Classification of 7-dimensional einstein nilradicals

Abstract

The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety ${\mathfrak{N}_n}\left( \mathbb{C} \right)$ of n-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which GL _n (ℂ)-orbits in ${\mathfrak{N}_n}\left( \mathbb{C} \right)$ have a critical point of the squared norm of the moment map. In this paper, we give a classification result of such distinguished orbits for n = 7. The set ${{{{\mathfrak{N}_n}\left( \mathbb{C} \right)}} \left/ {{{\text{G}}{{\text{L}}_7}\left( \mathbb{C} \right)}} \right.}$ is formed by 148 nilpotent Lie algebras and 6 one-parameter families of pairwise non-isomorphic nilpotent Lie algebras. We have applied to each Lie algebra one of three main techniques to decide whether it has a distinguished orbit or not.