An algorithm to determine the isomorphism classes of 4-dimensional complex Lie algebras

BY

Yoshio Agaoka

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

Abstract

We give an algorithm to determine the isomorphism classes of 4-dimensional complex Lie algebras from the representation theoretic viewpoint. For this purpose, we give the GL(V)-irreducible decomposition of the polynomial ring of the space ∧^{2}V^{∗} ⊗ V (V = C^{4}) up to degree three, and show that intrinsic concepts defined by these covariants are sufficient to distinguish the isomorphism classes. As an application, we describe the variety of 4-dimensional Lie algebras and their degenerations in a comparatively simple form, by introducing a new family of normal forms of 4dimensional Lie algebras that are just fitted for these purposes.

Introduction.

In this paper, we study the set of 4-dimensional complex Lie algebras from the representation theoretic viewpoint. In particular, we give an algorithm to determine the isomorphism class of a given abstract 4-dimensional Lie algebra, in terms of a finite number of covariants and invariants of the group GL(V).

Many results are already known for 4-dimensional (real and complex) Lie algebras, such as the classification, the degeneration and deformation, and the number of varieties consisting of Lie algebras, etc. (cf. [7], [9], [12], [18], [19], [23], [25], [29].) But, in spite of these results, several important problems are still left unsolved concerning the isomorphism classes of Lie algebras. For example, it is in general a hard algebraic problem to determine the explicit isomorphism class of a given abstract Lie algebra 𝔤, i.e., which normal form in the classification table is isomorphic to a given 𝔤. Of course, the dimensions of [𝔤, 𝔤] and [[𝔤, 𝔤], [𝔤, 𝔤]], etc. give some necessary conditions to determine this isomorphism class.

02000 Mathematics Subject Classification. 17B05, 13A50, 14L30.

0Key words. Variety of Lie algebras, covariant, invariant, deformation, degeneration.

But, these conditions are not in general enough to determine it. In this paper, we give a finite number of intrinsic concepts of 4-dimensional complex Lie algebras 𝔤, by which we can determine their isomorphism classes uniquely without constructing the explicit isomorphisms. (See the examples at the end of §6.)

For this purpose, we consider the set of all Lie algebra structures on a fixed 4dimensional complex vector space V. By fixing a basis of V, the structure constant {c_{ij}^{k}} of 𝔤 can be naturally considered as an element of ∧^{2}V^{∗} ⊗ V, and we identify it with the Lie algebra 𝔤 itself. Since {c_{ij}^{k}} satisfies the Jacobi identities that are the quadratic polynomial relations of {c_{ij}^{k}}, the set of all Lie algebras constitutes an algebraic set of the space ∧^{2}V^{∗} ⊗ V, which splits into four irreducible varieties. (cf. [9], [12], [18], [25]. See also Proposition 8.) The group GL(V) naturally acts on this space, and it is clear that the orbit decomposition of this algebraic set is equivalent to the classification of Lie algebras.

But to obtain a method to determine the isomorphism classes of Lie algebras, we take another standpoint, and consider the polynomial ring of ∧^{2}V^{∗} ⊗ V and its GL(V)- irreducible decomposition. The above mentioned “intrinsic concepts” on 𝔤 are usually expressed as the vanishing of some irreducible components of the polynomial ring (= covariants). For example, the space of linear polynomials of {c_{ij}^{k}} splits into two GL(V)- irreducible components, and the vanishing of one component is equivalent to the “unimodularity” of 𝔤, the vanishing of the other component is equivalent to the condition that the Lie bracket is expressed as [X, Y] = f(X)Y − f(Y)X for some f ∈ 𝔤^{∗}. Both concepts play fundamental roles in determining the isomorphism classes. The above mentioned values dim [𝔤, 𝔤] and dim [[𝔤, 𝔤], [𝔤, 𝔤]] also can be characterized by the vanishing or non-vanishing of some covariants. (For details, see Proposition 4.) As a result, we can show that “intrinsic concepts” to determine the isomorphism classes of 𝔤 are all expressible as the vanishing of covariants of {c_{ij}^{k}} up to degree three. And in terms of these concepts, we explicitly give an algorithm to determine the isomorphism classes of 4-dimensional complex Lie algebras, which is the main result of this paper (Theorem 11 and Figure 2).

As a by-product of this method, we can also show several facts on the set of 4dimensional Lie algebras, in particular, degenerations and the variety of Lie algebras. (For the definition of degeneration, see §5.) Degenerations of 4-dimensional Lie algebras are already completely determined in [7]. But the final results given there are quite complicated. In this paper, we introduce a new family of normal forms of 4-dimensional complex Lie algebras that are just fitted to describe the varieties and the degenerations. And in terms of these normal forms, we summarize the results on these subjects in a comparatively simple form. To give these nice normal forms is another main result of the present paper.

Now, we state the contents of this paper. In §1, we give a new classification table of 4-dimensional complex Lie algebras, consisting of ten normal forms (Table 1). These normal forms possess several nice properties, and after explaining them, we next summarize the fundamental quantities of these Lie algebras (Table 2). Among others, we state a remarkable property that except for one Lie algebra, the ratio of the eigenvalues of ad X does not depend on the choice of X ∈ 𝔤 if X is sufficiently generic (Proposition 2). This property leads us to define three fundamental invariants of 4-dimensional Lie algebras, which play a crucial role in determining the isomorphism classes. In §2, we give the GL(V)-irreducible decomposition of the polynomial ring of the space ∧^{2}V^{∗} ⊗ V up to degree three. We also explicitly give the generators of these irreducible components which we use in this paper. And we evaluate them for each normal form by using computers (Table 3). To know the vanishing or non-vanishing of each generator for a given Lie algebra is one crucial step in the actual determination of the isomorphism classes. In §3, we characterize “intrinsic concepts” on 𝔤 determined by the vanishing of these covariants. Some unfamiliar but important properties naturally appear. In §4, by using the ratio of the eigenvalues of ad X stated above, we define three fundamental invariants χ_{1} ∼ χ_{3} of 4-dimensional Lie algebras. Roughly speaking, these invariants serve as the coordinate of the moduli space of the variety of Lie algebras, because the parameters appearing in the normal forms of Lie algebras in Table 1 are uniquely determined by these invariants χ_{1} ∼ χ_{3} (Proposition 5).

In §5, by using the above results, we summarize some known facts and some new results on degenerations and the varieties of 4-dimensional Lie algebras. On account of the nice properties of our normal forms, these results are expressed in a comparatively simple form. In particular, we give the defining equations of four irreducible varieties of Lie algebras, and the explicit orbit decompositions of them, including the degenerations (Proposition 9, Figure 1). We know that there exists one principal line of degenerations in each variety. But, several “singular” Lie algebras make the situation a little complicated. In the final section (§6), we give an algorithm to determine the isomorphism classes of 4-dimensional complex Lie algebras, which is the main subject of this paper. The results are summarized in Theorem 11 and Figure 2. Roughly speaking, dimensions of [𝔤, 𝔤] and [[𝔤, 𝔤], [𝔤, 𝔤]], three invariants χ_{1} ∼ χ_{3}, and three kinds of covariants are enough to determine the isomorphism classes.

In the appendices, we summarize several related topics. In Appendix A, we construct several isomorphisms to the previously known normal forms. Here, we adopt two classification tables in [7], [23], and show that they just coincide with our normal forms in Table 1, after a change of the basis. In Appendix B, we review the results on 3-dimensional complex Lie algebras in [5], where we studied 3-dimensional Lie algebras from the same standpoint of this paper. But the normal forms listed in [5] are not so fitted to describe degenerations. And for this reason, we re-summarize the simplified results here, by introducing a new family of norma forms.

Finally, we add some comments for higher dimensional case. Theoretically, we can continue to develop our method to higher dimensional Lie algebras. But unfortunately, calculations of covariants and invariants, and the irreducible decomposition of the polynomial ring become complicated as the dimension becomes large, and we do not know what kind of concepts will be required in order to distinguish the isomorphism classes. This is mainly due to the lack of our knowledge on the GL(V)-irreducible decomposition of the space S^{p}(∧^{2}V^{∗} ⊗ V), especially due to the lack of decomposition formulas of the “plethysm” {1^{2}} ⊗ {λ} (cf. §2, [22], [4]).

As an application of this paper, we can describe the existence (or non-existence) of left invariant symplectic structures on 4-dimensional complex Lie groups in an intrinsic way.

For details, see [6]. In addition, we can apply the representation theoretic method of this paper to other kind of geometric problems on multi-tensor spaces, such as the Grassmann space ∧^{p}V^{∗} (p ≥ 3), the space of curvature like tensors on V, etc. We will treat these problems in the forthcoming papers.

§1. Normal forms.

In this section, we give a simple classification table of 4-dimensional complex Lie algebras, fitted to describe an algorithm to determine the isomorphism classes and several properties on the varieties of Lie algebras, such as degenerations, orbit decompositions, etc.

There are already several classifications of 4-dimensional Lie algebras. But unfortunately, it seems that all of them are not fitted to describe the above subjects. In fact, for these classifications, a family of normal forms which contains a parameter often degenerates to singular Lie algebras at some special points. (For example, see Remark (6) below.) And this phenomenon makes a description of the above subjects quite complicated. Here, based on previously known classifications, we give a new classification table of 4-dimensional complex Lie algebras fitted for our purposes. Explicit relations to other normal forms are given in Appendix A.

Proposition 1. Any 4-dimensional complex Lie algebra is isomorphic to one of the following Lie algebras in Table 1 (α and β represent complex parameters):

Table 1

non-trivial bracket operations

L_{0}

L_{1}  [X_{1}, X_{2}] = X_{3}
L_{2}  [X_{1}, X_{2}] = X_{3}, [X_{1}, X_{3}] = X_{4}
L_{3}  [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = X_{3}, [X_{1}, X_{4}] = X_{4}
L_{4}(α)  [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = X_{3}, [X_{1}, X_{4}] = X_{3} + αX_{4}
L_{4}(∞) [X_{1}, X_{2}] = X_{2}
L_{5}  [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = X_{3}, [X_{1}, X_{4}] = 2X_{4}, [X_{2}, X_{3}] = X_{4}
L_{6}  [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = −X_{3}, [X_{2}, X_{3}] = X_{1}
L_{7}(α, β) [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = X_{2} + αX_{3}, [X_{1}, X_{4}] = X_{3} + βX_{4}
L_{8}(α)  [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = X_{2} + αX_{3}, [X_{1}, X_{4}] = (α + 1)X_{4}, [X_{2}, X_{3}] = X_{4}
L_{9}  [X_{1}, X_{2}] = X_{2}, [X_{3}, X_{4}] = X_{4}

These Lie algebras are not isomorphic to each other except for the following cases :

• L_{7}(α, β) ≅ L_{7}(α′, β′) if and only if two ratios 1 : α : β and 1 : α′ : β′ coincide after a suitable change of ordering.

• L_{8}(α) ≅ L_{8}(α′) if and only if α = α′ or αα′ = 1.

(As for the Lie algebra L_{4}(α) (α ∈ C ∪ {∞}), L_{4}(α) is isomorphic to L_{4}(α′) if and only if α = α′.)

Outline of the proof. We can show that the above table exhausts all 4-dimensional complex Lie algebras by constructing the isomorphisms to other known normal forms. (For details, see Appendix A.) By calculating the dimensions of [L_{i}, L_{i}] and the GL(V)- orbit of L_{i}, we have immediately L_{i} ≇ L_{j} for i ≠ j. (See Table 2 below.) The remaining special isomorphisms for L_{4}, L_{7} and L_{8} can be checked directly. q.e.d.

We must explain the typical features of these normal forms. But in advance, we state one notational remark.

A Lie algebra is by definition a pair (V, [ , ]), where V is a vector space and [ , ] is an element of ∧^{2}V^{∗} ⊗ V satisfying the Jacobi identity. In this paper, we often express the vector space V as 𝔤 if a Lie algebra structure [ , ] on V is explicitly or implicitly given. But sometimes, in case the underlying vector space V is fixed, we often identify the Lie bracket [ , ] with 𝔤, and we consider 𝔤 as an element of ∧^{2}V^{∗} ⊗ V. This is clearly an abuse of notation. But, the author believes that the readers can correctly understand the situation without any confusion. (In this paper, V always means C^{4}, except Appendix B.)

Remark. (1) Among 4-dimensional Lie algebras, the following ones are expressed as sums of lower dimensional Lie algebras:

L_{0} ≅ C^{4},  L_{1} ≅ M_{1} ⊕ C,
L_{4}(0) ≅ M_{2} ⊕ C,  L_{4}(∞) ≅ M_{3}(0) ⊕ C ≅ 𝔞𝔣𝔣(1, C) ⊕ C^{2},
L_{6} ≅ 𝔤𝔩(2, C) ≅ 𝔰𝔩(2, C) ⊕ C, L_{7}(α, 0) ≅ M_{3}(α) ⊕ C, (α ≠ 0),
L_{9} ≅ 𝔞𝔣𝔣(1, C) ⊕ 𝔞𝔣𝔣(1, C).

Here, the Lie algebra 𝔞𝔣𝔣(1, C) means the non-abelian 2-dimensional Lie algebra. For the definition of the 3-dimensional Lie algebras M_{i} appeared above, see Table 5 in Appendix B. It should be remarked that the set of decomposable Lie algebras does not form a “closed” subset of ∧^{2}V^{∗} ⊗V in the usual topology because the limitting Lie algebra lim_{α→0} L_{7}(α, 0) = L_{7}(0, 0) is not decomposable. See also Remark (1) after Proposition 6 in §5.

(2) Nilpotent Lie algebras are exhausted by L_{0}, L_{1}, L_{2}. All 4-dimensional Lie algebras are solvable except for the unimodular Lie algebra L_{6} ≅ 𝔤𝔩(2, C). As we see later, four Lie algebras L_{6} ∼ L_{9} constitute the “principal part” of the set of 4-dimensional Lie algebras (cf. Proposition 8), and the remaining Lie algebras L_{3}, L_{4}, L_{5} are intermediate degenerate Lie algebras.

(3) For the Lie algebra L_{7}(α, β), we often say that two “unordered ratios” 1 : α : β and 1 : α′ : β′ coincide if these ratios coincide after a suitable change of ordering. In this case, two Lie algebras L_{7}(α, β) and L_{7}(α′, β′) are isomorphic, as stated in Proposition 1. For example, it is easy to see that L_{7}(α, β) with α + β = 1 is isomorphic to L_{7}(γ, γ + 1) for some γ ∈ C.

(4) It is convenient to use the symbolical notation L_{7}(∞, 1) = L_{7}(0, 0) in considering the degeneration of Lie algebras (cf. §5, Figure 1). In fact, from the above remark, we have L_{7}(α, 1) ≅ L_{7}(\frac{1}{α}, \frac{1}{α}) for α ≠ 0, and hence lim_{α→∞} L_{7}(α, 1) ≅ L_{7}(0, 0). In the similar way, we may consider L_{7}(∞, ∞) = L_{7}(1, 0) and L_{7}(∞, −∞) = L_{7}(−1, 0).

(5) We may say that the Lie algebra L_{4}(α) converges to L_{4}(∞) as α → ∞ in spite of its appearance. To check this fact, we consider the family of Lie algebras L′_{4}(k, l) ((k, l) ≠ (0, 0)) defined by

[Y_{1}, Y_{2}] = kY_{2}, [Y_{1}, Y_{3}] = kY_{3}, [Y_{1}, Y_{4}] = kY_{3} + lY_{4}.

