Four-dimensional Lie algebras revisited

We discuss a system of 16 quadratic equations in 24 variables that arises in the study of Lie algebras. The solutions are the Lie algebra structures on a 4-dimensional vector space. There are four irreducible components of dimension 11. We compute their degrees and Hilbert polynomials, and thereby answer a 1999 question by Kirillov and Neretin. This is joint work with Laurent Manivel and Svala Sverrisdottir.