Indefinite Einstein metrics on nice Lie groups

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 $\ge 8$.

arXiv comment29 pages, 5 tables. v2: presentation improved, definition of sigma-compatible metrics replaced with the more general definition of sigma-diagonal metric. v3: misprints corrected

References in corpus (6)

• Einstein nilpotent Lie groups
• Ricci-flat and Einstein pseudoriemannian nilmanifolds
• Classification of Nilsoliton metrics in dimension seven
• Construction of nice nilpotent Lie groups
• Indefinite Einstein metrics on simple Lie groups
• Compact pseudo-Riemannian homogeneous Einstein manifolds of low dimension

Cited by in corpus (7)

• Einstein nilpotent Lie groups
• Ricci-flat and Einstein pseudoriemannian nilmanifolds
• Construction of nice nilpotent Lie groups
• Indefinite nilsolitons and Einstein solvmanifolds
• A construction of Einstein solvmanifolds not based on nilsolitons
• A non-Standard Indefinite Einstein Solvmanifold
• Nice pseudo-Riemannian nilsolitons