Then, we have

L′_{4}(k, l) ≅ \{ \begin{matrix}L_{4}(∞) & k = 0, \\ L_{4}(\frac{l}{k}) & k ≠ 0. \end{matrix}

In particular, L′_{4}(k, l) ≅ L′_{4}(k′, l′) if and only if (k′, l′) = (ck, cl) for some c ≠ 0, and we see that the parameter space of L_{4}(α) can be naturally identified with the 1-dimensional complex projective space P^{1}(C). From these facts, we have lim_{α→∞} L_{4}(α) ≅ lim_{α→∞} L′_{4}(1, α) ≅ lim_{α→∞} L′_{4}(\frac{1}{α}, 1) ≅ L′_{4}(0, 1) ≅ L_{4}(∞).

(6) For most classifications, the Lie algebra L′_{7}(α, β) defined by

[Y_{1}, Y_{2}] = Y_{2}, [Y_{1}, Y_{3}] = αY_{3}, [Y_{1}, Y_{4}] = βY_{4}.

is adopted as one normal form. Clearly, the bracket operation of L′_{7}(α, β) is simpler than that of L_{7}(α, β), and this Lie algebra is isomorphic to L_{7}(α, β) if α ≠ β, α ≠ 1 and β ≠ 1. But, for the remaining cases, L′_{7}(α, β) is isomorphic to other Lie algebras:

L′_{7}(α, β) ≅ \{ \begin{matrix}L_{4}(∞) & α = β = 0, \\ L_{3} & α = β = 1, \\ L_{4}(\frac{1}{α}) & α = β ≠ 0, 1, \\ L_{4}(β) & α = 1, β ≠ 1, \\ L_{4}(α) & β = 1, α ≠ 1. \end{matrix}

For most classifications, the Lie algebras L_{7}(α, α) (≅ L_{7}(1, \frac{1}{α})) and L_{7}(1, 0), not appearing in this family {L′_{7}(α, β)} are treated as other separate normal forms. But actually, by calculating the dimension of GL(V)-orbits of these Lie algebras, we know that the above Lie algebras L_{3} and L_{4}(α) are singular. (For example, the Lie algebra L_{3} is a degeneration of L_{7}(1, 1). See Table 2 and Figure 1 in §5.) On the contrary, the dimension of the GL(V)-orbit of L_{7}(α, β) is constant for any α, β. And hence, L_{7}(α, α) and L_{7}(1, 0) should be included in the continuous family of Lie algebras, instead of L_{3} and L_{4}(α). Hence, to describe deformations and degenerations, the family of Lie algebras L_{7}(α, β) are nicely fitted as normal forms than L′_{7}(α, β). (Symbolically, in the language of matrices, we may say that L′_{7}(α, β) corresponds to a “diagonal” matrix and L_{7}(α, β) corresponds to a matrix with a non-trivial “Jordan block”. Clearly, the latter one is generic in the set of matrices with multiple eigenvalues.)

As for the Lie algebra L_{8}(α), it is isomorphic to the Lie algebra L′_{8}(α) defined by

[Y_{1}, Y_{2}] = Y_{2}, [Y_{1}, Y_{3}] = αY_{3}, [Y_{1}, Y_{4}] = (α + 1)Y_{4}, [Y_{2}, Y_{3}] = Y_{4},

if α ≠ 1. And the bracket operation of this Lie algebra is simpler than that of L_{8}(α). But L′_{8}(α) degenerates to L_{5} in the case α = 1. A similar phenomenon occurs for the Lie algebra L_{4}(α). The Lie algebra defined by

[Y_{1}, Y_{2}] = Y_{2}, [Y_{1}, Y_{3}] = Y_{3}, [Y_{1}, Y_{4}] = αY_{4}

is isomorphic to L_{4}(α) if α ≠ 1, and to L_{3} if α = 1. By the same reason as above, it is better to adopt the Lie algebras L_{8}(α) and L_{4}(α) as normal forms.

Next, as one peculiar feature of 4-dimensional Lie algebras, we consider the ratio of the eigenvalues of ad X (X ∈ 𝔤). The following proposition is quite important, especially in defining the invariants of 4-dimensional Lie algebras in §4.

Proposition 2. Assume 𝔤 is not isomorphic to L_{9}. Then, the ratio of the eigenvalues of ad X does not depend on the choice of X if X is sufficiently generic.

Proof. We can easily show this fact by using the classification table (Table 1). For example, for the Lie algebra L_{7}(α, β), we have

ad (aX_{1} + bX_{2} + cX_{3} + dX_{4}) = ( \begin{matrix}0 & 0 & 0 & 0 \\ −b − c & a & a & 0 \\ −cα − d & 0 & aα & a \\ −dβ & 0 & 0 & aβ \end{matrix} ) ,

and the eigenvalues of this matrix are {0, a, aα, aβ}. Hence, if a ≠ 0, the ratio of the eigenvalues of ad (aX_{1} + bX_{2} + cX_{3} + dX_{4}) is always equal to 0 : 1 : α : β. We can easily calculate the ratio for the remaining Lie algebras. q.e.d.

Remark. (1) For the Lie algebra L_{9}, the eigenvalues of ad (aX_{1} + bX_{2} + cX_{3} + dX_{4}) are given by {0, 0, a, c}, and this ratio essentially depends on X. Among 4-dimensional comlex Lie algebras, L_{9} is uniquely characterized by this property. We also remark that the ratios for real solvable 4-dimensional Lie algebras are listed up in [34; p.180 ∼ 181].

(2) If possible, it is desirable to prove Proposition 2 without the help of the classification. But unfortunately, we do not know such a proof at present.

In Table 2, we summarize fundamental quantities of L_{i}, including the ratios of the eigenvalues of adX. We remark that our normal forms are selected such that the dimension of the GL(V)-orbit of L_{i} does not depend on the parameters (α and β), as we stated above. Perhaps, this is the most important feature of our normal forms. But instead, the dimensions of 𝔤^{(1)} and 𝔤^{(2)} may vary for singular α and β, where 𝔤^{(1)} = [𝔤, 𝔤], 𝔤^{(2)} = [[𝔤, 𝔤], [𝔤, 𝔤]]. In the following, we denote by 𝒪(𝔤) the GL(V)-orbit of 𝔤 in ∧^{2}V^{∗} ⊗ V. Note that the dimension of 𝒪(𝔤) can be calculated by the formula:

dim 𝒪(𝔤) = dim 𝔤^{∗} ⊗ 𝔤 − dim Der(𝔤),

where Der(𝔤) is the space of derivations of 𝔤, i.e.,

Table 2

dim 𝔤^{(1)}  dim 𝔤^{(2)} dim 𝒪(𝔤)  ratio
L_{0}  0  0  0  0 : 0 : 0 : 0
L_{1}  1  0  6  0 : 0 : 0 : 0
L_{2}  2  0  9  0 : 0 : 0 : 0
L_{3}  3  0  4  0 : 1 : 1 : 1
L_{4}(α)  \{ \begin{matrix}1 & α = ∞ \\ 2 & α = 0 \\ 3 & α ≠ 0, ∞ \end{matrix}  0  8  0 : 1 : 1 : α (∗)_{1}
L_{5}  3  1  9  0 : 1 : 1 : 2
L_{6}  3  3  12  0 : 0 : 1 : −1
L_{7}(α, β) \{ \begin{matrix}2 & α = 0 or β = 0 \\ 3 & α, β ≠ 0 \end{matrix}  0  10  0 : 1 : α : β
L_{8}(α)  \{ \begin{matrix}2 & α = 0 \\ 3 & α ≠ 0 \end{matrix}  \{ \begin{matrix}0 & α = 0 \\ 1 & α ≠ 0 \end{matrix}  11  0 : 1 : α : α + 1
L_{9}  2  0  12  (∗)_{2}

𝔤^{(1)} = [𝔤, 𝔤], 𝔤^{(2)} = [𝔤^{(1)}, 𝔤^{(1)}]. (∗)_{1}: We consider 0 : 1 : 1 : ∞ = 0 : 0 : 0 : 1. (∗)_{2}: Two eigenvalues are 0. But the remaining two eigenvalues essentially depend on the choice of X ∈ L_{9}.

Der(𝔤) = { A ∈ 𝔤^{∗} ⊗ 𝔤 | A[X, Y] = [AX, Y] + [X, AY], ^{∀}X, Y ∈ 𝔤 }.

Theoretically, the value dim 𝒪(𝔤) may serve as one measure to determine the isomorphism class of 𝔤. But for general (un-normalized) 𝔤, the determination of dim𝒪(𝔤) requires many calculations, and we do not use this value as our device.

It should be noted that the dimensions of GL(V)-orbits are not stable by summations of Lie algebras. For example, the dimension of the GL(V)-orbit of the 3-dimensional complex Lie algebra M_{3}(α) is always 5 for α ∈ C. (See Table 5 in Appendix B.) But curiously, by adding a 1-dimensional abelian center, a singular parameter α appears. In fact, we have

dim 𝒪(M_{3}(α) ⊕ C) = \{ \begin{matrix}dim 𝒪(L_{4}(∞)) = 8 & α = 0, \\ dim 𝒪(L_{7}(α, 0)) = 10 & α ≠ 0. \end{matrix}

§2. Polynomial ring of ∧^{2}V^{∗} ⊗ V.

In order to give an algorithm to determine the isomorphism classes of Lie algebras, we need several intrinsic concepts of 𝔤, by which we can distinguish non-isomorphic Lie algebras. These intrinsic concepts which we use in this paper are all characterized by the vanishing of GL(V)-invariant sets of polynomials of structure constants up to degree three. For example, by fixing a basis {X_{1}, ⋯, X_{4}} of 𝔤 and by putting [X_{i}, X_{j}] = Σc_{ij}^{k}X_{k}, the unimodularity of 𝔤 is characterized by the vanishing of four linear polynomials Σ_{k} c_{ik}^{k} = 0 (i = 1 ∼ 4), as stated in Introduction. These four polynomials {Σ_{k} c_{ik}^{k}} constitute a GL(V)-invariant irreducible subspace of (∧^{2}V^{∗} ⊗ V)^{∗}. Other intrinsic concepts of 𝔤 which we use in this paper are also characterized by the vanishing of polynomials of some GL(V)-irreducible components of the polynomial ring Σ_{p} S^{p}(∧^{2}V^{∗} ⊗ V)^{∗}.

Hence, to explain the algorithm to determine the isomorphism classes, we must in advance describe the structure of the polynomial ring Σ_{p} S^{p}(∧^{2}V^{∗} ⊗ V)^{∗} from the GL(V)- representation theoretic viewpoint, though it is in general a hard problem.

In this section, we give the explicit GL(V)-irreducible decomposition of the space S^{p}(∧^{2}V^{∗} ⊗ V)^{∗} for p = 1 ∼ 3, and calculate their generators. In addition, we evaluate these generators for each normal form in Table 1. By this method, we can obtain several nice devices to determine the isomorphism classes of 𝔤. Intrinsic concepts defined by the vanishing of these GL(V)-invariant sets of polynomials are explained in detail in the next section.

We will calculate the generator of each GL(V)-irreducible component of S^{p}(∧^{2}V^{∗}⊗V)^{∗} by the method stated in [1; p.115 ∼ 116]. For this purpose, we fix a volume form Ω of V once for all, and rewrite the space as follows. Since V is 4-dimensional, we can naturally identify two spaces ∧^{2}V^{∗} and ∧^{2}V by using the volume form Ω. Hence S^{p}(∧^{2}V^{∗} ⊗ V)^{∗} is isomorphic to S^{p}(∧^{2}V ⊗V)^{∗}. As GL(V)-modules, irreducible components of S^{p}(∧^{2}V^{∗}⊗V)^{∗} and the corresponding components in S^{p}(∧^{2}V ⊗ V)^{∗} are isomorphic to each other by multiplying some powers of det g (g ∈ GL(V)). For our purpose, we only use the concepts determined by the vanishing of polynomials, or the concepts determined by the ratio of polynomials with the same degree. Hence, our arguments do not depend on the choice of the volume form of V.

Now, we give the explicit GL(V)-irreducible decomposition of the space of polynomials on ∧^{2}V ⊗ V up to degree three. We express the GL(V)-irreducible representation space corresponding to the partition λ = (λ_{1}, ⋯, λ_{4}) by the symbol S_{λ} = S_{λ}(V^{∗}). (For the representation theory of the group GL(V), see [22], [1], etc.) For example, the symbol S_{2} expresses the space of symmetric 2-forms S^{2}V^{∗}. Strictly speaking, this space S^{2}V^{∗} should be expressed as S_{0,0,0,−2}. But, for simplicity, we use the above dual notation throughout this paper.

Then, in the case of degree = 1, by using Littlewood-Richardson’s rule, we obtain the irreducible decomposition immediately: (∧^{2}V ⊗ V)^{∗} = S_{11} ⊗ S_{1} = S_{21} + S_{111}. For higher degree cases, we use the formula S^{p}(∧^{2}V ⊗ V)^{∗} ≅ Σ_{λ} S_{λ}(∧^{2}V^{∗}) ⊗ S_{λ}(V^{∗}), where λ runs all over the partitions such that |λ| = p and depth ≤ 4. The decomposition of the plethysm S_{λ}(∧^{2}V^{∗}) (= {1^{2}} ⊗ {λ}, in the classical notation) for small |λ| is given for example in [3], [8]. (Or one can calculate it by using the software “SYMMETRICA”:

http://www.mathe2.uni-bayreuth.de.) As a result, we have:

Proposition 3. GL(V)-irreducible decompositions of S^{p}(∧^{2}V ⊗ V)^{∗} for p = 1 ∼ 3 are given by

p = 1 : S_{21} + S_{111},

p = 2 : S_{42} + 2S_{321} + 2S_{3111} + 2S_{222} + S_{2211}, p = 3 : S_{63} + 2S_{531} + S_{522} + 2S_{5211} + S_{441} + 3S_{432} + 3S_{4311} + 5S_{4221} + 3S_{333} + 4S_{3321}

+ 3S_{3222}.

Remind that the partitons λ with depth > 4 do not appear in the above decomposition because dim V = 4. The coefficient of S_{λ} implies its multiplicity. In the following, we often call the irreducible component S_{λ} in S^{p}(∧^{2}V ⊗ V)^{∗} covariant.

Remark. Theoretically, we can continue to decompose the space S^{p}(∧^{2}V ⊗V)^{∗} for large p. But unfortunately, closed decomposition formulas of S^{p}(∧^{2}V ⊗ V)^{∗} for general p are not known yet even in the case dim V = 4. We often encounter this type of difficulty in considering multi-tensor spaces, and we must find some new method to understand these spaces (cf. [1], [2], [13], [32]).

