Negative eigenvalues of the Ricci operator of solvable metric Lie algebras

TL;DR

This paper establishes a precise condition under which the Ricci operator of solvable metric Lie algebras has multiple negative eigenvalues, highlighting implications for non-unimodular and non-abelian nilpotent cases.

Contribution

It provides a necessary and sufficient condition for negative eigenvalues of the Ricci operator in solvable metric Lie algebras, advancing understanding of their geometric properties.

Findings

Ricci operator of certain solvable Lie algebras has at least two negative eigenvalues

Non-unimodular solvable metric Lie algebras exhibit this property

Non-abelian nilpotent metric Lie algebras also have at least two negative eigenvalues

Abstract

In this paper we get a necessary and sufficient condition for the Ricci operator of a solvable metric Lie algebra to have at least two negative eigenvalues. In particular, this condition implies that the Ricci operator of every non-unimodular solvable metric Lie algebra or every non-abelian nilpotent metric Lie algebra has this property.