Journal of Symplectic Geometry
Volume 6 (2008)
A groupoid approach to quantization
Pages: 61 – 125
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.
2010 Mathematics Subject Classification
Primary 46L65. Secondary 22A22, 53D17, 53D50.