Lie Groupoid C * -Algebras and Weyl Quantization

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.