On Ricci negative Lie groups

TL;DR

This paper reviews known results and open questions about Lie groups with negative Ricci curvature, introduces a convex cone of derivations related to nilpotent Lie algebras, and explores conditions for Ricci negativity.

Contribution

It provides an overview of Ricci negative Lie groups and introduces a new convex cone of derivations that parametrizes solvable Lie algebras with Ricci negative metrics.

Findings

Introduction of the convex cone C(n) of derivations for nilpotent Lie algebras

Connection between derivations and Ricci negative metrics on solvable Lie groups

Discussion of open questions and conjectures in the solvable case

Abstract

We give an overview of what is known on Lie groups admitting a left-invariant metric of negative Ricci curvature, including many natural questions and conjectures in the solvable case. We also introduce an open and convex cone C(n) of derivations attached to each nilpotent Lie algebra n, which is defined as the image of certain moment map and parametrizes a set of solvable Lie algebras with nilradical n admitting Ricci negative metrics.