Diagram involutions and homogeneous Ricci-flat metrics

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 $leq6$, every nice nilpotent Lie group of dimension $leq7$ 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 ${rm SL}(n)$, ${rm SO}(p,q)$, ${rm Sp}(n,mathbb R)$. Most of these metrics are shown not to be flat.