Lobachevskii Journal of Mathematics Vol. 3, 1999, 5–17 \begin{matrix}c \\ ◯ \end{matrix}Y. Agaoka

Y. Agaoka

ON THE VARIETY OF 3-DIMENSIONAL LIE ALGEBRAS

(submitted by B.N.Shapukov)

Abstract. It is known that a 3-dimensional Lie algebra is unimodular or solvable as a result of the classification. We give a simple proof of this fact, based on a fundamental identity for 3-dimensional Lie algebras, which was first appeared in [21]. We also give a representation theoretic meaning of the invariant of 3-dimensional Lie algebras introduced in [15], [22], by calculating the GL(V)-irreducible decomposition of polynomials on the space ∧^{2}V^{∗} ⊗ V up to degree 3. Typical four covariants naturally appear in this decomposition, and we show that the isomorphism classes of 3-dimensional Lie algebras are completely determined by the GL(V)-invariant concepts in ∧^{2}V^{∗} ⊗ V defined by these four covariants. We also exhibit an explicit algorithm to distinguish them.

Introduction

In this survey paper, we consider a variety consisting of 3-dimensional Lie algebras from the representation theoretic viewpoint, and give some perspective to the understanding of the set of 3-dimensional Lie algebras.

As is well known, the set of Lie algebra structure on a fixed vector space V constitutes a GL(V)-invariant algebraic set in ∧^{2}V^{∗} ⊗ V, whose defining equations are the quadratic polynomials called the Jacobi identity: 𝔖 [[X, Y], Z] = 0. (The symbol 𝔖 implies the cyclic sum.) In general, this algebraic set is not irreducible, and the number of irreducible components is known for low dimensional cases (cf. [7], [11], [15], [19]).

In the case of dim V = 3, it is decomposed into two irreducible components, one of which consists of unimodular Lie algebras, and the other of which consists of solvable Lie algebras. In this paper, we give a simple proof of this fact without using the classification of Lie algebras, but instead, by showing the fundamental identity

𝔖 (Tr ad X) · [Y, Z] = 0,

which is a peculiar phenomenon for 3-dimensional case.

The second purpose of this paper is to describe the set of 3-dimensional Lie algebras from GL(V)-invariant representation theoretic viewpoint. We first give a GL(V)-irreducible decomposition of the set of polynomials on ∧^{2}V^{∗} ⊗V up to degree 3, by using Littlewood-Richardson’s rule. And next, we show that among the generators of these irreducible components, there exist four fundamental generators (= covariants), by which we can describe several concepts in a GL(V)-invariant way. For example, we can give a representation theoretic meaning of the invariant of 3-dimensional Lie algebras introduced in [15], [22]. And in addition, we can state an algorithm to distinguish the isomorphism classes of 3-dimensional Lie algebras, including the mutual relation of GL(V)-orbits and their degenerations, in terms of these four covariants.

Of course, we already know the classification of 3-dimensional Lie algebras, and their individual structures are well known. But by this GL(V)- invariant viewpoint, we can obtain a systematic understanding of the set of 3-dimensional Lie algebras, which cannot be read from a mere classification, constituting a dense forest of mutually unrelated normal forms. The explicit description of these results are summarized at the end of § 4. We remark that our viewpoint in this paper is quite effective in considering several GL(V)- invariant problems in geometry. For other explicit examples, see [4], [5], [6].

Throughout this paper, we assume that Lie algebras and vector spaces are defined over the field of complex numbers C, unless otherwise stated.

The author is grateful to the referee who carefully read the manuscript and gave valuable comments. He is also grateful to Professor Umehara who showed him a preprint [21] and gave several useful advices on the subject of this paper.

§ 1. Fundamental identity.

We fix a basis X_{1}, ⋯, X_{n} of an n-dimensional vector space V, and consider a skew symmetric bilinear map [ , ] : V ×V → V defined by [X_{i}, X_{j}] = ∑ c_{ij}^{k}X_{k} (c_{ij}^{k} = −c_{ji}^{k}). Then, the bilinear map [ , ] defines a Lie algebra structure on V if and only if it satisfies the Jacobi identity

∑_{l} (c_{ij}^{l}c_{lk}^{m} + c_{jk}^{l}c_{li}^{m} + c_{ki}^{l}c_{lj}^{m}) = 0.

This implies that the set of Lie algebra structures on V are defined by quadratic polynomial relations on c_{ij}^{k}.

In the case of dim V = 3, the Jacobi identity has another peculiar expression, from which the fundamental identity stated in Introduction follows.

Theorem 1. The following identity holds for any [ , ] ∈ ∧^{2}V^{∗} ⊗ V where V is a 3-dimensional vector space.

𝔖 [[X_{1}, X_{2}], X_{3}] = 𝔖 (Tr adX_{1}) · [X_{2}, X_{3}].

