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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Einstein solvmanifolds and the pre-Einstein derivation


Author: Y. Nikolayevsky
Journal: Trans. Amer. Math. Soc. 363 (2011), 3935-3958
MSC (2000): Primary 53C30, 53C25
Published electronically: March 10, 2011
MathSciNet review: 2792974
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Abstract: An Einstein nilradical is a nilpotent Lie algebra which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable extension. For every nilpotent Lie algebra, we construct an (essentially unique) derivation, the pre-Einstein derivation, the solvable extension by which may carry an Einstein inner product. Using the pre-Einstein derivation, we then give a variational characterization of Einstein nilradicals. As an application, we prove an easy-to-check convex geometry condition for a nilpotent Lie algebra with a nice basis to be an Einstein nilradical and also show that a typical two-step nilpotent Lie algebra is an Einstein nilradical.


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Additional Information

Y. Nikolayevsky
Affiliation: Department of Mathematics, La Trobe University, Victoria, 3086, Australia
Email: y.nikolayevsky@latrobe.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05045-2
PII: S 0002-9947(2011)05045-2
Keywords: Einstein solvmanifold, Einstein nilradical
Received by editor(s): March 31, 2008
Received by editor(s) in revised form: March 10, 2009
Published electronically: March 10, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.