Non-solvable Lie groups with negative Ricci curvature

Abstract

Until a couple of years ago, the only known examples of Lie groups admitting left-invariant metrics with negative Ricci curvature were either solvable or semisimple. We use a general construction from a previous article of the second named author to produce a great amount of examples with compact Levy factor. Given a compact semisimple real Lie algebra $\mathfrak u$ and a real representation $\pi$ satisfying some technical properties, the construction returns a metric Lie algebra $\mathfrak l(\mathfrak u,\pi)$ with negative Ricci operator. In this paper, when $\mathfrak u$ is assumed to be simple, we prove that $\mathfrak l(\mathfrak u,\pi)$ admits a metric having negative Ricci curvature for all but finitely many finite-dimensional irreducible representations of $\mathfrak u\otimes_{\mathbb R} \mathbb C$, regarded as a real representation of $\mathfrak u$. We also prove in the last section a more general result where the nilradical is not abelian, as it is in every $\mathfrak l(\mathfrak u,\pi)$.