Proof. We may assume that X_{1}, X_{2}, X_{3} are linearly independent because both expressions of the above equality reduce to zero in case X_{1}, X_{2}, X_{3} are linearly dependent. We express [X_{i}, X_{j}] = ∑ c_{ij}^{k}X_{k} as above. Then, by direct calculations, we can easily show that both sides of the above equality are equal to

\begin{aligned}& (c_{12}^{2}c_{23}^{1} − c_{12}^{1}c_{23}^{2} + c_{13}^{3}c_{23}^{1} − c_{13}^{1}c_{23}^{3})X_{1} + (c_{12}^{1}c_{13}^{2} − c_{12}^{2}c_{13}^{1} + c_{13}^{3}c_{23}^{2} − c_{13}^{2}c_{23}^{3})X_{2} \\ + & (c_{12}^{1}c_{13}^{3} − c_{12}^{3}c_{13}^{1} + c_{12}^{2}c_{23}^{3} − c_{12}^{3}c_{23}^{2})X_{3}. \end{aligned}

But, we give here a representation theoretic proof, using the Schur functions. (For the definition and the fundamental properties of Schur functions, see [17].) We fix a volume form ω on V throughout. Then, we have a natural isomorphism V → ∧^{2}V^{∗} defined by X → X⌋ω, and the bilinear map [ , ] = {c_{ij}^{k}} may be considered as an element of V ⊗ V under this isomorphism.

In terms of the above basis of V, both sides of the equality in Theorem 1 are expressed as quadratic polynomials of c_{ij}^{k}, and they are considered as elements of S^{2}(V ⊗V)^{∗}. The group SL(V) which preserves the volume form ω naturally acts on S^{2}(V ⊗ V)^{∗}, and in the case dim V = 3, from Littlewood-Richardson’s rule, we have the following irreducible decomposition of S^{2}(V ⊗ V)^{∗}:

S^{2}(V ⊗ V)^{∗} = S^{2}(V^{∗} ⊗ V^{∗})
= S^{2}(S^{2}V^{∗} + ∧^{2}V^{∗})
= S^{2}(S^{2}V^{∗}) + S^{2}V^{∗} ⊗ ∧^{2}V^{∗} + S^{2}(∧^{2}V^{∗})
= (S_{4} + S_{22}) + (S_{31} + S_{211}) + S_{22}
= S_{4} + S_{31} + 2 S_{22} + S_{211}.

Here, S_{λ} implies the dual of the irreducible representation space of SL(V) corresponding to the partition λ. (For example, the symbols S_{1} and S_{2} correspond to V^{∗} and S^{2}V^{∗}, respectively. Strictly speaking, the dual space V^{∗} must be represented as S_{0···0−1}. But we use this dual notations for simplicity. For details, see [17], [3].) Since the multiplicity of S_{211} in S^{2}(V ⊗ V)^{∗} is one, and the space S_{211} is isomorphic to S_{1} as an SL(V)-module, we have dim Hom_{SL(V)}(S^{2}(V ⊗ V), V) = 1. For c = [ , ] ∈ ∧^{2}V^{∗} ⊗ V ≅ V ⊗ V, we define two elements Φ(c), Ψ(c) ∈ V by

\begin{aligned}𝔖 [[X_{1}, X_{2}], X_{3}] & = ω(X_{1}, X_{2}, X_{3})Φ(c), \\ 𝔖 (Tr ad X_{1}) · [X_{2}, X_{3}] & = ω(X_{1}, X_{2}, X_{3})Ψ(c). \end{aligned}

Then, we have Φ, Ψ ∈ Hom_{SL(V)}(S^{2}(V ⊗V), V), and they must be proportional. The proportional constant does not depend on a specific [ , ], and hence, by considering the bilinear map defined by

[X_{1}, X_{2}] = X_{2}, [X_{2}, X_{3}] = X_{1}, [X_{3}, X_{1}] = 0,

we can easily show that this constant is equal to 1. q.e.d.

By this theorem, we obtain the fundamental identity for 3-dimensional Lie algebras:

𝔖 (Tr ad X_{1}) · [X_{2}, X_{3}] = 0.

From this identity, we know that Tr ad X = 0 for any X ∈ 𝔤 if dim [𝔤, 𝔤] = 3. Hence, we obtain the following result without the help of the classification of 3-dimensional Lie algebras. (In the following, we often express the space V as 𝔤 if a Lie algebra structure on V is explicitly or implicitly given.)

Corollary 2. Let 𝔤 be a 3-dimensional Lie algebra. Then 𝔤 satisfies Tr ad X = 0 for any X ∈ 𝔤, or dim [𝔤, 𝔤] ≤ 2. Namely, 𝔤 is unimodular or solvable in the case dim 𝔤 = 3.

Remark. (1) The above theorem and corollary hold over any field k of characteristic ≠ 2. As far as the author knows, the above fundamental identity was first appeared in the (unpublished) paper [21].

