The graph groupoid of an odd-dimensional Vaksman–Soibelman quantum sphere

Quantum spheres are among the most studied examples of compact quantum spaces. In particular, the ( 2 n + 1 ) -dimensional quantum sphere of Vaksman and Soibelman is described by a C*-algebra that is the Cuntz–Krieger algebra of a directed graph, as proved by Hong and Szymański in 2002. A few years earlier, in 1997, Sheu proved that the same C*-algebra is isomorphic to the groupoid C*-algebra of an étale groupoid. Here we show that the path groupoid of the directed graph of Hong and Szymański is isomorphic to the groupoid discovered by Sheu.