Journal of Lie Theory

no. 2
Einstein Solvmanifolds and Two-Step Nilpotent Lie Algebras with a Special Nice Basis
Hamid-Reza Fanaï1; Zeinab Khodaei2
1Dept. of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran
2Dept. of Mathematics, Institute for Advanced Studies (IASBS), P.O. Box 45195-1159, Zanjan, Iran
Journal of Lie Theory, Volume 28 (2018) no. 2, pp. 343-356
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].
DOI: 10.5802/jolt.1005
Classification: 22E60, 53C25
Keywords: Nice basis, two-step nilpotent Lie algebra, Einstein solvmanifolds

Hamid-Reza Fanaï  1 ; Zeinab Khodaei  2

1 Dept. of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran
2 Dept. of Mathematics, Institute for Advanced Studies (IASBS), P.O. Box 45195-1159, Zanjan, Iran 
Hamid-Reza Fanaï; Zeinab Khodaei. Einstein Solvmanifolds and Two-Step Nilpotent Lie Algebras with a Special Nice Basis. Journal of Lie Theory, Volume 28 (2018) no. 2, pp. 343-356. doi: 10.5802/jolt.1005
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     author = {Hamid-Reza Fana{\"\i} and Zeinab Khodaei},
     title = {Einstein {Solvmanifolds} and {Two-Step} {Nilpotent} {Lie} {Algebras} with a {Special} {Nice} {Basis}},
     journal = {Journal of Lie Theory},
     pages = {343--356},
     year = {2018},
     volume = {28},
     number = {2},
     doi = {10.5802/jolt.1005},
     zbl = {1392.53065},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1005/}
}
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