CONFORMAL PROPERTIES OF INDEFINITE BI-INVARIANT METRICS

• Published in: Transformation groups 2021-09, Vol.26 (3), p.859-892
• Main Authors: FRANCIS-STAITE, KELLI, LEISTNER, THOMAS
• Format: Article
• Language: English
• Subjects: Algebra; Invariants; Lie Groups; Mathematics; Mathematics and Statistics; Questions; Topological Groups
• ISSN: 1083-4362; 1531-586X
• DOI: 10.1007/s00031-020-09561-9

An indecomposable Lie group with Riemannian bi-invariant metric is always simple and hence Einstein. Indefinite bi-invariant metrics are not necessarily Einstein, not even on simple Lie groups. We study the question of whether a semi-Riemannian bi-invariant metric is conformal to an Einstein metric. We obtain results for all three cases in the structure theorem by Medina and Revoy for indecomposable metric Lie algebras: the case of simple Lie algebras, and the cases of double extensions of metric Lie algebras by ℝ or a simple Lie algebra. Simple Lie algebras are conformally Einstein precisely when they are Einstein, or when equal to sl 2 ℂ and conformally flat. Double extensions of metric Lie algebras by simple Lie algebras of rank greater than one are never conformally Einstein, and neither are double extensions of Lorentzian oscillator algebras, whereas the oscillator algebras themselves are conformally Einstein. Our results give a complete answer to the question of which metric Lie algebras in Lorentzian signature and in signature (2, n − 2) are conformally Einstein.