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Journal of Lie Theory 28 (2018), No. 2, 343--356
Copyright Heldermann Verlag 2018

Einstein Solvmanifolds and Two-Step Nilpotent Lie Algebras with a Special Nice Basis

Hamid-Reza Fanaï
Dept. of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran

Zeinab Khodaei
Dept. of Mathematics, Institute for Advanced Studies (IASBS), P.O. Box 45195-1159, Zanjan, Iran

Consider a two-step nilpotent Lie algebra n with a special nice basis as introduced by Y. Nikolayevsky [Einstein solvmanifolds and the pre-Einstein derivation, Trans. Amer. Math. Soc. 363 (2011) 3935--3958] endowed with an inner product which makes the basis orthonormal. We describe necessary and sufficient conditions for the existence of a rank-one Einstein metric solvable extension of n. Since every two-step 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) 2235--2251]) 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) 37--39].

Keywords: Nice basis, two-step nilpotent Lie algebra, Einstein solvmanifolds.

MSC: 22E60, 53C25

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