Structure of locally conformally symplectic Lie algebras and solvmanifolds

Angella, Daniele; Bazzoni, Giovanni; Parton, Maurizio

doi:10.2422/2036-2145.201708_015

Cited by 9 publications

(39 citation statements)

References 41 publications

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“…The center of a Lie algebra with a LCS structure was studied in [2], in particular the authors characterized the center of a nilpotent LCS Lie algebra and they proved that the dimension of the center is at most 2. Note that the nilpotency condition implies that the LCS structure is of the first kind.…”

Section: Center Of Lcs Lie Algebras Of the Second Kindmentioning

confidence: 99%

“…We also use Theorem 3. Table 1 the unimodular 4-dimensional Lie algebras admitting a LCS structure of the second kind (see [2]).…”

Section: Examples Of Lie Algebras With Lcs Structures Of the Second Kindmentioning

confidence: 99%

“…As in the case of manifolds we have that a LCS structure (ω, θ) on a Lie algebra g can be of the first kind or of the second kind. Indeed, let us denote by g ω the set of infinitesimal automorphisms of the LCS structure, that is, (2) g ω = {X ∈ g : L X ω = 0} = {X ∈ g : ω([X, Y ], Z)+ω(Y, [X, Z]) = 0 for all Y, Z ∈ g} where L denotes the Lie derivative. Note that g ω ⊂ g is a Lie subalgebra, thus the restriction of θ to g ω is a Lie algebra morphism called the Lee morphism.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand not much is known about Lie algebras with a LCS structure of the second kind. In [2] the authors study three different type of constructions of LCS Lie algebras. One corresponds to exacts LCS Lie algebras.…”

Section: Introductionmentioning

confidence: 99%

“…In this work we deal with this problem and inspired by ideas of [2] we provide another construction of LCS Lie algebras of the second kind which is different from those given in [2]. After we provide this construction in Theorem 3.3 two question arise naturally: Question 1: Given a LCS structure of the second kind on a Lie algebra, can it be obtained from our construction?…”

Section: Introductionmentioning

confidence: 99%

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On a certain class of locally conformal symplectic structures of the second kind

Origlia

2020

Differential Geometry and its Applications

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We study locally conformal symplectic (LCS) structures of the second kind on a Lie algebra. We show a method to build new examples of Lie algebras admitting LCS structures of the second kind starting with a lower dimensional Lie algebra endowed with a LCS structure and a suitable representation. Moreover, we characterize all LCS Lie algebras obtained with our construction. Finally, we study the existence of lattices in the associated simply connected Lie groups in order to obtain compact examples of manifolds admitting this kind of structure.2010 Mathematics Subject Classification. 53D05, 22E60, 22E25, 53C15, 22E40.There is another way to distinguish LCS structures, to do this, we can deform the de Rham differential d to obtain the adapted differential operatorfor any differential form α ∈ Ω * (M). Since θ is d-closed, this operator satisfies d 2 θ = 0, thus it defines the Morse-Novikov cohomology H * θ (M) of M relative to the closed 1-form θ (see [13,14]). Note that if θ is exact then H * θ (M) ≃ H * dR (M). It is known that if M is a compact oriented n-dimensional manifold, then H 0 θ (M) = H n θ (M) = 0 for any non exact closed 1-form θ (see for instance [6,7]). For any LCS structure (ω, θ) on M, the 2-form ω defines a cohomology class [ω] θ ∈ H 2 θ (M), since d θ ω = dω − θ ∧ ω = 0. The LCS structure (ω, θ) is said to be exact if ω is d θ -exact or [ω] θ = 0, i.e., ω = dη − θ ∧ η for some 1-form η, and it is non-exact if [ω] θ = 0. It was proved in [18] that if the LCS structure (ω, θ) is of the first kind on M then ω is d θ -exact, i.e., [ω] θ = 0. But the converse is not true. Recently in [3] other cohomologies for LCS manifolds were introduced, inspired by the almost Hermitian setting. More precisely, the authors define the LCS-Bott-Chern cohomology and the LCS-Aeppli cohomology on any compact LCS manifold, and compute them for some LCS solvmanifolds in low dimensions.

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Section: Center Of Lcs Lie Algebras Of the Second Kindmentioning

confidence: 99%

“…We also use Theorem 3. Table 1 the unimodular 4-dimensional Lie algebras admitting a LCS structure of the second kind (see [2]).…”

Section: Examples Of Lie Algebras With Lcs Structures Of the Second Kindmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On a certain class of locally conformal symplectic structures of the second kind

Origlia

2020

Differential Geometry and its Applications

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Locally conformally Kähler solvmanifolds: a survey

Andrada

Origlia

2019

Complex Manifolds

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A Hermitian structure on a manifold is called locally conformally Kähler (LCK) if it locally admits a conformal change which is Kähler. In this survey we review recent results of invariant LCK structures on solvmanifolds and present original results regarding the canonical bundle of solvmanifolds equipped with a Vaisman structure, that is, a LCK structure whose associated Lee form is parallel.2010 Mathematics Subject Classification. 53B35, 53A30, 22E25.

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