  Let V be the vector space of all skew-symmetric (non-associative) complex
algebras of dimension n and L the algebraic subset of V of all Lie algebras. We
consider the moment map for the action of GL(n) on the projective space P(V)
and study the critical points of the functional F:square norm of the moment
map, in order to understand the stratification of L defined by the negative
gradient flow of F.
  We obtain a description of the critical points which lie in L in terms of
those which are nilpotent, as well as the minima and maxima of F on L. A
characterization of the critical points modulo isomorphism, as the finite union
of categorical quotients of suitable actions is considered, and some
applications to the study of L are given.
