## Einstein solvmanifolds and the pre-Einstein derivation

HTML articles powered by AMS MathViewer

- by Y. Nikolayevsky PDF
- Trans. Amer. Math. Soc.
**363**(2011), 3935-3958 Request permission

## 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

- D. V. Alekseevskiĭ,
*Classification of quaternionic spaces with transitive solvable group of motions*, Izv. Akad. Nauk SSSR Ser. Mat.**39**(1975), no. 2, 315–362, 472 (Russian). MR**0402649** - D. V. Alekseevskiĭ,
*Homogeneous Riemannian spaces of negative curvature*, Mat. Sb. (N.S.)**96(138)**(1975), 93–117, 168 (Russian). MR**0362145** - D. V. Alekseevskiĭ and B. N. Kimel′fel′d,
*Structure of homogeneous Riemannian spaces with zero Ricci curvature*, Funkcional. Anal. i PriloŽen.**9**(1975), no. 2, 5–11 (Russian). MR**0402650** - David Birkes,
*Orbits of linear algebraic groups*, Ann. of Math. (2)**93**(1971), 459–475. MR**296077**, DOI 10.2307/1970884 - Armand Borel and Harish-Chandra,
*Arithmetic subgroups of algebraic groups*, Ann. of Math. (2)**75**(1962), 485–535. MR**147566**, DOI 10.2307/1970210 - Isabel Dotti Miatello,
*Ricci curvature of left invariant metrics on solvable unimodular Lie groups*, Math. Z.**180**(1982), no. 2, 257–263. MR**661702**, DOI 10.1007/BF01318909 - Patrick Eberlein,
*The moduli space of 2-step nilpotent Lie algebras of type $(p,q)$*, Explorations in complex and Riemannian geometry, Contemp. Math., vol. 332, Amer. Math. Soc., Providence, RI, 2003, pp. 37–72. MR**2016090**, DOI 10.1090/conm/332/05929 - Patrick Eberlein,
*Geometry of 2-step nilpotent Lie groups*, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 67–101. MR**2090766** - Patrick Eberlein,
*Riemannian 2-step nilmanifolds with prescribed Ricci tensor*, Geometric and probabilistic structures in dynamics, Contemp. Math., vol. 469, Amer. Math. Soc., Providence, RI, 2008, pp. 167–195. MR**2478470**, DOI 10.1090/conm/469/09166 - A. G. Èlašvili,
*Stationary subalgebras of points of general position for irreducible linear Lie groups*, Funkcional. Anal. i Priložen.**6**(1972), no. 2, 65–78 (Russian). MR**0304555** - Carolyn S. Gordon and Megan M. Kerr,
*New homogeneous Einstein metrics of negative Ricci curvature*, Ann. Global Anal. Geom.**19**(2001), no. 1, 75–101. MR**1824172**, DOI 10.1023/A:1006767203771 - L. Yu. Galitski and D. A. Timashev,
*On classification of metabelian Lie algebras*, J. Lie Theory**9**(1999), no. 1, 125–156. MR**1680007** - Jens Heber,
*Noncompact homogeneous Einstein spaces*, Invent. Math.**133**(1998), no. 2, 279–352. MR**1632782**, DOI 10.1007/s002220050247 - Jorge Lauret,
*Ricci soliton homogeneous nilmanifolds*, Math. Ann.**319**(2001), no. 4, 715–733. MR**1825405**, DOI 10.1007/PL00004456 - Jorge Lauret,
*Finding Einstein solvmanifolds by a variational method*, Math. Z.**241**(2002), no. 1, 83–99. MR**1930986**, DOI 10.1007/s002090100407 - Jorge Lauret,
*Degenerations of Lie algebras and geometry of Lie groups*, Differential Geom. Appl.**18**(2003), no. 2, 177–194. MR**1958155**, DOI 10.1016/S0926-2245(02)00146-8 - Jorge Lauret,
*On the moment map for the variety of Lie algebras*, J. Funct. Anal.**202**(2003), no. 2, 392–423. MR**1990531**, DOI 10.1016/S0022-1236(02)00108-8 - Jorge Lauret,
*Einstein solvmanifolds are standard*, Ann. of Math. (2)**172**(2010), no. 3, 1859–1877. MR**2726101**, DOI 10.4007/annals.2010.172.1859 - Lauret J., Will C.,
*Einstein solvmanifolds: Existence and non-existence questions*, preprint 2006, arXiv: math.DG/0602502. - Eugene M. Luks,
*What is the typical nilpotent Lie algebra?*, Computers in nonassociative rings and algebras (Special Session, 82nd Annual Meeting Amer. Math. Soc., San Antonio, Tex., 1976) Academic Press, New York, 1977, pp. 189–207. MR**0453830** - John Milnor,
*Curvatures of left invariant metrics on Lie groups*, Advances in Math.**21**(1976), no. 3, 293–329. MR**425012**, DOI 10.1016/S0001-8708(76)80002-3 - G. D. Mostow,
*Fully reducible subgroups of algebraic groups*, Amer. J. Math.**78**(1956), 200–221. MR**92928**, DOI 10.2307/2372490 - Yuri Nikolayevsky,
*Einstein solvmanifolds with a simple Einstein derivation*, Geom. Dedicata**135**(2008), 87–102. MR**2413331**, DOI 10.1007/s10711-008-9264-y - Yuri Nikolayevsky,
*Einstein solvmanifolds with free nilradical*, Ann. Global Anal. Geom.**33**(2008), no. 1, 71–87. MR**2369187**, DOI 10.1007/s10455-007-9077-5 - —,
*Einstein solvmanifolds attached to two-step nilradicals*, preprint, 2008, arXiv: math.DG/0805.0646. - Yu. G. Nikonorov,
*On Einstein extensions of nilpotent metric Lie algebras [translation of MR2485371]*, Siberian Adv. Math.**17**(2007), no. 3, 153–170. MR**2647826**, DOI 10.3103/S1055134407030017 - Tracy L. Payne,
*The existence of soliton metrics for nilpotent Lie groups*, Geom. Dedicata**145**(2010), 71–88. MR**2600946**, DOI 10.1007/s10711-009-9404-z - V. L. Popov,
*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 (Russian). MR**0262416** - R. W. Richardson and P. J. Slodowy,
*Minimum vectors for real reductive algebraic groups*, J. London Math. Soc. (2)**42**(1990), no. 3, 409–429. MR**1087217**, DOI 10.1112/jlms/s2-42.3.409 - D. Rand, P. Winternitz, and H. Zassenhaus,
*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**961578**, DOI 10.1016/0024-3795(88)90210-8 - È. B. Vinberg and V. L. Popov,
*Invariant theory*, Algebraic geometry, 4 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 137–314, 315 (Russian). MR**1100485** - Cynthia Will,
*Rank-one Einstein solvmanifolds of dimension 7*, Differential Geom. Appl.**19**(2003), no. 3, 307–318. MR**2013098**, DOI 10.1016/S0926-2245(03)00037-8 - Joseph A. Wolf and Roger Zierau,
*Riemannian exponential maps and decompositions of reductive Lie groups*, Topics in geometry, Progr. Nonlinear Differential Equations Appl., vol. 20, Birkhäuser Boston, Boston, MA, 1996, pp. 349–353. MR**1390323**

## 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