Quantum Symmetries of Graph C*-algebras Having Maximal Permutational Symmetry

TL;DR

This paper classifies the quantum symmetries of finite graph C*-algebras, showing they are limited to three families of compact matrix quantum groups, and rules out certain automorphism groups for graphs without isolated vertices.

Contribution

It establishes a complete classification of quantum symmetries for graph C*-algebras within a specific category, identifying exactly three families of quantum groups and excluding some automorphisms.

Findings

Exactly three families of quantum groups can be realized as symmetries.

No non-scalar matrix quantum automorphism group exists for graphs without isolated vertices.

Classification within the specified category is complete and restrictive.

Abstract

Quantum symmetry of a graph $C^{*}$-algebra $C^{*}(\Gamma)$ corresponding to a finite graph $\Gamma$ has been explored by several mathematicians within different categories in the past few years. In this article, we establish that there are exactly three families of compact matrix quantum groups, containing the symmetric group on the set of edges of the underlying graph $\Gamma$, that can be achieved as the quantum symmetries of graph $C^*$-algebras in the category introduced by Joardar and Mandal. Moreover, we demonstrate that there does not exist any graph $C^*$-algebra associated with a finite graph $\Gamma$ without isolated vertices having $A_{u^t}(F^{\Gamma})$ as the quantum automorphism group of $C^*(\Gamma)$ for a non-scalar matrix $F^{\Gamma}$.