})^{I} = H^{n+1,1}

where λ = −(n + 2)/(n + 1) H^{n+1,2} = ℝ (e_{1}, e_{2}) ↦ aB_{n} (H^{n+1,2})^{I} = 0

(18)  H^{2,1} = ℝ  e_{1} ↦ aλB_{1}, e_{2} ↦ aA_{1}

(H^{2,1})^{I} = 0 H^{2,2} = ℝ (e_{1}, e_{2}) ↦ aA_{1} (H^{2,2})^{I} = 0 where λ ≠ −2/3 (H^{2,2})^{I} = H^{2,2} where λ = −2/3

(19)  H^{2,2} = ℝ  (e_{1}, e_{2}) ↦ aA_{1}

(H^{2,2})^{I} = 0

(20) λ = −(n + 1)/(n − 1)

H^{2,1} = ℝ e_{1} ↦ aλB_{1}, e_{2} ↦ aA_{1} (H^{2,1})^{I} = 0 H^{2,2} = ℝ (e_{1}, e_{2}) ↦ aA_{1} (H^{2,2})^{I} = 0 H^{n+1,1} = ℝ e_{1} ↦ 0, e_{2} ↦ aB_{n} (H^{n+1,1})^{I} = 0 H^{n+1,2} = ℝ (e_{1}, e_{2}) ↦ aB_{n} (H^{n+1,2})^{I} = 0

(21) All cohomology groups vanish.

(22)  H^{0,2} = ℝ  (e_{1}, e_{2}) ↦ ae_{1}

(H^{0,2})^{I} = 0 H^{2,1} = ℝ e_{1} ↦ aA_{1}^{1}, e_{2} ↦ 0 (H^{2,1})^{I} = 0

(23) dim G_{k} = 2 H^{0,2} = ℝ  (e_{1}, e_{2}) ↦ ae_{1}

(H^{0,2})^{I} = 0 H^{1,1} = ℝ e_{1} ↦ aA_{1}^{0}, e_{2} ↦ 0 (H^{1,1})^{I} = 0

dim G_{k} = 1 for k > n ≥ 1 H^{0,2} = ℝ (e_{1}, e_{2}) ↦ ae_{1} (H^{0,2})^{I} = 0 H^{1,1} = ℝ e_{1} ↦ aA_{1}^{0}, e_{2} ↦ 0 (H^{1,1})^{I} = 0 H^{n+1,1} = ℝ e_{1} ↦ aA_{n+1}^{n}, e_{2} ↦ aA_{n+2}^{n} (H^{n+1,1})^{I} = H^{n+1,1} H^{n+1,2} = ℝ (e_{1}, e_{2}) ↦ aA_{n+1}^{n} (H^{n+1,2})^{I} = 0

(24) dim G_{k} = 1 for k ≥ 1

H^{0,2} = ℝ (e_{1}, e_{2}) ↦ ae_{1} (H^{0,2})^{I} = 0 H^{1,1} = ℝ^{2} e_{1} ↦ aA_{1}^{0} + bA_{2}^{0},

e_{2} ↦ bA_{1}^{0}

(H^{1,1})^{I} = ℝ e_{1} ↦ bA_{2}^{0}, e_{2} ↦ bA_{1}^{0} H^{1,2} = ℝ (e_{1}, e_{2}) ↦ aA_{1}^{0} (H^{1,1})^{I} = 0

G_{k} = 0 for k > n ≥ 1 and dim G_{k} = 1, n = 1, …, n H^{0,2} = ℝ (e_{1}, e_{2}) ↦ ae_{1} (H^{0,2})^{I} = 0 H^{1,1} = ℝ^{2} e_{1} ↦ aA_{1}^{0} + bA_{2}^{0},

e_{2} ↦ bA_{1}^{0}

(H^{1,1})^{I} = ℝ e_{1} ↦ bA_{2}^{0}, e_{2} ↦ bA_{1}^{0} H^{1,2} = ℝ (e_{1}, e_{2}) ↦ bA_{1}^{0} (H^{1,1})^{I} = 0 H^{n+1,1} = ℝ e_{1} ↦ 0, e_{2} ↦ aA_{n+2}^{n} (H^{n+1,1})^{I} = 0 H^{n+1,2} = ℝ (e_{1}, e_{2}) ↦ aA_{n+2}^{n} (H^{n+1,2})^{I} = 0

By Proposition 5.3, L ≅ ∏ G_{p} for the algebras (3), (λ ≠ −1, 0), (4), (6),

(7) (λ ≠ 0), (16), (19), and (21), since the appropriate cohomology groups vanish. A straightforward but lengthy computation shows that the algebras (5), (20), and (22) satisfy the hypothesis of Proposition 5.1; therefore, these algebras are also flat.

