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On the moment map for the variety of Lie algebras

Jorge Lauret
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Abstract

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.

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Jorge Lauret. 2002-06-05. On the moment map for the variety of Lie algebras. https://arxiv.org/abs/math/0206046

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