(2) In the case of dim V ≥ 4, in contrast with the case of dim V = 3, the set of quadratic polynomials appearing in the Jacobi identity is not irreducible, and constitutes two GL(V)-irreducible components of S^{2}(∧^{2}V^{∗} ⊗ V)^{∗}, whose dimensions are 1/6 · n(n − 1)(n + 1)(n − 3) and 1/2 · n(n − 1). And hence, the argument in the proof of Theorem 1 does not hold for dim V ≥ 4. For example, we can easily check that 𝔖 (Tr ad X_{1}) · [X_{2}, X_{3}] ≠ 0 for the 4-dimensional Lie algebra defined by

[X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = X_{3}, [X_{1}, X_{4}] = 2 X_{4}, [X_{2}, X_{3}] = X_{4}.

It seems that the fundamental identity of the above type directly related to the Jacobi identity does not exist in the case dim V ≥ 4.

Next, we consider the set of 3-dimensional Lie algebras in ∧^{2}V^{∗} ⊗ V. We already know the following result.

Theorem 3 (cf. [7], [11], [15], [19]). The set of 3-dimensional Lie algebra structures on V constitutes two irreducible varieties in ∧^{2}V^{∗} ⊗ V, one of which is the closure of the set of simple Lie algebras and the other of which is the set of solvable Lie algebras.

In the following, we denote these varieties by Σ_{simp} and Σ_{solv}, respectively. They are both stable under the natural action of GL(V). Clearly, each GL(V)- orbit in Σ_{simp} or Σ_{solv} corresponds to an isomorphism class of Lie algebras.

Actually, two cases in Corollary 2 just correspond to two varieties in Theorem 3. The variety Σ_{simp} is a linear subspace of ∧^{2}𝔤^{∗} ⊗ 𝔤 defined by the condition Tr ad X = 0 (X ∈ 𝔤), i.e., Σ_{simp} is just the set of unimodular Lie algebras. The defining equation of the variety Σ_{solv} is given by dim [𝔤, 𝔤] ≤ 2, which is a cubic polynomial relation on c_{ij}^{k}. It is known that both varieties are of the same dimension 6.

§ 2. GL(V)-irreducible components of S^{p}(∧^{2}V^{∗} ⊗ V)^{∗}.

As stated above, the set of 3-dimensional Lie algebras is the union of two GL(V)-invariant irreducible varieties, and hence, their defining equations form GL(V)-invariant subspaces of the polynomial ring ∑ S^{p}(∧^{2}V^{∗} ⊗ V)^{∗}. In this section, in order to describe several GL(V)-invariant concepts in Σ_{simp} and Σ_{solv}, including the representation theoretic meaning of their defining equations, we first give a list of irreducible decomposition of S^{p}(∧^{2}V^{∗} ⊗ V)^{∗} and the generator of each component up to degree 3.

For simplicity, we fix a volume form ω of V as in the proof of Theorem 1, and identify two spaces ∧^{2}V^{∗} ⊗ V and V ⊗ V. Then, as GL(V)-modules, these spaces are isomorphic to each other by multiplying an invariant of SL(V) corresponding to a suitable power of det g (g ∈ GL(V)). This argument also holds for polynomials on ∧^{2}V^{∗} ⊗ V ≅ V ⊗ V for each fixed degree. In the following, we only consider algebraic concepts determined by the vanishing (or by the ratio) of these polynomials, and hence, we may use the space S^{p}(V ⊗V)^{∗} instead of S^{p}(∧^{2}V^{∗} ⊗ V)^{∗}.

The explicit decomposition of S^{p}(V ⊗ V)^{∗} is easily obtained by using Littlewood-Richardson’s rule as in § 1, and the generator of each irreducible component can be calculated by the method stated in [3; p.115 ∼ 116]. (By this method, the highest element in S_{λ} automatically appears.) Most generators are expressed as a product of lower degree generators, and we marked the symbol “∗” to the generators that cannot be expressed as a product form. These marked generators correspond to covariants (or invariants) in classical invariant theory (cf. [9], [16], [20]). In the following, we express c_{ij}^{k} as a matrix form: c_{i1} = c_{23}^{i}, c_{i2} = c_{31}^{i}, c_{i3} = c_{12}^{i} (i = 1 ∼ 3).

