On the Variety of Four Dimensional Lie Algebras

摘要

Lie algebras of dimension n are defined by their structure constants, which can be seen as sets of N = n(2)(n - 1)/2 scalars (if we take into account the skew-symmetry condition) to which the Jacobi identity imposes certain quadratic conditions. Up to rescaling, we can consider such a set as a point in the projective space PN-1. Suppose n = 4, hence N = 24. Take a random subspace of dimension 12 in P-23, over the complex numbers. We prove that this subspace will contain exactly 1033 points giving the structure constants of some four-dimensional Lie algebras. Among those, 660 will be isomorphic to bfgl(2), 195 will be the sum of two copies of the Lie algebra of one-dimensional affine transformations, 121 will have an abelian three-dimensional derived algebra, and 57 will have for derived algebra the three dimensional Heisenberg algebra. This answers a question of Kirillov and Neretin.