Journal of Lie Theory Volume 9 (1999) 193–202 C ◯1999 Heldermann Verlag

Degenerations of nilpotent Lie algebras

Dietrich Burde

Communicated by K.–H. Neeb

Abstract. In this paper we study degenerations of nilpotent Lie algebras. If λ, μ are two points in the variety of nilpotent Lie algebras, then λ is said to degenerate to μ, λ →_{deg} μ, if μ lies in the Zariski closure of the orbit of λ. It is known that all degenerations of nilpotent Lie algebras of dimension n < 7 can be realized via a one-parameter subgroup. We construct degenerations between characteristically nilpotent filiform Lie algebras. As an application it follows that for any dimension n ≥ 7 there exist examples of degenerations of nilpotent Lie algebras which cannot be realized via a one–parameter subgroup.

1. Introduction

Let V be a vector space of dimension n over a field K. An n– dimensional Lie algebra 𝔤 may be considered as an element λ of the affine variety Hom(Λ^{2}V, V) via the bilinear skew–symmetric mapping λ : 𝔤 ⊗ 𝔤 ↦ 𝔤 defining the Lie bracket on 𝔤. The set of Lie algebra structures is an algebraic subset L_{n} of the variety Hom(Λ^{2}V, V) and the linear reductive group GL_{n}(K) acts on L_{n} by (g ∗ μ)(x, y) = g(μ(g^{−1}(x), g^{−1}(y))). The orbits under this action are the isomorphism classes. We say that λ degenerates to μ, if μ is in \overline{O(λ)}, the Zariski closure of the orbit of λ. We denote this by λ →_{deg} μ. The degeneration is nontrivial if μ lies in the boundary of O(λ). In this paper we consider the GL_{n}(K)–stable subvariety N_{n} of L_{n} consisting of nilpotent Lie algebras. In particular, we deal with the open subset F_{n} of N_{n} consisting of filiform nilpotent Lie algebras. If not otherwise declared we will assume K = ℂ. The determination and classification of orbit closures is a difficult problem. An important tool is the study of degenerations via a one–parameter subgroup, in short 1–PSG. Here there are many results known for the action of a linear reductive group on an algebraic variety (e.g., the theorems of Hilbert, Kraft and Mumford). In [6] there is discussed whether every degeneration of A–modules can be obtained via a 1–PSG. (See the remark following the theorem of Hilbert–Mumford–Birkes, p. 232. Here A is a finitely generated associative ℂ–algebra.) It is shown that for two A–modules M, N ∈ mod_{A,V} there is a 1–PSG g : ℂ^{∗} → GL(V) with

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag lim_{t→0} g_{t} · M = N if and only if there is a filtration on M such that the associated graded module is isomorphic to N. In fact, there exist degenerations of A–modules which cannot be realized via a 1–PSG. This question is more difficult for the variety of nilpotent Lie algebras. It is known that all degenerations of nilpotent Lie algebras of dimension n < 7 can be realized via a one–parameter subgroup. In higher dimensions this was not known. In [4] a criterion is proved deciding whether a nilpotent Lie algebra is a degeneration of some other nilpotent Lie algebra via a one–parameter subgroup. It is the analog of Kraft’s criterion: If μ is a degeneration of λ via a one–parameter subgroup g_{t}, then μ is the associated graded Lie algebra given by the filtration on λ induced by g_{t}. Also the converse holds. We will establish nontrivial degenerations between characteristically nilpotent Lie algebras. By the above criterion such a degeneration cannot be realized via a one–parameter subgroup: If λ, μ are characteristically nilpotent Lie algebras, then they do not admit such a gradation. It is natural to look for nilpotent Lie algebras of maximal nilpotence index, i.e., filiform Lie algebras. We construct column–degenerations of filiform nilpotent Lie algebras. These are the main degenerations λ →_{deg} μ which are possible such that λ and μ are both filiform nilpotent.

