Abstract
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).
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Alekseevskii D.V.: Classification of quaternionic spaces with transitive solvable group of motions. Math. USSR – Izv. 9, 297–339 (1975)
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)
Cairns, G., Jessup, B.: A special family of positively graded Lie algebras (2005). Preprint
Dotti Miatello I.: Ricci curvature of left-invariant metrics on solvable unimodular Lie groups. Math. Z. 180, 257–263 (1982)
Goze M., Hakimjanov Y.: Sur les algèebres de Lie nilpotentes admettant un tore de dèrivations. Manuscripta Math. 84, 115–124 (1994)
Goze, M., Khakimdjanov, Y.: Nilpotent and solvable Lie algebras. In: Handbook of Algebra, vol. 2, pp. 615–663, North-Holland, Amsterdam (2000)
Gordon C., Kerr M.: New homogeneous Einstein metrics of negative Ricci curvature. Ann. Global Anal. Geom. 19, 75–101 (2001)
Heber J.: Noncompact Homogeneous Einstein spaces. Invent. Math. 133, 279–352 (1998)
Heinzner P., Stötzel H.: Semistable points with respect to real forms. Math. Ann. 338, 1–9 (2007)
Lauret J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319, 715–733 (2001)
Lauret J.: Finding Einstein solvmanifolds by a variational method. Math. Z. 241, 83–99 (2002)
Lauret, J.: Minimal metrics on nilmanifolds. In: Diff. Geom. and its Appl., Proc. Conf. Prague 2004, pp. 77–94 (2005)
Lauret J.: A canonical compatible metric for geometric structures on nilmanifolds. Ann. Global Anal. Geom. 30, 107–138 (2006)
Lauret, J.: Einstein solvmanifolds are standard (2007). Preprint, math/0703472
Lauret, J., Will, C.: Einstein solvmanifolds: existence and non-existence questions (2006). Preprint, math.DG/0602502
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)
Nikolayevsky Y.: Einstein solvable Lie algebras with free nilradical. Ann. Global Anal. Geom. 33, 71–87 (2008)
Nikolayevsky, Y.: Nilradicals of Einstein solvmanifolds (2006). Preprint, math.DG/0612117
Payne, T.: The existence of soliton metrics for nilpotent Lie groups (2005). Preprint
Schützdeller, P.: Convexity properties of moment maps of real forms acting on Kählerian manifolds. Dissertation, Ruhr-Universität Bochum (2006)
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)
Will C.: Rank-one Einstein solvmanifolds of dimension 7. Diff. Geom. Appl. 19, 307–318 (2003)
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Nikolayevsky, Y. Einstein solvmanifolds with a simple Einstein derivation. Geom Dedicata 135, 87–102 (2008). https://doi.org/10.1007/s10711-008-9264-y
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DOI: https://doi.org/10.1007/s10711-008-9264-y