Einstein solvmanifolds attached to two-step nilradicals

TL;DR

This paper classifies two-step nilpotent Einstein nilradicals with a two-dimensional center, identifying specific matrix pencil conditions that determine their Einstein property, advancing understanding in geometric Lie algebra structures.

Contribution

It provides a classification of two-step nilpotent Einstein nilradicals with a two-dimensional center based on matrix pencil invariants, filling a gap in the existing literature.

Findings

Lie algebra is Einstein nilradical if its defining matrix pencil has no nilpotent blocks.

No elementary divisors of very high multiplicity are present in the matrix pencil.

Connection established between duality of Lie algebras and Einstein nilradical property.

Abstract

A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra, which can serve as the nilradical of an Einstein metric solvable Lie algebra, is called an Einstein nilradical. Despite a substantial progress towards the understanding of Einstein nilradicals, there is still a lack of classification results even for some well-studied classes of nilpotent Lie algebras, such as the two-step ones. In this paper, we give a classification of two-step nilpotent Einstein nilradicals in one of the rare cases when the complete set of affine invariants is known: for the two-step nilpotent Lie algebras with the two-dimensional center. Informally speaking, we prove that such a Lie algebra is an Einstein nilradical, if it is defined by a matrix pencil having no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity. We also discuss the connection between the property of a two-step nilpotent Lie algebra and its dual to be an Einstein nilradical.