2. Preliminaries

A Lie algebra 𝔤 over a field K determines a multiplication table relative to each basis of 𝔤. If [e_{i}, e_{j}] = ∑_{k=1}^{n} γ_{ij}^{k}e_{k}, then (γ_{ij}^{k}) ∈ K^{n^{3}} is called a structure for 𝔤. A point in L_{n} is a Lie algebra structure which can be identified with the bilinear skew–symmetric mapping λ : 𝔤 ⊗ 𝔤 → 𝔤 defining the Lie bracket on 𝔤. Since the Jacobi identity and the antisymmetry are defined by polynomial conditions, i.e., by (n^{3} − n)/6 algebraic equations, L_{n} is an affine algebraic subvariety of Hom(Λ^{2}V, V). GL_{n}(K) acts on L_{n} via change of basis, i.e. by (g ∗ μ)(x, y) = g(μ(g^{−1}(x), g^{−1}(y))). An orbit O(μ) under this action consists of all structures in a single isomorphism class.

Definition 1. A Lie algebra λ is said to degenerate to another Lie algebra μ, if μ is represented by a structure which lies in the Zariski closure of the GL_{n}(K)–orbit of a structure which represents λ. In this case the entire orbit O(μ) lies in the closure of O(λ). We denote this by λ →_{deg} μ.

Degeneration (for isomorphism classes) is transitive: If λ →_{deg} μ and μ →_{deg} ν, then λ →_{deg} ν. Let K = ℂ. It is known that the usual analytic topology on ℂ^{n^{3}} leads to the same degenerations as does the Zariski topology. Therefore the following condition will imply that λ →_{deg} μ:

∃ g_{t} ∈ GL_{n}(ℂ(t)) such that lim_{t→0} g_{t} ∗ λ = μ

Here ℂ(t) is the field of fractions of the polynomial ring ℂ[t].

Example 1. Any n–dimensional Lie algebra λ degenerates to the abelian Lie algebra K^{n} which corresponds to 0 ∈ Hom(Λ^{2}V, V): Let g_{t} = t^{−1}E_{n}, where E_{n} is the identity matrix. Then we have (g_{t} ∗ λ)(x, y) = t^{−1}λ(tx, ty) = tλ(x, y), hence lim_{t→0} g_{t} ∗ λ = 0.

Let Z(λ) denote the center of the Lie algebra λ and [λ, λ] the commutator subalgebra. To construct degenerations we have to respect the following necessary conditions:

Lemma 1. Let λ →_{deg} μ be a nontrivial degeneration. Then it follows that: (1) dim Der(λ) < dim Der(μ)

(2) dim O(λ) > dim O(μ)

(3) dim[λ, λ] ≥ dim[μ, μ]

(4) dim Z(λ) ≤ dim Z(μ)

(5) rank(λ) ≤ rank(μ) If λ is solvable of step k, then μ is solvable of step ≤ k. The same holds for nilpotent. In that case, dim λ^{(i)} ≥ dim μ^{(i)} where λ^{(1)} = λ, λ^{(i+1)} = [λ, λ^{(i)}].

Note that dim O(λ) = (dim λ)^{2} − dim Der(λ). The proof can be found more or less in the literature, see [1], [7]. The main argument relies on the following fact: Let B be a Borel subgroup of GL_{n}(K) and λ, μ ∈ N_{n}. If λ →_{deg} μ and λ lies in a B–stable closed subset R ⊂ N_{n}, then μ must also be represented by a structure in R. We remark that a Borel subgroup plays an important role for degenerations. We will use also Borel subgroups in Proposition 3. The following fact is proved in [4]: Let G be a complex reductive algebraic group acting rationally on some algebraic set X. Let B be a Borel subgroup of G. Then \overline{G ∗ x} = G ∗ \overline{(B ∗ x)} for all x ∈ X.

Definition 2. A degeneration λ →_{deg} μ is called a one–parameter subgroup degeneration (1–PSG) if it can be realized by a group homomorphism g : K^{∗} → GL_{n}(K) such that μ = lim_{t→0} g_{t} ∗ λ.

The notion of a 1–PSG degeneration is independent of the choice of a basis. We have the following criterion [4]:

Proposition 1. If λ →_{deg} μ via a 1–PSG then μ is the associated ℤ–graded Lie algebra given by the filtration on λ induced by g_{t}. Conversely, if μ is the associated graded Lie algebra given by some filtration of λ then λ →_{deg} μ via a 1–PSG.

