Relèvement d'une algébroïde de Courant

Boumaiza, Mohamed; Zaalani, Nadhem

doi:10.1016/j.crma.2009.01.001

Cited by 8 publications

(10 citation statements)

References 6 publications

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“…Note that the original definition was the following: let Λ Γ be the graph of the partial multiplication m in Γ, i.e. 3 . It is shown in [7] that Λ * is the graph of a groupoid multiplication on T * Γ, which is exactly the multiplication defined above.…”

Section: Pontryagin Bundle Over a Lie Groupoidmentioning

confidence: 99%

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Glanon groupoids

Lean

Stiénon

2015

Math. Ann.

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We introduce the notion of Glanon groupoids, which are Lie groupoids equipped with multiplicative generalized complex structures. It combines symplectic groupoids, holomorphic Lie groupoids and holomorphic Poisson groupoids into a unified framework. Their infinitesimal, Glanon Lie algebroids are studied. We prove that there is a bijection between Glanon Lie algebroids and source-simply connected and source-connected Glanon groupoids. As a consequence, we recover various integration theorem and obtain the integration theorem for holomorphic Poisson groupoids.Comment: Improved versio

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Section: Pontryagin Bundle Over a Lie Groupoidmentioning

confidence: 99%

“…In [3], Boumaiza-Zaalani proved that the tangent bundle of a Courant algebroid is naturally a Courant algebroid. In this section, we study the Courant algebroid structure on P T M in terms of the isomorphism Σ : T P M → P T M defined in Proposition 2.9.…”

Section: Tangent Courant Algebroidmentioning

confidence: 99%

Glanon groupoids

Lean

Stiénon

2015

Math. Ann.

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“…In this section we describe our main results and examples. Proofs of the results will be presented later, in Section 4, after describing the tangent prolongation of a Courant algebroid [5], and the theory of double structures. Definition 3.1.…”

Section: Pseudo-dirac Structuresmentioning

confidence: 99%

“…For f ∈ C ∞ (M ), we introduce the notation f 5 Then we have the following rules Pradines [47], and later Konieczna-Urbański [32], Mackenzie [42,41] and Grabowski-Rotkiewicz [19], studied the duals D * x and D * y , proving that they form the total space for double vector bundles themselves. q B/M −−−→ M is defined to be a subbundle K ⊆ T B such that dq B/M : K → T M is a fibrewise isomorphism [16].…”

Section: T B B T M M Bmentioning

confidence: 99%

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Pseudo-Dirac structures

Li-Bland

2014

Indagationes Mathematicae

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A Dirac structure is a Lagrangian subbundle of a Courant algebroid, L ⊂ E, which is involutive with respect to the Courant bracket. In particular, L inherits the structure of a Lie algebroid. In this paper, we introduce the more general notion of a pseudo-Dirac structure: an arbitrary subbundle, W ⊂ E, together with a pseudo-connection on its sections, satisfying a natural integrability condition. As a consequence of the definition, W will be a Lie algebroid. Allowing non-isotropic subbundles of E incorporates non-skew tensors and connections into Dirac geometry. Novel examples of pseudo-Dirac structures arise in the context of quasi-Poisson geometry, Lie theory, generalized Kähler geometry, and Dirac Lie groups, among others. Despite their greater generality, we show that pseudo-Dirac structures share many of the key features of Dirac structures. In particular, they behave well under composition with Courant relations.

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Pre-Courant algebroids

Bruce

Grabowski

2019

Journal of Geometry and Physics

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Pre-Courant algebroids are 'Courant algebroids' without the Jacobi identity for the Courant-Dorfman bracket. We examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof. In particular, we define symplectic almost Lie 2-algebroids and show how they correspond to pre-Courant algebroids. We give the definition of (sub-)Dirac structures and study the naïve quasi-cochain complex within the setting of supergeometry. Moreover, the framework of supermanifolds allows us to economically define and work with pre-Courant algebroids equipped with a compatible non-negative grading. VB-Courant algebroids are natural examples of what we call weighted pre-Courant algebroids and our approach drastically simplifies working with them.Dedicated to the memory of James Alfred Bruce MSC (2010): 17A32; 17B99; 53D17; 58A50.The Jacobiator associated with the Courant-Dorfman pre-bracket on a symplectic almost Lie 2-algebroid has some interesting properties which we state in the following two propositions.Proposition 3.7. Let J Θ be the Jacobiator associated with a symplectic almost Lie 2-algebroid. Then J Θ is totally (graded) skew-symmetric, i.e.,for all σ, ψ and λ ∈ A 1 (F 2 ).Proof. Directly from the definition of the Jacobiator and application of the Jacobi identity for the Poisson bracket, we have J Θ (σ, ψ, λ) + (−1) ( σ+1)( ψ+1) J Θ (ψ, σ, λ) = (−1) ψ {{Σ, {σ, ψ}}, λ}, where we use the shorthand Σ = 1 2 {Θ, Θ}. As {ψ, σ} = ψ, σ is a function on M , the right-hand side of the above vanishes (see Definition 3.6). Directly from the Jacobi identity we haveUsing the skew-symmetry of the Poisson bracket and the Jacobi identity again, we arrive at J Θ (σ, ψ, λ) + (−1) ( ψ+1)( λ+1) J Θ (σ, λ, ψ) = (−1) ψ ({Σ, {σ, {ψ, λ}}} − {σ, {Σ, {ψ, λ}}}) = 0,

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Relèvement d'une algébroïde de Courant

Cited by 8 publications

References 6 publications

Glanon groupoids

Glanon groupoids

Pseudo-Dirac structures

Pre-Courant algebroids

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