Moduli of Einstein and non-Einstein nilradicals

TL;DR

This paper investigates the distribution and construction of Einstein and non-Einstein nilradicals among nilpotent Lie groups, especially two-step ones, using Geometric Invariant Theory to generate new examples and families.

Contribution

It introduces new continuous families of Einstein and non-Einstein nilradicals and demonstrates the existence of arbitrarily large-dimensional families of non-Einstein nilradicals.

Findings

Existence of large families of non-Einstein nilradicals.

Construction of many new families of nilradicals.

Focus on two-step nilpotent Lie groups.

Abstract

In this note we are concerned with the distribution of Einstein and non-Einstein nilradicals among all nilpotent Lie groups. A nilpotent Lie group is called an Einstein, resp. non-Einstein, nilradical if it is a nilpotent Lie group which does, resp. does not, admit a left-invariant Ricci soliton metric. Using techniques from Geometric Invariant Theory, we construct many new (continuous) families of both kinds of nilpotent Lie groups. Moreover, it is shown by example that there exist families of non-Einstein nilradicals of arbitrarily large dimension; the dimension of these families of Lie groups depends only on the dimension of the underlying Lie groups. In this work our attention is focused on the class of two-step nilpotent Lie groups.