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 $\mathfrak{g}{⋊}_D\mathbb{R}$, 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.

arXiv comment23 pages, 1 figure, 1 ancillary file. Presentation improved and bibliography updated

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Cited by in corpus (1)

• A non-Standard Indefinite Einstein Solvmanifold