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 𝔩(𝔲, π) admits a metric having negative Ricci curvature for all but finitely many finite-dimensional irreducible representations of 𝔲⨂_ℝℂ, 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).

直到几年前，唯一已知的承认具有负 Ricci 曲率的左不变度量的李群例子要么是可解的，要么是半简单的。我们使用第二位作者之前文章中的一般结构来产生大量例子具有紧凑的 Levi 因子。给定一个紧凑的半单实李代数 𝔲 和一个满足某些技术属性的实数表示π ，该构造返回一个带有负 Ricci 算子的度量李代数 (𝔲, π )。在本文中，当 u 被假设为简单时，我们证明 𝔩(𝔲, π ) 承认一个具有负 Ricci 曲率的度量，除了有限多个有限维不可约表示 𝔲⨂ _ℝℂ，被视为𝔲的真实表示。我们还在最后一节中证明了一个更一般的结果，其中 nilradical 不是阿贝尔的，因为它在每个 (𝔲, π ) 中都是如此。