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Einstein solvmanifolds with a simple Einstein derivation. (English) Zbl 1145.53040
Summary: The structure of a solvable Lie group admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras).
53C30Homogeneous manifolds (differential geometry)
53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)
17B30Solvable, nilpotent Lie (super)algebras
[1]Alekseevskii D.V.: Classification of quaternionic spaces with transitive solvable group of motions. Math. USSR – Izv. 9, 297–339 (1975) · Zbl 0324.53038 · doi:10.1070/IM1975v009n02ABEH001479
[2]Ancochéa-Bermúdez J.M., Goze M.: Le rang du systeme linéaire des racines d’une algèbre de Lie rigide résoluble complexe. Commun. Algebra 20, 875–887 (1992) · Zbl 0748.17005 · doi:10.1080/00927879208824380
[3]Cairns, G., Jessup, B.: A special family of positively graded Lie algebras (2005). Preprint
[4]Dotti Miatello I.: Ricci curvature of left-invariant metrics on solvable unimodular Lie groups. Math. Z. 180, 257–263 (1982) · Zbl 0471.53033 · doi:10.1007/BF01318909
[5]Goze M., Hakimjanov Y.: Sur les algèebres de Lie nilpotentes admettant un tore de dèrivations. Manuscripta Math. 84, 115–124 (1994) · Zbl 0823.17009 · doi:10.1007/BF02567448
[6]Goze, M., Khakimdjanov, Y.: Nilpotent and solvable Lie algebras. In: Handbook of Algebra, vol. 2, pp. 615–663, North-Holland, Amsterdam (2000)
[7]Gordon C., Kerr M.: New homogeneous Einstein metrics of negative Ricci curvature. Ann. Global Anal. Geom. 19, 75–101 (2001) · Zbl 0990.53054 · doi:10.1023/A:1006767203771
[8]Heber J.: Noncompact Homogeneous Einstein spaces. Invent. Math. 133, 279–352 (1998) · Zbl 0906.53032 · doi:10.1007/s002220050247
[9]Heinzner P., Stötzel H.: Semistable points with respect to real forms. Math. Ann. 338, 1–9 (2007) · Zbl 1129.32015 · doi:10.1007/s00208-006-0063-1
[10]Lauret J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319, 715–733 (2001) · Zbl 0987.53019 · doi:10.1007/PL00004456
[11]Lauret J.: Finding Einstein solvmanifolds by a variational method. Math. Z. 241, 83–99 (2002) · Zbl 1015.53028 · doi:10.1007/s002090100407
[12]Lauret, J.: Minimal metrics on nilmanifolds. In: Diff. Geom. and its Appl., Proc. Conf. Prague 2004, pp. 77–94 (2005)
[13]Lauret J.: A canonical compatible metric for geometric structures on nilmanifolds. Ann. Global Anal. Geom. 30, 107–138 (2006) · Zbl 1102.53021 · doi:10.1007/s10455-006-9015-y
[14]Lauret, J.: Einstein solvmanifolds are standard (2007). Preprint, math/0703472
[15]Lauret, J., Will, C.: Einstein solvmanifolds: existence and non-existence questions (2006). Preprint, math.DG/0602502
[16]Millionschikov, D.: Graded filiform Lie algebras and symplectic nilmanifolds. In: Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, vol. 212, pp. 259–279 (2004)
[17]Nikolayevsky Y.: Einstein solvable Lie algebras with free nilradical. Ann. Global Anal. Geom. 33, 71–87 (2008) · Zbl 1156.53032 · doi:10.1007/s10455-007-9077-5
[18]Nikolayevsky, Y.: Nilradicals of Einstein solvmanifolds (2006). Preprint, math.DG/0612117
[19]Payne, T.: The existence of soliton metrics for nilpotent Lie groups (2005). Preprint
[20]Schützdeller, P.: Convexity properties of moment maps of real forms acting on Kählerian manifolds. Dissertation, Ruhr-Universität Bochum (2006)
[21]Vergne M.: Cohomologie des algèbres de Lie nilpotentes. Application à l’ étude de la varié;té des algèbres de Lie nilpotentes. Bull. Soc. Math. France 98, 81–116 (1970)
[22]Will C.: Rank-one Einstein solvmanifolds of dimension 7. Diff. Geom. Appl. 19, 307–318 (2003) · Zbl 1045.53032 · doi:10.1016/S0926-2245(03)00037-8