The remaining algebras to be considered are (1)–(3), (7), (17), (18), (23), and (24). If the cohomological elements c, η_{0}…. , η_{n−1} are known modulo the actions of Aut(∏ G_{p}) and H, then we may determine all complete filtered Lie algebras with dim G_{−1} = 2 provided that all higher obstructions vanish.

Lemma 5.5. There exist exactly two complete filtered Lie algebras L having graded algebra (1). These algebras are determined by c ∈ H^{0,2} with c = 0 if L is graded and c ≠ 0 if L is nongraded.

Proof. The only nonzero cohomology group is H^{0,2}. Let c ∈ H^{0,2} have generator c(e_{1}, e_{2}) = ae_{1}. The hypothesis of Theorem 3.2 are satisfied; hence, c induces a complete filtered Lie algebra bracket on ∏ G_{p}. The group of n-maps acts trivially on c since H^{i,0} = 0 for i ≥ 0. An automorphism α on G_{−1} is given by a matrix of the form

(\begin{matrix}r & s \\ 0 & t \end{matrix}) .

An easy computation yields α_{∗}c(e_{1}, e_{2}) = (a/t)e_{1}. An appropriate choice of α will send c to any other nonzero element in H^{0,2}. Therefore, if c ≠ 0, there exists exactly one nongraded algebra L.

The proofs of the next two lemmas are similar to the proof of Lemma 5.5.

Lemma 5.6. The two complete filtered Lie algebras having graded algebra (3) (λ = −1) are characterized by c ≠ 0 and c = 0 (graded case), where c ∈ H^{1,2}.

Lemma 5.7. For the algebra (7) (λ = 0), let c ∈ H^{1,2} have the generator c(e_{1}, e_{2}) = βA_{0}. Then there are three distinct complete filtered Lie algebras determined by β > 0, β < 0, and β = 0 (graded case).

Lemma 5.8. The complete filtered Lie algebras having graded algebra (2) are parameterized by (β_{−1}, …, β_{n}), β_{i} ∈ ℝ. Furthermore, L_{β_{−1},... ,β_{n}} ≅ L_{γ_{−1},... ,γ_{n}} if there exists a λ ∈ ℝ such that

(β_{−1}, …, β_{n}) = (λ^{n+2}γ_{−1}, …, λγ_{n}).

Proof. Let c ∈ H^{n+1,2} and η_{i} ∈ Hom(G_{i}, H^{n+1,1}) for 0 ≤ i ≤ n be given by

c(e_{1}, e_{2}) = β_{−1}A_{n},
η_{i}(A_{i}, e_{1}) = 0,
η_{i}(A_{i}, e_{2}) = β_{i}A_{n}.

One quickly checks that the hypothesis of Theorem 3.3 hold. Applying Proposition 4.2, we see that the n-maps have no effect on the c, η_{0}, …, η_{n−1}.

It remains to show how Aut(∏ G_{p}) acts on the c, η_{0}, …, η_{n−1}. Let α be as in Lemma 5.5. Then

α_{∗}c(e_{1}, e_{2}) = (β_{−1}/t)A_{n},
α^{∗}η_{i}(A_{i}, e_{2}) = (β_{i}/t^{n−i−1})A_{n}.

Therefore, if L = L_{β_{−1},... ,β_{n}} is the algebra determined by the c and η_{i}, then any other algebra determined by

\overline{c}(e_{1}, e_{2}) = λ^{n+2}β_{−1}A_{n},
\overline{η}_{i}(A_{i}, e_{1}) = 0,
\overline{η}_{i}(A_{i}, e_{2}) = λ^{n−i+1}β_{i}A_{n},

for some λ ∈ ℝ, must be isomorphic to L.

