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02 Invariants of solvable rigid Lie algebras up to

dimension 8

Rutwig Campoamor-Stursberg
Depto. Geometŕıa y Topoloǵıa
Fac. CC. Matemáticas U.C.M.

E-28040 Madrid ( Spain)e-mail: rutwig@nfssrv.mat.ucm.es

Abstract

The invariants of all complex solvable rigid Lie algebras up to di-

mension eight are computed. Moreover we show, for rank one solvable

algebras, some criteria to deduce to non-existence of non-trivial invari-

ants or the existence of fundamental sets of invariants formed by rational

functions of the Casimir invariants of the associated nilradical.

1 Introduction

The importance of Casimir invariants of Lie algebras has increased its impor-
tance in representation theory and physics since the classical results due to
Casimir and Racah [1, 2]. They allow for example a labelling of irreducible
representations of Lie algebras, and in physical applications, the eigenvalues of
invariant functions of the symmetry group provide characterizations of a phys-
ical system, such as energy spectra. Casimir invariants of Lie algebras, which
are elements of the centre of the universal enveloping algebra A of a Lie algebra,
are commonly computed by using coadjoint orbits [3, 4]. For semi-simple Lie
algebras the question has been solved long ago [2], but for solvable Lie alge-
bras the question remains open. Here we obtain invariants which are not more
Casimir invariants in the classical sense, since they may be transcendental func-
tions. Their number and precise structure is also unknown, up to certain special
classes which have been treated in recent literature [5, 6, 7, 8].

In this paper we determine explicitely all invariant functions of complex
solvable rigid Lie algebras up to dimension 8. The interest of rigid structures
is justified, for example, by the fact that the Lie algebra of the homogeneous
Galilei group deforms onto the Lorentz algebra, which is rigid for being semi-
simple [9]. The classification of eight dimensional solvable rigid Lie algebras is
a consequence of the root theory for solvable algebras developed in [10], and it
allows theoretically the full determination of solvable rigid Lie algebras in any
fixed dimension. Analyzing the structure of the toral subalgebras of such Lie
algebras, we will be able to isolate properties that are valid in any dimension. For

1

http://arXiv.org/abs/math/0202006v2


example, we will find sufficiency criteria to ensure the non-existence of nontrivial
invariants for certain classes of rank one Lie algebras, or conditions to obtain the
invariants of a solvable rigid Lie algebra as rational functions of the invariants
of its nilradical. This points out that, for decomposable solvable Lie algebras,
the key to the determination of invariants depends more on the way a torus acts
on the nilradical than on the structure of the latter. This will also enable us to
find explicitly the invariants of solvable rigid Lie algebras by extrapolation of
the solutions in lower dimensional cases. This procedure is of interest, since it
indicates the existence of certain rigid structures that depend heavily on their
dimension. On the other side, nilradicals of rigid solvable Lie algebras are non-
abelian, and their study enlarges the efforts made for these algebras in the last
years [5, 7]. The rigid case allows us to show that for solvable Lie algebras the
results will not be so harmonious as in the semi-simple case, since there are
Lie algebras whose number of functionally independent invariants depends not
on the dimension of a Cartan subalgebra, but on the dimension of the algebra.
This is a succint indication that the non-decomposable case is, without doubt,
the most difficult to solve. Probably the study of specific classes of Lie algebras
is the only manner to approach the general solvable case, as classifications of
these algebras only exist up to dimension six, and more advances seem not to
be very probable ( among other reasons, because they imply the classification
of nilpotent Lie algebras, classified only up to dimension seven). Resuming, the
rigid case should be a first step to obtain interesting results on decomposable
solvable Lie algebras, by studying the subtori of a maximal torus of derivations.

Any Lie algebra considered in this work is finite dimensional over the field
C. Moreover, any n-dimensional Lie algebra g = (Cn, µ) is identified with its
law µ in the variety Ln [11]. Moreover, we convene that nonwritten brackets
are either zero or obtained by antisymmetry.

2 Decomposability of rigid Lie algebras

A Lie algebra g is a vector space V endowed with an alternating bilinear product
which satisfies the Jacobi identity. If M denotes the algebraic set consisting of
all Lie algebra laws on V , then the linear group GL (V ) acts on the space of all
alternating bilinear forms over V by

(f ∗ µ) = f−1 (µ (f, f)) , f ∈ GL(V ), µ ∈ O(V ) (1)

showing that M is stable under this action. Therefore, the orbits O(µ) of the
linear group GL (V ) on M correspond to the isomorphism classes of Lie algebra
laws on V . This allows to identify a Lie algebra g with the pair (V, µ), where µ

is an element of the orbit O(µ). We say that a Lie algebra g = (V, µ) is rigid if
the orbit O(µ) is open in M. Thus, roughly speaking, a Lie algebra g is rigid if
any Lie algebra g′ close to g is isomorhic to g. This led to the famous rigidity
theorem of Nijenhuis and Richardson [12], though, as pointed out by Richardson
in [13], there are rigid Lie algebras whose second adjoint cohomology group is

2



non-zero. As known, semi-simple Lie algebras have vanishing second adjoint
cohomology groups, as a consequence of the Whitehead lemmas, and by the
Levi decomposition theorem, the study of rigid Lie algebras reduces mainly to
analyze solvable Lie algebras. Now rigidity imposes strong structural conditions
in the solvable case, which simplify its description. The most important property
in this sense is their decomposability.

