A criterion for hypersymmetry on discrete groupoids

TL;DR

This paper characterizes hypersymmetry in discrete groupoids with Fell bundles by linking it to the symmetry properties of isotropy subgroups, establishing a precise criterion for hypersymmetry.

Contribution

It provides a new criterion for hypersymmetry in discrete groupoids based on the symmetry of isotropy subgroups and characterizes hypersymmetry as equivalent to rigid symmetry.

Findings

Hypersymmetry in discrete groupoids is characterized by isotropy subgroup symmetry.

Hypersymmetry equals rigid symmetry for discrete groupoids.

A criterion for hypersymmetry using Fell bundles with constant fibers.

Abstract

Given a Fell bundle $\mathscr C\overset{q}{\to}\Xi$ over the discrete groupoid $\Xi$, we study the symmetry of the associated Hahn algebra $\ell^{\infty,1}(\Xi\!\mid\!\mathscr C)$ in terms of the isotropy subgroups of $\Xi$. We prove that $\Xi$ is symmetric (resp. hypersymmetric) if and only if all of the isotropy subgroups are symmetric (resp. hypersymmetric). We also characterize hypersymmetry using Fell bundles with constant fibers, showing that for discrete groupoids, 'hypersymmetry' equals 'rigid symmetry'.