
Journal of Lie Theory 26 (2016), No. 1, 001010 Copyright Heldermann Verlag 2016 On the Variety of Four Dimensional Lie Algebras Laurent Manivel Institut de Mathématiques de Marseille, Université Technopôle ChâteauGombert, 39 rue Frédéric JoliotCurie, 13453 Marseille Cedex 13, France laurent.manivel@math.cnrs.fr [Abstractpdf] Lie algebras of dimension $n$ are defined by their structure constants, which can be seen as sets of $N=n^2(n1)/2$ scalars (if we take into account the skewsymmetry 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 {\bf P}$^{N1}$. Suppose $n=4$, hence $N=24$. Take a random subspace of dimension $12$ in ${\bf P}^{23}$, over the complex numbers. We prove that this subspace will contain exactly $1033$ points giving the structure constants of some fourdimensional Lie algebras. Among those, $660$ will be isomorphic to ${\bf{gl}}_2$, $195$ will be the sum of two copies of the Lie algebra of onedimensional affine transformations, $121$ will have an abelian threedimensional derived algebra, and $57$ will have for derived algebra the three dimensional Heisenberg algebra. This answers a question of Kirillov and Neretin. Keywords: Classification of Lie algebras, irreducible component, degree, resolution of singularities. MSC: 14C17, 14M99, 17B05 [ Fulltextpdf (297 KB)] for subscribers only. 