A curve of nilpotent Lie algebras which are not Einstein nilradicals

TL;DR

This paper constructs two continuous families of 9-dimensional 2-step nilpotent Lie algebras that are not Einstein nilradicals, providing new examples that challenge the classification of Einstein solvmanifolds.

Contribution

It introduces two explicit curves of non-isomorphic 9-dimensional 2-step nilpotent Lie algebras that are not Einstein nilradicals, expanding known examples.

Findings

Two curves of non-isomorphic 9-dimensional 2-step nilpotent Lie algebras are not Einstein nilradicals.

These examples demonstrate the diversity of nilpotent Lie algebras outside Einstein nilradicals.

The results suggest limitations in the classification of Einstein solvmanifolds.

Abstract

The only known examples of noncompact Einstein homogeneous spaces are standard solvmanifolds (special solvable Lie groups endowed with a left invariant metric), and according to a long standing conjecture, they might be all. The classification of Einstein solvmanifolds is equivalent to the one of Einstein nilradicals, i.e. nilpotent Lie algebras which are nilradicals of the Lie algebras of Einstein solvmanifolds. Up to now, there have been found very few examples of graded nilpotent Lie algebras that can not be Einstein nilradicals. In particular, in each dimension, there are only finitely many known. We exhibit in the present paper two curves of pairwise non-isomorphic 9-dimensional 2-step nilpotent Lie algebras which are not Einstein nilradicals.