Next, of all irreducible components of S^{p}(∧^{2}V ⊗V)^{∗} (p ≤ 3), we give here generators p_{λ} of S_{λ} for nine components. We can determine the isomorphism classes of 4-dimensional Lie algebras in terms of these nine irreducible components. (We omit to give the generators of the remaining irreducible components because we do not use them in this paper.) As a result, for the components with multiplicity > 1, at most one component is selected here, and we express it simply as S_{λ} because there is no danger of confusion. By fixing a basis {X_{1}, ⋯, X_{4}} of V, we have the following list:

• p = 1:

p_{21} = c_{34}^{1} ∈ S_{21},

p_{111} = c_{14}^{1} + c_{24}^{2} + c_{34}^{3} = −Tr ad X_{4} ∈ S_{111},

• p = 2:

p_{321} = c_{24}^{1}c_{34}^{2} − c_{34}^{1}c_{24}^{2} ∈ S_{321},

p_{3111} = c_{12}^{1}c_{34}^{1} − c_{13}^{1}c_{24}^{1} + c_{14}^{1}c_{23}^{1} ∈ S_{3111},

p_{222} = c_{14}^{1}c_{24}^{2} + c_{14}^{1}c_{34}^{3} − c_{24}^{1}c_{14}^{2} − c_{34}^{1}c_{14}^{3} + c_{24}^{2}c_{34}^{3} − c_{34}^{2}c_{24}^{3} = 1/2 · {(Tr ad X_{4})^{2} − Tr (ad X_{4})^{2}} ∈ S_{222},

• p = 3:

p_{441} = c_{14}^{1}c_{34}^{1}c_{34}^{2} + c_{24}^{1}(c_{34}^{2})^{2} − (c_{34}^{1})^{2}c_{14}^{2} − c_{34}^{1}c_{24}^{2}c_{34}^{2} ∈ S_{441},

p_{4221} = | \begin{matrix}c_{23}^{1} & c_{23}^{2} & c_{23}^{3} \\ c_{24}^{1} & c_{24}^{2} & c_{24}^{3} \\ c_{34}^{1} & c_{34}^{2} & c_{34}^{3} \end{matrix} | ∈ S_{4221},

p_{333} = | \begin{matrix}c_{14}^{1} & c_{14}^{2} & c_{14}^{3} \\ c_{24}^{1} & c_{24}^{2} & c_{24}^{3} \\ c_{34}^{1} & c_{34}^{2} & c_{34}^{3} \end{matrix} | = −1/6 · \{(Tr ad X_{4})^{3} − 3 Tr adX_{4} · Tr (ad X_{4})^{2} + 2 Tr (ad X_{4})^{3}\}

∈ S_{333},

p_{3222} = | \begin{matrix}c_{12}^{1} & c_{12}^{2} & c_{12}^{3} \\ c_{23}^{1} & c_{23}^{2} & c_{23}^{3} \\ c_{34}^{1} & c_{34}^{2} & c_{34}^{3} \end{matrix} | + | \begin{matrix}c_{12}^{1} & c_{12}^{2} & c_{12}^{4} \\ c_{24}^{1} & c_{24}^{2} & c_{24}^{4} \\ c_{34}^{1} & c_{34}^{2} & c_{34}^{4} \end{matrix} | + | \begin{matrix}c_{13}^{1} & c_{13}^{3} & c_{13}^{4} \\ c_{24}^{1} & c_{24}^{3} & c_{24}^{4} \\ c_{34}^{1} & c_{34}^{3} & c_{34}^{4} \end{matrix} | − | \begin{matrix}c_{13}^{1} & c_{13}^{2} & c_{13}^{3} \\ c_{23}^{1} & c_{23}^{2} & c_{23}^{3} \\ c_{24}^{1} & c_{24}^{2} & c_{24}^{3} \end{matrix} |

− | \begin{matrix}c_{14}^{1} & c_{14}^{2} & c_{14}^{4} \\ c_{23}^{1} & c_{23}^{2} & c_{23}^{4} \\ c_{24}^{1} & c_{24}^{2} & c_{24}^{4} \end{matrix} | − | \begin{matrix}c_{14}^{1} & c_{14}^{3} & c_{14}^{4} \\ c_{23}^{1} & c_{23}^{3} & c_{23}^{4} \\ c_{34}^{1} & c_{34}^{3} & c_{34}^{4} \end{matrix} | + 2 | \begin{matrix}c_{23}^{2} & c_{23}^{3} & c_{23}^{4} \\ c_{24}^{2} & c_{24}^{3} & c_{24}^{4} \\ c_{34}^{2} & c_{34}^{3} & c_{34}^{4} \end{matrix} | ∈ S_{3222}.

These generators can be obtained by applying the method satated in [1; p.115 ∼ 116]. We give one example. We denote by {ω_{1}, ⋯, ω_{4}} the dual basis of {X_{1}, ⋯, X_{4}}. Then, the bracket [ , ] ∈ ∧^{2}V^{∗} ⊗ V is expressed as Σ_{i<j} c_{ij}^{k}ω_{i} ∧ ω_{j} ⊗ X_{k}. In terms of the volume form Ω = ω_{1} ∧ ⋯ ∧ ω_{4}, this bracket is transformed into the element Σ_{i<j} a_{ijk}X_{i} ∧ X_{j} ⊗ X_{k} ∈ ∧^{2}V ⊗ V, where a_{12k} = c_{34}^{k}, a_{13k} = −c_{24}^{k}, a_{14k} = c_{23}^{k}, a_{23k} = c_{14}^{k}, a_{24k} = −c_{13}^{k}, a_{34k} = c_{12}^{k}. Then, the generator of the space S_{321} ⊂ S^{2}(∧^{2}V ⊗ V)^{∗} is given by

∑_{σ∈𝔖_{3},τ∈𝔖_{2}} (−1)^{σ}(−1)^{τ}a_{σ(1)τ(1)σ(2)}a_{1τ(2)σ(3)} = a_{121}a_{132} − a_{122}a_{131}
= c_{24}^{1}c_{34}^{2} − c_{34}^{1}c_{24}^{2}
= p_{321}.

Here, 𝔖_{n} denotes the symmetric group with degree n and (−1)^{σ} denotes the sign of σ ∈ 𝔖_{n}. Other generators can be calculated in the same way. But it requires many hard computations on polynomials, and for most cases, we used computers to calculate p_{λ}. Of course, the polynomial p_{λ} essentially depends on the choice of a basis of V.

Note that the set of sixteen polynomials appearing in the Jacobi identity splits into two irreducible components of S^{2}(∧^{2}V ⊗ V)^{∗}: S_{2211} and one component of 2S_{3111}. (We have dim S_{2211} = 6 and dim S_{3111} = 10.)

Among the above generators, three polynomials p_{111}, p_{222}, p_{333} play an important role, especially in §4 where we define the invariants of 4-dimensional Lie algebras. In addition, the cubic polynomial

φ_{1} = 8p_{333} − 4p_{111}p_{222} + p_{111}^{3} ∈ S_{333}

also plays an important role. (Actually, it gives a part of the defining equations of the variety Σ_{3}. See Proposition 9 in §5.) We express the eigenvalues of ad X (X ∈ 𝔤) as {0, ε_{1}, ε_{2}, ε_{3}}. Then, by putting X_{4} = X in the above list of generators, we have immediately

p_{111} = −(ε_{1} + ε_{2} + ε_{3}),
p_{222} = ε_{1}ε_{2} + ε_{2}ε_{3} + ε_{3}ε_{1},
p_{333} = −ε_{1}ε_{2}ε_{3}.

(In particular, the characteristic polynomial of ad X is expressed as |λI − ad X| = λ(λ^{3} + p_{111}λ^{2} + p_{222}λ + p_{333}).) By substituting these values to the above φ_{1}, we have

φ_{1} = (ε_{1} + ε_{2} − ε_{3})(ε_{1} + ε_{3} − ε_{2})(ε_{2} + ε_{3} − ε_{1}).

Hence, the vanishing of this polynomial also expresses an intrinsic property of 𝔤 (cf. Proposition 4). The following sextic polynomial

φ_{2} = 3(3p_{333} − p_{111}p_{222})^{2} + 4(p_{222}^{3} − p_{111}^{2}p_{222}^{2} + p_{111}^{3}p_{333}) ∈ S_{666} ⊂ S^{6}(∧^{2}V ⊗ V)^{∗}

also possesses an intrinsic meaning. In fact, by substituting p_{111}, p_{222} and p_{333} in the same way as above, we have

φ_{2} = −(ε_{1} − ε_{2})^{2}(ε_{2} − ε_{3})^{2}(ε_{3} − ε_{1})^{2},

which implies that φ_{2} may be considered as a sort of discriminant. (But in this paper, we do not explicitly use this polynomial φ_{2}.)

As stated above, the generator p_{λ} ∈ S_{λ} depends on the choice of a basis {X_{1}, ⋯, X_{4}}. Hence, the vanishing or non-vanishing of p_{λ} has no intrinsic meaning. But the vanishing of all polynomials in S_{λ} possesses an intrinsic meaning of 𝔤. In the following, we express this situation symbolically as “S_{λ} = 0”, and often say that the covariant S_{λ} vanishes for 𝔤. Our next task is to clarify the intrinsic property of 𝔤 defined by “S_{λ} = 0”. But, before stating these properties, we give a table summarizing the vanishing or non-vanishing of S_{λ} for each Lie algebra L_{i} in the next page (Table 3). The symbol “0” in Table 3 implies “S_{λ} = 0”, and the symbol “∗” implies that there exists a non-vanishing polynomial in S_{λ}. To check these results, we have only to evaluate the generator p_{λ} of S_{λ} for a generic basis of L_{i}. In actual calculations, we used computers. We use Table 3 frequently in the subsequent sections.

§3. Intrinsic concepts determined by the vanishing of covariants.

In this section, we state several intrinsic concepts of 𝔤 determined by the vanishing of covariants appeared in §2. The results are summarized in the following proposition. Most concepts appearing in this proposition are actually used in the algorithm to determine the isomorphism classes of Lie algebras which we will state in §6. Here, the symbol ⟨φ_{i}⟩ = 0 means the vanishing of the irreducible component of S^{p}(∧^{2}V ⊗ V)^{∗} generated by the polynomial φ_{i}, and we denote by {0, ε_{1}, ε_{2}, ε_{3}} the eigenvalues of ad X for generic X ∈ 𝔤, as before.

Proposition 4. (1) S_{21} = 0 if and only if there exists an element f ∈ 𝔤^{∗} such that [X, Y] = f(X)Y − f(Y)X.

(2) S_{111} = 0 if and only if 𝔤 is unimodular.

(3) S_{3111} = 0 if and only if dα ∧ dα = 0 for any α ∈ 𝔤^{∗}. (The condition dα ∧ dα = 0 is equivalent to the decomposability of dα for the 4-dimensional case).

(4) S_{222} = 0 if and only if ε_{1}ε_{2} + ε_{2}ε_{3} + ε_{3}ε_{1} = 0.

(5) S_{441} = 0 if and only if dim ⟨X, Y, [X, Y], [X, [X, Y]]⟩ ≤ 3 for any X, Y ∈ 𝔤.

(6) S_{4221} = 0 if and only if dim ⟨[X, Y], [Y, Z], [Z, X]⟩ ≤ 2 for any X, Y, Z ∈ 𝔤. This condition is also equivalent to [[𝔤, 𝔤], [𝔤, 𝔤]] = 0.

(7) S_{333} = 0 if and only if some ε_{i} = 0. This condition is also equivalent to rank ad X ≤ 2 for any X ∈ 𝔤.

4-dimensional Lie algebras 13

Table 3

S_{21}  S_{111}  S_{321}  S_{3111}  S_{222}
L_{0}  0  0  0  0  0
L_{1}  ∗  0  0  0  0
L_{2}  ∗  0  ∗  0  0
L_{3}  0  ∗  0  0  ∗
L_{4}(α)  ∗  \{ \begin{matrix}0 & α = −2 \\ ∗ & α ≠ −2 \end{matrix}  \{ \begin{matrix}0 & α = ∞ \\ ∗ & α ≠ ∞ \end{matrix}  0  \{ \begin{matrix}0 & α = −1/2, ∞ \\ ∗ & α ≠ −1/2, ∞ \end{matrix}
L_{5}  ∗  ∗  ∗  ∗  ∗
L_{6}  ∗  0  ∗  0  ∗
L_{7}(α, β) ∗ \{ \begin{matrix}0 & α + β = −1 \\ ∗ & α + β ≠ −1 \end{matrix}  ∗  0  \{ \begin{matrix}0 & αβ + α + β = 0 \\ ∗ & αβ + α + β ≠ 0 \end{matrix}
L_{8}(α)  ∗  \{ \begin{matrix}0 & α = −1 \\ ∗ & α ≠ −1 \end{matrix}  ∗  \{ \begin{matrix}0 & α = −1 \\ ∗ & α ≠ −1 \end{matrix} \{ \begin{matrix}0 & α^{2} + 3α + 1 = 0 \\ ∗ & α^{2} + 3α + 1 ≠ 0 \end{matrix}
L_{9}  ∗  ∗  ∗  ∗  ∗
S_{441}  S_{4221}  S_{333}  S_{3222}  φ_{1}  φ_{2}
L_{0}  0  0  0  0  0  0
L_{1}  0  0  0  0  0  0
L_{2}  ∗  0  0  0  0  0
L_{3}  0  0  ∗  0  ∗  0
L_{4}(α)  0  0  \{ \begin{matrix}0 & α = 0, ∞ \\ ∗ & α ≠ 0, ∞ \end{matrix}  0 \{ \begin{matrix}0 & α = 0, 2 \\ ∗ & α ≠ 0, 2 \end{matrix}  0
L_{5}  0  ∗  ∗  0  0  0
L_{6}  ∗  ∗  0  ∗  0  ∗
L_{7}(α, β) ∗  0  \{ \begin{matrix}0 & α or β = 0 \\ ∗ & α, β ≠ 0 \end{matrix}  0  (∗)_{1}  (∗)_{2}
L_{8}(α)  ∗ \{ \begin{matrix}0 & α = 0 \\ ∗ & α ≠ 0 \end{matrix} \{ \begin{matrix}0 & α = 0, −1 \\ ∗ & α ≠ 0, −1 \end{matrix}  0  0  \{ \begin{matrix}0 & α = 0, 1 \\ ∗ & α ≠ 0, 1 \end{matrix}
L_{9}  ∗  0  0  0  ∗  ∗

(∗)_{1} : \{ \begin{matrix}0 & |α − β| = 1 or α + β = 1, \\ ∗ & |α − β| ≠ 1, α + β ≠ 1, \end{matrix}

(∗)_{2} : \{ \begin{matrix}0 & α = β or α = 1 or β = 1, \\ ∗ & α ≠ β, α ≠ 1, β ≠ 1. \end{matrix}

(8) S_{3222} = 0 if and only if dim [[𝔤, 𝔤], [𝔤, 𝔤]] ≤ 1 (or equivalently ≤ 2). This condition is also equivalent to the solvability of 𝔤.

(9) S_{21} = S_{111} = 0 if and only if 𝔤 is abelian.

(10) S_{111} = S_{222} = S_{333} = 0 if and only if 𝔤 is nilpotent.

(11) S_{321} = S_{222} = 0 if and only if dim [𝔤, 𝔤] ≤ 1. This condition is also equivalent to rank ad X ≤ 1 for any X ∈ 𝔤.

(12) S_{4221} = S_{333} = 0 if and only if dim [𝔤, 𝔤] ≤ 2.

(13) ⟨φ_{1}⟩ = 0 if and only if ε_{i} = ε_{j} + ε_{k} for some distinct i, j, k.

(14) ⟨φ_{2}⟩ = 0 if and only if ε_{i} = ε_{j} for some i ≠ j.