p = 1 : S_{2} + S_{11}

∗ S_{2} : c_{11}
∗ S_{11} : c_{12} − c_{21}

p = 2 : S_{4} + S_{31} + 2 S_{22} + S_{211}

S_{4} : c_{11}^{2} = S_{2}^{2}
S_{31} : c_{11}(c_{12} − c_{21}) = S_{2}S_{11}
∗ S_{22_{a}} : c_{11}c_{22} − c_{12}c_{21}
S_{22_{b}} : (c_{12} − c_{21})^{2} = S_{11}^{2}
∗ S_{211} : c_{11}c_{23} − c_{11}c_{32} + c_{12}c_{31} − c_{13}c_{21}

p = 3 : S_{6} + S_{51} + 2 S_{42} + S_{411} + 2 S_{33} + 2 S_{321} + 2 S_{222}

S_{6} : c_{11}^{3} = S_{2}^{3}

S_{51} : c_{11}^{2}(c_{12} − c_{21}) = S_{2}^{2} S_{11} S_{42_{a}} : c_{11}(c_{12} − c_{21})^{2} = S_{2} S_{11}^{2} S_{42_{b}} : c_{11}(c_{11}c_{22} − c_{12}c_{21}) = S_{2} S_{22_{a}} S_{411} : c_{11}(c_{11}c_{23} − c_{11}c_{32} + c_{12}c_{31} − c_{13}c_{21}) = S_{2} S_{211} S_{33_{a}} : (c_{12} − c_{21})^{3} = S_{11}^{3} S_{33_{b}} : (c_{12} − c_{21})(c_{11}c_{22} − c_{12}c_{21}) = S_{11}S_{22_{a}}

∗ S_{321_{a}}: | \begin{matrix}c_{11} & c_{11} & c_{21} \\ c_{21} & c_{12} & c_{22} \\ c_{31} & c_{13} & c_{23} \end{matrix} |

S_{321_{b}}: (c_{12} − c_{21})(c_{11}c_{23} − c_{11}c_{32} + c_{12}c_{31} − c_{13}c_{21}) = S_{11} S_{211}

∗ S_{222_{a}}: | \begin{matrix}c_{11} & c_{12} & c_{13} \\ c_{21} & c_{22} & c_{23} \\ c_{31} & c_{32} & c_{33} \end{matrix} |

∗ S_{222_{b}}: c_{11}(c_{23} − c_{32})^{2} + c_{22}(c_{13} − c_{31})^{2} + c_{33}(c_{12} − c_{21})^{2} +(c_{12} − c_{21})(c_{23}c_{31} − c_{13}c_{32}) + (c_{23} − c_{32})(c_{31}c_{12} − c_{21}c_{13})

+(c_{31} − c_{13})(c_{12}c_{23} − c_{32}c_{21})

Note that the dimension of the irreducible representation space corresponding to the partition {a, b, c} is given by

dim S_{abc} = \frac{1}{2}(a − b + 1)(a − c + 2)(b − c + 1)

in the case dim V = 3. In the following, for each irreducible component S_{λ}, we often express its generator by using the same letter S_{λ}. (If the multiplicity is greater than one, we distinguish these generators by the symbols S_{λ_{a}}, S_{λ_{b}}, etc.)

In viewing the above list, we know that there appear seven fundamental covariants (= essential generators marked by “∗”) up to degree 3. We conjecture that the polynomial ring ∑ S^{p}(V ⊗ V)^{∗} is generated by these seven fundamental covariants, namely, each generator of the component of S^{p}(V ⊗ V)^{∗} is expressed as a product of these seven covariants. By direct calculations, we can show that this conjecture is correct up to degree 8. But it seems that the complete determination of covariants and their syzygies are hard to solve as in classical invariant theory, though it is an important and fundamental problem. (For the spaces S^{2}V, ∧^{2}V and V ⊕ ∧^{2}V, we know the complete answer to this problem. (cf. [1], [2], [8]. See also [4].)) If the above conjecture is correct, all GL(V)-invariant concepts in the space V ⊗ V (V = C^{3}) can be described by these seven covariants.

We remark that these seven polynomials are not functionally independent on V ⊗ V. In fact, these generators satisfy the following sextic identity:

S_{2} S_{11}^{2} S_{222_{a}} + S_{211}^{2} S_{22_{a}} + S_{321_{a}}^{2} = S_{2} S_{22_{a}} S_{222_{b}} + S_{11} S_{211} S_{321_{a}}.

The space S^{6}(V ⊗ V)^{∗} contains the irreducible components S_{642} with multiplicity seven. But there exist eight generators corresponding to S_{642}:

\begin{matrix}S_{2} S_{11}^{2} S_{222_{a}}, S_{2} S_{11}^{2} S_{222_{b}}, S_{2} S_{22_{a}} S_{222_{a}}, S_{2} S_{22_{a}} S_{222_{b}}, \\ S_{11}^{2} S_{211}^{2}, S_{11} S_{211} S_{321_{a}}, S_{211}^{2} S_{22_{a}}, S_{321_{a}}^{2}. \end{matrix}

And the difference 8 − 7 = 1 implies the existence of the above sextic identity. This is the unique non-trivial polynomial relation among seven fundamental covariants because the rank of the (7, 9)-matrix (∂S_{λ}/∂c_{ij}) is 6 at a generic point of V ⊗ V, where S_{λ} are the fundamental covariants.

§ 3. Four covariants for 3-dimensional Lie algebras.

