
Journal of Lie Theory 28 (2018), No. 2, 343356 Copyright Heldermann Verlag 2018 Einstein Solvmanifolds and TwoStep Nilpotent Lie Algebras with a Special Nice Basis HamidReza Fanaï Dept. of Mathematical Sciences, Sharif University of Technology, P.O. Box 111559415, Tehran, Iran fanai@sharif.ir Zeinab Khodaei Dept. of Mathematics, Institute for Advanced Studies (IASBS), P.O. Box 451951159, Zanjan, Iran z.khodaei@iasbs.ac.ir Consider a twostep nilpotent Lie algebra n with a special nice basis as introduced by Y. Nikolayevsky [Einstein solvmanifolds and the preEinstein derivation, Trans. Amer. Math. Soc. 363 (2011) 39353958] endowed with an inner product which makes the basis orthonormal. We describe necessary and sufficient conditions for the existence of a rankone Einstein metric solvable extension of n. Since every twostep nilpotent Lie algebra attached to a graph (as introduced by S. G. Dani, M. G. Mainkar [Anosov automorphisms on compact nilmanifolds associated with graphs, Trans. Amer. Math. Soc. 357 (2005) 22352251]) has such a nice basis, this note generalizes a recent result of H.R. Fanaï [Einstein solvmanifolds and graphs, C. R. Acad. Sci. Paris, Ser. I 344 (2007) 3739]. Keywords: Nice basis, twostep nilpotent Lie algebra, Einstein solvmanifolds. MSC: 22E60, 53C25 [ Fulltextpdf (262 KB)] for subscribers only. 