# Conformal properties of indefinite bi-invariant metrics

## TL;DR

This paper investigates when indefinite bi-invariant metrics on Lie groups are conformally equivalent to Einstein metrics, providing a complete classification in certain Lorentzian and (2,n-2) signatures.

## Contribution

It offers a comprehensive analysis of conformally Einstein properties of metric Lie algebras across various cases, extending known results to indefinite metrics.

## Key findings

- Simple Lie algebras are conformally Einstein iff they are Einstein or isomorphic to sl(2,C) and conformally flat.
- Double extensions by higher rank simple Lie algebras are not conformally Einstein.
- Oscillator algebras are conformally Einstein, but their double extensions are not.

## Abstract

An indecomposable Lie group with Riemannian bi-invariant metric is always simple and hence Einstein. For indefinite metrics this is no longer true, not even for 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 $\mathbb{R}$ or a simple Lie algebra. Simple Lie algebras are conformally Einstein precisely when they are Einstein, or when equal to $\mathfrak{sl}_2\mathbb{C}$ 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.