Diagram involutions and homogeneous Ricci-flat metrics

Abstract

We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\le 6$, every nice nilpotent Lie group of dimension $\le 7$ and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups $\mathrm{SL}(n)$, $\mathrm{SO}(p,q)$, $\mathrm{Sp}(n,\mathbb{R})$. Most of these metrics are shown not to be flat.

v2: Corollary 5.7 added, stating that the Ricci-flat metrics can be taken to be nonflat; two remarks added (2.8,5.10); two references added; two misprints corrected in Table 2 and p.29. The paper contains 3 tables; 32 pages. To appear in Manuscripta Mathematica