Classification of orbit closures of 4–dimensional complex Lie algebras

Let g be a Lie algebra of dimension n over a field K . Then g determines a multiplication table relative to each basis {e1, . . . , en}. If [ei, ej] = ∑n k=1 γ k i,jek , then (γ k i,j) ∈ K n is called a structure for g and the γ i,j the structure constants of g. The elements of Ln(K) are exactly the Lie algebra structures. They form an affine algebraic variety and the group GLn(K) acts on Ln(K) by (g ∗ μ)(x, y) = g(μ(g(x), g(y))) . The orbits under this action are the isomorphism classes. We say that λ degenerates to μ or μ is a degeneration of λ , if μ is in the Zariski closure of the orbit of λ . We denote this by μ ∈ O(λ) or λ →deg μ . The degeneration is nontrivial if μ lies in the boundary of O(λ) . The classification of orbit closures of a given Lie algebra in general is not known. All the orbit closures of a given dimension have been determined only in low dimensions for nilpotent Lie algebras [GRH], [SEE]. Special kinds of degenerations, namely contractions, have been studied by physicists [LEV]. It is useful to study degenerations which can be realized via one–parameter subgroups. It has been asked whether every degeneration in Ln can be realized via a 1–PSG. This turns out to be true for n ≤ 3 but does not hold for n = 4 [STE]. In the case of nilpotent Lie algebras, however, it is true for all n < 7 . Nevertheless it does not hold in general: recently the first author discovered counterexamples for any dimension n ≥ 7 [BUR]. In this paper we classify all possible degenerations of Lie algebra structures in L4(C), i.e. we determine the Zariski closure of all Lie algebras λ ∈ L4(C).