Abstract
Abstract We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice basis is orthogonal and a more general class associated to order two permutations of the nice basis. We obtain classifications in dimension 8 and, under the assumption that the root matrix is surjective, dimension 9; moreover, we prove that Einstein nilpotent Lie groups of nonzero scalar curvature exist in every dimension ≥ 8 {\geq 8} .
Highlights
The construction of Einstein Riemannian metrics is a classical problem
Homogeneous Einstein manifolds with positive scalar curvature are compact; both necessary and sufficient conditions on a compact homogeneous space for the existence of an Einstein metric are known
A classification up to dimension 8 was obtained in [17] under the assumption that the Nikolayevsky derivation has eigenvalues of multiplicity one. This condition implies that the Lie algebra has a nice basis as defined in [19]; nice bases can be characterized by the condition that all diagonal metrics have diagonal Ricci operator
Summary
We recall the definition of nice Lie algebra and the basic related facts; we refer to [4] for further details. To each nice nilpotent Lie algebra we can associate a directed graph ∆ as follows:. It is clear that equivalent nice Lie algebras determine isomorphic nice diagrams. The correspondence between nice nilpotent Lie algebras and nice diagrams is not one to one. Nice Lie algebras associated to a nice diagram ∆ are parametrized as follows. Equivalence classes of nice nilpotent Lie algebras with diagram ∆ are parametrized by elements of. Each nice diagram ∆ has an associated root matrix defined as follows. There is a different well-known method to attach a nice nilpotent Lie algebra to an undirected graph, where each node and edge define an element of the nice basis, and nonzero Lie brackets correspond to pairs of nodes connected by an edge The resulting Lie algebra is always two-step, and admits no Einstein metric of nonzero scalar curvature (see [5])
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