Classification of orbit closures of 4-dimensional complex Lie algebras

Abstract

Let Ln(C) be the variety of complex n-dimensional Lie algebras. The group GLn(C) acts on it via change of basis. An orbit O(μ) under this action consists of all structures isomorphic to μ. The aim of this paper is to give a complete classification of orbit closures of 4-dimensional Lie algebras, i.e., determining all μ∈O(λ)where λ∈L4(C). Starting with a classification of complex Lie algebras of dimension n≤4, we study the behavior of several Lie algebra invariants under degeneration, i.e., under transition to the orbit closure. As a corollary, we will show that all degenerations in L3(C) can be realized via a one-parameter subgroup, but this is not the case in L4(C). © 1999 Academic Press.