Example 2. Let K = ℂ. Every degeneration λ →_{deg} μ of nilpotent Lie algebras of dimension n < 7 can be obtained via one–parameter subgroups [4], [7]. Every degeneration in L_{3} can be realized via a 1–PSG, but not in L_{4} [1].

Definition 3. A Lie algebra 𝔤 is called characteristically nilpotent if all its derivations are nilpotent.

Lemma 2. Let 𝔤 be a characteristically nilpotent Lie algebra. Then 𝔤 is nilpotent and admits no nontrivial ℤ–gradation.

Proof. The first part follows by Engel’s theorem since all inner derivations ad(x) are nilpotent. Suppose that 𝔤 = ⨁_{i∈ℤ} 𝔤_{i} is a gradation. Then we can construct a derivation by D(x_{i}) = ix_{i} for all x_{i} ∈ 𝔤_{i} which clearly is not nilpotent. This is a contradiction.

Characteristically nilpotent algebras form a relatively large subclass of nilpotent Lie algebras. In [3] the following result is proved:

Proposition 2. For n ≥ 8 any irreducible component of the variety F_{n} contains a nonempty Zariski–open subset of characteristically nilpotent Lie algebras.

Finally, let us mention the connection between degenerations and deformations. Gerstenhaber’s definition of a deformation is as follows: Let V be a vector space over K and λ ∈ Hom(Λ^{2}V, V) be a Lie product. If λ_{t} := λ+tϕ_{1}+t^{2}ϕ_{2}+t^{3}ϕ_{3}+… is a Lie product on V ⊗_{K} K((t)) where K((t)) is the formal power series field and ϕ_{i} ∈ Hom(Λ^{2}V, V), then λ_{t} is called a deformation of λ.

A more general definition is given in [2]. This paper deals with the connection of degenerations and deformations. It is proved that every nontrivial degeneration λ →_{deg} μ defines a nontrivial deformation of μ. The converse is not true. However, there are special classes of deformations which do define degenerations. In the following we will consider infinitesimal deformations of filiform Lie algebras which define, under certain conditions, degenerations between filiform Lie algebras.

3. Degenerations of filiform Lie algebras

We will construct degenerations between filiform Lie algebras in any dimension n ≥ 7 which cannot be realized via 1–PSGs. Let L = L(n) be the standard graded filiform Lie algebra generated by e_{1}, …, e_{n} with Lie brackets [e_{1}, e_{i}] = e_{i+1} for i = 2, …, n − 1. Denote by H^{2}(L, L) the second Lie algebra cohomology of L with adjoint coefficients and by {F_{k}H^{2}(L, L)}_{k∈ℤ} the canonical filtration of this space [5]. Let ψ be an integrable 2–cocycle and define the Lie algebra L_{ψ} by [x, y]_{ψ} = [x, y]_{L} + ψ(x, y). There is a canonical basis for H^{2}(L, L). Define the index set I_{n}^{0} := {(k, s) ∈ ℕ^{2} | 2 ≤ k ≤ [n/2], 2k + 1 ≤ s ≤ n} and let

I_{n} = \{ \begin{matrix}I_{n}^{0}, & n ≡ 1(2) \\ I_{n}^{0} ∪ \{(\frac{n}{2}, n)\}, & n ≡ 0(2) \end{matrix}

The space F_{1}H^{2}(L, L) has a canonical basis consisting of the cohomology classes of the 2–cocycles ψ_{k,s} for (k, s) ∈ I_{n}. The 2–cocycles are defined by ψ_{k,s}(e_{i}, e_{i+1}) = δ_{ik}e_{s}. We have

dim F_{1}H^{2}(L, L) = \{ \begin{matrix}\frac{(n − 3)^{2}}{4} , & n ≡ 1(2) \\ \frac{(n − 4)(n − 2) + 4}{4}, & n ≡ 0(2) \end{matrix}

The following result is known [5]:

Lemma 3. Every filiform nilpotent Lie algebra of dimension n ≥ 3 is isomorphic to an infinitesimal deformation L_{ψ} of L where ψ is an integrable 2–cocycle whose cohomology class lies in F_{1}H^{2}(L, L).

It follows that we can obtain a special form for the structure constants γ_{ij}^{k} of a filiform Lie algebra:

Lemma 4. Let 𝔤 be a complex filiform nilpotent Lie algebra of dimension n. Then there exists a basis {e_{1}, …, e_{n}} such that

(a) [e_{1}, e_{i}] = e_{i+1} for i = 2, …, n − 1

(b) The structure constants in [e_{i}, e_{j}] = ∑_{k} γ_{ij}^{k}e_{k}, 2 ≤ i < j can be written as

γ_{ij}^{k} = ∑_{ℓ=0}^{[(j−i−1)/2]}(−1)^{ℓ}\binom{j − i − ℓ − 1}{ℓ}α_{i+ℓ}^{k−j+i+2ℓ+1}

where the constants α_{i}^{j} are zero for all pairs (i, j) not in I_{n}.

There are (n−3)^{2}/4 structure constants α_{i}^{j} if n is odd, and \frac{1}{4}(n−2)(n− 4) + 1 otherwise. The formula above yields a convenient way to describe filiform Lie algebras. The Jacobi identity is not satisfied automatically, unless n < 8. However, the polynomial conditions are much simpler with respect to the above basis. We may represent such a filiform Lie algebra by a diagram of the structure constants: Let N = [\frac{n}{2}]:

α_{2}^{5} α_{2}^{6} · · · · · · α_{2}^{n-1} α_{2}^{n}
α_{3}^{7} α_{3}^{8} ⋯ α_{3}^{n}
· · · · · ·
α_{N}^{n}

Denote the columns of that diagram from the left by A_{n−4}, A_{n−3}, …, A_{2}, A_{1} .

Example 3. Let 𝔤 be a complex filiform Lie algebra of dimension 9. Then there is a basis {e_{1}, …, e_{9}} such that

\begin{aligned}[e_{1}, e_{i}] & = e_{i+1}, i ≥ 2 \\ [e_{2}, e_{3}] & = α_{1}e_{5} + α_{2}e_{6} + α_{3}e_{7} + α_{4}e_{8} + α_{5}e_{9} \\ [e_{2}, e_{4}] & = α_{1}e_{6} + α_{2}e_{7} + α_{3}e_{8} + α_{4}e_{9} \\ [e_{2}, e_{5}] & = (α_{1} − α_{6})e_{7} + (α_{2} − α_{7})e_{8} + (α_{3} − α_{8})e_{9} \\ [e_{2}, e_{6}] & = (α_{1} − 2α_{6})e_{8} + (α_{2} − 2α_{7})e_{9} \\ [e_{2}, e_{7}] & = (α_{1} − 3α_{6} + α_{9})e_{9} \\ [e_{3}, e_{4}] & = α_{6}e_{7} + α_{7}e_{8} + α_{8}e_{9} \\ [e_{3}, e_{5}] & = α_{6}e_{8} + α_{7}e_{9} \\ [e_{3}, e_{6}] & = (α_{6} − α_{9})e_{9} \\ [e_{4}, e_{5}] & = α_{9}e_{9} \end{aligned}

where 𝔤 depends on 9 parameters {α_{k}^{s} | (k, s) ∈ I_{9}} = {α_{1}, …, α_{9}}. The Jacobi identity for 𝔤 is equivalent to α_{9}(2α_{1} + α_{6}) − 3α_{6}^{2} = 0. The diagram is given by

α^{1} α^{2} α^{3} α^{4} α^{5}

α^{6} α^{7} α^{8}

α_{9}

Let λ be a complex filiform Lie algebra of dimension n with Lie brackets given in Lemma 4, i.e., λ is an infinitesimal deformation of L, the standard graded filiform. The question is whether there exists a degeneration λ →_{deg} L. In general, the answer is no – see the Lie algebra of example 1.5. in [2], which is the filiform Lie algebra λ with n = 7. The bracket [e_{2}, e_{4}] given there should be equal to βe_{6} + γe_{7} . However, under certain conditions one can degenerate “by columns”, and iterating this process will yield a degeneration to L.