Now, we impose the condition S_{211} = 0 corresponding to the Jacobi identity, i.e., we assume that the space V possesses a Lie algebra structure. (The expression S_{λ} = 0 implies that not only the generator itself, but all polynomials in the space S_{λ} vanish.) In this section, we give a geometric meaning defined by the fundamental covariants listed in § 2. But in advance, we first review a classification of 3-dimensional complex Lie algebras for later use (cf. [10], [14], [19]).

non-trivial bracket operations

L_{0} = C^{3}

L_{1} = Heisenberg  [X_{1}, X_{2}] = X_{3}
L_{2} = 𝔞𝔣𝔣(1, C) ⊕ C^{1} [X_{1}, X_{2}] = X_{2}
L_{3}  [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = X_{2} + X_{3}
L_{4}(α)  [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = αX_{3}
L_{5} = 𝔰𝔩(2, C)  [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = −X_{3}, [X_{2}, X_{3}] = X_{1}

For the Lie algebra L_{4}(α), it is known that L_{4}(α) ≅ L_{4}(α′) if and only if α = α′ or αα′ = 1. In addition, we have L_{4}(0) = L_{2}, and hence, we may assume |α| ≥ 1. Among them, we can easily check that L_{4}(−1) is isomorphic to the complex Euclidean Lie algebra

𝔢(2, C) = \{( \begin{matrix}0 & a & b \\ −a & 0 & c \\ 0 & 0 & 0 \end{matrix} )\} .

At a first glance, the Lie algebra L_{3} seems to be isolated from other Lie algebras. But this appearance is not correct, and it is continuously deformable from the Lie algebra L_{4}(α) as follows. We define a Lie algebra M(α) by

[X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = X_{2} + α X_{3}.

Then, we can easily show that

M(α) ≅ \{ \begin{matrix}L_{4}(α) & (α ≠ 0, 1) \\ L_{2} & (α = 0) \\ L_{3} & (α = 1) \end{matrix} ,

which implies that the Lie algebra L_{3} is adjacent to L_{4}(α) in spite of its appearance.

Now, for each Lie algebra 𝔤, we substitute the value c_{ij}^{k} to seven fundamental covariants listed in § 2, where c_{ij}^{k} is the structure constant with respect to a generic basis of 𝔤. Then, it follows that the equalities

S_{211} = S_{321_{a}} = S_{222_{b}} = 0

hold for any 3-dimensional Lie algebra, and hence there remain four fundamental covariants S_{2}, S_{11}, S_{22_{a}} and S_{222_{a}}. The value for these covariants are summarized in the following table. (We assume that the complex number α satisfies |α| ≥ 1, α ≠ ±1.)

S_{2} S_{11} S_{22_{a}} S_{222_{a}}  dimension of ; GL(V)-orbit
L_{0}  0 0 0  0  0
L_{1}  ∗ 0 0  0  3
L_{2}  ∗ ∗ 0  0  5
L_{3}  ∗ ∗ ∗  0  5
L_{4}(α) ∗ ∗ ∗  0  5
L_{4}(1) 0 ∗ ∗  0  3
L_{4}(−1) ∗ 0 ∗  0  5
L_{5}  ∗ 0 ∗  ∗  6

Here, the symbol “0” implies that the generator S_{λ} vanishes with respect to any basis of 𝔤, and the symbol “∗” implies that S_{λ} does not vanish for a generic basis. (In this table, we also exhibit the dimension of each GL(V)-orbit.)

We summarize the GL(V)-invariant concepts defined by these covariants as follows. (As stated before, we express V as 𝔤.)

Proposition 4. (1) The condition S_{2} = 0 implies that there exists φ ∈ 𝔤^{∗} such that [X, Y] = φ(X)Y − φ(Y)X. These Lie algebras constitute a 3-dimensional linear subspace contained in the variety Σ_{solv}.

(2) The condition S_{11} = 0 implies that 𝔤 is unimodular, and this is the defining equation of the variety Σ_{simp}. (We remark that the condition S_{11} = 0 automatically implies S_{211} = 0. Of course, combined linear conditions S_{2} = S_{11} = 0 imply that 𝔤 is abelian.)

(3) The condition S_{22_{a}} = 0 holds if and only if dim [𝔤, 𝔤] ≤ 1.

(4) The condition S_{222_{a}} = 0 holds if and only if dim [𝔤, 𝔤] ≤ 2, and this condition is also equivalent to the solvability of 𝔤. In particular, the defining equations of Σ_{solv} are given by S_{211} = S_{222_{a}} = 0.

(5) Nilpotent Lie algebras are characterized by two conditions S_{11} = S_{22_{a}} = 0, i.e., a 3-dimensional Lie algebra 𝔤 is nilpotent if and only if it is unimodular and dim [𝔤, 𝔤] ≤ 1. (Actually, it is 2-step nilpotent.)

Proof. The results in (1) and (2) follow immediately from the explicit irreducible decomposition (V ⊗ V)^{∗} = S_{2} ⊕ S_{11}. The statements in (3) and (4) follow from the fact that S_{22_{a}} and S_{222_{a}} are the principal minor and the determinant of the matrix (c_{ij}), respectively (cf. [1], [2]). The result (5) can be easily obtained by using the classification stated above. q.e.d.

