Non-solvable Lie groups with negative Ricci curvature

LAURET, EMILIO A.; WILL, CYNTHIA E.

TRANSFORMATION GROUPS

BIRKHAUSER BOSTON INC

Lugar: Boston; Año: 2022 vol. 27 p. 163 - 179

1083-4362

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 amount of examples with compact Levi factor.Given a compact semisimple real Lie algebra $mathfrak u$ and a real representation $pi$ satisfying some technical properties, the construction returnsa 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 $mathfrak uotimes_{mathbb R}mathbb C$, 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 $mathfrak l(mathfrak u,pi)$.