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Curvature spectra of simple Lie groups

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. We observe that the result of Meyberg (in Abh. Math. Semin. Univ. Hamb. 54:177–189, 1984), describing the spectrum of Ω in complex simple Lie groups, easily leads to an analogous description for real simple Lie groups. In particular, 1 is not an eigenvalue of Ω in any real or complex simple Lie group G except those locally isomorphic to SL( $n,\mathbb {C}$ ) or one of its real forms. As shown in our recent paper (Derdzinski and Gal in Indiana Univ. Math. J., to appear), the last conclusion implies that, on such simple Lie groups G, 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 This in turn leads to a proof of a known, yet unpublished, fact: namely, that a semisimple real or complex Lie algebra with no simple ideals of dimension 3 is essentially determined by its Cartan three-form. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

Curvature spectra of simple Lie groups

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References (10)

Publisher
Springer Journals
Copyright
Copyright © 2013 by Mathematisches Seminar der Universität Hamburg and Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Algebra; Differential Geometry; Combinatorics; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/s12188-013-0085-z
Publisher site
See Article on Publisher Site

Abstract

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. We observe that the result of Meyberg (in Abh. Math. Semin. Univ. Hamb. 54:177–189, 1984), describing the spectrum of Ω in complex simple Lie groups, easily leads to an analogous description for real simple Lie groups. In particular, 1 is not an eigenvalue of Ω in any real or complex simple Lie group G except those locally isomorphic to SL( $n,\mathbb {C}$ ) or one of its real forms. As shown in our recent paper (Derdzinski and Gal in Indiana Univ. Math. J., to appear), the last conclusion implies that, on such simple Lie groups G, 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.

Journal

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Oct 31, 2013

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