Definition 1 A Lie algebra g is called decomposable if it can be written as

g = s ⊕ t ⊕ n

where s is a Levi subalgebra, n the nilradical and t an abelian subalgebra whose
elements are ad-semisimple and which satisfies [s + t, t] = 0.

The abelian subalgebra t of Derg defined by

t = {adX, X ∈ t}

is called, following Malcev [14], an exterior torus on g. It is called maximal
torus, if it is maximal for the inclusion. As proven by Malcev, all maximal tori
are pairwise conjugated, thus their dimension constitutes an invariant of the
algebra, called the rank of g and noted by r(g).

By the Levi decomposition theorem, we can restrict ourselves to analyze the
solvable Lie algebras. Now semi-simple Lie algebras are rigid by the Nijenhuis-
Richardson theorem [12], so that it seems reasonable to begin with the study of
solvable rigid Lie algebras in order to obtain criteria that are not dependent on
the dimension. Now the structure of solvable rigid Lie algebras is known, and it
simplifies the question is some manner. The central result is due to Carles [15]:

Theorem 1 A rigid Lie algebra g is algebraic. In particular, it is decomposable.

In particular, if the Levi subalgebra s is zero, then g is a solvable Lie algebra.
In order to study such algebras, Ancochea and Goze developed a root theory
[10] for rigid solvable algebras. We recall this theory briefly:

Let r be a complex solvable decomposable Lie algebra, and t be a maximal
torus. Let X be a non-zero vector such that adµ0X belongs to t .

Definition 2 We say that X ∈ t is regular if the dimension of

V0 (X) = {Y ∈ g | [X, Y ] = 0}

is minimal that is, dimV0 (X) ≤ dimV0 (Z) for all Z such that adZ belongs to
t.

Choose a regular vector X and let be p = dimV0 (X). Consider a basis
(X, Y1, ..., Yn−p, X1, ..., Xp−1) of eigenvectors of adX such that (X, X1, ..., Xp−1)
is a basis of V0 (X), (Y1, ..., Yn−p, X1, ..., Xk0) is a basis of the maximal nilpotent
ideal n of g, and (Xk0+1, ..., Xp−1) are vectors such that adXi ∈ T .

3



Definition 3 Suppose that g is not nilpotent. The root system of g associated
to (X, Y1, ..., Yn−p, X1, ..., Xp−1) is the linear system (S) defined by the following
equations :

xi + xj = xk if the Xk-component of [Xi, Xj] is non-zero.
yi + yj = yk if the Yk-component of [Yi, Yj ] is non-zero.
xi + yj = yk if the Yk-component of [Xi, Yj ] is non-zero.
yi + yj = xk if the Xkcomponent of [Yi, Yj ] is non-zero.

Theorem 2 If rank (S) 6= dim (n) − 1, then g is not rigid.

See [10] for a proof. There are two important corollaries which will be of
importance for understanding the structure of solvable rigid laws.

Corollary 1 If r = n ⊕ t is rigid, then t is a maximal torus over n.

Corollary 2 If g is rigid then there is regular vector X such that adgX is
diagonal and its eigenvalues are integers.

These properties determine if a given Lie algebra is rigid or not. For example,
let us suppose that all elements of V0 (X) are semi-simple. If

rank (S) 6= dimD1 (g) − 1

where D1 (g) is the derived subalgebra of g, then g is not rigid.

Remark 1 Even if the roots can be choosen in Z, this does not in general imply
that the Lie algebra is rational. In [11] various examples of this have been worked
out.

3 Basic results on invariants. Notation

As known, the dual representation ad∗ to the adjoint representation ad of a
Lie algebra g is called the coadjoint representation [3]. The problem of finding
its invariants is indeed reduced to that of solving a system of linear first order
partial differential equations. If B = {X1, .., Xn} is a basis of the n-dimensional
Lie algebra g and {x1, .., xn} a coordinate system on the dual space, then the in-

finitesimal generators of the action are denoted by X̃i. If moreover the structure
constants of g are given by [Xi, Xj ] = Ck

i,jXk over the basis B, the it follows
that a function F ∈ C∞ (g∗) is an invariant of the coadjoint representation if
and only if the two following conditions are satisfied:

1. X̃i =
∑

j(−Ck
i,j)xk

∂
∂xj

and
[
X̃i, X̃j

]
= Ck

i,jX̃k,

2. F satisfies the system of linear first order PDE X̃iF = 0, 1 ≤ i ≤ n

4



Solutions to this system are usually found by integration of the correspond-
ing system of characteristic equations or other standard integration procedures
[16].A maximal set of functionally independent solutions will be called a fun-
damental set of invariants. The determination of the invariants as solutions of
the associated system of linear first order PDEs is the most common method to
determine the invariants of Lie algebras [4, 8, 17, 18], and the one we will apply
here. The cardinal of a fundamental set of invariants is N = dim (g)− r, where
r is the rank of the commutator table considered as a matrix [19]. Since a Lie
algebra law is an alternated tensor of type (2, 1), this rank does not depend on
the basis chosen. Moreover, by antisymmetry, this rank is even, and we obtain
that N ≡ dim (g) (mod2). In particular, an odd dimensional Lie algebras has
nontrivial invariants. From a result due to Dixmier [3, 4], it is known that if the
Lie algebra is algebraic, then we can find a maximal set of functionally inde-
pendent solutions formed by rational functions. In view of Carles’ algebraicity
theorem, we in particular obtain the following result.