Lemma 5.9. Let ∏ G_{p} be the graded algebra

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & 0 & b \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

as in (17). If λ ≠ −1, 0, −1/2, then ∏ G_{p} is flat. If λ = −1/2, then there is exactly one nongraded algebra characterized by c ≠ 0, c ∈ H^{1,2}. If λ = 0 there is exactly one nongraded algebra characterized by η_{0}(B_{0}) ≠ 0, η_{0} ∈ Hom(G_{0}, H^{1,1}).

Proof. Let

c(e_{1}, e_{2}) = βA_{0},
η_{0}(A_{0}, e_{1}) = λγB_{0}, η_{0}(A_{0}, e_{2}) = γA_{0},
η_{0}(B_{0}, e_{1}) = λδB_{0}, η_{0}(B_{0}, e_{2}) = δA_{0}.

If λ = −1/2 and we apply the structural equations of Theorem 3.3, we may assume that c(e_{1}, e_{2}) = βA_{0}. If λ ≠ −1, 0, or −1/2, then η_{0}(A_{0}, e_{1}) = λγB_{0} and η_{0}(A_{0}, e_{2}) = γA_{0}. If λ = 0, then η_{0}(A_{0}, e_{2}) = γA_{0} and η_{0}(B_{0}, e_{2}) = δA_{0}. The actions of the n-maps in the first case show that there are no nongraded algebras if λ ≠ 1, 0, or −1/2. If λ = −1/2, the n-maps act trivially. If λ = 0, we may assume that γ = 0. Finally, notice that α ∈ Aut(∏ G_{p}) is given on G_{−1} by a matrix of the form

(\begin{matrix}r & s \\ 0 & t \end{matrix}) .

If either λ = 0 or −1/2, any nonzero element may be sent to any other nonzero element by the appropriate choice of α.

Lemma 5.10. Let

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (0) − ⋯

be as in (17). If λ ≠ 0 or −2, then all algebras are graded. If λ = 0, then there exists one nongraded algebra characterized by η_{0}(B_{0}) ≠ 0. If λ = −2, there is exactly one nongraded algebra characterized by η_{0}(A_{0}) ≠ 0.

Proof. Let

c(e_{1}, e_{2}) = αA_{0} + βB_{0},
η_{0}(A_{0}, e_{1}) = λγB_{0}, η_{0}(A_{0}, e_{2}) = γA_{0} + δB_{0},
η_{0}(B_{0}, e_{1}) = λσB_{0}, η_{0}(B_{0}, e_{2}) = σA_{0} + τB_{0}.

The equations of Theorem 3.3 allow for two cases. If λ = 0, then

η_{0}(A_{0}, e_{2}) = γA_{0} + δB_{0},
η_{0}(B_{0}, e_{2}) = σA_{0} + γB_{0}.

If λ ≠ 0, then

c(e_{1}, e_{2}) = \frac{λγ^{2}}{(λ + 1)^{2}} B_{0},

η_{0}(A_{0}, e_{1}) = λγB_{0}, η_{0}(A_{0}, e_{2}) = γA_{0} + δB_{0},

η_{0}(B_{0}, e_{1}) = 0, η_{0}(B_{0}, e_{2}) = \frac{λ − 1}{λ + 1}γB_{0}.

Applying Proposition 4.2, we may assume that η_{0}(B_{0}, e_{2}) = σA_{0} if λ = 0. If λ ≠ 0, the c and the η_{i}’s vanish except for the case λ = −2, where η_{0}(A_{0}, e_{2}) = δB_{0}. The automorphism group of ∏ G_{p} sends any nonzero element to any other nonzero element.

The proofs of the next four lemmas are similar to the proofs above.

Lemma 5.11. Let

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (0) − ⋯

be the algebra given in (17), where n ≥ 1 and G_{k} = 0 for k > n. If λ ≠ −(n + 2)/(n + 1), then L ≅ ∏ G_{p}. If λ = −(n + 2)/(n + 1), then there exists exactly one nongraded algebra determined by η_{0}(A_{0}) ≠ 0.

Lemma 5.12. Algebra (18) is flat if λ ≠ −1, 0, or −2/3. If λ = 0 or λ = −2/3, then there is exactly one nongraded example in each case that is determined by η_{0}(B_{0}) ≠ 0 and c ≠ 0, respectively.