Definition 4. Let λ be as above and k ∈ {2, …, n − 5}. Suppose that the columns A_{n−4}, … A_{k+2} of the diagram of λ have only zero entries, but A_{k+1} has a nonzero entry. Let T_{n}(ℂ(t)) denote the Borel subgroup of GL_{n}(ℂ(t)) consisting of lower–triangular matrices and define a matrix g_{t,k}^{−1} ∈ T_{n}(ℂ(t)) by

\begin{aligned}g_{t,k}^{−1}(e_{1}) & = te_{1} \\ g_{t,k}^{−1}(e_{i}) & = t^{n+i−4−k}(e_{i} + ∑_{j=1}^{k−1} f_{j}(s)e_{i+j+1}), i ≥ 2 \end{aligned}

with polynomial functions f_{i}(s) ∈ ℂ(α_{k}^{l})[s] in the variable s = 1 − t and coefficients in ℂ(α_{k}^{l}). The inverse matrix g_{t,k} is called the column degeneration matrix of level k.

This matrix will realize a degeneration λ →_{deg} μ_{0,k} to a filiform Lie algebra μ_{0,k} as follows: Under certain conditions on the polynomials f_{i} the Lie algebra μ_{t,k} := g_{t,k} ∗ λ will have the same diagram as λ except for the entries of the first column A_{k+1} multiplied by t. In that case μ_{0,k} := lim_{t→0} μ_{t,k} (which is just μ_{t,k} for t = 0) is a degeneration of λ. Then the columns A_{n−4}, …, A_{k+1} of the diagram of μ_{0,k} are zero. There are of course necessary conditions on such polynomials f_{i}, e.g., the properties of Lemma 1. In particular, dim Der(λ) < dim Der(μ_{0,k}) must be satisfied. If k = 2 then these conditions are obvious and we obtain nontrivial degenerations between filiform Lie algebras of any dimension n ≥ 7:

Proposition 3. Let λ be a complex filiform Lie algebra of dimension n ≥ 7 with basis {e_{1}, …, e_{n}}, α = α_{2}^{n−2}, β = α_{2}^{n−1}, γ = α_{2}^{n}, δ = α_{3}^{n} and defining brackets

\begin{aligned}λ(e_{1}, e_{i}) & = e_{i+1}, i ≥ 2 \\ λ(e_{2}, e_{3}) & = αe_{n−2} + βe_{n−1} + γe_{n} \\ λ(e_{2}, e_{4}) & = αe_{n−1} + βe_{n} \\ λ(e_{2}, e_{5}) & = (α − δ)e_{n} \\ λ(e_{3}, e_{4}) & = δe_{n} \end{aligned}

and let s = 1 − t and f(s) = \frac{γ}{2δ} s with δ ≠ 0. Define g_{t} ∈ T_{n}(ℂ(t)) by

\begin{aligned}g_{t}(e_{1}) & = \frac{1}{t}e_{1} \\ g_{t}(e_{i}) & = ∑_{j=0}^{[(n−i)/2]}(−1)^{j} \frac{f(s)^{j}}{t^{n+i+2j−6}} e_{i+2j}, i ≥ 2 \end{aligned}

Then μ_{t} := g_{t} ∗ λ is a filiform Lie algebra with brackets

\begin{aligned}μ_{t}(e_{1}, e_{i}) & = e_{i+1}, i ≥ 2 \\ μ_{t}(e_{2}, e_{3}) & = tαe_{n−2} + βe_{n−1} + γe_{n} \\ μ_{t}(e_{2}, e_{4}) & = tαe_{n−1} + βe_{n} \\ μ_{t}(e_{2}, e_{5}) & = t(α − δ)e_{n} \\ μ_{t}(e_{3}, e_{4}) & = tδe_{n} \end{aligned}

Therefore λ degenerates to μ := lim_{t→0} μ_{t}: α β γ ; δ ⟶_{deg}  0 β γ ; 0

Proof. Note that the Jacobi identity for λ is satisfied. The inverse of g_{t} is given by

\begin{aligned}g_{t}^{−1}(e_{1}) & = te_{1} \\ g_{t}^{−1}(e_{i}) & = t^{n+i−6}(e_{i} + f(s)e_{i+2}), i ≥ 2 \end{aligned}

We have to compute the Lie brackets of μ_{t}. The crucial ones are the following:

μ_{t}(e_{1}, e_{i}) = g_{t}(λ(g_{t}^{−1}(e_{1}), g_{t}^{−1}(e_{i}))) = g_{t}(λ(te_{1}, t^{n+i−6}e_{i} + f(s)t^{n+i−6}e_{i+2}))

