A Tannaka Theorem for Proper Lie Groupoids

TL;DR

This paper extends Tannaka duality to proper Lie groupoids by introducing smooth Euclidean fields, a new category that generalizes smooth vector bundles, enabling a duality theorem in this broader context.

Contribution

It introduces smooth Euclidean fields and proves a Tannaka duality theorem for proper Lie groupoids, expanding the applicability of Tannaka theory.

Findings

Established a Tannaka duality theorem for proper Lie groupoids.

Defined smooth Euclidean fields as a finite-dimensional analogue of Hilbert fields.

Demonstrated the duality using smooth actions on these fields.

Abstract

By replacing the category of smooth vector bundles over a manifold with the category of what we call smooth Euclidean fields, which is a proper enlargement of the former, and by considering smooth actions of Lie groupoids on smooth Euclidean fields, we are able to prove a Tannaka duality theorem for proper Lie groupoids. The notion of smooth Euclidean field we introduce here is the smooth, finite dimensional analogue of the usual notion of continuous Hilbert field.