Proposition 1 A rigid Lie algebra n ⊕ t having non-trivial invariants admits
a fundamental set of invariants formed by rational functions.

The converse is easily seen to be false. For example, take the non-nilpotent
Lie algebra N

αβγδ
6,1 of [7] with brackets [X1, N1] = αN1, [X1, N2] = γN2, [X1, N4] =

N4, [X2, N1] = βN1, [X2, N2] = δN2, [X2, N3] = N3. It is easy to verify that it is
non-rigid, as the torus generated by X1, X2 is non-maximal, since the nilradical
of N

αβγδ
6,1 is abelian of rank four, and has a fundamental set of invariants formed

by the rational functions

{
n

β
3 nα

4

n1
,

n
(βγ−αδ)
3

n
γ
1

}
.

4 Some sufficiency criteria

In this paragraph we analyze some results which allow to deduce the inexistence
of nontrivial invariants of a solvable rigid law r = n⊕t by analyzing the structure
of the torus t.

Proposition 2 Let dim (n ⊕ t) = 2n. Suppose that the rank of n is one and that
the nilradical n has a one dimensional center. If there exists a basis {Y1, .., Y2n}
such that

1. t is generated by Yn,

2. Y2n ∈ Z (n), and

3. [Yj , Y2n−j ] = Y2n, 2 ≤ j ≤ n − 1

then r = n ⊕ t has no nontrivial invariants.

Corollary 3 For n ≥ 3, any solvable Lie algebra r = n ⊕ t whose torus t has
the eigenvalues (1, 2, .., n− 1, n + 1, .., 2n) has only trivial invariant functions.

5



In [7] the author points out the importance of finding characterizations
of those solvable Lie algebras with non-abelian nilradical non admitting non-
trivial invariant functions. Even in the decomposable case such a characteriza-
tion seems not to be realizable, as the following example shows: Consider the
nilpotent Lie algebra n of dimension seven given by the brackets [X1, X2] =
X4, [X1, X3] = X5, [X1, X6] = X7, [X2, X3] = X6, [X2, X5] = X7, [X3, X4] =
−X7. This algebra is of rank three. Take the one dimensional tori t1 and t2
whose action over the basis {X1, .., X7} is, respectively, given by the sequences
(1, 0, 0, 1, 1, 0, 1) and (1,−1, 0, 0, 1,−1, 0). Both solvable non-nilpotent Lie alge-
bras n + t1 and n + t2 have the same non-abelian nilradical, but the first algebra
has no non-trivial invariants, while the second admits the polynomial invariant
y7, since its center is nonzero. This shows that, even if characterizations of solv-
able Lie algebras having only trivial solutions exist, they cannot be based on
the structure of the nilradical, but on the way the torus (in the decomposable
case) acts on this nilradical.
This example also provides an obvious criterion to guarantee the existence of
polynomial solutions of a solvable Lie algebra:

Proposition 3 Let n be a nilpotent Lie algebra of rank r ≥ 1 and t a maximal
torus over n. If there exists a toral subalgebra t′ ⊂ t such that the action of t′ on
a central ideal I is zero, then the solvable Lie algebra n⊕ t′ has at least dim (I)
polynomial solutions. Moreover, if the ideal I is generated by {Y1, .., Yt}, these
solution can be chosen as {y1, .., yt}.

5 Solvable rigid Lie algebras up to dimension six

The determination of solvable rigid laws in dimensions up to six does not provide
much information about the variety, since the number of these laws is very
reduced. In dimension 2 and 4 ( there are no solvable rigid laws in dimension 3
[11]) the solvable rigid algebras are simply the semidirect product of abelian Lie
algebras with a maximal torus, and, by the results of [5, 17], have no nontrivial
invariants. For dimension 5 we have only the law r = h1 ⊕ t, where h1 is the
3-dimensional Heisenberg Lie algebra. Here we find a rational invariant, which
also follows from the analysis undertaken in [17]. Finally, there are three solvable
rigid laws in dimension 6. As known, invariants of solvable Lie algebras up to
this dimension have been determined in [7, 8].

6 Solvable rigid Lie algebras in dimension 7

As follows from the parity of the dimension, seven dimensional rigid solvable
Lie algebras have at least one nontrivial invariant function. It can easily be
seen that any such algebra has at most rank three, and that their nilradical
is non-abelian. Their classification follows easily from the classification of low
dimensional Lie algebras [20].
In keeping the notation used in previous sections, we illustrate the determination

6



Table 1: Invariants of solvable rigid Lie algebras in dimension 6.
Algebra Brackets Invariants
r16 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5 none

[Y1, Yi] = Yi+1, i = 2, 3, 4
[Y2, Y3] = Y5

r26 [V1, Yi] = iYi, i = 1, 2, 3, 4 none
[V2, Yi] = Yi, i = 2, 3, 4
[Y1, Yi] = Yi+1, i = 2, 3

r36 [V1, Yi] = iYi, i = 1, 2, 3 none
[V2, Y2] = Y2,

[V3, Y3] = Y3

of the invariants of a solvable Lie algebra considering the rank one Lie algebra
r37 given by the brackets [V1, Yi] = iYi, i = 1, 3, 4, 5, 6, 7; [Y1, Yi] = Yi+1, i =
3, 4, 5, 6; [Y3, Y4] = Y7 over the basis {Y1, Y3, Y4, Y5, Y6, Y7, V1}. It is trivial to

see that the system
{

Ỹi.F = 0, i = 1, 3, 4, 5, 6, 7; Ṽ1.F = 0
}

reduces to the two

following equations:

