Einstein solvmanifolds and the pre-Einstein derivation

Abstract

Abstract. An Einstein nilradical is a nilpotent Lie algebra which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable extension. For every nilpotent Lie algebra, we construct an (essentially unique) derivation, the pre-Einstein derivation, the solvable extension by which may carry an Einstein inner product. Using the pre-Einstein derivation, we then give a variational characterization of Einstein nilradicals. As an application, we prove an easy-to-check convex geometry condition for a nilpotent Lie algebra with a nice basis to be an Einstein nilradical and also show that a typical two-step nilpotent Lie algebra is an Einstein nilradical.

On distinguished orbits of reductive representations

“…In addition to the aforementioned motivation to study distinguished orbits, a second reason comes from the intriguing interplay between the Ricci flow on nilpotent Lie groups and the gradient flow of the norm squared of the moment map associated to the natural action of GL n (R) on V = Λ 2 (R n ) * ⊗ R n ; the distinguished orbits of nilpotent Lie brackets are in 1-1 correspondence with nilsoliton metrics on simply connected nilpotent Lie groups (see, for instance, [Lau2]). Nikolayevsky proves in [Nik2] many theorems on Einstein nilradicals (nilpotent Lie algebras admitting a nilsoliton metric) by using the results given in [RS]. Among these results, we can highlight the Nikolayevsky nice basis criterium, which provides an easy-to-check convex geometry condition for a nilpotent Lie algebra with a nice basis to admit a nilsoliton metric.…”

“…In this section, we formulate and prove a generalization of Nikolayevsky's nice basis criterium ([Nik2,Theorem 3.]). We begin with an elementary proof of the convexity of m a (A ⋅ v) where A is a connected abelian Lie group without compact factor acting linearly by symmetric operators on a real vector space V (with respect some inner product ⟨⋅, ⋅⟩ on V ).…”

“…In [Lau1], Jorge Lauret noted that there is an intriguing relationship between the geometry of nilpotent Lie groups and the geometric invariant theory (GIT) applied to actions of reductive subgroups of GL m (R) on Λ 2 (R m ) * ⊗R m . For instance, the recent advances in the study of Einstein solvmanifolds and, more generally, Solvsolitons, have come from using powerful tools that are given by GIT (see [Lau2,Nik2]). By using this fact, it has been proposed in [Lau1] a way to study the problem of finding "the best metric" which is compatible with a fixed geometric structure on a simply connected nilpotent Lie group.…”

“…The first infinite families of symplectic Lie algebras appear in dimension 6. In a forthcoming paper, we develop analogous results to those in [Nik2] for the symplectic case and these are used to classify minimal metrics on 6-dimensional symplectic Lie algebras.…”

“…By following Nikolayevsky's ideas given in [Nik1,Nik2], we introduce the notion of nice space of a real reductive representation and we show that such criterium is a general fact of the theory of real reductive representations which are rational. To do this, we give an elementary proof of the convexity of m a (A ⋅ v) where A is a connected abelian Lie group with out compact factor acting linearly and by symmetric operators on a finite dimensional real vector space V .…”