We remark that the algebra given by

(\begin{matrix}a & b \\ 0 & 0 \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & a & b \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

in (23) is graded by Proposition 5.1.

Lemma 5.13. Let

(\begin{matrix}a & b \\ 0 & c \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & a & b \\ 0 & ⋯ & 0 & 0 & 0 \end{matrix}) − (\begin{matrix}0 & ⋯ & 0 & 0 & b \\ 0 & ⋯ & 0 & 0 & 0 \end{matrix}) − ⋯

be the algebra in (23) with dim G_{k} = 1 for k > n. The nongraded algebras are parameterized by (β_{0}, … β_{n}), β_{i} ∈ ℝ; where L_{β_{0},...,β_{n}} ≅ L_{γ_{0},...,γ_{n}} if there exists λ ∈ ℝ such that

(β_{0}, …, β_{n}) = (λ^{n+1}γ_{0}, …, λγ_{n}).

Lemma 5.14. Given the algebra

(\begin{matrix}a & b \\ 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & 0 & b \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

in (24), there exists one nongraded algebra determined by η_{0}(A_{1}^{0}) ≠ 0.

Lemma 5.15. Let

(\begin{matrix}a & b \\ 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (0) − ⋯

be as in (24) where G_{k} = 0 for k > n. The nongraded algebras are parameterized by (β_{0}, … β_{n}), β_{i} ∈ ℝ; and L_{β_{0},...,β_{n}} ≅ L_{γ_{0},...,γ_{n}} if there exists λ ∈ ℝ such that (β_{0}, …, β_{n}) = (λ^{n+1}γ_{0}, …, λγ_{n}).

Proof. Let c(e_{1}, e_{2}) = αA_{n+2}^{n}, η_{0}(A_{0}, e_{2}) = βA_{n+2}^{n}, and η_{i}(B_{i}, e_{2}) = γ_{i}A_{n+2}^{n} for 0 ≤ i < n. Equations (1) through (7) of Theorem 3.3 are satisfied. Let D be the n-derivation D given by

De_{1} = A_{2}^{0}, De_{2} = A_{1}^{0}, DA_{1}^{0} = 0,
DA_{i+2}^{i} = (i + 2)A_{i+3}^{i+1} (0 ≤ i < n),
DA_{n+2}^{n} = 0.

The n-map exp D is given by

(exp D)(e_{1}) = e_{1} + A_{2}^{0} + ⋯ A_{n+2}^{n},
(exp D)(e_{2}) = e_{2} + A_{1}^{0},
(exp D)(A_{1}^{0}) = 0,

(exp D)(A_{i+2}^{i}) = A_{i+2}^{i} + (i + 2)A_{i+3}^{i+1} + \frac{(i + 2)(i + 3)}{2!}A_{i+4}^{i+2} + · · · .

The n-maps act on c as in Proposition 4.1, allowing us to assume that c = 0. The action of the n-maps on H^{n+1,1} also allows us to assume that β = 0. Now apply Aut(∏ G_{p}) as in Lemma 5.8.

Theorem 5.16. Table 3 is a complete list of all nongraded algebras for ∏ G_{p}, where dim G_{−1} = 2.

Table 3. Nongraded Algebras with dim G_{−1} = 2.

(1) One nongraded algebra.

(2) Nongraded algebras L_{β_{−1}···β_{n}}, β_{i} ∈ ℝ where L_{β_{−1}···β_{n}} ≅ L_{γ_{−1}···γ_{n}} if there exists λ ∈ ℝ such that (β_{−1}, …, β_{n}) = (λ^{n+2}γ_{−1}, …, λγ_{n}).

(3) One nongraded algebra (λ = −1).

(7) Two nongraded algebras (λ = 0).

(17) One nongraded algebra in each case (λ = −1/2, 0).

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & 0 & b \\ 0 & 0 & 0 & 0 \end{matrix}) − ⋯

One nongraded algebra in each case (λ = −2, 0).

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (0) − ⋯

One nongraded algebra in each case (λ = −(n + 2)/(n + 1)) and G_{k} = 0 for k > n.

(\begin{matrix}λa & b \\ 0 & (λ + 1)a \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (0) − ⋯

(18) One nongraded algebra in each case (λ = 0, −2/3).