X̃ ′

1.F = (y6∂y5 + y7∂y6).F = 0 (2)

Ṽ ′

1 .F = (5y5∂y5 + 6y6∂y6 + 7y7∂y7).F = 0 (3)

From the characteristic equations of (6.1) we easily obtain the set of invariants{
y7, y

2
6 − 2y5y7

}
. Since the reduced system is complete, it suffices to apply

the usual procedures of systems of two equations in three variables [16]: taking

u = y7 and v = y2
6−2y5y7 we obtain Ṽ ′

1 (u) = 7u, Ṽ ′

1 (v) = 12v and the equation

∂F (u, v)

∂u
+

12v

7u

∂F (u, v)

∂v
= 0 (4)

which provides the solution
{

v7

u12 =
(2y5y7−y2

6)7

y12
7

}
. As the rank of the commuta-

tor table ( interpreted as matrix ) is six, the preceding solution constitutes a
fundamental set of invariants for r37.

Proposition 4 Any solvable rigid Lie algebra r of dimension 7 and rank one has
a fundamental set of invariants formed by a quotients of invariant polynomials
of the nilradical.

The proof is straightforward, as follows from table 2. Now this result can
be generalized to arbitrary dimension, under an additional assumption. From
the analysis of the existing lists of rigid Lie algebras, it has been conjectured
in the middle 80’s that any solvable rigid Lie algebra r has necessarily a trivial
center [15, 21, 23, 24]. For rank r ≥ 2 this fact is easily proven by deforming
the torus, while for rank one the question remains a conjecture. The difficulty

7



Table 2: Invariants of solvable rigid Lie algebras in dimension 7.
Algebra Brackets Invariants

r17 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 6 I1 =
(3y2

6y3+y3
5−3y4y5y6)

2

y5
6

[Y1, Yi] = Yi+1, i = 2, 3, 4, 5
[Y2, Yi] = Yi+2, i = 3, 4.

r27 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 7 I1 =
(6y3y5y7−2y3

5−3y7y2
4+6y2

7y1)7

y15
7

[Y1, Yi] = Yi+1, i = 2, 3, 4
[Y2, Yi] = Yi+2, i = 3, 5
[Y3, Y4] = −Y7

r37 [V1, Yi] = iYi, i = 1, 3, 4, 5, 6, 7 I1 =
(2y5y7−y2

6)7

y12
7

[Y1, Yi] = Yi+1, i = 3, 4, 5, 6,

[Y3, Y4] = Y7

r47 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5 I1 =
(2y5y′

3+y2
4−2y3y5)

5

y8
5

[V1, Y
′

3 ] = 3Y ′

3 ,
[Y1, Yi] = Yi+1, i = 2, 3, 4,

[Y2, Y3] = [Y2, Y
′

3 ] = Y5.

r57 [V1, Yi] = iYi, i = 1, 3, [V1, Vi] = (i − 2)Yi, i = 4, 5 I1 =
(3y2y2

5+y3
4−3y3y4y5)

2

(2y3y5−y2
4)3

[V2, Yi] = Yi, i = 2, 3, 4, 5,

[Y1, Yi] = Yi+1, i = 2, 3, 4.

r67 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, I1 =
y2
4(2y1y5+y2

3−2y2y4)
3

y2
5

[V2, Yi] = Yi, i = 2, 3, 4,

[V2, Y5] = 2Y5,

[Y1, Yi] = Yi+1, i = 2, 3,

[Y2, Y3] = Y5.

r77 [V1, Yi] = Yi, i = 1, 3, [V1, Vi] = 2Yi, i = 2, 5 I1 =
−2y2

5v2+v1y2
5+2y2y3y5+y1y4y5−y2y2

4

y2
5

[V2, Yi] = Yi, i = 3, 4, 5,

[Y1, Yi] = Yi+1, i = 3, 4,

[Y2, Y3] = Y5

r87 [V1, Yi] = iYi, i = 1, 3 I1 = v1y3−v2y3+y1y2

y3

[V2, Yi] = Yi, i = 2, 3,

[V ′

2 , Y4] = Y4,

[Y1, Y2] = Y3.

8



of this case is deeply related with the classification of complete Lie algebras and
cohomological problems [15].
In any case, if r = n ⊕ t is a solvable rigid Lie algebra of rank one and X a
central element of the nilradical n, the infinitesimal generator X̃ is of the form
X̃ = [T, X ] ∂

∂t
, and therefore an invariant function F does not depend on t. The

following proposition is an easy provable consequence of the structure of the
remaining system X̃iF = 0, since this system is formed by the linear system of
PDE which gives the invariants of the nilradical, to which the equation which
describes the action of the torus over n must be added:

Proposition 5 Let r = n ⊕ t be a solvable rigid Lie algebra of rank one and
trivial center. Then r admits a fundamental set of invariants formed by quotients
of invariant polynomials of the nilradical n.

7 Solvable rigid Lie algebras in dimension 8

The first classification of eight dimensional solvable rigid Lie algebras appeared
in 1986, though the corrected complete list has recently been published in [21].
It is based on the study of the eigenvalues of a regular operator ad(X), and is
also related with the classification of seven dimensional nilpotent Lie algebras
[20]. The rank theorem allowed to complete the list and to detect the absence of
two laws. The list we present here is basically the one given in [21], up to minor
changes relative to the choice of regular vectors. This is also the first dimension
where even dimensional solvable rigid laws admit nontrivial invariants. It is
convenient to separate these algebras by rank.

