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 large number of examples with compact Levi factor. Given a compact semisimple real Lie algebra u and a real representation π satisfying some technical properties, the construction returns a metric Lie algebra (u,π) with negative Ricci operator. In this paper, when u is assumed to be simple, we prove that (u,π) admits a metric having negative Ricci curvature for all but finitely many finite-dimensional irreducible representations of (Formula presented.), regarded as a real representation of u. We also prove in the last section a more general result where the nilradical is not abelian, as it is in every (u,π).