The graph groupoid of a quantum sphere

TL;DR

This paper demonstrates that the path groupoid of a specific directed graph model of quantum spheres is isomorphic to a groupoid previously identified by Sheu, linking two different descriptions of quantum sphere C*-algebras.

Contribution

It establishes an isomorphism between the graph path groupoid and Sheu's groupoid, unifying two frameworks for quantum sphere C*-algebras.

Findings

The path groupoid of Hong and Szymański's graph is isomorphic to Sheu's groupoid.

This connection clarifies the structure of quantum sphere C*-algebras.

The result bridges graph-based and groupoid-based descriptions of quantum spaces.

Abstract

Quantum spheres are among the most studied examples of compact quantum spaces, described by C*-algebras which are Cuntz-Krieger algebras of a directed graph, as proved by Hong and Szyma\'nski in 2002. About five years earlier, in 1997, Sheu proved that the C*-algebra of a quantum sphere is a groupoid C*-algebra. Here we show that the path groupoid of the directed graph of Hong and Szyma\'nski is isomorphic to the groupoid discovered by Sheu.