As follows from table 3., any solvable rigid algebra of rank one, up to the
algebras r18, r

6
8 and r98, has two invariants. For the first two ones this follows at

once from proposition 5., while for the third the reasoning is similar.
The tables for rigid Lie algebras of rank at least two show a rather interest-

ing fact, namely, that the algebra has nontrivial invariants if and only if it is
isomorphic to either r198 or r328 . If we analyze the nilradicals of these algebras,
we see that the one corresponding to r198 is isomorphic to the filiform Lie algebra
L6 [22], while the second is isomorphic to the five dimensional Heisenberg Lie
algebra h2. The interest of this is that these nilpotent Lie algebras are ”ex-
treme”, in the sense that L6 corresponds to the simplest algebra of maximal
nipotence index, while the Heisenberg algebra is the most simple metabelian
Lie algebra, which are the ”most nilpotent”. There is another important obser-
vation: while the rank of hp increases with p, the rank of Ln does not depend
on the dimension, and equals 2 [22].

8 Higher dimensions

Although the root theory introduced in [10] allows a complete classification of
solvable rigid laws in dimension nine, this has not been established due to the
great number of isomorphism classes, but only for particular cases [23, 24]. In

9



Table 3: Invariants of solvable rigid Lie algebras in dimension 8 and rank 1.
Algebra Brackets Invariants
r18 [V1, Yi] = iYi, i = 1, 2, 3, 5, 6, 7, 8, none

[Y1, Yi] = Yi+1, i = 2, 6, 7
[Y2, Y3] = Y5, [Y2, Y5] = Y7,

[Y2, Y6] = Y8, [Y3, Y5] = Y8.

r28 [V1, Yi] = iYi, i = 1, 3, 4, 5, 6, 7, 8 I1 =
y7
8

(y2
7−2y6y8)4

[Y1, Yi] = Yi+1, i = 3, 4, 5, 6, 7 I2 =
(y2

6−2y5y7+2y4y8)
2

y3
8

[Y3, Yi] = Yi+3, i = 4, 5.

r38 [V1, Yi] = iYi, i = 1, 3, 4, 5, 6, 7, 9 I1 =
y7
9

y9
7

[Y1, Yi] = Yi+1, i = 3, 4, 5, 6 I2 =
(2y2

9y1−y2
6y7+2y4y6y9−y2

5y9+2y2
7y5−2y3y7y9)

7

y7y7
9

[Y3, Yi] = Yi+3, i = 4, 6,

[Y4, Y5] = −Y9.

r48 [V1, Yi] = iYi, i = 1, 4, 5, 6, 7, 8, 9 I1 =
(y2

8−2y7y9)
9

y16
9

[Y1, Yi] = Yi+1, i = 4, 5, 6, 7, 8 I2 =
y8
9

(3y6y2
9+y3

8−3y7y8y9)3

[Y4, Y5] = Y9.

r58 [V1, Yi] = iYi, i = 2, 3, 4, 5, 6, 7, 8 I1 =
y8
7

y8
7

[Y2, Yi] = Yi+2, i = 3, 4, 5, 6 I2 =
(2y2

8y4−y2
6y8+2y6y2

7−2y5y7y8)
7

y4
7y14

8

[Y3, Yi] = Yi+3, i = 4, 5.

r68 [V1, Yi] = iYi, i = 2, 3, 4, 6, 7, 8, 10 none
[Y2, Yi] = Y2+i, i = 4, 6, 8,

[Y3, Yi] = Yi+3, i = 4, 7,

[Y4, Y6] = Y10.

r78 [V1, Yi] = iYi, i = 2, 3, 5, 6, 7, 8, 9 I1 =
y8
9

y9
8

[Y2, Yi] = Yi+2, i = 3, 5, 6, 7,

[Y3, Yi] = Yi+3, i = 5, 6. I2 =
(2y5y2

9−y2
7y9+2y7y2

8−2y6y8y9)
8

y5
8y16

9

r88 [V1, Yi] = iYi, i = 2, 3, 5, 7, 8, 9, 11 I1 =
y18
11

(y2
9−2y7y11)11

[Y2, Yi] = Yi+2, i = 3, 5, 7, 9 I2 =
(y2

9−2y7y11)3

(3y5y2
11+y3

9−3y7y9y11−
3
2y2

8y11)2

[Y3, Yi] = Yi+3, i = 5, 8.

r98 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 6 none
[V1, Y

′

3 ] = 3Y ′

3

[Y1, Yi] = Yi+1, i = 2, 3, 4, 5,

[Y2, Yi] = Yi+2, i = 3, 4,

[Y2, Y
′

3 ] = Y5, [Y3, Y
′

3 ] = Y6

r108 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 6, I1 =
(2y6y′

4+y2
5−2y4y6)

3

y5
6

[V1, Y
′

4 ] = 4Y ′

4 I2 =
(3y2

6y3+y3
5−3y4y5y6)

