We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural GL(n,R) action, whose orbits parametrize Lie groups with a left-invariant metric; we show that the Ricci operator can be identified with the moment map relative to a natural symplectic structure. From this description we deduce that the Ricci operator is the derivative of the scalar curvature s under gauge transformations of the metric, and show that Lie algebra derivations with nonzero trace obstruct the existence of Einstein metrics with s≠0. Using the notion of nice Lie algebra, we give the first example of a left-invariant Einstein metric with s≠0 on a nilpotent Lie group. We show that nilpotent Lie groups of dimension ≤6 do not admit such a metric, and a similar result holds in dimension 7 with the extra assumption that the Lie algebra is nice.

Conti, D., Rossi, F. (2019). Einstein nilpotent Lie groups. JOURNAL OF PURE AND APPLIED ALGEBRA, 223(3), 976-997 [10.1016/j.jpaa.2018.05.010].

Einstein nilpotent Lie groups

Conti, D
;
Rossi, FA
2019

Abstract

We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural GL(n,R) action, whose orbits parametrize Lie groups with a left-invariant metric; we show that the Ricci operator can be identified with the moment map relative to a natural symplectic structure. From this description we deduce that the Ricci operator is the derivative of the scalar curvature s under gauge transformations of the metric, and show that Lie algebra derivations with nonzero trace obstruct the existence of Einstein metrics with s≠0. Using the notion of nice Lie algebra, we give the first example of a left-invariant Einstein metric with s≠0 on a nilpotent Lie group. We show that nilpotent Lie groups of dimension ≤6 do not admit such a metric, and a similar result holds in dimension 7 with the extra assumption that the Lie algebra is nice.
Articolo in rivista - Articolo scientifico
Ricci tensor, moment map,Einstein pseudoriemannian metrics, nilpotent Lie groups
English
2019
223
3
976
997
partially_open
Conti, D., Rossi, F. (2019). Einstein nilpotent Lie groups. JOURNAL OF PURE AND APPLIED ALGEBRA, 223(3), 976-997 [10.1016/j.jpaa.2018.05.010].
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/207918
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