Quantum spheres as graph C*-algebras: a review

TL;DR

The paper addresses realizing the Vaksman–Soibelman quantum spheres $S^{2n+1}_q$ as graph C*-algebras by developing a self-contained, graph-based framework. It constructs the coordinate algebras $\mathcal{A}(S^{2n+1}_q)$ with a $\mathbb{T}^{n+1}$-grading and analyzes both the $q=0$ and $0<q<1$ cases, using conditional expectations à la MK22 to relate the universal C*-algebras to graph C*-algebras. A key contribution is the explicit graph realization for $q=0$ via Leavitt path algebras of $\Sigma_n$ and $\widetilde{\Sigma}_n$, and the subsequent demonstration that $C(S^{2n+1}_q)\cong C(S^{2n+1}_0)$ for all $0<q<1$, establishing independence of the C*-algebra from the deformation parameter. This work provides a transparent, pedagogical route to linking quantum spheres with graph C*-algebras, enabling combinatorial methods in the study of quantum homogeneous spaces and their C*-algebraic invariants.

Abstract

In this survey, we discuss the description of Vaksman-Soibelman quantum spheres using graph C*-algebras, following the seminal work of Hong and Szymański. We give a slightly different proof of the isomorphism with a graph C*-algebra, borrowing the idea of Mikkelsen and Kaad of using conditional expectations to prove the desired result.

Figures (1)

• Figure 1: The graph $\Sigma_n$.

Theorems & Definitions (63)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof
• Definition 2.5
• Proposition 2.7
• proof