Proof. (1), (2), (9). We put f(X) = \frac{1}{3} TradX, and define a new bracket operation [ , ]′ by [X, Y]′ = [X, Y]−f(X)Y +f(Y)X. Then, we have [X, Y] = [X, Y]′+{f(X)Y −f(Y)X}, and this gives the GL(V)-irreducible decomposition of the space ∧^{2}V^{∗} ⊗ V. In fact, we already know that the space ∧^{2}V^{∗} ⊗V splits into two irreducible components (Proposition 3). And it is easy to see that the trace of the adjoint map Y ↦ [X, Y]′ is zero for any X. Hence, [ , ]′ gives the traceless part, and f(X)Y − f(Y)X gives the contracted part of [ , ] ∈ ∧^{2}V^{∗} ⊗ V, respectively. By definition, the condition S_{111} = 0 is equivalent to the unimodularity of 𝔤, i.e., the vanishing of f. And hence, the remaining condition S_{21} = 0 is equivalent to [ , ]′ = 0, i.e., [X, Y] is expressed as f(X)Y − f(Y)X. Clearly, combined conditions S_{21} = S_{111} = 0 are equivalent to c_{ij}^{k} = 0, which implies that 𝔤 is abelian.

(3) The condition S_{3111} = 0 is equivalent to the vanishing of ten polynomials c_{12}^{i}c_{34}^{j} − c_{13}^{i}c_{24}^{j} + c_{14}^{i}c_{23}^{j} + c_{34}^{i}c_{12}^{j} − c_{24}^{i}c_{13}^{j} + c_{23}^{i}c_{14}^{j}. Since dα(X_{i}, X_{j}) = −α([X_{i}, X_{j}]) = −Σ c_{ij}^{k}α(X_{k}) for α ∈ 𝔤^{∗} and X_{i} ∈ 𝔤, this condition is equivalent to (dα ∧ dα)(X_{1}, ⋯, X_{4}) = 0.

(4), (7), (13), (14). These are clear from the definition of p_{222}, p_{333}, φ_{1}, φ_{2}. For the second statement in (7), we can directly check that the Lie algebras satisfying the condition rank ad X ≤ 2 for any X ∈ 𝔤 are exhausted by L_{0}, L_{1}, L_{2}, L_{4}(0), L_{4}(∞), L_{6}, L_{7}(α, 0), L_{8}(0), L_{8}(−1), L_{9}, and by using Table 3, we can show that these Lie algebras are just characterized by the condition S_{333} = 0.

(5) We can rewrite the polynomial p_{441} in the form

p_{441} = | \begin{matrix}c_{34}^{1} & Σ_{k} c_{4k}^{1}c_{34}^{k} \\ c_{34}^{2} & Σ_{k} c_{4k}^{2}c_{34}^{k} \end{matrix} | .

Then, it is easy to check that this polynomial is equal to the determinant of the matrix (X_{3}, X_{4}, [X_{4}, X_{3}], [X_{4}, [X_{4}, X_{3}]]) in case X_{1} ∼ X_{4} are linearly independent. Hence, the condition S_{441} = 0 is equivalent to dim ⟨X, Y, [X, Y], [X, [X, Y]]⟩ ≤ 3 for any X, Y ∈ 𝔤.

(6) The polynomial p_{4221} is equal to the principal minor of the (4, 3)-matrix ([X_{2}, X_{3}], [X_{2}, X_{4}], [X_{3}, X_{4}]). Hence, the condition S_{4221} = 0 is equivalent to dim ⟨[X, Y], [Y, Z], [Z, X]⟩ ≤ 2 for any X, Y, Z ∈ 𝔤. In view of Table 2 and Table 3, we can check that this condition is just equivalent to [[𝔤, 𝔤], [𝔤, 𝔤]] = 0.

(8), (10), (11), (12). These statements can be directly verified by using Table 1 ∼ Table 3. q.e.d.

Remark. (1) In the above proof, we often used the classification table. But, it is desirable to prove this proposition in an intrinsic way without the help of the classification.

(2) Among the above conditions, the condition S_{3111} = 0 in (3) plays a crucial role in considering the existence or non-existence of left invariant symplectic strucutres on 4-dimensional complex Lie groups. For details, see [6].

(3) Unfortunately, we do not know what does the single condition S_{321} = 0 geometrically means.

(4) As we stated before, 4-dimensional Lie algebras are unimodular or solvable. The same fact also holds for 3-dimensional Lie algebras. In the 3-dimensional case, all Lie algebras satisfy the so-called “fundamental identity”: 𝔖_{X,Y,Z} (Tr ad X) · [Y, Z] = 0, where 𝔖 implies the cyclic sum. And by using this identity, we can show the above fact directly without the help of the classification (cf. [5; p.6 ∼ 8]). But, we do not know whether there exists a similar “fundamental identity” for 4-dimensional Lie algebras, by which we can directly prove the above fact. (From the polynomial viewpoint, the Jacobi identity implies S_{111} = 0 or S_{3222} = 0, as a result of the classification.)

§4. Invariants of 4-dimensional Lie algebras.

To distinguish the isomorphism classes of Lie algebras, we need more delicate additional devices. For example, as stated in Proposition 1, two Lie algebras L_{4}(α) and L_{4}(α′) are isomorphic if and only if α = α′. Hence, we must extract the value α from the Lie algebra structure of L_{4}(α) in order to determine the isomorphism class. For this purpose, we introduce three fundamental invariants of 4-dimensional Lie algebras taking values in C ∪ {∞} as follows:

χ_{1}(𝔤) = \frac{p_{222}}{p_{111}^{2}} , χ_{2}(𝔤) = \frac{p_{333}}{p_{111}^{3}} , χ_{3}(𝔤) = \frac{p_{333}^{2}}{p_{222}^{3}} .

Clearly, these invariants satisfy the relation χ_{1}(𝔤)^{3}χ_{3}(𝔤) = χ_{2}(𝔤)^{2}. We express the eigenvalues of adX (X ∈ 𝔤) as {0, ε_{1}, ε_{2}, ε_{3}} as before. Then, by substituting p_{111} = −(ε_{1}+ε_{2}+ε_{3}), p_{222} = ε_{1}ε_{2} + ε_{2}ε_{3} + ε_{3}ε_{1}, p_{333} = −ε_{1}ε_{2}ε_{3} into the above, we know that χ_{i}(𝔤) are essentially determined by the ratio of ε_{i}, which indicates that χ_{i}(𝔤) are the intrinsic invariants of 𝔤. But, there are some exceptional cases. For the nilpotent Lie algebras L_{0}, L_{1} and L_{2}, the invariants χ_{i}(𝔤) are undetermined because p_{111} = p_{222} = p_{333} = 0 for these cases. In addition, the following special cases are also undetermined because both denominators and numerators simultaneously vanish.

\{ \begin{matrix}χ_{1}(𝔤) & : & L_{7}(α, β) (α + β = −1, αβ = 1), \\ χ_{2}(𝔤) & : & L_{6}, L_{7}(−1, 0), L_{8}(−1), \\ χ_{3}(𝔤) & : & L_{4}(∞), L_{7}(0, 0). \end{matrix}

As for the Lie algebra L_{9}, the ratio of ε_{i} has not an intrinsic meaning, and hence χ_{1}(𝔤) is also undetermined. But, since one ε_{i} is always zero, we have p_{333} = 0. Hence, we put χ_{2}(𝔤) = χ_{3}(𝔤) = 0 for this Lie algebra.

We can easily calculate the explicit values of χ_{i}(L_{j}) in view of Table 2. But the calculation of χ_{i}(𝔤) for a general (un-normalized) Lie algebra 𝔤 requires not a little computations on matrices. Namely, we first calculate the values Tr (ad X)^{k} (k = 1 ∼ 3) for generic X ∈ 𝔤. Then, by the definition of p_{kkk}, we have

p_{111} = −Tr ad X,
p_{222} = \frac{1}{2}\{(Tr ad X)^{2} − Tr (ad X)^{2}\},
p_{333} = −\frac{1}{6}\{(Tr ad X)^{3} − 3 Tr ad X · Tr (ad X)^{2} + 2 Tr (ad X)^{3}\},

and theoretically, χ_{i}(𝔤) are determined from these values. (See the examples at the end of §6.)

In general, the value χ_{3}(𝔤) is automatically determined by χ_{1}(𝔤) and χ_{2}(𝔤). This third invariant χ_{3}(𝔤) is mainly used in case S_{111} = 0, i.e., for unimodular Lie algebras. (See Proposition 5 (3).)

We summarize the explicit values of χ_{i}(L_{j}) in Table 4. The symbol “ − ” in this table implies that it is undetermined.

Table 4

χ_{1}(𝔤)  χ_{2}(𝔤)  χ_{3}(𝔤)
L_{3}  \frac{1}{3}  \frac{1}{27}  \frac{1}{27}
L_{4}(α)  \frac{2α+1}{(α+2)^{2}}  \frac{α}{(α+2)^{3}}  \{ \begin{matrix}− & α = ∞ \\ \frac{α^{2}}{(2α+1)^{3}} & α ≠ ∞ \end{matrix}
L_{5}  \frac{5}{16}  \frac{1}{32}  \frac{4}{125}
L_{6}  ∞  −  0
L_{7}(α, β) \frac{αβ+α+β}{(α+β+1)^{2}} (∗)_{1}  \frac{αβ}{(α+β+1)^{3}} (∗)_{2}  \frac{α^{2}β^{2}}{(αβ+α+β)^{3}} (∗)_{3}
L_{8}(α)  \frac{α^{2}+3α+1}{4(α+1)^{2}}  \{ \begin{matrix}− & α = −1 \\ \frac{α}{8(α+1)^{2}} & α ≠ −1 \end{matrix}  \frac{α^{2}(α+1)^{2}}{(α^{2}+3α+1)^{3}}
L_{9}  −  0  0

(∗)_{1} : undetermined in case α + β = −1, αβ = 1. (∗)_{2} : undetermined in case (α, β) = (−1, 0), (0, −1). (∗)_{3} : undetermined in case α = β = 0. If β = −α − 1, we have χ_{3}(𝔤) = −\frac{α^{2}(α+1)^{2}}{(α^{2}+α+1)^{3}} .

In terms of these invariants, the parameters α and β in L_{4}, L_{7}, L_{8} are essentially uniquely determined as follows.

Proposition 5. (1) L_{4}(α) is isomorphic to L_{4}(α′) if and only if χ_{1}(L_{4}(α)) = χ_{1}(L_{4}(α′)) and χ_{2}(L_{4}(α)) = χ_{2}(L_{4}(α′)). In this case, the parameter α is given by

α = \{ \begin{matrix}∞ & χ_{1} = χ_{2} = 0, \\ −2 & χ_{1} = ∞, \\ 1 & χ_{1} = \frac{1}{3}, \\ \frac{2χ_{2}(1−3χ_{1})}{χ_{1}^{2}+3χ_{1}χ_{2}−4χ_{2}} & otherwise. \end{matrix}

For this Lie algebra, χ_{1} and χ_{2} satisfy the relation 3(3χ_{2} − χ_{1})^{2} + 4(χ_{1}^{3} − χ_{1}^{2} + χ_{2}) = 0 in case they have finite values.

(2) L_{7}(α, β) (α + β ≠ −1, α, β ≠ 0) is isomorphic to L_{7}(α′, β′) (α′ + β′ ≠ −1, α′, β′ ≠ 0) if and only if χ_{1}(L_{7}(α, β)) = χ_{1}(L_{7}(α′, β′)) and χ_{2}(L_{7}(α, β)) = χ_{2}(L_{7}(α′, β′)). The parameters α and β are determined from χ_{1}, χ_{2} by two conditions α + β = a − 1 and αβ = χ_{2}a^{3}, where a is a complex number satisfying the condition χ_{2}a^{3} − χ_{1}a^{2} + a − 1 = 0.

(3) L_{7}(α, −(α + 1)) (α ≠ 0, −1) is isomorphic to L_{7}(α′, −(α′ + 1)) (α′ ≠ 0, −1) if and only if χ_{3}(L_{7}(α, −(α + 1))) = χ_{3}(L_{7}(α′, −(α′ + 1))). The parameter α is determined from the equation χ_{3} = −α^{2}(α + 1)^{2}/(α^{2} + α + 1)^{3} if χ_{3} ≠ ∞. In case χ_{3} = ∞, we have α = (−1 ± \sqrt{3}i)/2, both of which define the isomorphic Lie algebras.

(4) L_{7}(α, 0) is isomorphic to L_{7}(α′, 0) if and only if χ_{1}(L_{7}(α, 0)) = χ_{1}(L_{7}(α′, 0)). The parameter α is determined from the equation χ_{1} = α/(α + 1)^{2} if χ_{1} ≠ ∞. In case χ_{1} = ∞, we have α = −1.

(5) L_{8}(α) is isomorphic to L_{8}(α′) if and only if χ_{1}(L_{8}(α)) = χ_{1}(L_{8}(α′)). The parameter α is determined from the equation χ_{1} = (α^{2} + 3α + 1)/(4(α + 1)^{2}) if χ_{1} ≠ ∞. In case χ_{1} = ∞, we have α = −1.

Proof. We prove the “if” part of this proposition. The “only if” part is clear from the definition of χ_{i}(𝔤).

(1) Assume α ≠ 1, −2, ∞. Then, from the definition of χ_{1}(𝔤) and χ_{2}(𝔤), we have

\begin{matrix}2χ_{2}(1 − 3χ_{1}) = \frac{2α(α−1)^{2}}{(α+2)^{5}} , \\ χ_{1}^{2} + 3χ_{1}χ_{2} − 4χ_{2} = \frac{2(α−1)^{2}}{(α+2)^{5}} \end{matrix}

for 𝔤 = L_{4}(α). Hence, the value α is uniquely determined from χ_{1}(𝔤) and χ_{2}(𝔤). For the remaining cases, we can easily check that α is also uniquely determined from the values χ_{1}(𝔤), χ_{2}(𝔤).

(2) Assume χ_{1}(L_{7}(α, β)) = χ_{1}(L_{7}(α′, β′)) and χ_{2}(L_{7}(α, β)) = χ_{2}(L_{7}(α′, β′)). From the condition α, β ≠ 0, we have χ_{2} ≠ 0, and it is easy to see that the solutions of the cubic equation χ_{2}t^{3} − χ_{1}t^{2} + t − 1 = 0 are α + β + 1, (α + β + 1)/α and (α + β + 1)/β if χ_{1} = (αβ + α + β)/(α + β + 1)^{2} and χ_{2} = αβ/(α + β + 1)^{3}. Replacing α, β by α′, β′ and considering the same cubic equation, we know that two sets of solutions

\{α + β + 1, \frac{α + β + 1}{α}, \frac{α + β + 1}{β}\} and \{α′ + β′ + 1, \frac{α′ + β′ + 1}{α′}, \frac{α′ + β′ + 1}{β′}\}

must coincide. There are six combinations of correspondence between these two sets, and by checking them, it follows that (α′, β′) is equal to one of the following:

(α, β), (β, α), (\frac{1}{α}, \frac{β}{α}) , (\frac{β}{α}, \frac{1}{α}) , (\frac{1}{β} , \frac{α}{β}) , (\frac{α}{β}, \frac{1}{β} ) .

For any case, two unordered ratios 1 : α : β and 1 : α′ : β′ coincide, and hence we have L_{7}(α, β) ≅ L_{7}(α′, β′).