The explicit description of the GL(V)-invariant varieties determined by these conditions will be described in detail in the next section.

§ 4. Varieties of 3-dimensional Lie algebras.

In this section, we study GL(V)-invariant varieties consisting of 3-dimensionl Lie algebras and the orbit structures of them, in terms of four fundamental covariants listed up in § 3.

In the following, we denote by 𝒪(L_{k}) the GL(V)-orbit of the Lie algebra L_{k}, and by \overline{𝒪(L_{k})} its Zariski closure. The orbit decompositions in the following proposition are more or less well known for many Lie algebraists. But, these decompositions are easily proved by using the results stated in the previous sections. For example, we have L_{4}(1) ∉ \overline{𝒪(L_{2})} because S_{22_{a}} ≠ 0 for L_{4}(1). (Remind that the defining equations of Σ_{simp} and Σ_{solv} are S_{11} = 0 and S_{211} = S_{222_{a}} = 0, respectively. The dimension of each orbit is stated in § 3.)

Proposition 5. (1) The varieties Σ_{simp} and Σ_{solv} are the union of the following GL(V)-orbits:

\begin{aligned}Σ_{simp} & = 𝒪(L_{5}) ∪ 𝒪(L_{4}(−1)) ∪ 𝒪(L_{1}) ∪ 𝒪(L_{0}), \\ Σ_{solv} & = ∪_{α}𝒪(L_{4}(α)) ∪ 𝒪(L_{3}) ∪ 𝒪(L_{2}) ∪ 𝒪(L_{1}) ∪ 𝒪(L_{0}) (|α| ≥ 1). \end{aligned}

And hence, we have Σ_{simp} ∩ Σ_{solv} = 𝒪(L_{4}(−1)) ∪ 𝒪(L_{1}) ∪ 𝒪(L_{0}).

(2) The GL(V)-orbit decompositions of \overline{𝒪(L_{k})} are given by

\overline{𝒪(L_{0})} = 𝒪(L_{0}) = \{0\},
\overline{𝒪(L_{1})} = 𝒪(L_{1}) ∪ 𝒪(L_{0}),
\overline{𝒪(L_{2})} = 𝒪(L_{2}) ∪ 𝒪(L_{1}) ∪ 𝒪(L_{0}),
\overline{𝒪(L_{3})} = 𝒪(L_{3}) ∪ 𝒪(L_{4}(1)) ∪ 𝒪(L_{1}) ∪ 𝒪(L_{0}),
\overline{𝒪(L_{4}(α))} = 𝒪(L_{4}(α)) ∪ 𝒪(L_{1}) ∪ 𝒪(L_{0}), (|α| ≥ 1, α ≠ ±1),
\overline{𝒪(L_{4}(1))} = 𝒪(L_{4}(1)) ∪ 𝒪(L_{0}),
\overline{𝒪(L_{4}(−1))} = 𝒪(L_{4}(−1)) ∪ 𝒪(L_{1}) ∪ 𝒪(L_{0}) = Σ_{simp} ∩ Σ_{solv},
\overline{𝒪(L_{5})} = 𝒪(L_{5}) ∪ 𝒪(L_{4}(−1)) ∪ 𝒪(L_{1}) ∪ 𝒪(L_{0}) = Σ_{simp}.

A generic element of Σ_{simp} is represented by L_{5} (= 𝔰𝔩(2, C)) and the Lie algebras L_{4}(−1), L_{1}, L_{0} are degenerations of L_{5}. (For the concept “degeneration”, see [12], [19].) These Lie algebras are distinguished in Σ_{simp} by the rank of the matrix (c_{ij}), as stated in [15]. The other variety Σ_{solv} does not admit a dense orbit, in contrast with Σ_{simp}, and it mainly consists of infinite family of 5-dimensional orbits, which implies the existence of a 1-dimensional moduli space. (Remind that dim Σ_{simp} = dim Σ_{solv} = 6.)

Concerning this moduli space, an invariant χ(𝔤) ∈ C ∪ {∞} was introduced in the papers [15], [22]. (In the following, we mainly use the notation in [22]. The invariant J defined in [15] is related to χ(𝔤) in [22] by the equality χ(𝔤) = 2/(1 − J).) This invariant is quite useful in classifying 3-dimensional Lie algebras in Σ_{solv}, and roughly speaking, it gives a coordinate of the above moduli space. In our viewpoint, this invariant χ(𝔤) can be expressed as the ratio of two covariants belonging to the same partition {22}:

χ(𝔤) = \frac{S_{22_{b}}}{S_{22_{a}}} = \frac{S_{11}^{2}}{S_{22_{a}}} = \frac{(c_{12} − c_{21})^{2}}{c_{11}c_{22} − c_{12}c_{21}} . (∗)

