Curvature spectra of simple Lie groups

TL;DR

This paper investigates the curvature spectra of simple Lie groups, revealing eigenvalue properties of the curvature operator and implications for Einstein metrics, with specific focus on classical and exceptional groups.

Contribution

It extends Meyberg's results to classify eigenvalues of the curvature operator on simple Lie groups and links these spectral properties to the uniqueness of Einstein metrics.

Findings

1 is not an eigenvalue of the curvature operator except for specific Lie groups

Nonzero multiples of the Killing form are isolated among Einstein metrics on most simple Lie groups

Semisimple Lie algebras without 3-dimensional simple ideals are characterized by their Cartan three-form

Abstract

The Killing form \beta\ of a real (or complex) semisimple Lie group G is a left-invariant pseudo-Riemannian (or, respectively, holomorphic) Einstein metric. Let \Omega\ denote the multiple of its curvature operator, acting on symmetric 2-tensors, with the factor chosen so that \Omega\beta=2\beta. The result of Meyberg [8], describing the spectrum of \Omega\ in complex simple Lie groups G, easily implies that 1 is not an eigenvalue of \Omega\ 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 \beta\ are isolated among left-invariant Einstein metrics. Meyberg's theorem also allows us to understand the kernel of \Lambda, 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 This in turn leads to a proof of a known, yet unpublished, fact: namely, that a semisimple real or complex Lie algebra with no simple ideals of dimension 3 is essentially determined by its Cartan three-form.