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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
DOI: https://doi.org/10.1090/S0002-9947-2011-05045-2
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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  • [Al1] Alekseevskii D.V., Classification of quaternionic spaces with transitive solvable group of motions, Izv. Akad. Nauk SSSR Ser. Mat., 39 (1975), 315 - 362. MR 0402649 (53:6465)
  • [Al2] -, Homogeneous Riemannian spaces of negative curvature, Mat. Sb. (N.S.), 96(138) (1975), 93 - 117. MR 0362145 (50:14587)
  • [AK] Alekseevskii D.V., Kimel'fel'd B.N., Structure of homogeneous Riemannian spaces with zero Ricci curvature, Funct. Anal. Appl., 9 (1975), 97 - 102. MR 0402650 (53:6466)
  • [Bir] Birkes D., Orbits of linear algebraic groups, Ann. of Math. (2), 93 (1971), 459 - 475. MR 0296077 (45:5138)
  • [BHC] Borel A., Harish-Chandra, Arithmetic Subgroups of Algebraic Groups, Ann. of Math. (2), 75 (1962), 485 - 535. MR 0147566 (26:5081)
  • [DM] Dotti Miatello I., Ricci curvature of left-invariant metrics on solvable unimodular Lie groups, Math. Z., 180 (1982), 257 - 263. MR 0661702 (84a:53044)
  • [Eb1] Eberlein P., The moduli space of $ 2$-step nilpotent Lie algebras of type $ (p,q)$, Explorations in complex and Riemannian geometry, Contemp. Math., 332, Amer. Math. Soc., Providence, RI, 2003, 37 - 72. MR 2016090 (2004j:17015)
  • [Eb2] -, Geometry of $ 2$-step nilpotent Lie groups, Modern Dynamical Systems and applications, Cambridge Univ. Press, Cambridge, 2004, 67 - 101. MR 2090766 (2005m:53081)
  • [Eb3] -, Riemannian $ 2$-step nilmanifolds with prescribed Ricci tensor, Geometric and Probabilistic Structures in Dynamics, Contemp. Math., 469, Amer. Math. Soc., Providence, RI, 2008, 167 - 195. MR 2478470 (2010c:53074)
  • [Ela] Elashvili A.G., Stationary subalgebras of points of general position for irreducible linear Lie groups, Funct. Anal. Appl., 6 (1972), 139 - 148. MR 0304555 (46:3690)
  • [GK] Gordon C., Kerr M., New homogeneous Einstein metrics of negative Ricci curvature, Ann. Global Anal. Geom., 19 (2001), 75 - 101. MR 1824172 (2002f:53067)
  • [GT] Galitski L., Timashev D., On classification of metabelian Lie algebras, J. Lie Theory, 9 (1999), 125 - 156. MR 1680007 (2000f:17015)
  • [Heb] Heber J., Noncompact homogeneous Einstein spaces, Invent. Math., 133 (1998), 279 - 352. MR 1632782 (99d:53046)
  • [La1] Lauret J., Ricci soliton homogeneous nilmanifolds, Math. Ann., 319 (2001), 715 - 733. MR 1825405 (2002k:53083)
  • [La2] -, Finding Einstein solvmanifolds by a variational method, Math. Z., 241 (2002), 83 - 99. MR 1930986 (2003g:53064)
  • [La3] -, Degenerations of Lie algebras and geometry of Lie groups, Differential Geom. Appl., 18 (2003), 177 - 194. MR 1958155 (2004c:17008)
  • [La4] -, On the moment map for the variety of Lie algebras, J. Funct. Anal., 202 (2003), 392 - 423. MR 1990531 (2004d:14067)
  • [La5] -, Einstein solvmanifolds are standard, Ann. of Math. (2) 172 (2010), 1859 - 1877. MR 2726101
  • [LW] Lauret J., Will C., Einstein solvmanifolds: Existence and non-existence questions, preprint 2006, arXiv: math.DG/0602502.
  • [Luk] Luks E.M., What is the typical nilpotent Lie algebra? Computers in nonassociative rings and algebras, 189 - 207. Academic Press, New York, 1977. MR 0453830 (56:12083)
  • [Mil] Milnor J., Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293 - 329. MR 0425012 (54:12970)
  • [Mos] Mostow G.D., Fully reducible subgroups of algebraic groups, Amer. J. Math., 78 (1956), 200 - 221. MR 0092928 (19:1181f)
  • [Ni1] -, Einstein solvmanifolds with a simple Einstein derivation, Geom. Dedicata, 135 (2008), 87 - 102. MR 2413331
  • [Ni2] -, Einstein solmanifolds with free nilradical, Ann. Global Anal. Geom., 33 (2008), 71 - 87. MR 2369187 (2008m:53120)
  • [Ni3] -, Einstein solvmanifolds attached to two-step nilradicals, preprint, 2008, arXiv: math.DG/0805.0646.
  • [Nik] Nikonorov Yu.G., On Einstein extensions of nilpotent metric Lie algebras, Siberian Adv. Math., 17 (2007), 153 - 170. MR 2647826
  • [Pay] Payne T., The existence of soliton metrics for nilpotent Lie groups, Geom. Dedicata, 145 (2010), 71-88. MR 2600946
  • [Pop] Popov V.L., Criteria for the stability of the action of a semisimple group on the factorial of a manifold, Izv. Akad. Nauk SSSR Ser. Mat., 34 (1970), 523 - 531. MR 0262416 (41:7024)
  • [RS] Richardson R., Slodowy P., Minimum vectors for real reductive algebraic groups, J. London Math. Soc. (2), 42 (1990), 409 - 429. MR 1087217 (92a:14055)
  • [RWZ] Rand D., Winternitz P., Zassenhaus H., On the identification of a Lie algebra given by its structure constants. I. Direct decompositions, Levi decompositions, and nilradicals, Linear Algebra Appl., 109 (1988), 197 - 246. MR 0961578 (89i:17001)
  • [VP] Vinberg E.B., Popov V.L., Invariant theory. In: Algebraic geometry. IV. Encyclopaedia of Mathematical Sciences, 55, Springer-Verlag, Berlin, 1994. MR 1100485 (92d:14010)
  • [Wil] Will C., Rank-one Einstein solvmanifolds of dimension 7, Differential Geom. Appl., 19 (2003), 307 - 318. MR 2013098 (2004j:53060)
  • [WZ] Wolf J.A., Zierau R., Riemannian exponential maps and decompositions of reductive Lie groups, Topics in geometry, Progr. Nonlinear Differential Equations Appl., 20, Birkhäuser Boston, Boston, 1996, 349-353. MR 1390323 (97c:53080)

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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: https://doi.org/10.1090/S0002-9947-2011-05045-2
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.

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