A groupoid approach to quantization

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.

Double Lie algebroids and representations up to homotopy

“…Infinitesimal ideal systems [5,9] may also be understood in terms of 2-representations. It is proved in [2] that infinitesimal ideal systems are in bijective correspondence with double subbundles of (T A; A, T M; M) of the form (F; A, F …”

Quantization of the symplectic groupoid

“…The second step in geometric quantization is the choice of a polarization. We require the compatibility with the groupoid structure expressed by the following definition given in [7] in order to construct a convolution product between polarized sections.…”

“…Very recently, E. Hawkins in [7] revived the subject. According to his proposal, geometric quantization should produce a C * -algebra on the space of states.…”

“…Since a multiplicative polarization is an ordinary polarization that is compatible with the groupoid structures, the main difficulties of geometric quantization are still there. In [7] it is observed that the ideal case of multiplicative real polarization is given by a groupoid fibration, i.e. the space of leaves is itself a groupoid and the convolution algebra should be in that case its convolution algebra.…”

The modular class as a quantization invariant: a study case

“…In [7] Hawkins proposed a framework in order to discuss geometric quantization of the symplectic groupoid G ; in particular he defined a notion of multiplicative polarization F ⊂ T G by imposing a compatibility with the groupoid structure. As usual in geometric quantization, the existence of polarizations is difficult to assess and in general requiring a smooth space of leaves puts severe constraints.…”