On Ricci negative solvmanifolds and their nilradicals

Deré, Jonas; Lauret, Jorge

doi:10.1002/mana.201700455

Cited by 9 publications

(16 citation statements)

References 31 publications

(53 reference statements)

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“…) of u C such that, for every j, V (j) contains a weight η (j) in a Weyl chamber associated to ∆ (j) . It follows that η := η (1)…”

Section: Weyl Chamber Approachmentioning

confidence: 99%

“…In the sequel, when we consider a particular complex simple Lie algebra, we will use the positive root system chosen in [6, §C. [1][2].…”

Section: Explicit Examplesmentioning

confidence: 99%

“…In the homogeneous case, while the scalar and sectional curvature behavior are settled, the question of when a homogeneous manifold admits a left-invariant metric with negative Ricci curvature seems far from being understood (see e.g. the introductions of [1] or [10]).…”

Section: Introductionmentioning

confidence: 99%

“…The second sort of examples are solvable Lie groups. This is the most developed case (see [2], [10], [9], [1]) probably due to the relationship with Einstein solvmanifolds. Recall that any non-flat Einstein solvmanifold is an example of a Lie group with a metric of negative Ricci curvature.…”

Section: Introductionmentioning

confidence: 99%

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Non-Solvable Lie Groups With Negative Ricci Curvature

Will

2020

Transformation Groups

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Until a couple of years ago, the only known examples of Lie groups admitting left-invariant metrics with negative Ricci curvature were either solvable or semisimple. We use a general construction from a previous article of the second named author to produce a large amount of examples with compact Levi factor. Given a compact semisimple real Lie algebra u and a real representation π satisfying some technical properties, the construction returns a metric Lie algebra l(u, π) with negative Ricci operator. In this paper, when u is assumed to be simple, we prove that l(u, π) admits a metric having negative Ricci curvature for all but finitely many finite-dimensional irreducible representations of u⊗ R C, regarded as a real representation of u. We also prove in the last section a more general result where the nilradical is not abelian, as it is in every l(u, π). * Supported by CONICET, FONCYT, and the Alexander von Humboldt Foundation (return fellowship).

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“…) of u C such that, for every j, V (j) contains a weight η (j) in a Weyl chamber associated to ∆ (j) . It follows that η := η (1)…”

Section: Weyl Chamber Approachmentioning

confidence: 99%

“…In the sequel, when we consider a particular complex simple Lie algebra, we will use the positive root system chosen in [6, §C. [1][2].…”

Section: Explicit Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Non-Solvable Lie Groups With Negative Ricci Curvature

Will

2020

Transformation Groups

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“…This is the perspective taken, amongst many others, in the papers [22,45,47]. Recently Deré and Lauret [23] use nice convexity properties of the moment map for the variety of nilpotent Lie algebras to investigate which nilpotent Lie algebras admit a Ricci negative solvable extension. This motivated us to investigate convexity properties of gradient maps associated to real reductive representations.…”

Section: Introductionmentioning

confidence: 99%

Convexity properties of gradient maps associated to real reductive representations

Biliotti

2020

Journal of Geometry and Physics

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Let G be a connected real reductive Lie group acting linearly on a finite dimensional vector space V over R. This action admits a Kempf-Ness function and so we have an associated gradient map. If G is Abelian we explicitly compute the image of G orbits under the gradient map, generalizing a result proved by Kac and Peterson [38]. A similar result is proved for the gradient map associated to the natural G action on P(V ). We also investigate the convex hull of the image of the gradient map restricted on the closure of G orbits. Finally, we give a new proof of the Hilbert-Mumford criterion for real reductive Lie groups avoiding any algebraic result.2010 Mathematics Subject Classification. 22E45,53D20; 14L24.

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On Ricci Negative Lie Groups

Lauret

Will

2021

Geometry, Lie Theory and Applications

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On Ricci negative solvmanifolds and their nilradicals

Cited by 9 publications

References 31 publications

Non-Solvable Lie Groups With Negative Ricci Curvature

Non-Solvable Lie Groups With Negative Ricci Curvature

Convexity properties of gradient maps associated to real reductive representations

On Ricci Negative Lie Groups

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