Smooth Cartan triples and Lie twists over Hausdorff \'etale Lie groupoids

TL;DR

This paper develops a framework to recover Lie structures on twists over Hausdorff étale groupoids using functional-analytic data, extending Connes' theorem and characterizing smooth extensions to Lie groupoids.

Contribution

It introduces Lie twists over Hausdorff Lie groupoids and provides necessary and sufficient conditions for their smoothness, linking algebraic and geometric structures.

Findings

Characterization of when a smooth structure extends to a Lie groupoid

Conditions for a twist to be a Lie twist with smooth sections

Application to Cartan pairs of C*-algebras and normalisers

Abstract

We describe how to recover a Lie structure on a twist over a Hausdorff \'etale groupoid from functional-analytic data in the spirit of Connes' reconstruction theorem for manifolds. We first characterise when a smooth structure on the unit space of a Hausdorff \'etale groupoid can be extended to a Lie-groupoid structure on the whole groupoid. We introduce Lie twists over Hausdorff Lie groupoids, building on Kumjian's notion of a twist over a topological groupoid. We establish necessary and sufficient conditions on a family of sections of a twist over a Lie groupoid under which the twist can be made into a Lie twist so that all the specified sections are smooth. We use these results in the setting of twists over \'etale groupoids to describe conditions on a Cartan pair of C*-algebras and a family of normalisers of the subalgebra, under which Renault's Weyl twist for the pair can be made into a Lie twist for which the given normalisers correspond to smooth sections.