Diagram involutions and homogeneous Ricci-flat metrics

Conti, Diego; Barco, Viviana del; Rossi, Federico A.

doi:10.1007/s00229-020-01225-y

Cited by 13 publications

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References 29 publications

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“…Write * instead of * g . By (7), since Id ∈ D h , τ belongs to D g . Conversely, for every φ ∈ D g , we have that…”

Section: The Uniqueness Problem For Ad-invariant Metricsmentioning

confidence: 98%

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Uniqueness of ad-invariant metrics

Conti,

del Barco,

Rossi

2021

Preprint

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We consider Lie algebras admitting an ad-invariant metric, and we study the problem of uniqueness of the ad-invariant metric up to automorphisms. This is a common feature in low dimensions, as one can observe in the known classification of nilpotent Lie algebras of dimension ≤ 7 admitting an ad-invariant metric. We prove that uniqueness of the metric on a complex Lie algebra g is equivalent to uniqueness of ad-invariant metrics on the cotangent Lie algebra T * g; a slightly more complicated equivalence holds over the reals. This motivates us to study the broader class of Lie algebras such that the ad-invariant metric on T * g is unique.We prove that uniqueness of the metric forces the Lie algebra to be solvable, but the converse does not hold, as we show by constructing solvable Lie algebras with a one-parameter family of inequivalent ad-invariant metrics. We prove sufficient conditions for uniqueness expressed in terms of both the Nikolayevsky derivation and a metric counterpart introduced in this paper.Moreover, we prove that uniqueness always holds for irreducible Lie algebras which are either solvable of dimension ≤ 6 or real nilpotent of dimension ≤ 10.

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“…Write * instead of * g . By (7), since Id ∈ D h , τ belongs to D g . Conversely, for every φ ∈ D g , we have that…”

Section: The Uniqueness Problem For Ad-invariant Metricsmentioning

confidence: 98%

“…Now write S g = D g + Span {ψ}, and consider the affine line φ t = Id + tψ. Suppose that for some t we have Id ∈ D h φ t ; then φ t ∈ D g by (7), which implies t = 0. Since ψ / ∈ D g , {h φt } is a section for the action of Aut(g), and thus non-weakly-solitary Aut(g)-orbits are isolated.…”

Section: The Uniqueness Problem For Ad-invariant Metricsmentioning

confidence: 99%

Uniqueness of ad-invariant metrics

Conti,

del Barco,

Rossi

2021

Preprint

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“…The Lie algebra of Example 2.4 admits nilsoliton metrics of types (Nil1) (see [6]), (Nil2) (the one in the example) and (Nil4) (see [21]). The Lie algebra of Example 2.5 admits a nilsoliton metric of type (Nil1) (see [6]) and a nilsoliton metric of type (Nil4), namely the indefinite one appearing in the examples. Note that it cannot be a Riemannian nilsoliton (see [12]).…”

Section: Nilsolitonsmentioning

confidence: 99%

“…The first equation was proved in the proof of Theorem 2.1, and for the second we have: tr Ric 2 = Ric, Ric tr = Ric, λId + D tr = λ tr Ric + Ric, D tr = λ tr Ric .Thus, the indefinite nilsoliton equation corresponds to four different situations:(Nil1) λ = 0, D = 0. This is the Ricci-flat case, examples of which exist in abundance (see e.g [6]…”

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confidence: 99%

Indefinite nilsolitons and Einstein solvmanifolds

Conti,

Rossi

2021

Preprint

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A nilsoliton is a nilpotent Lie algebra g with a metric such that Ric = λId + D, with D a derivation. For indefinite metrics, this determines four different geometries, according to whether λ and D are zero or not. We illustrate with examples the greater flexibility of the indefinite case compared to the Riemannian setting. We determine the algebraic properties that D must satisfy when it is nonzero.For each of the four geometries, we show that under suitable assumptions it is possible to extend the nilsoliton metric to an Einstein solvmanifold of the form g ⋊ R k . Conversely, we introduce a large class of indefinite Einstein solvmanifolds of the form g ⋊ R k that determine a nilsoliton metric on g by restriction. We show with examples that, unlike in the Riemannian case, one cannot establish a correspondence between the full classes of Einstein solvmanifolds and nilsolitons.

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“…The other possible signature (corresponding to changing the sign of X) is (3,4); summing up, S σ (4, 3) = {1, 237, 457, 12345}, S σ (3, 4) = {67, 1236, 1456, 234567}.…”

Section: Ricci-flat Metricsmentioning

confidence: 99%

Ricci-flat and Einstein pseudoriemannian nilmanifolds

Conti

Rossi

2019

Complex Manifolds

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This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups of dimension ≤ 8 are obtained. Some related open questions are presented. MSC 2010 : 53C25; 53C50, 53C30, 22E25.

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Diagram involutions and homogeneous Ricci-flat metrics

Cited by 13 publications

References 29 publications

Uniqueness of ad-invariant metrics

Uniqueness of ad-invariant metrics

Indefinite nilsolitons and Einstein solvmanifolds

Ricci-flat and Einstein pseudoriemannian nilmanifolds

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