Abstract
We classify solvable Lie groups with a free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for the Abelian and the two-step ones, there are only six others: \({\mathfrak{f}}(2,3), {\mathfrak{f}}(2,4), {\mathfrak{f}}(2,5), {\mathfrak{f}}(3,3), {\mathfrak{f}}(4,3), {\mathfrak{f}}(5,3) (here {\mathfrak{f}}(m,p)\) is a free p-step Lie algebra on m generators). The reason for that is the inequality-type restrictions on the eigenvalue type of an Einstein nilradical obtained in the paper.
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Nikolayevsky, Y. Einstein solvmanifolds with free nilradical. Ann Glob Anal Geom 33, 71–87 (2008). https://doi.org/10.1007/s10455-007-9077-5
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DOI: https://doi.org/10.1007/s10455-007-9077-5