Einstein solvmanifolds and the pre-Einstein derivation

TL;DR

This paper introduces the pre-Einstein derivation as a key tool to classify Einstein nilradicals and their solvable extensions, simplifying the process of identifying Einstein solvmanifolds.

Contribution

It constructs the pre-Einstein derivation for nilpotent Lie algebras and provides a variational characterization to determine Einstein nilradicals.

Findings

A convex geometry condition for Einstein nilradicals with a nice basis.

Most two-step nilpotent Lie algebras are Einstein nilradicals.

The pre-Einstein derivation is essentially unique for each nilpotent Lie algebra.

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.