Next, we determine the values of parameters α, β from χ_{1} and χ_{2}. We assume that the invariants are expressed as χ_{1} = (α_{0}β_{0} + α_{0} + β_{0})/(α_{0} + β_{0} + 1)^{2}, χ_{2} = α_{0}β_{0}/(α_{0} + β_{0} + 1)^{3} for some α_{0}, β_{0} (α_{0} + β_{0} ≠ −1, α_{0}, β_{0} ≠ 0). And let a be a solution of the cubic equation χ_{2}t^{3}−χ_{1}t^{2}+t−1 = 0. Then, as we see above, a is equal to one of α_{0}+β_{0}+1, (α_{0}+β_{0}+1)/α_{0} and (α_{0} + β_{0} + 1)/β_{0}. First, we take a solution a = α_{0} + β_{0} + 1. Then, the equations α + β = a − 1 and αβ = χ_{2}a^{3} are equivalent to α + β = α_{0} + β_{0} and αβ = α_{0}β_{0}, which implies that (α, β) = (α_{0}, β_{0}) or (β_{0}, α_{0}). If we take a different solution a = (α_{0} + β_{0} + 1)/α_{0}, then we have α + β = a − 1 = (β_{0} + 1)/α_{0} and αβ = χ_{2}a^{3} = β_{0}/α_{0}^{2}. In this case, we have (α, β) = (\frac{β_{0}}{α_{0}} , \frac{1}{α_{0}} ) or (\frac{1}{α_{0}} , \frac{β_{0}}{α_{0}} ), and hence, L_{7}(α, β) ≅ L_{7}(\frac{β_{0}}{α_{0}} , \frac{1}{α_{0}} ) ≅ L_{7}(α_{0}, β_{0}). By using the third solution a = (α_{0} + β_{0} + 1)/β_{0}, we obtain the same conclusion. Hence, we may say that the parameters α and β are essentially determined from χ_{1} and χ_{2}, by the procedure stated in (2).

(3) Assume χ_{3}(L_{7}(α, −(α+1))) = χ_{3}(L_{7}(α′, −(α′+1))) ≠ ∞. Then, we have −\frac{α^{2}(α+1)^{2}}{(α^{2}+α+1)^{3}}

= −\frac{α′^{2}(α′+1)^{2}}{(α′^{2}+α′+1)^{3}} , and by solving this sextic equation, we have

α′ = α, \frac{1}{α}, −(α + 1), \frac{−1}{α + 1}, −\frac{α + 1}{α}, \frac{−α}{α + 1}.

Hence, for any case, the unordered ratios 1 : α : −(α + 1) and 1 : α′ : −(α′ + 1) coincide, which implies L_{7}(α, −(α+1)) ≅ L_{7}(α′, −(α′+1)). If χ_{3}(L_{7}(α, −(α+1))) = ∞, then we have α = (−1±\sqrt{3}i)/2, and this Lie algebras is isomorphic to L_{7}((−1+\sqrt{3}i)/2, (−1−\sqrt{3}i)/2).

(4) Assume χ_{1}(L_{7}(α, 0)) = χ_{1}(L_{7}(α′, 0)) ≠ 0, ∞. Then, from this condition, we have easily α = α′ or αα′ = 1, and hence L_{7}(α, 0) is isomorphic to L_{7}(α′, 0). If χ_{1}(L_{7}(α, 0)) = 0 (resp. ∞), then we have α = 0 (resp. −1), and L_{7}(α, 0) is also uniquely determined.

(5) If χ_{1}(L_{8}(α)) = χ_{1}(L_{8}(α′)) ≠ ∞, then we have α = α′ or αα′ = 1 from this condition, which implies L_{8}(α) ≅ L_{8}(α′). In case χ_{1}(L_{8}(α)) = ∞, we have α = −1, and L_{8}(α) is also uniquely determined. q.e.d.

Remark. The invariant χ_{3}(𝔤) for the unimodular Lie algebra L_{7}(α, −(α+1)) resembles the j-invariant of the elliptic curve y^{2} = x(x+1)(x−α), where j = 2^{8} \frac{(α^{2}+α+1)^{3}}{α^{2}(α+1)^{2}} (cf. [16], [26; p.140]). The invariant χ_{3}(L_{7}(α, −(α + 1))) is a rational function of α, and it is invariant under the action of the symmetric group 𝔖_{3} consisting of transformations α ↦ α′ given in the proof of (3). This is essentially the unique rational function of α possessing this property.

In §3, we explained the intrinsic properties of 𝔤 defined by the vanishing of φ_{1}, φ_{2}. These properties can be also expressed in terms of the invariants χ_{1}(𝔤) and χ_{2}(𝔤). In fact, for Lie algebras having finite values of χ_{1}(𝔤), χ_{2}(𝔤), we can easily show that the condition ⟨φ_{1}⟩ = 0 is equivalent to 4χ_{1} = 8χ_{2} + 1, and ⟨φ_{2}⟩ = 0 is equivalent to 3(3χ_{2} − χ_{1})^{2} + 4(χ_{1}^{3} − χ_{1}^{2} + χ_{2}) = 0. Note that the second condition appears in Proposition 5 (1) above.

We can also describe several intrinsic properties of 𝔤 in terms of these invariants. For example, we can verify that the rank of the exterior differential map d : ∧^{2}𝔤^{∗} ⟶ ∧^{3}𝔤^{∗} is 3 for “generic” Lie algebras, and the set of “singular” Lie algebras satisfying rank d ≤ 2 constitutes a reducible algebraic subset of ∧^{2}V^{∗} ⊗ V. We can characterize this algebraic set in terms of the covariants S_{λ} and invariants χ_{i} appeared in §2 and §4. (For details, see [6].) This result plays an essential role in considering left invariant symplectic structures on 4-dimensional complex Lie groups.

Our invariants χ_{1}(𝔤) and χ_{2}(𝔤) are essentially related to the (i, j)-invariants introduced in [7; p.734]. In fact, it is easy to see that any (i, j)-invariant of [7] can be expressed as a rational function of χ_{1} and χ_{2} because it is equal to

\frac{(ε_{1}^{i} + ε_{2}^{i} + ε_{3}^{i})(ε_{1}^{j} + ε_{2}^{j} + ε_{3}^{j})}{ε_{1}^{i+j} + ε_{2}^{i+j} + ε_{3}^{i+j}}

in our notation. For example, we have

(1, 1)-invariant = \frac{1}{1 − 2χ_{1}} ,  (2, 1)-invariant = \frac{1 − 2χ_{1}}{1 − 3χ_{1} + 3χ_{2}} ,
(3, 1)-invariant = \frac{1 − 3χ_{1} + 3χ_{2}}{1 − 4χ_{1} + 2χ_{1}^{2} + 4χ_{2}} , (2, 2)-invariant = \frac{(1 − 2χ_{1})^{2}}{1 − 4χ_{1} + 2χ_{1}^{2} + 4χ_{2}} ,

etc.

§5. Varieties of 4-dimensional Lie algebras and their degenerations.

As a by-product of the results in the previous sections, we can describe the varieties and the degenerations of 4-dimensional Lie algebras. It is convenient to summarize these results before exhibiting an algorithm to determine the isomorphism classes because they give one basis of the understanding of the algorithm. The results in Proposition 6 and Proposition 8 are essentially already known. But, these results are summarized in a comparatively simple form, on account of our normal forms that are fitted to describe degenerations.

We first recall the definition of degeneration. We say that a Lie algebra 𝔤_{1} degenerates to 𝔤_{2} if 𝔤_{1} ≇ 𝔤_{2} and 𝔤_{2} ∈ \overline{𝒪(𝔤_{1})}, where \overline{𝒪(𝔤_{1})} denotes the Zariski closure of the GL(V)-orbit 𝒪(𝔤_{1}). In this case, we have dim 𝒪(𝔤_{1}) > dim 𝒪(𝔤_{2}). Degenerations of 4-dimensional complex Lie algebras are already completely determined in [7; p.736]. Here, we re-summarize the results in terms of our normal forms. In the following, the symbol 𝔤_{1} ⟶^{deg} 𝔤_{2} implies that 𝔤_{1} degenerates to 𝔤_{2}. (We sometimes drop the symbol “deg” on the arrow if there is no danger of confusion.) Note that the notion of degeneration is transitive, i.e., if 𝔤_{1} ⟶^{deg}

𝔤_{2} and 𝔤_{2} ⟶^{deg} 𝔤_{3}, then we have 𝔤_{1} ⟶^{deg} 𝔤_{3}.

Proposition 6 (cf. [7; p.736]). Essential degenerations of 4-dimensional complex Lie algebras are exhausted by the following, i.e., all degenerations are obtained by composing the following degenerations :

L_{4}(1)  ⟶ L_{3}  ⟶ L_{0},
L_{6}  ⟶ L_{8}(−1),
L_{7}(α, β) ⟶ L_{2}  ⟶ L_{1} ⟶ L_{0},
L_{7}(α, 1) ⟶ L_{4}(α) ⟶ L_{1} (α ∈ C ∪ {∞}),
L_{8}(α) ⟶ L_{7}(α, α + 1),
L_{8}(1) ⟶ L_{5} ⟶ L_{4}(2),
L_{9}  ⟶ L_{7}(α, 0),
L_{9}  ⟶ L_{8}(0).

(Note that we use the notational convention L_{7}(∞, 1) = L_{7}(0, 0) as stated in §1.)

Outline of the proof. We explicitly construct a curve in ∧^{2}V^{∗} ⊗ V which expresses a degeneration for each case. For example, the Lie algebra

[X_{1}, X_{2}] = X_{3}, [X_{1}, X_{3}] = X_{4}, [X_{1}, X_{4}] = t^{3}αβX_{2} − t^{2}(αβ + α + β)X_{3} + t(α + β + 1)X_{4}

is isomorphic to L_{7}(α, β) in case t ≠ 0, and to L_{2} in case t = 0. The Lie algebra

[X_{1}, X_{2}] = X_{2} + X_{4}, [X_{1}, X_{3}] = X_{2} + αX_{3}, [X_{1}, X_{4}] = (α + 1)X_{4}, [X_{2}, X_{3}] = tX_{4}

is isomorphic to L_{8}(α) in case t ≠ 0, and to L_{7}(α, α + 1) in case t = 0. These facts show that there exist degenerations L_{7}(α, β) ⟶^{deg} L_{2} and L_{8}(α) ⟶^{deg} L_{7}(α, α + 1). Remaining degenerations can be checked in a similar way.

On the other hand, we need several devices to show the non-existence of degenerations. First, as we stated before, if dim 𝒪(𝔤_{1}) ≤ dim 𝒪(𝔤_{2}), then 𝔤_{1} cannot degenerate to 𝔤_{2}. In case dim 𝒪(𝔤_{1}) > dim 𝒪(𝔤_{2}), we show the non-existence of degenerations by using the covariants and invariants which we introduced in §2 and §4. For example, for the Lie algebra L_{2}, we have S_{111} = S_{222} = 0 from Table 3. But, for the Lie algebra L_{4}(α), two covariants S_{111} and S_{222} cannot simultaneously vanish, which implies that a degeneration L_{2} ⟶^{deg} L_{4}(α) does not exist for any α. (Note that dim 𝒪(L_{2}) = 9 > dim 𝒪(L_{4}(α)) = 8.) As another example, we consider the case L_{7}(α, β) ⟶^{deg} L_{3}. In case α + β + 1 = 0, we can show the non-existence of degenerations in the same way as above by using the covariant S_{111}. In case α + β + 1 ≠ 0, we use two invariants χ_{1} and χ_{2} to check the non-existence of degenerations. Note that in this case, χ_{1} and χ_{2} are well-defined for both Lie algebras. If there exists a degeneration L_{7}(α, β) ⟶^{deg} L_{3}, then the values of invariants of L_{7}(α, β) must coincide with that of L_{3} because L_{3} is contained in the Zariski closure of the GL(V)-orbit of L_{7}(α, β). Hence, from Table 4, we obtain two conditions

\begin{matrix}(α + β)^{2} − 3αβ − (α + β) + 1 = 0, \\ (α + β)^{3} + 3(α + β)^{2} − 27αβ + 3(α + β) + 1 = 0. \end{matrix}

From these conditions, we have immediately α = β = 1, which implies that L_{7}(α, β) cannot degenerate to L_{3} in case (α, β) ≠ (1, 1). (We already know the existence of a degeneration L_{7}(1, 1) ⟶^{deg} L_{3}.) For the remaining cases not listed in Proposition 6, we can similarly prove the non-existence of degenerations. Note that for this purpose, we have only to use the covariants S_{111}, S_{222}, S_{333} (and three invariants χ_{1}, χ_{2}, χ_{3}), except for the cases L_{7}(α, β) ⟶^{deg} L_{5} and L_{9} ⟶^{deg} L_{8}(α) (α ≠ 0). For these two exceptional cases, we use the covariant S_{4221} to show the non-existence of degenerations. q.e.d.

Remark. (1) Two Lie algebras L_{6} and L_{9} are sums of lower dimensional Lie algebras. But their degenerate Lie algebras are not necessarily expressed as sums of Lie algebras such as L_{8}(−1), L_{8}(0), L_{2} (cf. Remark (1) after Proposition 1).

(2) Nilpotent Lie algebras are all contained in \overline{𝒪(L_{2})} (cf. Proposition 10 (7)).

As a corollary of Proposition 6, we can show several facts on the variety of 4-dimensional Lie algebras. But, before stating them, we first calculate the cohomology space of 𝔤 for later use.

Proposition 7. The dimensions of the second cohomology space H^{2}(𝔤, 𝔤) with coefficients in the adjoint representation are given as follows :

\begin{aligned}& dim H^{2}(𝔤, 𝔤) \\ L_{0} & 24 \\ L_{1} & 13 \\ L_{2} & 6 \\ L_{3} & 8 \\ L_{4}(α) & \{ \begin{matrix}7 & α = 0 \\ 6 & α = ∞ \\ 5 & α = 2 \\ 4 & α ≠ 0, 2, ∞ \end{matrix} \\ L_{5} & 3 \end{aligned}  \begin{aligned}& dim H^{2}(𝔤, 𝔤) \\ L_{6} & 0 \\ L_{7}(α, β) & \{ \begin{matrix}5 & (α, β) = (−1, 0), (0, −1) \\ 3 & α ≠ −1, β = 0 or α ≠ −1, β = α + 1 \\ 2 & α + β ≠ 1, |α − β| ≠ 1, α, β ≠ 0 \end{matrix} \\ L_{8}(α) & \{ \begin{matrix}2 & α = 0, −1 \\ 1 & α ≠ 0, −1 \end{matrix} \\ L_{9} & 0 \end{aligned}

We can check this result by direct calculations. By the deformation theory of Lie algebras [24], the dimension of the space H^{2}(𝔤, 𝔤) indicates the degree of freedom of non-trivial infinitesimal deformations of 𝔤. In particular, if H^{2}(𝔤, 𝔤) = 0 (such as L_{6}, L_{9}), then 𝔤 is rigid, i.e., the orbit space 𝒪(𝔤) is a Zariski open set of ∧^{2}V^{∗} ⊗ V, and its closure \overline{𝒪(𝔤)} is irreducible.

