Indefinite Einstein metrics on simple Lie groups

TL;DR

This paper characterizes the space of left-invariant Einstein metrics on certain simple Lie groups, revealing the structure and dimensions of the set containing the Killing form, and showing its algebraic and orbit properties.

Contribution

It explicitly describes the connected component of Einstein metrics containing the Killing form on specific Lie groups, detailing its algebraic variety structure and orbit decomposition.

Findings

The set of Einstein metrics includes the Killing form connection.

The connected component is an algebraic variety with specific real dimensions.

On SU(n), the Killing form is isolated among normalized Einstein metrics.

Abstract

The set E of Levi-Civita connections of left-invariant pseudo-Riemannian Einstein metrics on a given semisimple Lie group always includes D, the Levi-Civita connection of the Killing form. For the groups SU(l,j) (or SL(n,R), or SL(n,C) or, if n is even, SL(n/2,IH)), with 0<=j<=l and j+l>2 (or, n>2), we explicitly describe the connected component C of E, containing D. It turns out that C, a relatively-open subset of E, is also an algebraic variety of real dimension 2lj (or, real/complex dimension [n^2/2] or, respectively, real dimension 4[n^2/8]), forming a union of (j + 1)(j + 2)/2 (or, [n^2]+1 or, respectively, [n/4] + 1) orbits of the adjoint action. In the case of SU(n) one has 2lj=0, so that a positive-definite multiple of the Killing form is isolated among suitably normalized left-invariant Riemannian Einstein metrics on SU(n).