# Quantum Symmetries of Graph C*-Algebras

## TL;DR

This paper computes the quantum automorphism groups of finite graphs and shows they act maximally on the associated graph C*-algebras, revealing that quantum symmetries are preserved in the algebraic structure.

## Contribution

It establishes that the quantum automorphism group of a finite directed graph acts maximally on its graph C*-algebra, linking graph symmetries with algebraic automorphisms.

## Key findings

- Quantum automorphism groups act maximally on graph C*-algebras.
- Quantum symmetries of graphs coincide with those of their C*-algebras.
- Comparison of different definitions of quantum automorphism groups for small graphs.

## Abstract

The study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.