On the moment map for the variety of Lie algebras

We consider the moment map m : PVn↦iu(n) for the action of GL(n) on Vn=Λ2(Cn)∗⊗Cn. The critical points of the functional Fn=||m||2 : PVn↦R are studied, in order to understand the stratification of Ln⊂PVn defined by the negative gradient flow of Fn, where Ln is the algebraic subset of all Lie algebras. We obtain a description of the critical points which lie in Ln in terms of those which are nilpotent, as well as the minima and maxima of Fn : Ln↦R. A characterization of the critical points modulo isomorphism, as the union of categorical quotients of suitable actions is considered, and some applications to the study of Ln are given.