A Serre-Swan Theorem for Coisotropic Algebras

TL;DR

This paper establishes a Serre-Swan Theorem for coisotropic algebras, linking algebraic modules to geometric vector bundles in Poisson geometry and deformation quantization.

Contribution

It proves a Serre-Swan Theorem for coisotropic algebras, connecting algebraic modules with geometric vector bundles in the context of manifolds, submanifolds, and distributions.

Findings

Proves an equivalence of categories for regular projective modules and vector bundles.

Establishes a correspondence between algebraic and geometric structures in coisotropic settings.

Extends classical Serre-Swan Theorem to coisotropic algebras in Poisson geometry.

Abstract

Coisotropic algebras are used to formalize coisotropic reduction in Poisson geometry as well as in deformation quantization and find applications in various other fields as well. In this paper we prove a Serre-Swan Theorem relating the regular projective modules over the coisotropic algebra built out of a manifold $M$, a submanifold $C$ and an integrable smooth distribution $D \subseteq TC$ with vector bundles over this geometric situation and show an equivalence of categories for the case of a simple distribution.