Classiﬁcation of Deformation Quantization Algebroids on Complex Symplectic Manifolds

Polesello, Pietro

doi:10.2977/prims/1216238303

Cited by 8 publications

(8 citation statements)

References 22 publications

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“…We then classify such weak quantizations and then give a sufficient condition for the existence of an usual quantization of a given Poisson structure. We compare our results with previous works [14,17,18,16].…”

Section: Introductionsupporting

confidence: 70%

Weak quantization of Poisson structures

Calaque

Halbout

2011

Journal of Geometry and Physics

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Abstract. In this paper we prove that any Poisson structure on a sheaf of Lie algebroids admits a weak deformation quantization, and give a sufficient condition for such a Poisson structure to admit an actual deformation quantization. We also answer the corresponding classification problems. In the complex symplectic case, we recover in particular some results of Nest-Tsygan and Polesello-Schapira.We begin the paper with a recollection of known facts about deformation theory of cosimplicial differential graded Lie algebras.

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Section: Introductionsupporting

confidence: 70%

Weak quantization of Poisson structures

Calaque

Halbout

2011

Journal of Geometry and Physics

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“…We first classify deformations of the trivial gerbe, i.e. deformations of the structure sheaf as a stack, on a symplectic manifold M, C ∞ or complex (Theorem 4.2.1; this result is very close to the main theorem of [34]). More precisely, we first reduce the classification problem to classifying certain Q-algebras, using the term of A. Schwarz (or curved DGAs, as they are called in [4]).…”

Section: Introductionmentioning

confidence: 89%

Deformation quantization of gerbes

Bressler

Gorokhovsky

Nest

et al. 2007

Advances in Mathematics

134

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This is the first in a series of articles devoted to deformation quantization of gerbes. We introduce basic definitions, interpret deformations of a given stack as Maurer-Cartan elements of a differential graded Lie algebra (DGLA), and classify deformations of a given gerbe in terms of Maurer-Cartan elements of the DGLA of Hochschild cochains twisted by the cohomology class of the gerbe. We also classify all deformations of a given gerbe on a symplectic manifold, as well as provide a deformation-theoretic interpretation of the first Rozansky-Witten class.

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“…In contrast, in differential setups, larger sheaf categories are often taken as starting point, with no well-behaved category of quasi-coherent sheaves a priori available. The main body of [32] deals with abelian deformations of sheaf categories, and Morita theory of such categories was further developed in work by D'Agnolo and Polesello [10,11,43]. In the future, it would be interesting to understand to what extent the descent method used in this paper may be of use in non-algebraic contexts, to capture notions like DQ modules [25] or cohesive modules [2].…”

Section: Broader Contextmentioning

confidence: 99%

Non-commutative deformations and quasi-coherent modules

Van

Liu

Lowen

2016

Sel. Math. New Ser.

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We identify a class of quasi-compact semi-separated (qcss) twisted presheaves of algebras A for which well-behaved Grothendieck abelian categories of quasi-coherent modules Qch(A) are defined. This class is stable under algebraic deformation, giving rise to a 1-1 correspondence between algebraic deformations of A and abelian deformations of Qch(A). For a qcss presheaf A, we use the GerstenhaberSchack (GS) complex to explicitly parameterize the first-order deformations. For a twisted presheaf A with central twists, we descibe an alternative category QPr(A) of quasi-coherent presheaves which is equivalent to Qch(A), leading to an alternative, equivalent association of abelian deformations to GS cocycles of qcss presheaves of commutative algebras. Our construction applies to the restriction O of the structure sheaf of a scheme X to a finite semi-separating open affine cover (for which we have Qch(O) ∼ = Qch(X )). Under a natural identification of GS cohomology of O and

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Classiﬁcation of Deformation Quantization Algebroids on Complex Symplectic Manifolds

Cited by 8 publications

References 22 publications

Weak quantization of Poisson structures

Weak quantization of Poisson structures

Deformation quantization of gerbes

Non-commutative deformations and quasi-coherent modules

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