Representations of orbifold groupoids

TL;DR

This paper demonstrates that all orbifold groupoids can be faithfully represented on finite-dimensional Hilbert spaces, linking them to translation groupoids of compact group actions, thus advancing their structural understanding.

Contribution

It introduces a faithful representation of orbifold groupoids on finite-dimensional Hilbert spaces and establishes their Morita equivalence to translation groupoids of compact group bundles.

Findings

Every orbifold groupoid admits a faithful finite-dimensional Hilbert space representation.

Orbifold groupoids are Morita equivalent to translation groupoids of compact topological group actions.

The results unify effective and ineffective orbifold representations.

Abstract

Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent to the translation groupoid of an action of a bundle of compact topological groups.