Classification of Orbit Closures of 4-Dimensional Complex Lie Algebras

Let n( ) be the variety of complexn-dimensional Lie algebras. The groupGLn( ) acts on it via change of basis. An orbitO(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 where λ 4( ). Starting with a classification of complex Lie algebras of dimensionn ≤ 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 3( ) can be realized via a one-parameter subgroup, but this is not the case in 4( ).