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A groupoid approach to quantization. (English) Zbl 1154.46041

Summary: Many interesting \(C^*\)-algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution \(C^*\)-algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, including geometric quantization of symplectic manifolds and the \(C^*\)-algebra of a Lie groupoid. I sketch a few new examples, including twisted groupoid \(C^*\)-algebras as quantizations of bundle affine Poisson structures.

MSC:

46L65 Quantizations, deformations for selfadjoint operator algebras
53D17 Poisson manifolds; Poisson groupoids and algebroids
22A22 Topological groupoids (including differentiable and Lie groupoids)
53D50 Geometric quantization
58A05 Differentiable manifolds, foundations
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