## Journal of Symplectic Geometry

### A groupoid approach to quantization

Eli Hawkins

#### Abstract

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.

#### Article information

Source
J. Symplectic Geom., Volume 6, Number 1 (2008), 61-125.

Dates
First available in Project Euclid: 2 July 2008