The graph groupoid of a quantum sphere

TL;DR

The paper addresses identifying the groupoid model for the C*-algebra of a quantum sphere by proving that the path groupoid of Hong–Szymański's graph is isomorphic to Sheu's groupoid. It builds explicit combinatorial models of the path space via maps $\Phi$, $\iota$, $\varepsilon$ and shift- and tail-equivalence, to realize the isomorphism between the graph groupoid and Sheu's groupoid. In the $n=1$ case (the quantum SU(2) sphere), it provides an explicit isomorphism that reproduces Sheu's result, and for general $n$ it extends the construction to identify the graph groupoid with $\mathfrak{F}_n$, yielding $C(S^{2n+1}_q)\cong C^*(\mathfrak{F}_n)$. Thus the graph- and groupoid-based descriptions offer equivalent étale models for the quantum sphere C*-algebras, with applications to their K-theory, CW-structure, and module-cancellation phenomena.

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 Szymański 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 Szymański is isomorphic to the groupoid discovered by Sheu.

Figures (1)

• Figure 1: The graph of $S^{2n+1}_q$.

Theorems & Definitions (30)

• Example 3.1
• Theorem 3.2: KPRR97
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Example 4.1
• Example 4.2
• Proposition 4.3
• proof