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# Quantum symmetries of graph $C^*$-algebras

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Published:2018-01-24

• Simon Schmidt,
Saarland University, Fachbereich Mathematik, 66041 Saarbrücken, Germany
• Moritz Weber,
Saarland University, Fachbereich Mathematik, 66041 Saarbrücken, Germany
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## 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.
 Keywords: finite graph, graph automorphism, automorphism group, quantum automorphism, graph C*-algebra, quantum group, quantum symmetry
 MSC Classifications: 46LXX - unknown classification 46LXX05CXX - unknown classification 05CXX20B25 - Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51-XX]

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