2

y5
6

[Y1, Yi] = Yi+1, i = 2, 3, 4, 5
[Y2, Yi] = Yi+2, i = 3, 4,

[Y2, Y
′

4 ] = Y6

10



Table 4: continued
Algebra Brackets Invariants

r118 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 6, I1 =
(y′

5−y5)
6

y5
6

[V1, Y
′

5 ] = 5Y ′

5 , [Y1, Y
′

5 ] = Y6 I2 =
(6y2

6y3+2y3
5+3y2

5(y′

5−y5)−6y4y5y6−6y4(y
′

5−y5)y6)
2

y5
6

[Y1, Yi] = Yi+1, i = 2, 3, 4, 5,

[Y2, Y3] = Y ′

5 , [Y2, Y4] = Y6

r128 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 7, I1 =
(2y3y7−y2

5)7

y16
7

[V1, Y
′

3 ] = 3Y ′

3 , [Y1, Y2] = Y
′

3 I2 =
y30
7 (y4

5y2
7+2y3y2

5y3
7+4y2

3y4
7)7

(y2
7y10

5 −4y3y3
7y8

5+4y2
3y6

5y4
7−8y3

3y4
5y5

7+32y4
3y2

5y6
7−32y5

3y7
7)7

[Y1, Y
′

3 ] = Y4, [Y1, Y4] = Y5

[Y2, Y3] = Y5, [Y
′

2 , Y
′

3 ] = Y5

[Y2, Y5] = [Y
′

3 , Y4] = Y7.

r138 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 7, I1 =
(y5−y′

5)
7

y5
7

[V1, Y
′

5 ] = 5Y ′

5 , I2 =
(6y2

7y1−2y3
5+3y2

5(y5−y′

5)+6y3y5y7−3y7y2
4)7

y15
7

[Y1, Yi] = Yi+1, i = 2, 3, 4,

[Y2, Y3] = Y ′

5 , [Y3, Y4] = −Y7

[Y2, Y5] = [Y2, Y
′

5 ] = Y7,

r148 [V1, Yi] = iYi, i = 1, 3, 4, 5, 6, 7, I1 =
(2y5y7−y2

6)7

y12
7

[V1, Y
′

4 ] = 4Y ′

4 , I2 =
(3y2

7(y′

4−y4)−y3
6+3y5y6y7)

7

y18
7

[Y1, Yi] = Yi+1, i = 3, 4, 5, 6
[Y3, Y4] = [Y3, Y

′

4 ] = Y7.

r158 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, I1 =
(2y′

3y5+y2
4−2y4y′

4)
5

y8
5

[V1, Y
′

3 ] = 3Y ′

3 , [V1, Y
′

4 ] = 4Y ′

4 , I2 =
(y4−y′

4)
5

y4
5

[Y1, Yi] = Yi+1, i = 2, 3, 4,

[Y1, Y
′

3 ] = Y ′

4 , [Y1, Y
′

4 ] = [Y2, Y3] = Y5

r168 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, I1 =
(2y′

3y5−y2
4+2y3y5)

5

y8
5

[V1, Y
′

1 ] = Y ′

1 , [V1, Y
′

3 ] = 3Y ′

3 , I2 =
(y′

1y5−y2y4+y′

3y3−y1y5)
5

y6
5

[Y1, Yi] = Yi+1, i = 2, 4,

[Y1, Y
′

i ] = Yi+1, i = 1, 3,

[Y ′

1 , Yi] = Yi+1, i = 3, 4,

[Y ′

1 , Y2] = Y ′

3 ,

[Y2, Y3] = −[Y2, Y
′

3 ] = Y5

r178 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, I1 =
y′

5

y5

[V1, Y
′

i ] = iY ′

i , i = 3, 5, I2 =
(2y3y2

5−2y′

3y′

5y5−y2
4(y′

5−y5)+4y3y5y′

5)
5

y8
5(y′

5)
5

[Y1, Yi] = Yi+1, i = 2, 3, 4,

[Y1, Y
′

3 ] = Y4, [Y2, Y3] = Y ′

5 ,

[Y2, Y
′

3 ] = Y5 + 2Y ′

5

r188 [V1, Yi] = iYi, i = 1, 2, 3 I1 =
y′

3

y3

[V1, Y
′

i ] = iY ′

i , i = 1, 2, 3 I2 =
(−y”

1(y′

3)
2+y′

3(y
2
2+(y′

2)
2)−2y1(y

2
3−(y′

3)
2)+2y3(y

′

1y′

3−y′

2y2))
3

y3(y′

3)
6

[V1, Y
”
1 ] = Y ”

1 , [Y1, Y
′

1 ] = Y2,

[Y1, Y
”
1 ] = Y

′

2 , [Y ”
1 , Y ′

2 ] = Y ′

3 ,

[Y1, Y
′

2 ] = Y
′

3 , [Y ′

1 , Y2] = Y
′

3 ,

[Y ′

1 , Y ′

2 ] = [Y ”
1 , Y2] = Y3

11



Table 5: Invariants of solvable rigid Lie algebras in dimension 8 and rank 2.
Algebra Brackets Invariants

r198 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 6, I1 =
(y2

5−2y4y6)
3

(3y2
6y3+y3

5−3y4y5y6)2

[V2, Yi] = Yi, i = 2, 3, 4, 5, 6, I2 =
(−6y2y3

6+6y5y3y2
6+2y4

5−8y4y2
5y6+6y2

4y2
6)

(2y4y6−y2
5)2

[Y1, Yi] = Yi+1, i = 2, 3, 4, 5.

r208 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 6, none
[V2, Yi] = Yi, i = 3, 4, 5, 6,

[Y1, Yi] = Yi+1, i = 3, 4, 5,

[Y2, Yi] = Yi+2, i = 3, 4
r218 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 6, none

