The moment map for the variety of Leibniz algebras

Abstract

We consider the moment map [Formula: see text] for the action of [Formula: see text] on [Formula: see text], and study the functional [Formula: see text] restricted to the projectivizations of the algebraic varieties of all [Formula: see text]-dimensional Leibniz algebras [Formula: see text] and all [Formula: see text]-dimensional symmetric Leibniz algebras [Formula: see text], respectively. First, we give a description of the maxima and minima of the functional [Formula: see text], proving that they are actually attained at the symmetric Leibniz algebras. Then, for an arbitrary critical point [Formula: see text] of [Formula: see text], we characterize the structure of [Formula: see text] by virtue of the nonnegative rationality. Finally, we classify the critical points of [Formula: see text] for [Formula: see text], [Formula: see text], respectively.