(23) Let ∏ G_{p} be the algebra

(\begin{matrix}a & b \\ 0 & 0 \end{matrix}) − (\begin{matrix}0 & a & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & a & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − ⋯,

where dim G_{k} = 1 for k > n. Nongraded algebras L_{β_{0}···β_{n}} exist, where β_{i} ∈ ℝ and L_{β_{0}···β_{n}} ≅ L_{γ_{0}···γ_{n}} if there exists λ ∈ ℝ such that

(β_{0}, …, β_{n}) = (λ^{n+1}γ_{0}, …, λγ_{n}).

(24) One nongraded algebra for the graded algebra

(\begin{matrix}a & b \\ 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − ⋯

For the algebra

(\begin{matrix}a & b \\ 0 & 0 \end{matrix}) − (\begin{matrix}0 & 0 & b \\ 0 & 0 & 0 \end{matrix}) − ⋯ − (\begin{matrix}0 & ⋯ & 0 & b \\ 0 & ⋯ & 0 & 0 \end{matrix}) − (0) − ⋯,

where G_{k} = 0 for k > n, there exist nongraded algebras L_{β_{0}···β_{n}} with β_{i} ∈ ℝ and L_{β_{0}···β_{n}} ≅ L_{γ_{0}···γ_{n}} if there exists aλ ∈ ℝ such that (β_{0}, …, β_{n}) =

(λ^{n+1}γ_{0}, …, λγ_{n}).

6. Conclusion

Cartan first classified the pseudogroups on ℝ^{2} in [1] using a different approach, and Conn treats the structure of transitive Lie algebras in [2]. The methods used in this paper is that they can reasonably be applied to dimensions higher than two in many cases. Although the techniques here are useful in constructing examples of nongraded Lie algebras, they do not allow a complete classification. If higher obstructions in cohomology exist, then these techniques may fail. Volpert offers another method using the Spencer cohomology and spectral sequences to obtain examples of complete filtered Lie algebras in [14, 15]. Finally, complete filtered Lie algebras are the algebraic objects corresponding to pseudogroups and transitive differential geometry. The geometric meaning of the c’s and the η_{i}’s are only partially understood [3, 12, 13].

References

[1] Cartan, E., Les sous groupes continus de transformations, Ann. Sci. Ecole Norm. Sup. 25 (1908), 57–194.

[2] Conn, J., “Nonabelian Minimal Closed Ideals of Transitive Lie Algebras”, Mathematical Notes 25, Princeton University Press, 1981.

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

[4] Humphreys, J., “Introduction to Lie Algebras and Representation Theory,” Third edition. Springer-Verlag, New York etc. 1997.

[5]  Jacobson, N., “Lie Algebras,” John Wiley & Sons, 1962.

[6] Judson, T., Complete filtered Lie algebras and the Spencer cohomology, J. Algebra 125 (1989), 66–109.

[7] Kobayashi, S., and T. Nagano, On filtered Lie algebras and geometric structures III, J. Math. Mech. 14 (1965), 679–706.

[8] Kobayashi, S., “Transformation Groups in Differential Geometry,” Springer-Verlag, New York etc. 1995.

[9]  Koch, R. N., On filtered Lie algebras, Indiana Univ. J. 26 (1977), 115–123.

[10] —, Pseudogroups associated with the one dimensional foliation group (I), J. Math. Soc. Japan 23 (1971), 149–180.

[11] —, Pseudogroups associated with the one dimensional foliation group (II), J. Math. Soc. Japan 23 (1971), 181–212.

[12] Singer, I. M., and S. Sternberg, The infinite groups of Lie and Cartan, Part I (the transitive groups), J. Analyse Math. 15 (1965), 4–113.

[13] Sternberg, S., “Lectures on Differential Geometry,” Second edition. Chelsea Publishing Co., 1983.

[14] Volpert, K., Finite spectral sequences and Massey powers in the deformation theory of graded Lie algebras and associative algebras, J. Pure Appl. Algebra 87 (1993), 281–300.

[15] —, A spectral sequence connecting the Spencer cohomology of a transitive Lie algebra with its deformation cohomology, Algebras Groups Geom. 7 (1990), 313–344.

Thomas W. Judson Department of Mathematics and Computer Science University of Puget Sound 1500 North Warner Street Tacoma, Washington 98416 tjudson@ups.edu

Received May 8, 2001 and in final form October 11, 2001