Curvature spectra of simple Lie groups
The Killing form β of a real (or complex) semisimple Lie group G is a left-invariant pseudo-Riemannian (or, respectively, holomorphic) Einstein metric. Let Ω denote the multiple of its curvature operator, acting on symmetric 2-tensors, with the factor chosen so that Ωβ=2β. The result of Meyberg [8], describing the spectrum of Ω in complex simple Lie groups G, easily implies that 1 is not an eigenvalue of Ω in any real or complex simple Lie group G except those locally isomorphic to SU(p,q), or SL(n,R), or SL(n,C) or, for even n only, SL(n/2,H), where p\ge q\ge0 and p+q=n>2. Due to the last conclusion, on simple Lie groups G other the ones just listed, nonzero multiples of the Killing form β are isolated among left-invariant Einstein metrics. Meyberg's theorem also allows us to understand the kernel of Λ, which is another natural operator. This in turn leads to a proof of a known, yet unpublished, fact: namely, that a semisimple real or complex Lie algebra with no simple ideals of dimension 3 is essentially determined by its Cartan three-form.