Classification of 7-dimensional Einstein nilradicals

TL;DR

This paper provides a complete classification of 7-dimensional Einstein nilradicals by analyzing the orbits of complex nilpotent Lie algebras and identifying those with critical points of a specific moment map.

Contribution

It offers the first full classification of Einstein nilradicals in dimension 7, resolving a key problem in the theory of Einstein solvmanifolds.

Findings

Identified all critical orbits among 148 complex nilpotent Lie algebras in dimension 7.

Classified 6 non-isomorphic nilpotent Lie algebra curves.

Established criteria for the existence of Einstein nilradicals in this dimension.

Abstract

The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety of n-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which GL(n)-orbits in this variety have a critical point of the squared norm of the moment map. In dimension 7, there are 148 complex nilpotent Lie algebras and 6 curves of pairwise non-isomorphic nilpotent Lie algebras, and we give in this paper a complete classification of the aforementioned distinguished orbits.