From Proposition 6, the Lie algebras that cannot be expressed as degenerations of other Lie algebras are exhausted by L_{6}, L_{7}(α, β) (|α − β| ≠ 1, α ≠ 0, β ≠ 0), L_{8}(α) (α ≠ 0, −1) and L_{9}. (Note that the Lie algebra L_{7}(α, β) with α + β = 1 also should be excluded. But this Lie algebra is isomorphic to L_{7}(γ, γ + 1) for some γ. See Remark (3) after Proposition 1.) For two families of non-rigid Lie algebras L_{7}(α, β) and L_{8}(α), the number of parameters just coincide with the dimension of H^{2}(𝔤, 𝔤) for generic parameters. Hence, we obtain the following well known fact on the irreducible decomposition of the variety of Lie algebras.

Proposition 8 (cf. [9], [12], [18], [25]). The algebraic set of ∧^{2}V^{∗} ⊗ V consisting of all 4-dimensional Lie algebras splits into a union of four irreducible varieties Σ_{1} ∼ Σ_{4}. These varieties are the Zariski closures of the following GL(V)-orbits.

\begin{matrix}Σ_{1} = \overline{𝒪(L_{6})}, & Σ_{2} = \overline{∪_{α,β} 𝒪(L_{7}(α, β))}, \\ Σ_{3} = \overline{∪_{α} 𝒪(L_{8}(α))}, & Σ_{4} = \overline{𝒪(L_{9})}. \end{matrix}

In view of Table 3, we can characterize these varieties in terms of GL(V)-invariant sets of polynomials with degree at most three.

Proposition 9. The defining equations of the varieties Σ_{1} ∼ Σ_{4} are given by :

Σ_{1} : S_{111} = S_{333} = 0  (linear and cubic polynomials),
Σ_{2} : S_{3111} = S_{4221} = 0 (quadratic and cubic polynomials),
Σ_{3} : ⟨φ_{1}⟩ = S_{3222} = 0 (cubic polynomials),
Σ_{4} : S_{4221} = S_{333} = 0  (cubic polynomials),

in addition to the Jacobi identity.

Remark. By definition, a Lie algebra is defined by the vanishing of the quadratic polynomials of {c_{ij}^{k}} corresponding to the Jacobi identity. But by this proposition, {c_{ij}^{k}} also satisfies different types of additional polynomial relations, according to each variety Σ_{i}.

The orbit decompositions of Σ_{i} and their degenerations are summarized in Figure 1. For each variety Σ_{i}, we know that there exists one principal line of degenerations. But, there also appear several singular Lie algebras such as L_{3}, L_{5}, etc. and these degenerate Lie algebras make Figure 1 a little complicated. In particular, on account of these Lie algebras, it is hard to summarize all degenerations in “one” chart.

Note that the variety Σ_{2} mainly consists of the Lie algebras L_{7}(α, β). Among them, Lie algebras which satisfy dimH^{2}(𝔤, 𝔤) > 2 constitute a family {L_{7}(α, 0), L_{7}(α, α+1)} (cf. Proposition 7). And they just coincide with the ones that are situated in the intersection of other varieties Σ_{1}, Σ_{3}, Σ_{4}. Similar phenomenon occurs for the variety Σ_{3} = \overline{∪_{α} 𝒪(L_{8}(α))}.

As another corollary of Proposition 6, we can state the “variety” version of Proposition 4 in the following form. In view of Table 3 and Figure 1, we can easily verify these statements.

Proposition 10. (1) S_{21} = 0 if and only if 𝔤 ∈ \overline{𝒪(L_{3})}.

(2) 𝔤 is unimodular if and only if 𝔤 ∈ Σ_{1} ∪ \overline{∪_{α}𝒪(L_{7}(α, −α − 1))}.

(3) S_{3111} = 0 if and only if 𝔤 ∈ Σ_{1} ∪ Σ_{2}.

(4) S_{441} = 0 if and only if 𝔤 ∈ \overline{𝒪(L_{5})} ∪ \overline{∪_{α}𝒪(L_{4}(α))}.

(5) [[𝔤, 𝔤], [𝔤, 𝔤]] = 0 (i.e., S_{4221} = 0) if and only if 𝔤 ∈ Σ_{2} ∪ Σ_{4}.

(6) dim [[𝔤, 𝔤], [𝔤, 𝔤]] ≤ 1 (i.e., 𝔤 is solvable) if and only if 𝔤 ∈ Σ_{2} ∪ Σ_{3} ∪ Σ_{4}.

(7) 𝔤 is nilpotent (i.e., S_{111} = S_{222} = S_{333} = 0) if and only if 𝔤 ∈ \overline{𝒪(L_{2})}.

Figure 1

06910 11 12 dim 𝒪(𝔤)  L_{0} L_{1} L_{2} L_{7}(−1, 0) L_{8}(−1) L_{6}  Σ_{1}

[figure: 0; 4; 6; 8; 9; 10; dim 𝒪(𝔤); L_{3}; L_{4}(1); L_{7}(1, 1); L_{0}; L_{1}; L_{2}; L_{7}(α, β) ( \begin{matrix}α ≠ β \\ α, β ≠ 1 \end{matrix} ); L_{4}(α) ( \begin{matrix}α ∈ C ∪ \{∞\} \\ α ≠ 1 \end{matrix} ); L_{7}(α, 1) ( \begin{matrix}α ∈ C ∪ \{∞\} \\ α ≠ 1 \end{matrix} ); Σ_{2}; Note that L_{7}(∞, 1) = L_{7}(0, 0).]

24 Y. Agaoka Figure 1 (continued)

[figure: 0; 6; 8; 9; 10; 11; dim 𝒪(𝔤); L_{4}(0); L_{7}(1, 0); L_{8}(0); L_{0}; L_{1}; L_{2}; L_{7}(α, α + 1) (α ≠ 0, 1); L_{8}(α) (α ≠ 0, 1); L_{4}(2); L_{7}(2, 1); L_{8}(1); L_{5}; Σ_{3}]

[figure: 0; 6; 8; 9; 10; 11; 12; dim 𝒪(𝔤); L_{4}(0); L_{7}(1, 0); L_{8}(0); L_{0}; L_{1}; L_{2}; L_{7}(α, 0) (α ≠ 0, 1); L_{9}; L_{4}(∞); L_{7}(0, 0); Σ_{4}]

(8) dim [𝔤, 𝔤] ≤ 1 (i.e., S_{321} = S_{222} = 0) if and only if 𝔤 ∈ \overline{𝒪(L_{4}(∞))}.

(9) dim [𝔤, 𝔤] ≤ 2 (i.e., S_{4221} = S_{333} = 0) if and only if 𝔤 ∈ Σ_{4}.

In particular, from the result (2), we know that the set of all 4-dimensional “unimodular” Lie algebras splits into two varieties with dimensions 12 and 11.

§6. An algorithm to determine the isomorphism classes of 4-dimensional Lie algebras.

Now, in this section, we give an algorithm to determine the isomorphism classes of 4dimensional complex Lie algebras, by using the devices prepared in the previous sections. This algorithm is the main theme of the present paper.

First, among 4-dimensional Lie algebras, L_{0}, L_{6}, L_{9} and non-abelian nilpotent Lie algebras L_{1}, L_{2} can be characterized by simple methods. In fact, the Lie algebra L_{6} has a characteristic property dim [[𝔤, 𝔤], [𝔤, 𝔤]] = 3, and the Lie algebra L_{9} is uniquely characterized by the property that the ratio of the eigenvalues of ad X essentially depends on the choice of X. Nilpotent Lie algebras are characterized by the properties S_{111} = S_{222} = S_{333} = 0, which are also equivalent to ε_{1} = ε_{2} = ε_{3} = 0, where {0, ε_{1}, ε_{2}, ε_{3}} are the eigenvalues of ad X for generic X (cf. Proposition 4 (10)). Three nilpotent Lie algebras L_{0}, L_{1}, L_{2} are distinguished by the value dim [𝔤, 𝔤].

Next, we state a method to determine the isomorphism classes for the remaining Lie algebras L_{3}, L_{4}(α), L_{5}, L_{7}(α, β) and L_{8}(α). These Lie algebras are roughly classified into four classes by the values dim [𝔤, 𝔤] and dim [[𝔤, 𝔤], [𝔤, 𝔤]]:

dim [𝔤, 𝔤] = 1  : L_{4}(∞),
dim [𝔤, 𝔤] = 2  : L_{4}(0), L_{7}(α, 0), L_{8}(0),
dim [𝔤, 𝔤] = 3, dim [[𝔤, 𝔤], [𝔤, 𝔤]] = 0 : L_{3}, L_{4}(α) (α ≠ 0, ∞), L_{7}(α, β) (α, β ≠ 0),
dim [𝔤, 𝔤] = 3, dim [[𝔤, 𝔤], [𝔤, 𝔤]] = 1 : L_{5}, L_{8}(α) (α ≠ 0).

In particular, L_{4}(∞) is uniquely characterized by the property dim [𝔤, 𝔤] = 1. We give a method to determine the isomorphism classes for the remaining Lie algebras in terms of several covariants and invariants.

(i) We first consider three Lie algebras L_{3}, L_{4}(1) and L_{7}(1, 1). These Lie algebras constitute special degenerations: L_{7}(1, 1) ⟶^{deg} L_{4}(1) ⟶^{deg} L_{3}, and they are characterized by the properties χ_{1} = \frac{1}{3} and χ_{2} = \frac{1}{27} among the above remaining Lie algebras (in case the invariants have definite values). From Table 3, we have

S_{21} S_{441}
L_{3}  0  0
L_{4}(1)  ∗  0
L_{7}(1, 1) ∗  ∗

and hence, these three Lie algebras can be distinguished to each other by two covariants S_{21} and S_{441}.

(ii) Next, we consider the case dim [𝔤, 𝔤] = 2. Among these Lie algebras, there exist degenerations L_{8}(0) ⟶^{deg} L_{7}(1, 0) ⟶^{deg} L_{4}(0). The value of χ_{1} for these three Lie algebras is \frac{1}{4}. Hence, if χ_{1} ≠ \frac{1}{4}, it is isomorphic to the remaining Lie algebra L_{7}(α, 0) (α ≠ 1), and from Proposition 5 (4), the value of the parameter of L_{7}(α, 0) is uniquely determined by χ_{1}. For the above three Lie algebras with χ_{1} = \frac{1}{4}, we have from Table 3,

S_{3111} S_{441}
L_{4}(0) 0 0
L_{7}(1, 0) 0  ∗
L_{8}(0) ∗ ∗ .

Hence, they are distinguished by two covariants S_{3111} and S_{441}.

(iii) The case dim [𝔤, 𝔤] = 3 and dim [[𝔤, 𝔤], [𝔤, 𝔤]] = 1. Among these Lie algebras, there is a degeneration L_{8}(1) ⟶^{deg} L_{5}. For these two Lie algebras, the value of χ_{1} is equal to \frac{5}{16}, and hence, if χ_{1} ≠ \frac{5}{16}, it is isomorphic to L_{8}(α) (α ≠ 0, 1). In this case, from Proposition 5 (5), the value of the parameter α is uniquely determined by χ_{1}. To distinguish two Lie algebras L_{8}(1) and L_{5}, we have only to use the covariant S_{441} (cf. Table 3).

(iv) The case dim[𝔤, 𝔤] = 3 and [[𝔤, 𝔤], [𝔤, 𝔤]] = 0. Remaining Lie algebras are exhausted by L_{4}(α) (α ≠ 0, 1, ∞) and L_{7}(α, β) (α, β ≠ 0, (α, β) ≠ (1, 1)). From Table 3, two Lie algebras L_{4}(α) and L_{7}(α, β) are distinguished by the covariant S_{441}. And from Proposition 5 (1), (2) and (3), we can determine the values of the parameters of L_{4}(α) and L_{7}(α, β) in terms of χ_{1} ∼ χ_{3}.

By these procedures, in terms of covariants and invariants which we introduced in this paper, we can uniquely determine the isomorphism classes of 4-dimensional Lie algebras without constructing the explicit isomorphisms. Summarizing these results, we obtain the following main theorem of this paper.

Theorem 11. The isomorphism classes of 4-dimensional complex Lie algebras are determined uniquely in terms of the following quantities :

• dim [𝔤, 𝔤], dim [[𝔤, 𝔤], [𝔤, 𝔤]].

• S_{21}, S_{3111}, S_{441}.

• The ratio of the eigenvalues of ad X for generic X ∈ 𝔤, (i.e., three invariants χ_{1}, χ_{2} and χ_{3}.)

An algorithm to determine the isomorphism classes of 4-dimensional Lie algebras is summarized in Figure 2.

Concerning three covariants appeared in this theorem, remind the results in Proposition 4-dimensional Lie algebras 27

Figure 2 START

abel L_{0}

dim [[𝔤, 𝔤], [𝔤, 𝔤]] = 3 L_{6}

calculate the ratio of the eigenvalues of ad X

0 : ε_{1} : ε_{2} : ε_{3} ratio of the eigenvalues of

ad X depends on X

L_{9}

ε_{1} = ε_{2} = ε_{3} = 0 dim [𝔤, 𝔤] = 2

L_{2}
L_{1}

calculate χ_{1}, χ_{2}, χ_{3}

χ_{1} = \frac{1}{3}, χ_{2} = \frac{1}{27} S_{441} = 0

L_{7}(1, 1)
S_{21} = 0

L_{3}
L_{4}(1)

dim [𝔤, 𝔤] = 3

[[𝔤, 𝔤], [𝔤, 𝔤]] = 0 χ_{1} = \frac{5}{16}

\begin{matrix}L_{8}(α) \\ (α ≠ 0, 1) \end{matrix} if χ_{1} = \frac{α^{2}+3α+1}{4(α+1)^{2}}

S_{441} = 0

L_{5}
L_{8}(1)

S_{441} = 0 \begin{matrix}L_{4}(α) \\ (α ≠ 0, 1, ∞) \end{matrix} if \{ \begin{matrix}χ_{1} = \frac{2α+1}{(α+2)^{2}} \\ χ_{2} = \frac{α}{(α+2)^{3}} \end{matrix}

ε_{1} + ε_{2} + ε_{3} = 0 \begin{matrix}L_{7}(α, β)( \begin{matrix}α + β ≠ −1 \\ α, β ≠ 0 \\ (α, β) ≠ (1, 1) \end{matrix} ) \\ if & \{ \begin{matrix}χ_{1} = \frac{αβ+α+β}{(α+β+1)^{2}} \\ χ_{2} = \frac{αβ}{(α+β+1)^{3}} \end{matrix} \end{matrix}

\begin{matrix}L_{7}(α, −(α + 1)) (α ≠ 0, −1) \\ if χ_{3} = −\frac{α^{2}(α+1)^{2}}{(α^{2}+α+1)^{3}} \end{matrix}

dim [𝔤, 𝔤] = 2

L_{4}(∞)

χ_{1} = \frac{1}{4}

\begin{matrix}L_{7}(α, 0) \\ (α ≠ 1) \\ if χ_{1} = \frac{α}{(α+1)^{2}} \end{matrix}

S_{3111} = 0 L_{8}(0)

S_{441} = 0 L_{7}(1, 0)

L_{4}(0)

[figure] [figure] [figure] [figure] [figure] [figure]

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• S_{21} = 0 if and only if there exists an element f ∈ 𝔤^{∗} such that [X, Y] = f(X)Y − f(Y)X.

• S_{3111} = 0 if and only if dα ∧ dα = 0 for any α ∈ 𝔤^{∗}.

