On Ricci negative derivations

TL;DR

This paper investigates the space of diagonalizable derivations of nilpotent Lie algebras that lead to solvable extensions with negative Ricci curvature, confirming a conjecture in specific cases.

Contribution

It proves the Lauret-Will conjecture for dimension 5, Heisenberg, and standard filiform Lie algebras, advancing understanding of Ricci negative derivations.

Findings

Confirmed the conjecture in dimension 5

Validated the conjecture for Heisenberg algebras

Validated the conjecture for standard filiform Lie algebras

Abstract

Given a nilpotent Lie algebra, we study the space of all diagonalizable derivations such that the corresponding one-dimensional solvable extension admits a left-invariant metric with negative Ricci curvature. It has been conjectured by Lauret-Will that such a space coincides with an open and convex subset of derivations defined in terms of the moment map for the variety of nilpotent Lie algebras. We prove the validity of the conjecture in dimension 5, as well as for Heisenberg and standard filiform Lie algebras.