This invariant is well-defined for all 3-dimensional Lie algebras except nilpotent ones (= L_{0} and L_{1} ). (Remind that nilpotent Lie algebras are characterized by two conditions S_{11} = S_{22_{a}} = 0.) We can easily show that the value at the right end of (∗) does not depend on the choice of a basis of 𝔤. And hence in case c_{12} − c_{21} = c_{11}c_{22} − c_{12}c_{21} = 0 for some basis, we have only to change it such that χ(𝔤) has a definite value in order to calculate χ(𝔤). Of course, we can also calculate this invariant by using the definition

J = \frac{Tr (ad X)^{2}}{(Tr ad X)^{2}} = 1 − \frac{2}{χ(𝔤)}

in [15], which is independent of the choice of generic X ∈ 𝔤. Essentially, the invariant χ(𝔤) is determined by the ratio of two non-zero eigenvalues of ad X. In fact, denoting the eigenvalues of ad X by 0, α, β, we have the equality

χ(𝔤) = \frac{β}{α} + \frac{α}{β} + 2.

The explicit value of χ(𝔤) for each Lie algebra is given as follows:

χ(𝔤)
L_{2}  ∞
L_{3}  4
L_{4}(α) \frac{(α+1)^{2}}{α}
L_{5}  0

As we listed in § 3, there are three types of 5-dimensional GL(V)-orbits in Σ_{solv} : 𝒪(L_{2}), 𝒪(L_{3}) and 𝒪(L_{4}(α)) (|α| ≥ 1, α ≠ 1). For these Lie algebras, we have the following proposition, indicating that the invariant χ(𝔤) serves as a coordinate of the moduli space of these Lie algebras. This proposition is almost equivalent to Theorem in [21], or the normal form stated in [15; p.23] and [22; Theorem 2], though the unimodular Lie algebra L_{4}(−1) is excluded there.

Proposition 6. Let 𝔤 and 𝔤′ be two elements of 𝒪(L_{2}) ∪ 𝒪(L_{3}) ∪ 𝒪(L_{4}(α)) (|α| ≥ 1, α ≠ 1) = Σ_{solv} \ 𝒪(L_{4}(1)) ∪ 𝒪(L_{1}) ∪ 𝒪(L_{0}). Then, 𝔤 is isomorphic to 𝔤′ if and only if χ(𝔤) = χ(𝔤′).

This proposition follows immediately from the fact that the condition (α + 1)^{2}/α = (α′ + 1)^{2}/α′ is equivalent to α = α′ or αα′ = 1. (We remark that χ(L_{3}) = χ(L_{4}(1)) and the invariant χ(𝔤) does not distinguish two Lie algebras L_{3} and L_{4}(1).)

The set of Lie algebras appearing in this proposition is dense in Σ_{solv}, and the remaining Lie algebras L_{4}(1), L_{1} and L_{0} in Σ_{solv} are the degenerations of them.

It should be noted that in terms of this invariant χ(𝔤), Lie algebras in 𝒪(L_{2})∪𝒪(L_{3})∪𝒪(L_{4}(α)) (|α| ≥ 1, α ≠ ±1) = Σ_{solv}\𝒪(L_{4}(±1))∪𝒪(L_{1})∪𝒪(L_{0}) are expressed as

[e_{1}, e_{2}] = 0, [e_{2}, e_{3}] = e_{1}, [e_{3}, e_{1}] = −e_{1} + \frac{1}{χ(𝔤)}e_{2}

for a suitable basis e_{i} of 𝔤, as stated in [22]. (The sign of [e_{2}, e_{3}] in [22; Theorem 2] should be corrected to the above form, as Professor Umehara kindly teach this fact to the author.)

Combining these results, we can give a simple description of the set of 3dimensional Lie algebras from GL(V)-invariant viewpoint as follows: The variety Σ_{solv} is the closure of the family of 5-dimensional GL(V)-orbits, which are continuously deformable to each other. Any two closures of these orbits have a common intersection \overline{𝒪(L_{1})} = 𝒪(L_{1}) ∪ 𝒪(L_{0}). Among these 5-dimensional orbits, 𝒪(L_{3}) has a special feature, containing an exceptional orbit 𝒪(L_{4}(1)) in its closure. Another exceptional 5-dimensional orbit is 𝒪(L_{4}(−1)), which is contained in the other variety Σ_{simp}. Actually, the intersection Σ_{simp} ∩ Σ_{solv} just coincides with the closure \overline{𝒪(L_{4}(−1))}, as stated in Proposition 5.

