Curvature spectra of simple Lie groups
| Autor: | Derdzinski, Andrzej, Gal, Swiatoslaw R. |
|---|---|
| Rok vydání: | 2013 |
| Předmět: | |
| Zdroj: | Abhandlungen aus dem Mathematischen Seminar der Universit\"at Hamburg, 2013, Volume 83, Issue 2, pp 219--230 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s12188-013-0085-z |
| Popis: | 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 This in turn leads to a proof of a known, yet unpublished, fact: namely, that a semisimple real or complex Lie algebra with no simple ideals of dimension 3 is essentially determined by its Cartan three-form. Comment: 10 pages |
| Databáze: | arXiv |
| Externí odkaz: |
|