Curvature spectra of simple Lie groups (Q376737)

Some natural operators (in particular, operators \(\Omega\) and \(\Lambda\)) on semisimple -- real and complex -- Lie groups and Lie algebras are considered. Let \(\beta\) be the Killing form of a real/complex semisimple Lie group G. This \(\beta\) gives us a left-invariant pseudo-Riemannian (or, respectively, holomorphic) metric, which is also Einsteinian one. Let \(\Omega : [g^\ast]^{\odot 2} \to [g^\ast]^{\odot 2}\) be the specific multiple of the curvature operator of the metric \(\beta\) , acting on symmetric bilinear forms \(\sigma : g \to g\), so that \([ \Omega \sigma ](x, y) = 2 tr [(Ad x)(Ad y)\Sigma ]\) for \(\Sigma : g \to g\) with \(\sigma (x, y) = \beta (\Sigma x, y)\) for all \(x,y \in g\). Next let \(\Lambda : [g^\ast]^{\odot 2} \to [g^\ast]^{\wedge 4}\) be defined by \((\Lambda \sigma)(x,y,z,z^\prime)=\sigma ([x,y],[z,z^\prime])+\sigma([y,z],[x,z^\prime])+\sigma([z,x],[y,z^\prime])\). So \(\Lambda\) is a linear operator, sending symmetric bilinear forms \(\sigma\) on \(g\) to exterior 4-forms on \(g\). We have \(\Lambda \beta = 0\) and for semisimple Lie algebra \(g\) also we have \(\Omega \beta = 2 \beta\). The authors investigate these two operators \(\Omega, \Lambda\). They describe the spectra of \(\Omega\) and the subspace \(\text{ker} \Lambda\) in case of semisimple Lie algebras (real or complex). As a corollary they prove that if \(g,h\) are real or complex Lie algebra, the Cartan three-forms for \(g\) and \(h\) (in the real case it is supposed also that \(g\) has no summands of dimension 3) are isomorphic if and if \(h\) is isomorphic to \(g\). The Cartan three-form \(C \in [g^\ast]^{\wedge 3}\) of a Lie algebra g is defined by \(C = \beta ([\cdot, \cdot ], \cdot)\).