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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Einstein solvmanifolds and the pre-Einstein derivation
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by Y. Nikolayevsky PDF
Trans. Amer. Math. Soc. 363 (2011), 3935-3958 Request permission


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
  • MR Author ID: 246384
  • ORCID: 0000-0002-9528-1882
  • Email:
  • 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:
  • MathSciNet review: 2792974