A Construction of Einstein Solvmanifolds not Based on Nilsolitons

Abstract

We construct indefinite Einstein solvmanifolds that are standard, but not of pseudo-Iwasawa type. Thus, the underlying Lie algebras take the form \(\mathfrak {g}\rtimes _D\mathbb {R}\), where \(\mathfrak {g}\) is a nilpotent Lie algebra and D is a nonsymmetric derivation. Considering nonsymmetric derivations has the consequence that \(\mathfrak {g}\) is not a nilsoliton, but satisfies a more general condition. Our construction is based on the notion of nondiagonal triple on a nice diagram. We present an algorithm to classify nondiagonal triples and the associated Einstein metrics. With the use of a computer, we obtain all solutions up to dimension 5, and all solutions in dimension \(\le 9\) that satisfy an additional technical restriction. By comparing curvatures, we show that the Einstein solvmanifolds of dimension \(\le 5\) that we obtain by our construction are not isometric to a standard extension of a nilsoliton.