= t^{n+i−5}( ∑_{j=0}^{[(n−i−1)/2]}(−1)^{j} \frac{f(s)^{j}}{t^{n+i+2j−5}} e_{i+1+2j}

+ ∑_{j=0}^{[(n−i−3)/2]}(−1)^{j} \frac{f(s)^{j+1}}{t^{n+i+2(j+1)−5}} e_{i+1+2(j+1)}) = e_{i+1}

μ_{t}(e_{2}, e_{3}) = g_{t}(λ(g_{t}^{−1}(e_{2}), g_{t}^{−1}(e_{3})))

= g_{t}(λ(t^{n−4}e_{2} + f(s)t^{n−4}e_{4}, t^{n−3}e_{3} + f(s)t^{n−3}e_{5}))

= t^{2n−7}g_{t}(αe_{n−2} + βe_{n−1} + (γ + (α − 2δ)f(s))e_{n})

= t^{2n−7}(\frac{α}{t^{2n−8}} e_{n−2} − \frac{αf(s)}{t^{2n−6}}e_{n} + \frac{β}{t^{2n−7}} e_{n−1} + \frac{γ + (α − 2δ)f(s)}{t^{2n−6}}e_{n})

= tαe_{n−2} + βe_{n−1} + \frac{γ − γ(1 − t)}{t}e_{n}

= tαe_{n−2} + βe_{n−1} + γe_{n}

μ_{t}(e_{3}, e_{4}) = g_{t}(λ(g_{t}^{−1}(e_{3}), g_{t}^{−1}(e_{4})))
= g_{t}(λ(t^{n−3}e_{3} + f(s)t^{n−3}e_{5}, t^{n−2}e_{4} + f(s)t^{n−2}e_{6}))
= t^{2n−5}g_{t}(δe_{n}) = tδe_{n}

Corollary 1. Let α, β, γ, δ ≠ 0. Then the above degeneration λ →_{deg} μ cannot be realized via a 1–PSG. Proof. The degeneration is nontrivial since λ is not isomorphic to μ: The commutator subalgebra [μ, μ] is abelian, whereas [λ, λ] is not. It is easy to see, that μ is characteristically nilpotent: For β, γ ≠ 0 the derivations are stricly lower triangular matrices relative to the given basis and hence nilpotent. By Lemma 2, μ does not admit any nontrivial ℤ–gradation. Hence the corollary follows by Proposition 1.

To show that Proposition 3 can be generalized for k = 3 we state the following result:

Proposition 4. Let λ be a complex filiform Lie algebra of dimension n ≥ 8 with basis {e_{1}, …, e_{n}} and defining brackets

\begin{aligned}λ(e_{1}, e_{i}) & = e_{i+1}, i ≥ 2 \\ λ(e_{2}, e_{3}) & = αe_{n−3} + βe_{n−2} + γe_{n−1} + δe_{n} \\ λ(e_{2}, e_{4}) & = αe_{n−2} + βe_{n−1} + γe_{n} \\ λ(e_{2}, e_{5}) & = (α − ε)e_{n−1} + (β − κ)e_{n} \\ λ(e_{2}, e_{6}) & = (α − 2ε)e_{n} \\ λ(e_{3}, e_{4}) & = εe_{n−1} + κe_{n} \\ λ(e_{3}, e_{5}) & = εe_{n} \end{aligned}

Let s = 1 − t, ε ≠ 0 and define g_{t} ∈ T_{n}(ℂ(t)) by

\begin{aligned}g_{t}^{−1}(e_{1}) & = te_{1} \\ g_{t}^{−1}(e_{i}) & = t^{n+i−7}(e_{i} + f_{1}(s)e_{i+2} + f_{2}(s)e_{i+3}), i ≥ 2 \end{aligned}

where

\begin{aligned}f_{1}(s) & = \frac{γ}{2ε}s \\ f_{2}(s) & = \frac{−δ}{3ε} s^{2} + (\frac{2δε − γκ}{3ε^{2}})s. \end{aligned}

Then the Lie algebra μ_{t} := g_{t} ∗ λ has the same brackets as λ except for α, ε replaced by tα, tε. Hence λ degenerates to the limit algebra μ:

α β γ δ ; ε κ ⟶_{deg}  0 β γ δ ; 0 κ

Proof. The proof consists of the computation of the Lie brackets.

