2015/06/09 by Laurent Manivel, Manivel, Laurent
Chemistry · Mathematics · #Molecular spectroscopy and chirality #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.1506.02871
Lie algebras of dimension n are defined by their structure constants ,\nwhich can be seen as sets of N = n2 (n -- 1)/2 scalars (if we take into\naccount the skew-symmetry condition) to which the Jacobi identity imposes\ncertain quadratic conditions. Up to rescaling, we can consider such a set as a\npoint in the projective space PN--1. Suppose n =4, hence N = 24. Take\na random subspace of dimension 12 in P23 , over the complex numbers. We\nprove that this subspace will contain exactly 1033 points giving the\nstructure constants of some four dimensional Lie algebras. Among those, 660\nwill be isomorphic to gl\_2 , 195 will be the sum of two copies of the Lie\nalgebra of one dimensional affine transformations, 121 will have an abelian,\nthree-dimensional derived algebra, and 57 will have for derived algebra the\nthree dimensional Heisenberg algebra. This answers a question of Kirillov and\nNeretin.\n