• S_{441} = 0 if and only if dim ⟨X, Y, [X, Y], [X, [X, Y]]⟩ ≤ 3 for any X, Y ∈ 𝔤.

Of course, two values dim[𝔤, 𝔤] and dim[[𝔤, 𝔤], [𝔤, 𝔤]] are also characterized by the vanishing or non-vanishing of several covariants, as stated in Proposition 4.

Example. We give here two examples, showing the usefulness of this algorithm. We first consider the following Lie algebra:

\begin{matrix}[X_{1}, X_{2}] = −X_{1} − X_{2} + X_{3}, & [X_{1}, X_{3}] = −6X_{2} + 4X_{3}, \\ [X_{1}, X_{4}] = 2X_{1} − X_{2} + X_{4}, & [X_{2}, X_{3}] = 3X_{1} − 9X_{2} + 5X_{3}, \\ [X_{2}, X_{4}] = 4X_{1} − 2X_{2} + 2X_{4}, & [X_{3}, X_{4}] = 6X_{1} − 3X_{2} + 3X_{4}. \end{matrix}

We can easily check that [𝔤, 𝔤] = ⟨2X_{1} − X_{2}, 3X_{1} − X_{3}, X_{4}⟩ and [[𝔤, 𝔤], [𝔤, 𝔤]] = 0. And by putting X = aX_{1} + bX_{2} + cX_{3} + dX_{4}, we have

ad X = ( \begin{matrix}b − 2d & −a − 3c − 4d & 3b − 6d & 2a + 4b + 6c \\ b + 6c + d & −a + 9c + 2d & −6a − 9b + 3d & −a − 2b − 3c \\ −b − 4c & a − 5c & 4a + 5b & 0 \\ −d & −2d & −3d & a + 2b + 3c \end{matrix} ) .

Hence, after some calculations, we have TradX = 4(a+2b+3c), Tr(adX)^{2} = 6(a+2b+3c)^{2}, Tr (ad X)^{3} = 10(a + 2b + 3c)^{3}. From these values, we have

p_{111} = −Tr ad X = −4(a + 2b + 3c),

p_{222} = \frac{1}{2}\{(Tr ad X)^{2} − Tr (ad X)^{2}\} = 5(a + 2b + 3c)^{2},

p_{333} = −\frac{1}{6}\{(Tr ad X)^{3} − 3 Tr ad X · Tr (ad X)^{2} + 2 Tr (ad X)^{3}\} = −2(a + 2b + 3c)^{3},

and hence, χ_{1} = p_{222}/p_{111}^{2} = \frac{5}{16}, χ_{2} = p_{333}/p_{111}^{3} = \frac{1}{32}. In addition, four elements X_{1}, X_{4}, [X_{1}, X_{4}] = 2X_{1}−X_{2}+X_{4} and [X_{1}, [X_{1}, X_{4}]] = 3X_{1}−X_{3}+X_{4} are linearly independent, and we have S_{441} ≠ 0. Since the value of p_{111} is non-zero, this Lie algebra is not unimodular. Hence, by applying the algorithm in Figure 2, we know that this Lie algebra is isomorphic to L_{7}(α, β) with χ_{1} = \frac{5}{16}, χ_{2} = \frac{1}{32} and α + β ≠ −1, (α, β) ≠ (1, 1), α, β ≠ 0. By solving the equations

\frac{αβ + α + β}{(α + β + 1)^{2}} = \frac{5}{16}, \frac{αβ}{(α + β + 1)^{3}} = \frac{1}{32},

we have for example, (α, β) = (2, 1), and hence this Lie algebra is isomorphic to L_{7}(2, 1). As another example, we consider the following Lie algebra:

\begin{matrix}[X_{1}, X_{2}] = 4X_{1} + 3X_{2} − 6X_{3} + 2X_{4}, & [X_{1}, X_{3}] = 15X_{1} + 5X_{2} − 15X_{3} + 5X_{4}, \\ [X_{1}, X_{4}] = 50X_{1} + 15X_{2} − 48X_{3} + 16X_{4}, & [X_{2}, X_{3}] = 21X_{1} + 2X_{2} − 15X_{3} + 5X_{4}, \\ [X_{2}, X_{4}] = 93X_{1} + 21X_{2} − 81X_{3} + 27X_{4}, & [X_{3}, X_{4}] = 90X_{1} + 25X_{2} − 84X_{3} + 28X_{4}. \end{matrix}

Then, for X = aX_{1} + bX_{2} + cX_{3} + dX_{4}, we have

ad X = ( \begin{matrix}−4b − 15c − 50d & 4a − 21c − 93d & 15a + 21b − 90d & 50a + 93b + 90c \\ −3b − 5c − 15d & 3a − 2c − 21d & 5a + 2b − 25d & 15a + 21b + 25c \\ 6b + 15c + 48d & −6a + 15c + 81d & −15a − 15b + 84d & −48a − 81b − 84c \\ −2b − 5c − 16d & 2a − 5c − 27d & 5a + 5b − 28d & 16a + 27b + 28c \end{matrix} ) .

In this case, by putting p = a+2b+3c+4d, q = 3a+6b+8c+9d, we have TradX = p+q, Tr (ad X)^{2} = p^{2} + q^{2}, Tr (ad X)^{3} = p^{3} + q^{3}. Hence, the eigenvalues of ad X is given by {0, 0, p, q}, and the ratio essentially depends on X. (cf. Remark (1) after Proposition 2.) Thus, by the algorithm in Figure 2, it follows that this Lie algebra is isomorphic to L_{9}.

Appendix A. Relation to several normal forms.

In this Appendix A, we give isomorphisms between our normal forms in Table 1 and the normal forms of other two classifications explicitly. There are already several classifications of 4-dimensional real or complex Lie algebras. (For example, [7], [19], [21], [23], [25], [27], [28], [29], [31], [34], etc. But, as for the classification table in [25; p.209], it seems that it contains some mistakes. See also the comments in [7; p.732]. Among these classifications, S. Lie’s one [21] is of course the earliest one.) In this appendix, we state relations to normal forms in [7] which was essentially taken from [29], and [23] which is a classification over R. In particular, by checking the correspondence to [7] in detail, we see that the list of degenerations in Proposition 6 just coincides with the result in [7; p.736]. We remark that to find the isomorphic Lie algebra among several normal forms in several classifications is now an easy task for us on account of the algorithm in Figure 2. But, to find the explicit isomorphism is another problem, which requires many tedious calculations.

In Appendix A, {X_{1}, ⋯, X_{4}} denotes the basis of L_{i} in Table 1, and {e_{1}, ⋯, e_{4}} denotes the basis of the Lie algebras in [7] or [23].

(i) Correspondence to [7; p.733]. We use here the same symbols as in [7]. But, for three Lie algebras 𝔤_{2}, 𝔤_{3}, 𝔤_{8}, we replace the parameters α and β in [7] by λ and μ, respectively. We drop the parameter restrictions in [7] because singular cases often give good examples of deformations (cf. Figure 1). In the following list, we give the isomorphisms only for non-trivial cases. For the explicit bracket operations [e_{i}, e_{j}], see [7; p.733].

• C^{4} ≅ L_{0}.

• 𝔫_{3}(C) ⊕ C ≅ L_{1}.

• 𝔯_{2}(C) ⊕ C^{2} ≅ L_{4}(∞).

• 𝔯_{3}(C) ⊕ C ≅ L_{7}(1, 0),

e_{1} = X_{1}, e_{2} = X_{2}, e_{3} = X_{3}, e_{4} = X_{2} − X_{3} + X_{4}.

• 𝔯_{3,λ}(C) ⊕ C ≅ \{ \begin{matrix}L_{4}(∞), & λ = 0, \\ L_{4}(0), & λ = 1, \\ L_{7}(λ, 0), & λ ≠ 0, 1, \end{matrix}

\{ \begin{matrix}λ = 1 & : & e_{1} = X_{1}, e_{2} = X_{2}, e_{3} = X_{3}, e_{4} = X_{3} − X_{4}, \\ λ ≠ 0, 1 & : & e_{1} = X_{1}, e_{2} = X_{2}, e_{3} = X_{2} + (λ − 1)X_{3}, e_{4} = X_{2} − X_{3} + λX_{4}. \end{matrix}

• 𝔯_{2}(C) ⊕ 𝔯_{2}(C) ≅ L_{9}.

• 𝔰𝔩_{2}(C) ⊕ C ≅ L_{6},

e_{1} = X_{2}, e_{2} = 2X_{3}, e_{3} = 2X_{1}, e_{4} = X_{4}.

• 𝔫_{4}(C) ≅ L_{2}.

• 𝔤_{1}(α) ≅ \{ \begin{matrix}L_{3}, & α = 1, \\ L_{4}(α), & α ≠ 1, \end{matrix}

α ≠ 1 : e_{1} = X_{1}, e_{2} = X_{2}, e_{3} = X_{3}, e_{4} = X_{3} + (α − 1)X_{4}.

• 𝔤_{2}(λ, μ) ≅ L_{7}(α, β), (α + β ≠ −1),

e_{1} = \frac{1}{a}X_{1}, e_{2} = a^{2}X_{4}, e_{3} = a(X_{3} + βX_{4}), e_{4} = X_{2} + (α + β)X_{3} + β^{2}X_{4}.

Here, for given λ and μ, we define two complex numbers α, β by α + β = a − 1 and αβ = a^{3}λ, where a is a non-zero complex number satisfying λa^{3} − μa^{2} + a − 1 = 0. Since a ≠ 0, we have α + β ≠ −1. And from the definition of α and β, we can easily show the equalities

λ = \frac{αβ}{(α + β + 1)^{3}} = χ_{2}(𝔤), μ = \frac{αβ + α + β}{(α + β + 1)^{2}} = χ_{1}(𝔤).

Remark. In case λ ≠ 0, other solutions of the cubic equation λx^{3} − μx^{2} + x − 1 = 0 are given by a/α, a/β. And if we use a/α instead of a in the above isomorphism, then the solutions of α′ + β′ = \frac{a}{α} − 1, α′β′ = (\frac{a}{α})^{3} λ are (α′, β′) = (\frac{1}{α}, \frac{β}{α}), (\frac{β}{α}, \frac{1}{α}). But the unordered ratio 1 : α′ : β′ corresponding to this new solution coincides with the original ratio 1 : α : β, and hence we may use any solution of λx^{3} − μx^{2} + x − 1 = 0 in constructing the above isomorphism. (See the proof of Proposition 5 (2).) In case λ = 0, we can easily show that the same fact holds.

• 𝔤_{3}(λ) ≅ \{ \begin{matrix}L_{2}, & λ = 0, \\ L_{7}(α, −(α + 1)), & λ ≠ 0, (α ≠ 0, −1, α^{2} + α + 1 ≠ 0), \end{matrix}

λ ≠ 0 : e_{1} = kX_{1}, e_{2} = α^{2}(α + 1)^{2}X_{4},
e_{3} = α(α + 1)(α^{2} + α + 1){−X_{3} + (α + 1)X_{4}},
e_{4} = (α^{2} + α + 1)^{2}\{X_{2} − X_{3} + (α + 1)^{2}X_{4}\}, (k = −\frac{α^{2}+α+1}{α(α+1)} ≠ 0).

Here, in the case λ ≠ 0, the number α is a solution of the equation (x^{2} + x + 1)^{3} = λx^{2}(x + 1)^{2}. Clearly, we have α ≠ 0, −1 and α^{2} + α + 1 ≠ 0. In this case, the parameter λ satisfies the equality λ = \frac{(α^{2}+α+1)^{3}}{α^{2}(α+1)^{2}} = −\frac{1}{χ_{3}(𝔤)}.

Remark. Other solutions of the equation (x^{2} + x + 1)^{3} = λx^{2}(x + 1)^{2} are given by \frac{1}{α}, −(α + 1), \frac{−1}{α+1}, −\frac{α+1}{α}, \frac{−α}{α+1}, and it is easy to check that the unordered ratio 1 : α : −(α + 1) does not depend on the choice of these solutions. Hence, as above, we may use any solution of this equation in constructing the isomorphism. (See also the proof of Proposition 5 (3).)

• 𝔤_{4} ≅ L_{7}(ω, ω^{2}), (ω^{3} = 1, ω ≠ 1),

e_{1} = X_{1}, e_{2} = X_{4}, e_{3} = X_{3} + ω^{2}X_{4}, e_{4} = X_{2} − X_{3} + ωX_{4}.

• 𝔤_{5} ≅ L_{4}(1),

e_{1} = \frac{1}{3}X_{1}, e_{2} = 3X_{4}, e_{3} = X_{3}, e_{4} = X_{2}.

• 𝔤_{6} ≅ L_{5}.

• 𝔤_{7} ≅ L_{8}(−1),

e_{1} = X_{1}, e_{2} = X_{3}, e_{3} = X_{2} − X_{3}, e_{4} = −X_{4}.

• 𝔤_{8}(λ) ≅ L_{8}(α), λ = \frac{α}{(α+1)^{2}} (= 8χ_{2}(𝔤)), (α ≠ −1),

e_{1} = \frac{1}{α+1}X_{1}, e_{2} = (α + 1)X_{3}, e_{3} = X_{2} + αX_{3}, e_{4} = −(α + 1)X_{4}.

Remark. In case λ ≠ 0, the solutions of the equation \frac{x}{(x+1)^{2}} = λ are of the form α, \frac{1}{α} for each fixed λ. Since L_{8}(α) ≅ L_{8}(\frac{1}{α}), we may use any solution of \frac{x}{(x+1)^{2}} = λ in constructing the above isomorphism.

We add some comments. For the Lie algebra 𝔤_{2}(λ, μ), the parameters λ and μ just coincide with our invariants χ_{2}(𝔤), χ_{1}(𝔤), and they appear in the coefficients of the bracket [e_{1}, e_{4}] = λe_{2} − μe_{3} + e_{4} in a natural way (cf. [7; p.733]). Normal forms of 𝔤_{3}(λ) and 𝔤_{8}(λ) also possess this property. These facts imply that the normal forms in [7] (or [29]) are elegantly selected from the invariant theoretic viewpoint because the bracket operations are simply and uniquely expressed by their invariants. (In the 3-dimensional case, the normal form in [33] also possesses this property.)

But on the other hand, the normal forms in [7] are not necessarily fitted to describe deformations or degenerations of Lie algebras. For example, in view of the above isomorphism list, the Lie algebras 𝔯_{3}(C) ⊕ C, 𝔯_{3,λ}(C) ⊕ C (λ ≠ 0, 1), 𝔤_{2}(λ, μ), 𝔤_{3}(λ) (λ ≠ 0) and 𝔤_{4} should be gathered together to construct one family of Lie algebras because they are continuously deformable, possessing the same dimensional GL(V)-orbits. As another example, the family of Lie algebras 𝔤_{1}(α) contains a degenerate Lie algebra L_{3} in case α = 1. And as normal forms, it is desirable to adopt a family of Lie algebras such that 𝔤_{1}(α) corresponds to 𝔤_{5} ≅ L_{4}(1) in case α = 1. (Note that there exists a degeneration 𝔤_{5} ⟶^{deg} 𝔤_{1}(1).) In addition, this family should contain the Lie algebra 𝔯_{2}(C) ⊕ C^{2} because lim_{α→∞} 𝔤_{1}(α) ≅ 𝔯_{2}(C) ⊕ C^{2}. Our family of Lie algebras L_{4}(α) is selected to satisfy these conditions.

