# Diagram involutions and homogeneous Ricci-flat metrics

## TL;DR

This paper presents a combinatorial approach to constructing indefinite Ricci-flat metrics on certain nilpotent Lie groups, expanding known classes and providing new examples of Ricci-flat nilmanifolds with applications in geometry.

## Contribution

It introduces a novel combinatorial method for constructing Ricci-flat metrics and proves their existence on various classes of nilpotent Lie groups, including those associated with graphs and parabolic nilradicals.

## Key findings

- Constructed Ricci-flat metrics on nilpotent Lie groups of dimension ≤6 and certain higher-dimensional cases.
- Generated infinite families of Ricci-flat nilmanifolds related to classical Lie groups.
- Most constructed metrics are proven to be non-flat.

## 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 $\leq6$, 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.