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Smooth Cartan triples and Lie twists over Hausdorff étale Lie groupoids

Anna Duwenig, Aidan Sims

Abstract

We describe how to recover a Lie structure on a twist over a Hausdorff étale 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 étale 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 étale 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.

Smooth Cartan triples and Lie twists over Hausdorff étale Lie groupoids

Abstract

We describe how to recover a Lie structure on a twist over a Hausdorff étale 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 étale 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 étale 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.
Paper Structure (9 sections, 38 theorems, 119 equations)

This paper contains 9 sections, 38 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: Lie groupoids
  3. Étale groupoids with unit space a manifold
  4. Twists over Lie groupoids
  5. Lie twists over étale groupoids
  6. Smooth Cartan triples
  7. Differential Geometry
  8. Manifolds and submanifolds
  9. Principal bundles

Key Result

Lemma 2.6

Suppose that $G$ is a topological groupoid and that both $G$ and $G^{\hbox{$(0)$}}$ are manifolds. Let $\mathcal{A}=\{(W_{\alpha}, \psi_{\alpha})\}_{\alpha\in \mathfrak{A}}$ be a maximal smooth atlas of $G^{\hbox{$(0)$}}$. Assume that both $r$ and $s$ are local diffeomorphisms and let $\mathcal{U}$ For $\chi= (\alpha, B_{1}, B_{2})\in \mathfrak{C}$, let $V_{\chi}\coloneqq B_{1} { }\tensor*[_{}]{\

Theorems & Definitions (111)

  • Definition 2.1: cf. Sims:gpds
  • Definition 2.2: Dufour2005
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5: cf. Dufour2005
  • Lemma 2.6
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Definition 3.3
  • ...and 101 more