[V2, Yi] = Yi, i = 4, 5, 6,

[Y1, Yi] = Yi+1, i = 2, 4, 5, [Y2, Y4] = Y6

r228 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 6, none
[V2, Yi] = Yi, i = 2, 3, 4, [V2, Yi] = 2Yi, i = 5, 6,

[Y1, Yi] = Yi+1, i = 2, 3, 5,

[Y2, Yi] = Yi+2, i = 3, 4
r238 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 6, none

[V2, Y6] = Y6, [Y2, Y3] = Y5

[Y1, Yi] = Yi+1, i = 2, 3, 4,

r248 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 7, none
[V2, Yi] = Yi, i = 3, 4, 5, [V2, Y7] = 2Y7

[Y1, Yi] = Yi+1, i = 3, 4,

[Y2, Y3] = Y5, [Y3, Y4] = Y7

r258 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 7, none
[V2, Yi] = Yi, i = 2, 3, 4, [V2, Y5] = 2Y5,

[V2, Y7] = 3Y7, [Y1, Yi] = Yi+1, i = 2, 3,

[Y2, Yi] = Yi+2, i = 3, 5
r268 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 7, none

[V2, Yi] = Yi, i = 2, 3, 4, 5, [V2, Y7] = 2Y7,

[Y1, Yi] = Yi+1, i = 2, 3, 4,

[Y2, Y5] = −[Y3, Y4] = Y7

r278 [V1, Yi] = iYi, i = 1, 2, 3, 5, 6, none
[V2, Yi] = Yi, i = 2, 3, [V2, Y5] = 2Y5, [V2, Y6] = 3Y6

[V2, Y7] = 3Y7, [Y1, Yi] = Yi+1, i = 2, 6,

[Y2, Yi] = Yi+2, i = 3, 5
r288 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, 7, none

[V2, Y2] = Y2, [V2, Yi] = 2Yi, i = 3, 4,

[V2, Y5] = 3Y5, [V2, Y7] = 4Y7, [Y1, Y3] = y4

[Y3, Y4] = −Y7, [Y2, Y3] = Y5, [Y2, Y5] = Y7

[Y2, Yi] = Yi+2, i = 3, 5,

r298 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, none
[V1, Y

′

3 ] = 3Y ′

3 , [V2, Y
′

3 ] = Y ′

3 , [V2, Y5] = 2Y5

[V2, Yi] = Yi, i = 2, 3, 4, [Y1, Y2] = Y3

[Y1, Y3] = Y4, [Y1, Y
′

3 ] = Y4, [Y2, Y3] = Y5

12



Table 6: Invariants of solvable rigid Lie algebras in dimension 8 and rank 3.
Algebra Brackets Invariants
r308 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, none

[V2, Y5] = Y5, [V3, Yi] = Yi, i = 2, 3, 4,

[Y1, Yi] = Yi+1, i = 2, 3
r318 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, none

[V2, Yi] = Yi, i = 2, 3,

[V3, Yi] = Yi, i = 4, 5,

[Y1, Yi] = Yi+1, i = 2, 4

r328 [V1, Yi] = iYi, i = 1, 2, 3, 4, 5, I1 = v1y5+y1y4

y5

[V2, Yi] = Yi, i = 2, 4, 5, I2 = v2y5−v3y5+y2y3

y5

[V3, Yi] = Yi, i = 3, 4, 5,

[Y1, Y4] = [Y2, Y3] = Y5

Table 7: Invariants of solvable rigid Lie algebras in dimension 8 and rank 4.
Algebra Brackets Invariants
r338 [V1, Yi] = iYi, i = 1, 2, 3, 4, none

[V2, Y2] = Y2, [V3, Y3] = Y3, [V4, Y4] = Y4

contrast to other solvable Lie algebras, for the rigid case we know that the torus,
which is maximal, in fact determines the law on the nilradical. This information
can be used to derive formulae of the invariants for arbitrary dimension, when
the structure of the torus and its influence on the structure of the solutions
in low dimension is known. We will illustrate this method with the following
example: for m ≥ 2 let d2m+1 = g2m+1 ⊕ t be the solvable Lie algebra whose
brackets over the basis {X0, .., X3, Y1, .., Y2m−3, V1, ..Vm−1} are

[X0, Xi] = Xi+1, i = 1, 2; [X1, Y2m−3] = X3

[Y2i−1, Y2i] = X3, i = 1, .., m− 2; [V1, Xi] = Xi, i = 0, 2
[V1, X3] = 2X3; [V1, Y2m−3] = 2Y2m−3

[V1, Y2i] = −2Y2i, i = 1, .., m − 2
[V2, Xi] = Xi, i = 1, 2, 3; [V2, Y2i] = Y2i, 1 ≤ i ≤ m − 2
[Vi+2, Y2i−1] = Y2i−1, i = 1, .., m − 2, [Vi+2, Y2i] = −Y2i, i = 1, .., m− 2

Using the root system associated to this algebra, it is easily seen that it is rigid.
For m = 2 it is isomorphic to r77, and expressed in the basis above, a fundamental

set of invariants of this algebra is given by
{

−2x2
3v2+x2

3v1+x0x1x3−2x1x3y1+y1x2
2

x2
3

}
.