For k ≥ 4 it is no longer true that the conditions of Lemma 1 do not imply conditions on the structure constants of 𝔤. Let 𝔤 be as in example 3. The Jacobi identity is equivalent to α_{9}(2α_{1} + α_{6}) − 3α_{6}^{2} = 0. In order to obtain a column degeneration of level 4, the following conditions (given by Lemma 1) have to be satisfied:

α_{6}, 2α_{6} − α_{9} ≠ 0, α_{3}α_{9} = α_{6}α_{8}

Then the degeneration

α^{1} α^{2} α^{3} α^{4} α^{5} ; α^{6} α^{7} α^{8} ; α_{9} ⟶_{deg}  0 α^{2} α^{3} α^{4} α^{5} ; 0 α^{7} α^{8} ; 0

is given by the following matrix:

g_{t,4}^{−1} = ( \begin{matrix}t & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & t^{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & t^{4} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & f_{1}(s)t^{3} & 0 & t^{5} & 0 & 0 & 0 & 0 & 0 \\ 0 & f_{2}(s)t^{3} & f_{1}(s)t^{4} & 0 & t^{6} & 0 & 0 & 0 & 0 \\ 0 & f_{3}(s)t^{3} & f_{2}(s)t^{4} & f_{1}(s)t^{5} & 0 & t^{7} & 0 & 0 & 0 \\ 0 & 0 & f_{3}(s)t^{4} & f_{2}(s)t^{5} & f_{1}(s)t^{6} & 0 & t^{8} & 0 & 0 \\ 0 & 0 & 0 & f_{3}(s)t^{5} & f_{2}(s)t^{6} & f_{1}(s)t^{7} & 0 & t^{9} & 0 \\ 0 & 0 & 0 & 0 & f_{3}(s)t^{6} & f_{2}(s)t^{7} & f_{1}(s)t^{8} & 0 & t^{10} \end{matrix} )

Here s = 1 − t and f_{i} are polynomials in ℂ[s] as follows:

f_{1}(s) = \frac{α_{3}}{2α_{6}} s,

f_{2}(s) = \frac{−α_{4}}{3α_{6}} s^{2} + \frac{2α_{4}α_{6} − α_{3}α_{7}}{3α_{6}^{2}} s,

f_{3}(s) = \frac{α_{5}}{2(2α_{6} − α_{9})}s^{3} + \frac{2α_{3}^{2}α_{6} + α_{3}^{2}α_{9} − 12α_{6}^{2}α_{5} + 4α_{6}α_{4}α_{7}}{8α_{6}^{2}(2α_{6} − α_{9})} s^{2} +

\frac{α_{3}α_{7}^{2} − α_{3}^{2}α_{9} + 3α_{6}^{2}α_{5} − 2α_{6}α_{4}α_{7}}{2α_{6}^{2}(2α_{6} − α_{9})} s

References

[1] Steinhoff, C., Klassifikation und Degeneration von Lie Algebren, Diplomarbeit, Düsseldorf, 1997.

[2] Fialowsky, A., and O’Halloran, J., A comparison of deformations and orbit closure, Comm. in Algebra 18 (1990), 4121–4140.

[3] Goze, M., and Y. B. Hakimjanov, Sur les algèbres de Lie nilpotentes admettant un tore de derivations, Manuscripta Math. 84 (1994), 115– 224.

[4] Grunewald, F., and J. O’Halloran, Varieties of nilpotent Lie algebras of dimension less than six, J. Algebra 112 (1988), 315–325.

[5] Hakimjanov, Y. B., Characteristically nilpotent Lie algebras, Math. USSR 70 (1991), 65–78.

[6] Kraft, H. P., Geometric methods in representation theory, Representation of algebras, Springer Lecture Notes in Math. 944 (1982), 180–246.

[7] Seeley, C., Degenerations of 6–dimensional nilpotent Lie algebras over ℂ, Comm. Algebra 18 (1990), 3493–3505.

Dietrich Burde Mathematisches Institut Universität Düsseldorf D-40225 Düsseldorf Germany

Received February 27, 1998 and in final form July 22, 1998