Lie groupoid C*-algebras and Weyl quantization

TL;DR

This paper demonstrates that the reduced C*-algebra of any Lie groupoid provides a strict quantization of its dual Lie algebroid's Poisson structure, unifying various known quantization procedures.

Contribution

It generalizes Weyl's quantization to Lie groupoids, connecting it with Connes' tangent groupoid and Rieffel's quantization, and applies to gauge groupoids and transformation group C*-algebras.

Findings

C*-algebra of a Lie groupoid is a strict quantization of its Lie algebroid dual.

Recovers Connes' tangent groupoid and Weyl's quantization as special cases.

Quantizes semidirect product Poisson manifolds from group actions.

Abstract

For any Lie groupoid $G$, the vector bundle $g^*$ dual to the associated Lie algebroid $g$ is canonically a Poisson manifold. The (reduced) C*-algebra of $G$ (as defined by A. Connes) is shown to be a strict quantization (in the sense of M. Rieffel) of $g^*$. This is proved using a generalization of Weyl's quantization prescription on flat space. Many other known strict quantizations are a special case of this procedure; on a Riemannian manifold, one recovers Connes' tangent groupoid as well as a recent generalization of Weyl's prescription. When $G$ is the gauge groupoid of a principal bundle one is led to the Weyl quantization of a particle moving in an external Yang-Mills field. In case that $G$ is a Lie group (with Lie algebra $g$) one recovers Rieffel's quantization of the Lie-Poisson structure on $g^*$. A transformation group C*-algebra defined by a smooth action of a Lie group on a manifold $Q$ turns out to be the quantization of the semidirect product Poisson manifold $g^*x Q$ defined by this action.