Now observe that, for m ≥ 3, the nilradical of d2m+1 consists of a ”germ subal-
gebra” generated by {X0, .., X3, Y1}, to which three dimensional Heisenberg Lie
algebras whose derived subalgebra is the center (X3) are ”glued”. We claim

Proposition 6 For any m ≥ 3 the solvable Lie algebra d2m+1 has a fundamen-

13



tal set of invariants formed by the (m − 1) rational invariants given by

{
−2x2

3v2 + x2
3v1 + x0x1x3 − 2x1x3y2m−3 + y2m−3x

2
2

x2
3

,
x3vk+2 + y2k−1y2k

x3

}

for 1 ≤ k ≤ m − 2.

Proof. Over the coordinate system {x0, .., x3, y1, .., y2m−3, v1, .., vm} the
system is the following:

X̃0.F = (−x2∂x1 − x3∂x2 + x0∂v1) .F = 0 (5)

X̃1.F =
(
x2∂x0 − x3∂y2m−3 + x1∂v2

)
.F = 0 (6)

X̃2.F = (x3∂x0 + x2∂v1 + x2∂v2) .F = 0 (7)

X̃3.F = (2x3∂x0 + x3∂v2) .F = 0 (8)

Ỹ2i−1.F =
(
−x3∂y2i

+ y2i−1∂vi+1

)
.F = 0, 1 ≤ i ≤ m − 2 (9)

Ỹ2m.F =
(
x3∂y2i−1 + y2i

(
2∂v1 + ∂v2 − ∂vi+2

))
.F = 0, 1 ≤ i ≤ m − 2 (10)

Ỹ2m−3.F = (x3∂x1 + 2y2m−3∂v1) .F = 0 (11)

Equations (8.1), (8.2), (8.3), (8.4) and (8.7) correspond to the system of par-
tial differential equations associated to the seven dimensional Lie algebra r77,

with solution
−2x2

3v2+x2
3v1+x0x1x3−2x1x3y2m−3+y2m−3x2

2

x2
3

. Now this function is eas-

ily seen to be a solution of the system above. Equations (8.5) and (8.6) have,
for fixed i, the solution

x3vk+2+y2k−1y2k

x3
, which is also a solution of the whole

system. Observe that these solution have all the same structure, since they cor-
respond to the infinitesimal generators Ỹ2i−1, Ỹ2i of the Heisenberg subalgebras

generated by them, jointly with the element X̃3. The rank of the commutator
table is easily seen to be m − 1.

9 Conclusions

We have computed the invariants of complex solvable rigid Lie algebras up to
dimension eight. This is indeed the maximal dimension for which these algebras
have been classified, although certain partial classifications exist. Most of the
general results obtained for arbitrary dimension also remain valid in the real
case. However, there is a reason for studying complex rigid Lie algebras rather
than real ones. It is known that that any complex simple Lie algebra is in fact
defined over the rational Q, and from the tables above it also follows that any
solvable complex rigid Lie algebra is also rational. Since we have seen that the
eigenvalues for the action of a maximal torus of derivations can be chosen in
Z, Carles [15] conjectured that any rigid Lie algebra is rational. This would
justify in an elegant manner why simple Lie algebras are rational, since they
are rigid by cohomological reasons. Unfortunately this conjecture is false, as
pointed out by Goze and Ancochea in [11]. They gave examples of rigid Lie

14



algebras being not only non-rational, but even non-real. That these algebras
remained unknown relies on the fact that this pathology begins to appear from
dimension eleven on.
We have pointed out that, in the rigid case, useful information on the existence
or not of non-trivial invariants can be deduced from the action of the torus on
its nilradical, without paying closer attention to the structure of the latter. This
works since this structure is perfectly determined by the torus. The example
in section 8. also shows that we can determine a fundamental set of invariants
starting from solutions in low dimensional cases. This will be of particular im-
portance when we consider semidirect products of nilpotent Lie algebras and
non-maximal tori. Although in this case the algebras are not rigid, probably we
can deduce information about the invariants by analyzing the action of these
tori. This approach should provide at least sufficiency criteria for the non-
existence of non-trivial invariants, since the obtention of characterizations for
these algebras probably do not exist. Such sufficiency conditions will be of ex-
treme importance for the non-decomposable case.
The study of invariants of solvable rigid Lie algebras also seems to answer an-
other important question: it is well known that for semisimple Lie algebras the
cardinal of a fundamental set of invariants coincides with the rank of the alge-
bra, i.e., the dimension of a Cartan subalgebra [2]. It is therefore natural to
ask wheter there exists a link between the cardinal of a fundamental set and
the rank for the solvable case. If we consider the n + 3 dimensional solvable Lie
algebra Ln ⊕ t given by

[X1, Xi] = Xi+1, i = 2, .., n

[V1, Xi] = i.Xi, i = 1, .., n + 1

[V2, Xi] = Xi, i = 2, .., n + 1

we see that the rank is always two, and a Cartan subalgebra is generated by V1

and V2. Now the number of functionally independent invariants for this algebra
is N = n − 3, and therefore depends on the dimension of the nilradical, and
not on the dimension of a Cartan subalgebra. We may expect that for more
complicated examples, such as non-decomposable solvable Lie algebras, such a
formula will generally be not inferable.
Finally, we have seen by examples that a solvable Lie algebra having a fun-
damental set of invariants formed by rational functions needs not to be rigid.
However, there is a wide class of algebras having this property, namely those
satisfying dimg = dimDer (g), where Der (g) denotes the Lie algebra of deriva-
tions. Specifically, this is of interest for complete Lie algebras [25, 26], i.e.,
centerless Lie algebras whose derivations are all inner. This class is of great im-
portance for structure theory, and its invariants will surely provide interesting
information about their representations.

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15



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16