A duality property for complex Lie algebroids

Chemla, Sophie

doi:10.1007/s002090050520

Cited by 27 publications

(34 citation statements)

References 14 publications

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“…Its inverse functor is given by M → Hom OX (E, M). This equivalence of categories is well-known for D-modules ( [Bo], [Ka]) and was generalized to Lie algebroids in [Ch2]. In the case where X = R d and E = T R d , this equivalence of categories is particularly simple because we may choose dx 1 ∧· · ·∧dx d as a basis of the O R d -module Ω d X .…”

Section: Examples Of D(e)-modulesmentioning

confidence: 99%

Formality Theorem with Coefficients in a Module

Chemla

2008

Transformation Groups

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In this paper, X will denote a C ∞ manifold. In a very famous paper, Kontsevich [Ko] showed that the differential graded Lie algebra (DGLA) of polydifferential operators on X is formal. Calaque [C1] extended this theorem to any Lie algebroid. More precisely, given any Lie algebroid E over X, he defined the DGLA of E-polydifferential operators, Γ(X, E D * poly ), and showed that it is formal. Denote by Γ(X, E T * poly ) the DGLA of E-polyvector fields. Considering M , a module over E, we define Γ(X, E T * poly (M )) the Γ(X, E T * poly )-module of E-polyvector fields with values in M . Similarly, we define the Γ(X, E D * poly )-module of E-polydifferential operators with values in M , Γ(X, E D * poly (M )). We show that there is a quasi-isomorphism of L∞-modules over Γ(X, E T * poly ) from Γ(X, E T * poly (M )) to Γ(X, E D * poly (M )). Our result extends Calaque 's (and Kontsevich's) result.

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Section: Examples Of D(e)-modulesmentioning

confidence: 99%

Formality Theorem with Coefficients in a Module

Chemla

2008

Transformation Groups

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“…Also Chemla discussed, in the framework of complex Lie algebroids, the relation between Lie algebroids and certain algebras of differential operators, see e.g. [11]. Finally, one should note that although there are some similarities our setting and our construction are different from that in [36], where symplectic Lie algebroids are quantized and not the dual E * with its in general truly non-symplectic Poisson structure.…”

Section: V) Polmentioning

confidence: 99%

Deformation Quantization of Poisson Structures Associated to Lie Algebroids

Neumaier

Waldmann

2009

SIGMA

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Abstract. In the present paper we explicitly construct deformation quantizations of certain Poisson structures on E * , where E −→ M is a Lie algebroid. Although the considered Poisson structures in general are far from being regular or even symplectic, our construction gets along without Kontsevich's formality theorem but is based on a generalized Fedosov construction. As the whole construction merely uses geometric structures of E we also succeed in determining the dependence of the resulting star products on these data in finding appropriate equivalence transformations between them. Finally, the concreteness of the construction allows to obtain explicit formulas even for a wide class of derivations and selfequivalences of the products. Moreover, we can show that some of our products are in direct relation to the universal enveloping algebra associated to the Lie algebroid. Finally, we show that for a certain class of star products on E * the integration with respect to a density with vanishing modular vector field defines a trace functional.

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“…Such objects are called complex Lie algebroids by Chemla [6], but, as in [5], we will reserve this term for the "hybrid" concept defined below.…”

Section: Definition and First Examplesmentioning

confidence: 99%

Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

Leichtnam¹,

Tang²,

Weinstein³

2007

J. Eur. Math. Soc.

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Abstract. Let X be a complex manifold with strongly pseudoconvex boundary M. If ψ is a defining function for M, then − log ψ is plurisubharmonic on a neighborhood of M in X, and the (real) 2-form σ = i∂∂(− log ψ) is a symplectic structure on the complement of M in a neighborhood of M in X; it blows up along M.The Poisson structure obtained by inverting σ extends smoothly across M and determines a contact structure on M which is the same as the one induced by the complex structure. When M is compact, the Poisson structure near M is completely determined up to isomorphism by the contact structure on M. In addition, when − log ψ is plurisubharmonic throughout X, and X is compact, bidifferential operators constructed by Engliš for the Berezin-Toeplitz deformation quantization of X are smooth up to the boundary. The proofs use a complex Lie algebroid determined by the CR structure on M, along with some ideas of Epstein, Melrose, and Mendoza concerning manifolds with contact boundary.

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A duality property for complex Lie algebroids

Cited by 27 publications

References 14 publications

Formality Theorem with Coefficients in a Module

Formality Theorem with Coefficients in a Module

Deformation Quantization of Poisson Structures Associated to Lie Algebroids

Poisson geometry and deformation quantization near a strictly pseudoconvex boundary

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