Einstein solvmanifolds with a simple Einstein derivation

TL;DR

This paper provides a simple criterion to identify Einstein nilradicals with simple eigenvalues in solvable Lie groups and classifies filiform Einstein nilradicals, advancing understanding of Einstein metrics on solvable Lie groups.

Contribution

It introduces an easy-to-verify condition for Einstein nilradicals with simple eigenvalues and classifies filiform Einstein nilradicals, expanding the classification of Einstein solvmanifolds.

Findings

Established a necessary and sufficient condition for Einstein nilradicals with simple eigenvalues.

Classified filiform Einstein nilradicals within known Lie algebra frameworks.

Enhanced understanding of the structure of Einstein solvmanifolds.

Abstract

The structure of a solvable Lie groups admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras).