Quantum Symmetries of Graph C *-algebras

Schmidt, Simon ; Weber, Moritz

Canadian mathematical bulletin, Tome 61 (2018) no. 4, pp. 848-864

Résumé

The study of graph ${{C}^{*}}$ -algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. 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.

Détail

DOI : 10.4153/CMB-2017-075-4

Mots-clés : 46LXX, 05CXX, 20B25, finite graph, graph automorphism, automorphism group, quantum automorphism, graph C*-algebra, quantum group, quantum symmetry