Journal of Symplectic Geometry

A groupoid approach to quantization

Eli Hawkins

Full-text: Open access


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

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

First available in Project Euclid: 2 July 2008

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L65: Quantizations, deformations
Secondary: 53D17: Poisson manifolds; Poisson groupoids and algebroids 22A22: Topological groupoids (including differentiable and Lie groupoids) [See also 58H05] 53D50: Geometric quantization


Hawkins, Eli. A groupoid approach to quantization. J. Symplectic Geom. 6 (2008), no. 1, 61--125.

Export citation