A Construction of Einstein Solvmanifolds not Based on Nilsolitons

Abstract

AbstractWe construct indefinite Einstein solvmanifolds that are standard, but not of pseudo-Iwasawa type. Thus, the underlying Lie algebras take the form $$\mathfrak {g}\rtimes _D\mathbb {R}$$ g ⋊ D R , where $$\mathfrak {g}$$ g is a nilpotent Lie algebra and D is a nonsymmetric derivation. Considering nonsymmetric derivations has the consequence that $$\mathfrak {g}$$ 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 $$\le 9$$ ≤ 9 that satisfy an additional technical restriction. By comparing curvatures, we show that the Einstein solvmanifolds of dimension $$\le 5$$ ≤ 5 that we obtain by our construction are not isometric to a standard extension of a nilsoliton.