Einstein solvmanifolds and the pre-Einstein derivation
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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.References
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Additional Information
- Y. Nikolayevsky
- Affiliation: Department of Mathematics, La Trobe University, Victoria, 3086, Australia
- MR Author ID: 246384
- ORCID: 0000-0002-9528-1882
- Email: y.nikolayevsky@latrobe.edu.au
- Received by editor(s): March 31, 2008
- Received by editor(s) in revised form: March 10, 2009
- Published electronically: March 10, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 3935-3958
- MSC (2000): Primary 53C30, 53C25
- DOI: https://doi.org/10.1090/S0002-9947-2011-05045-2
- MathSciNet review: 2792974