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An Algebra of Deformation Quantization for Star-Exponentials on Complex Symplectic Manifolds

Authors:	Giuseppe Dito Pierre Schapira

Institution:	(1) Institut de Mathématiques de Bourgogne, Université de Bourgogne, B.P. 47870, 21078 Dijon Cedex, France;(2) Institut de Mathématiques, Université Pierre et Marie Curie, 175, rue du Chevaleret, 75013 Paris, France

Abstract:	The cotangent bundle T ^* X to a complex manifold X is classically endowed with the sheaf of k-algebras ${\mathcal{W}_{TX}}$ of deformation quantization, where k := ${\mathcal{W}_{\{pt\}}}$ is a subfield of ${\mathbb{C}\hbar, \hbar^{-1}]}$ . Here, we construct a new sheaf of k-algebras ${\mathcal{W}^t_{TX}}$ which contains ${\mathcal{W}_{TX}}$ as a subalgebra and an extra central parameter t. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If P is any section of order zero of ${\mathcal{W}_{TX}}$ , we show that ${{\rm exp}(t\hbar^{-1} P)}$ is well defined in ${\mathcal{W}^t_{T*X}}$ .

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