(Bi)modules, morphismes et réduction des star-produits: le cas symplectique, feuilletages et obstructions

Abstract

(Bi)modules, morphisms and reduction of star-products are studied in a framework of multidifferential operators along maps: morphisms deform Poisson maps and representations on functions spaces deform coisotropic maps. If a star-product is representable on a coisotropic submanifold, is is equivalent to a star-product for which the vanishing ideal is a left ideal. If the reduced phase space exists, a star-product with suitable Deligne class is representable and the reduced algebra is the commutant of this module (hence a bimodule). Obstructions to representability to third order are related to the Atiyah-Molino class of the foliation of the coisotropic submanifold, and the same kind of obstructions occurs for the quantisation of a Poisson map between symplectic manifolds. For vanishing Atiyah-Molino class, the representation and morphism problem is solvable.

133 pages, paper in French, Latex2e

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Cited by in corpus (7)

• Relative formality theorem and quantisation of coisotropic submanifolds
• Variations on Homological Reduction
• Deformation Quantization of Surjective Submersions and Principal Fibre Bundles
• Homology of formal deformations of proper etale Lie groupoids
• H-äquivariante Morita-Äquivalenz und Deformationsquantisierung
• Symplectic Connections of Ricci Type and Star Products
• Remarks on modules over deformation quantization algebras