Abstract view
Quantum symmetries of graph $C^*$algebras


Simon Schmidt,
Saarland University, Fachbereich Mathematik, 66041 Saarbrücken, Germany
Moritz Weber,
Saarland University, Fachbereich Mathematik, 66041 Saarbrücken, Germany
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
finite graph, graph automorphism, automorphism group, quantum automorphism, graph C*algebra, quantum group, quantum symmetry

MSC Classifications: 
46LXX, 05CXX, 20B25 show english descriptions
unknown classification 46LXX unknown classification 05CXX Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51XX]
46LXX  unknown classification 46LXX 05CXX  unknown classification 05CXX 20B25  Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51XX]