(ii) In the paper [23], Mubarakzyanov classified all 4-dimensional “real” Lie algebras. We can check that their complexifications are isomorphic to one of L_{i} in our list. We give here the explicit isomorphisms only for non-trivial cases. (For the definition of the Lie algebras g_{i}, see [23; p.115 ∼ 121]. With a slight modification of notations, the same list is quoted in [27], [28].) We assume that the coefficients α, β, γ, h, p in [23] represent complex numbers, and we also drop the parameter restrictions for these parameters by the same reason as (i).

• (4g_{1})^{C} ≅ L_{0}.

• (2g_{1} + g_{2})^{C} ≅ L_{4}(∞).

• (2g_{2})^{C} ≅ L_{9}.

• (g_{1} + g_{3,1})^{C} ≅ L_{1}.

• (g_{1} + g_{3,2})^{C} ≅ L_{7}(1, 0),

e_{1} = X_{2}, e_{2} = X_{3}, e_{3} = −X_{1}, e_{4} = X_{2} − X_{3} + X_{4}.

• (g_{1} + g_{3,3})^{C} ≅ L_{4}(0),

e_{1} = X_{2}, e_{2} = X_{3}, e_{3} = −X_{1}, e_{4} = X_{3} − X_{4}.

• (g_{1} + g_{3,4})^{C} ≅ \{ \begin{matrix}L_{4}(∞), & h = 0, \\ L_{4}(0), & h = 1, \\ L_{7}(h, 0), & h ≠ 0, 1, \end{matrix}

\{ \begin{matrix}h = 1 & : & e_{1} = X_{2}, e_{2} = X_{3}, e_{3} = −X_{1}, e_{4} = X_{3} − X_{4}, \\ h ≠ 0, 1 & : & e_{1} = X_{2}, e_{2} = X_{2} + (h − 1)X_{3}, e_{3} = −X_{1}, e_{4} = X_{2} − X_{3} + hX_{4}. \end{matrix}

• (g_{1} + g_{3,5})^{C} ≅ \{ \begin{matrix}L_{4}(∞), & p = ±i, \\ L_{7}(\frac{p−i}{p+i}, 0), & p ≠ ±i, \end{matrix}

\{ \begin{matrix}p = ±i & : & e_{1} = \frac{1}{2}X_{2} + \frac{1}{2}X_{3}, e_{2} = −\frac{p}{2}X_{2} + \frac{p}{2}X_{3}, e_{3} = −2pX_{1}, e_{4} = X_{4}, \\ p ≠ ±i & : & e_{1} = X_{3}, e_{2} = −(p + i)X_{2} + iX_{3}, e_{3} = −(p + i)X_{1}, \\ & & e_{4} = X_{2} − X_{3} + \frac{p−i}{p+i}X_{4}. \end{matrix}

• (g_{1} + g_{3,6})^{C} ≅ L_{6},

e_{1} = X_{2}, e_{2} = −X_{1}, e_{3} = −2X_{3}, e_{4} = X_{4}.

• (g_{1} + g_{3,7})^{C} ≅ L_{6},

e_{1} = iX_{1}, e_{2} = X_{2} − \frac{1}{2}X_{3}, e_{3} = iX_{2} + \frac{i}{2}X_{3}, e_{4} = X_{4}.

• g_{4,1}^{C} ≅ L_{2},

e_{1} = X_{4}, e_{2} = −X_{3}, e_{3} = X_{2}, e_{4} = X_{1}.

4-dimensional Lie algebras 33

• g_{4,2}^{C} ≅ \{ \begin{matrix}L_{4}(1), & α = 1, \\ L_{7}(α, 1), & α ≠ 1, \end{matrix}

\{ \begin{matrix}α = 1 & : & e_{1} = X_{2}, e_{2} = X_{3}, e_{3} = X_{4}, e_{4} = −X_{1}, \\ α ≠ 1 & : & e_{1} = X_{2} + (α − 1)X_{3}, e_{2} = X_{2}, e_{3} = X_{3} + (1 − α)X_{4}, e_{4} = −X_{1}. \end{matrix}

• g_{4,3}^{C} ≅ L_{7}(0, 0), e_{1} = X_{2}, e_{2} = X_{2} − X_{3}, e_{3} = X_{3} − X_{4}, e_{4} = −X_{1}.

• g_{4,4}^{C} ≅ L_{7}(1, 1), e_{1} = X_{2}, e_{2} = X_{3}, e_{3} = X_{4}, e_{4} = −X_{1}.

• g_{4,5}^{C} ≅ \{ \begin{matrix}L_{4}(∞), & β = γ = 0, \\ L_{3}, & β = γ = 1, \\ L_{4}(\frac{1}{β}), & β = γ ≠ 0, 1, \\ L_{4}(γ), & β = 1, γ ≠ 1, \\ L_{4}(β), & γ = 1, β ≠ 1, \\ L_{7}(β, γ), & β ≠ γ, β ≠ 1, γ ≠ 1, \end{matrix}

\{ \begin{matrix}β = γ = 1 & : & e_{1} = X_{2}, e_{2} = X_{3}, e_{3} = X_{4}, e_{4} = −X_{1}, \\ β = γ ≠ 0, 1 & : & e_{1} = βX_{3} + (1 − β)X_{4}, e_{2} = X_{3}, e_{3} = X_{2}, \\ & & e_{4} = −βX_{1}, \\ β = 1, γ ≠ 1 & : & e_{1} = X_{2}, e_{2} = X_{3}, e_{3} = X_{3} + (γ − 1)X_{4}, e_{4} = −X_{1}, \\ γ = 1, β ≠ 1 & : & e_{1} = X_{2}, e_{2} = X_{3} + (β − 1)X_{4}, e_{3} = X_{3}, e_{4} = −X_{1}, \\ β ≠ γ, β ≠ 1, γ ≠ 1 & : & e_{1} = X_{2}, e_{2} = X_{2} + (β − 1)X_{3}, \\ & & e_{3} = X_{2} + (γ − 1)X_{3} + (γ − 1)(γ − β)X_{4}, e_{4} = −X_{1}. \end{matrix}

• g_{4,6}^{C} ≅ \{ \begin{matrix}L_{4}(∞), & α = 0, p = ±i, \\ L_{7}(\frac{p−i}{p+i}, 0), & α = 0, p ≠ ±i, \\ L_{4}(\frac{p−i}{p+i}), & α = p + i ≠ 0, \\ L_{4}(\frac{p+i}{p−i}), & α = p − i ≠ 0, \\ L_{7}(\frac{p+i}{α}, \frac{p−i}{α}), & α ≠ 0, α ≠ p ± i, \end{matrix}

\{ \begin{matrix}α = 0, p = ±i & : & e_{1} = X_{4}, e_{2} = \frac{1}{2}X_{2} + \frac{1}{2}X_{3}, e_{3} = −\frac{p}{2}X_{2} + \frac{p}{2}X_{3}, \\ & & e_{4} = −2pX_{1}, \\ α = 0, p ≠ ±i & : & e_{1} = X_{2} − X_{3} + \frac{p−i}{p+i}X_{4}, e_{2} = X_{3}, \\ & & e_{3} = −(p + i)X_{2} + iX_{3}, e_{4} = −(p + i)X_{1}, \\ α = p + i ≠ 0 & : & e_{1} = X_{3}, e_{2} = X_{2} + X_{4}, \\ & & e_{3} = −iX_{2} − (p + i)X_{3} + iX_{4}, e_{4} = −(p + i)X_{1}, \\ α = p − i ≠ 0 & : & e_{1} = X_{3}, e_{2} = X_{2} + X_{4}, \\ & & e_{3} = iX_{2} − (p − i)X_{3} − iX_{4}, e_{4} = −(p − i)X_{1}, \\ α ≠ 0, α ≠ p ± i & : & e_{1} = X_{2}, e_{2} = −α^{2}X_{2} + kX_{4}, \\ & & e_{3} = α^{2}(α − p)X_{2} − αkX_{3} + ikX_{4}, e_{4} = −αX_{1}, \\ & & (k = (α − p)^{2} + 1 ≠ 0). \end{matrix}

• g_{4,7}^{C} ≅ L_{8}(1),

e_{1} = X_{4}, e_{2} = X_{2}, e_{3} = X_{3}, e_{4} = −X_{1}.

• g_{4,8}^{C} ≅ \{ \begin{matrix}L_{5}, & h = 1, \\ L_{8}(h), & h ≠ 1, \end{matrix}

\{ \begin{matrix}h = 1 & : & e_{1} = X_{4}, e_{2} = X_{2}, e_{3} = X_{3}, e_{4} = −X_{1}, \\ h ≠ 1 & : & e_{1} = (h − 1)X_{4}, e_{2} = X_{2}, e_{3} = X_{2} + (h − 1)X_{3}, e_{4} = −X_{1}. \end{matrix}

• g_{4,9}^{C} ≅ \{ \begin{matrix}L_{8}(\frac{p+i}{p−i}), & p ≠ i, \\ L_{8}(\frac{p−i}{p+i}), & p ≠ −i, \end{matrix}

\{ \begin{matrix}p ≠ i & : & e_{1} = (p − i)X_{4}, e_{2} = X_{3}, e_{3} = −(p − i)X_{2} − iX_{3}, \\ & & e_{4} = −(p − i)X_{1}, \\ p ≠ −i & : & e_{1} = (p + i)X_{4}, e_{2} = X_{3}, e_{3} = −(p + i)X_{2} + iX_{3}, \\ & & e_{4} = −(p + i)X_{1}. \end{matrix}

• g_{4,10}^{C} ≅ L_{9},

e_{1} = X_{2} + X_{4}, e_{2} = iX_{2} − iX_{4}, e_{3} = −X_{1} − X_{3}, e_{4} = iX_{1} − iX_{3}.

Appendix B. 3-dimensional complex Lie algebras.

In the survey paper [5], we summarized several known facts on the variety of 3dimensional complex Lie algebras, such as the classification, the irreducible decomposition of the variety, invariants, etc. from the same viewpoint of this paper. But, the normal forms listed in [5] are not fitted to describe deformations and degenerations, and the results are described in complicated forms there.

In this appendix, we introduce a new family of normal forms of 3-dimensional complex Lie algebras fitted for these purposes, and re-summarize the facts in [5] in a quite simple form. The final results are summarized in Figure 3 and Figure 4. (In this appendix, V always means the 3-dimensional complex vector space.)

We first give a new classification tabel of 3-dimensional complex Lie algebras, including the dimension of GL(V)-orbit 𝒪(𝔤):

Table 5

non-trivial bracket operations  dim 𝒪(𝔤)
M_{0}  0
M_{1}  [X_{1}, X_{2}] = X_{3}  3
M_{2}  [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = X_{3}  3
M_{3}(α) [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = X_{2} + αX_{3}  5
M_{4}  [X_{1}, X_{2}] = X_{2}, [X_{1}, X_{3}] = −X_{3}, [X_{2}, X_{3}] = X_{1}  6

Remark. (1) There are the following canonical isomorphisms to special Lie algebras:

M_{0} ≅ C^{3},  M_{1} ≅ Heisenberg Lie algebra,
M_{3}(0) ≅ 𝔞𝔣𝔣(1, C) ⊕ C, M_{3}(−1) ≅ 𝔢(2, C) ≅ \{( \begin{matrix}0 & a & b \\ −a & 0 & c \\ 0 & 0 & 0 \end{matrix} )\},
M_{4} ≅ 𝔰𝔩(2, C).

In addition, M_{3}(α) ≅ M_{3}(α′) if and only if α = α′ or αα′ = 1.

(2) The Lie algebra M_{3}(α) is isomorphic to the Lie algebra M′_{3}(α) defined by

[Y_{1}, Y_{2}] = Y_{2}, [Y_{1}, Y_{3}] = αY_{3}

if α ≠ 1. The bracket operation of M′_{3}(α) is simpler than that of M_{3}(α). But, as in the 4-dimensional case, M′_{3}(α) degenerates to M_{2} in case α = 1 (cf. Remark (6) after Proposition 1).

In terms of these normal forms, we can summarize several results on 3-dimensional complex Lie algebras in a simple form.

Proposition 12 (cf. [9], [12], [18], [25]). The algebraic set of ∧^{2}V^{∗}⊗V consisting of all 3-dimensional Lie algebras decomposes into two irreducible components Σ_{simp} = \overline{𝒪(M_{4})} and Σ_{solv} = \overline{∪_{α}𝒪(M_{3}(α))}, consisting of unimodular Lie algebras and of solvable Lie algebras, respectively. The orbit decompositions and degenerations are summarized in Figure 3.

Figure 3

[figure: 0; 3; 5; 6; dim 𝒪(𝔤); M_{2}; M_{3}(1); M_{0}; M_{1}; M_{3}(−1); M_{4}; M_{3}(α) (α ≠ ±1); = Σ_{simp}; = Σ_{solv}]

The variety Σ_{simp} is a 6-dimensional linear subspace of ∧^{2}V^{∗} ⊗ V defined by the condition ∑_{k} c_{ik}^{k} = 0 (= unimodularity). The other variety Σ_{solv} mainly consists of 1-dimensional family of 5-dimensional orbits M_{3}(α). The existence of the degenerate Lie algebra M_{2} makes the orbit decomposition of Σ_{solv} a little complicated.

To distinguish the family of Lie algebras M_{3}(α), we use the following invariant of 3dimensional Lie algebras. Let {0, ε_{1}, ε_{2}} be the eigenvalues of ad X for generic X ∈ 𝔤. Then the invariant χ(𝔤) ∈ C ∪ {∞} is defined by

χ(𝔤) = \frac{(ε_{1} + ε_{2})^{2}}{ε_{1}ε_{2}} .

(cf. [18], [33]. Note that the invariant J in [18] is related to χ(𝔤) through the equality χ(𝔤) = 2/(1 − J).) This invariant is well defined for all 3-dimensional Lie algebras except nilpotent ones (= M_{0}, M_{1}). The values of χ(𝔤) are summarized in Table 6:

Table 6

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

Then, the family of Lie algebras M_{3}(α) is distinguished in the following way. (Compare with the complicated statement in [5; p.14, Proposition 6].)

Proposition 13 (cf. [18], [33]). Assume 𝔤, 𝔤′ ∈ ∪_{α}𝒪(M_{3}(α)). Then, 𝔤 is isomorphic to 𝔤′ if and only if χ(𝔤) = χ(𝔤′).

In terms of this invariant and some GL(V)-invariant concepts on 𝔤, we can state an algorithm to determine the isomorphism classes of 3-dimensional Lie algebras in Figure 4.

As for real 3-dimensional Lie algebras, deformations and (a special kind of) degenerations are explained in detail in [10], [20].

Figure 4

START

abel M_{0} ≅ C^{3}

dim [𝔤, 𝔤] = 3 M_{4} ≅ 𝔰𝔩(2, C)

\begin{matrix}[X, Y] = f(X)Y − f(Y)X \\ for some f ∈ 𝔤^{∗} \end{matrix} M_{2}

dim [𝔤, 𝔤] = 1 & unimodular

M_{1} ≅ Heisenberg Lie algebra

M_{3}(α) if χ(𝔤) = \frac{(α+1)^{2}}{α}

Y
N
Y
N
Y
N
Y

N

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