article

Title: Indefinite Einstein metrics on simple Lie groups

Authors: Andrzej Derdzinski and Swiatoslaw R. Gal

Issue: Volume 63 (2014), Issue 1, 165-212

Abstract:

$The set $\mathcal{E}$ 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 $\SU(\ell,j)$ (or $\SL(n,\mathbb{R})$, or $\SL(n,\mathbb{C})$ or, if $n$ is even, $\SL(n/2,\mathbb{H})$), with $0\le j\le\ell$ and $j+\ell>2$ (or, $n>2$), we explicitly describe the connected component $\mathcal{C}$ of $\mathcal{E}$, containing $\mathrm{D}$. It turns out that $\mathcal{C}$, a relatively-open subset of $\mathcal{E}$, is also an algebraic variety of real dimension $2\ell j$ (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 $2\ell j=0$, so that a positive-definite multiple of the Killing form is isolated among suitably normalized left-invariant Riemannian Einstein metrics on $\SU(n)$.$