Rigidity on Quantum Symmetry for a Certain Class of Graph C*-algebras

Abstract

Quantum symmetry of graph $C^{\ast }$-algebras has been studied, under the consideration of different formulations, in the past few years. It is already known that the compact quantum group $(\underset{|E(\Gamma )|-times}{\underbrace{C(S^1)\ast C(S^1)\ast \cdots \ast C(S^1)}},\Delta )$ always acts on a graph $C^{\ast }$-algebra for a finite, connected, directed graph $\Gamma$ in the category introduced by Joardar and Mandal, where $|E(\Gamma )|:=$ number of edges in $\Gamma$. In this article, we show that for a certain class of graphs including Toeplitz algebra, quantum odd sphere, matrix algebra etc. the quantum symmetry of their associated graph $C^{\ast }$-algebras remains $(\underset{|E(\Gamma )|-times}{\underbrace{C(S^1)\ast C(S^1)\ast \cdots \ast C(S^1)}},\Delta )$ in the category as mentioned before. More precisely, if a finite, connected, directed graph $\Gamma$ satisfies the following graph theoretic properties : (i) there does not exist any cycle of length $\ge$ 2 (ii) there exists a path of length $(|V(\Gamma )|-1)$ which consists all the vertices, where $|V(\Gamma )|:=$ number of vertices in $\Gamma$ (iii) given any two vertices (may not be distinct) there exists at most one edge joining them, then the universal object coincides with $(\underset{|E(\Gamma )|-times}{\underbrace{C(S^1)\ast C(S^1)\ast \cdots \ast C(S^1)}},\Delta )$. Furthermore, we have pointed out a few counter examples whenever the above assumptions are violated.

Final version of the article appeared in Journal of Mathematical Physics. According to the guidelines of the journal: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Math. Phys. 65, 083501 (2024) and may be found at https://doi.org/10.1063/5.0177215

References in corpus (6)

• Quantum Symmetry Groups of Finite Spaces
• Quantum Group of Isometries in Classical and Noncommutative Geometry
• Quantum symmetry groups of C*-algebras equipped with orthogonal filtrations
• Noncommutative Balls and Mirror Quantum Spheres
• Quantum Symmetries of Graph C*-Algebras
• Quantum symmetry of graph C*-algebras at critical inverse temperature