Finally, summarizing these results, we give an algorithm to determine the isomorphism classes of 3-dimensional Lie algebras, which is quite useful in actual determination:

START ↓

abel ⟶^{Y} L_{0} = C^{3}

N ↓

dim [𝔤, 𝔤] = 3 ⟶^{Y} L_{5} = 𝔰𝔩(2, C)

N ↓

S_{2} = 0 ⟶^{Y} L_{4}(1)

N ↓ dim [𝔤, 𝔤] = 1 & unimodular

⟶^{Y}

L_{1} = Heisenberg

N ↓

\{ \begin{matrix}χ(𝔤) & = ∞ \\ χ(𝔤) & = 4 \\ χ(𝔤) & = 0 \\ χ(𝔤) & = (α + 1)^{2}/α \end{matrix} \begin{matrix}⟶ \\ ⟶ \\ ⟶ \\ ⟶ \end{matrix} \begin{matrix}L_{2} = 𝔞𝔣𝔣(1, C) ⊕ C^{1} \\ L_{3} \\ L_{4}(−1) = 𝔢(2, C) \\ L_{4}(α) (|α| ≥ 1, α ≠ ±1) \end{matrix}

Remind that the condition S_{2} = 0 is equivalent to [X, Y] = φ(X)Y − φ(Y)X for some φ ∈ 𝔤^{∗}, which is also equivalent to the condition c_{ij} + c_{ji} = 0 (i.e., Lie algebras corresponding to 𝔨 in [22]).

Viewing this algorithm, we once more emphasize that all isomorphism classes are completely characterized in terms of four covariants S_{2}, S_{11}, S_{22_{a}} and S_{222_{a}}. We may say that these covariants (including the invariant χ(𝔤) = S_{11}^{2}/S_{22_{a}}) serve as a complete measure in describing the structure of 3-dimensional Lie algebras, which shows the effectiveness of our GL(V)-invariant viewpoint stated in Introduction.

§ 5. Final remarks.

(1) As stated in the previous sections, two varieties Σ_{simp} and Σ_{solv} admit their own defining equations in addition to the Jacobi identity. This fact implies that the quadratic polynomials corresponding to the Jacobi identity essentially contain another higher or curiously lower degree polynomial relations in an implicit form. And such a phenomenon usually occurs in multi-tensor spaces (cf. [4], [5]). For 3-dimensional Lie algebras, this phenomenon follows from the identity in Theorem 1. But for higher dimensional cases, it seems that

Y. Agaoka 16

such a fundamental identity does not exist, and it is desirable to investigate another method to explain such a phenomenon.

(2) In the 3-dimensional case, four covariants are sufficient to describe Lie algebra structures on V. To apply our method to higher dimensional cases, we must know the GL(V)-irreducible decomposition of the polynomial ring ∑ S^{p}(∧^{2}V^{∗}⊗V)^{∗} at first. But this is a quite difficult problem in representation theory, related to “plethysm” and “3-tensor spaces”. (As for the concept “plethysm”, see [17]. The space ∧^{2}V^{∗} ⊗ V may be considered as a 3-tensor space because V and V^{∗} symbolically appear three times in ∧^{2}V^{∗} ⊗ V. See [4].) For example, in the 4-dimensional case, we know by direct calculations that the numbers of fundamental covariants in S^{p}(∧^{2}V^{∗} ⊗ V)^{∗} (p = 1 ∼ 4) are given by 2, 5, 14 and 28, respectively (cf. [6]). For sufficiently high dimensional case, these numbers become 2, 7, 40 and 255, respectively, and it seems almost impossible to obtain a formula for general p. We may say that we are just encountering the usual difficulty peculiar to classical invariant theory.

References

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[3] Y. Agaoka, On the curvature of Riemannian submanifolds of codimension 2, Hokkaido Math. J. 14 (1985), 107-135.

[4] Y. Agaoka, Invariant subvarieties of the 3-tensor space C^{2} ⊗ C^{2} ⊗ C^{2}, Mem. Fac. Integrated Arts and Sci., Hiroshima Univ., Ser.IV, Science Report, 20 (1994), 1-18.

[5] Y. Agaoka, On the decomposability of polynomial valued 2-forms, preprint.

[6] Y. Agaoka, Left invariant symplectic structures on 4-dimensional complex Lie groups, in preparation.

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17 3-DIMENSIONAL LIE ALGEBRAS

[18] Y. A. Neretin, An estimate of the number of parameters defining an n-dimensional algebra, Math. USSR Izv. 30 (1988), 283-294.

[19] A. L. Onishchik and E. B. Vinberg (ed.), Lie Groups and Lie Algebras III (Structure of Lie Groups and Lie Algebras), Encyclopaedia of Math. Sci. 41, Springer-Verlag, Berlin, Heidelberg, New York, 1994.

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[21] H. Tasaki and M. Umehara, On the variety of 3-dimensional Lie algebras, preprint.

[22] H. Tasaki and M. Umehara, An invariant on 3-dimensional Lie algebras, Proc. Amer. Math. Soc. 115 (1992), 293-294.

[23] E. A. Tevelev, Generic algebras, Transformation Groups 1 (1996), 127-151.

Department of Mathematics, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan E-mail address: agaoka@mis.hiroshima-u.ac.jp

Received May 5, 1999