Skip to main content Accessibility help

Quantum Symmetries of Graph C *-algebras

  • Simon Schmidt (a1) and Moritz Weber (a1)


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.



Hide All
[1] Banica, T., Quantum automorphism groups of homogeneous graphs. J. Funct. Anal. 224 (2005), 243280.!0.1016/j.jfa.2004.11.002
[2] Banica, T. and Bichon, J., Quantum automorphism groups of vertex-transitive graphs of order ≤ 11. J. Algebraic Combin. 26 (2007), no. 1, 83-105.
[3] Banica, T., Bichon, J., and Chenevier, G., Graphs having no quantum symmetry. Ann. Inst. Fourier (Grenoble) 57 (2007), 955971.
[4] Bhowmick, J., Goswami, D., and Skalski, A., Quantum isometry groups of 0-dimensional manifolds. Trans. Amer. Math. Soc. 363 (2011), no. 2, 901-921. http://dx.doi.Org/10.1090/S0002-9947-2010-05141-4
[5] Bichon, J., Quantum automorphism groups of finite graphs. Proc. Amer. Math. Soc. 131 (2003), 665673. http://dx.doi.Org/10.1090/S0002-9939-02-06798-9
[6] Bichon, J., Free wreath product by the quantum permutation group. Algebr. Represent. Theory 7 (2004), 343362. http://dx.doi.Org/10.1023/
[7] Chassaniol, A., Quantum automorphism group of the lexicographic product of finite regular graphs. J. Algebra 456 (2016), 2345. http://dx.doi.Org/10.1016/j.jalgebra.2O16.01.036
[8] Cuntz, J. and Krieger, W., A class of C*-algebras and topological Markov chains. Invent. Math. 56 (1980), no. 3, 251-268.
[9] Eilers, S., Restorff, G., Ruiz, E., and Sorensen, A., The complete classification of unital graph C*-algebras: Geometric and strong. arxiv:1611.07120
[10] Fulton, M. B., The quantum automorphism group and undirected trees. Ph.D. Thesis, Virginia Polytechnic Institute and State University, 2006.
[11] Goswami, D. and Bhowmick, J., Quantum isometry groups. Infosys Science Foundation Series in Mathematical Science, Springer, New Delhi, 2016. http://dx.doi.Org/10.1007/978-81-322-3667-2
[12] Joardar, S. and Mandai, A., Quantum symmetry of graph C* -algebras associated with connected graphs. 2017. arxiv:1 711.04253
[13] Neshveyev, S. and Tuset, L., Compact quantum groups and their representation categories. Cours Specialises, 20, Société Mathématique de France, Paris, 2013.
[14] Podles, P., Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups. Comm. Math. Phys. 170 (1995), 120. http://dx.doi.Org/10.1007/BF02099436
[15] Raeburn, I., Graph algebras. CBMS Regional Conference Series in Mathematics, 103, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005. http://dx.doi.Org/10.1090/cbms/103
[16] Speicher, R. and Weber, M., Quantum groups with partial commutation relations. arxiv:1603.09192
[17] Timmermann, T., An invitation to quantum groups and duality. From Hopf algebras to multiplicative unitaries and beyond. EMS Textbooks in Mathematics, European Mathematical Society (EMS), Zurich, 2008. http://dx.doi.Org/10.4171/043
[18] Wang, S., Quantum symmetry groups of finite spaces. Comm. Math. Phys. 195 (1998), 195211. http://dx.doi.Org/10.1OO7/sOO22OOO5O385
[19] Woronowicz, S. L., Compact matrix pseudogroups. Comm. Math. Phys. 111 (1987), 613665. http://dx.doi.Org/10.1007/BF01219077
[20] Woronowicz, S. L., A remark on compact matrix quantum groups. Lett. Math. Phys. 21 (1991), no. 1, 35-39. http://dx.doi.Org/10.1007/BF00414633
MathJax is a JavaScript display engine for mathematics. For more information see


Related content

Powered by UNSILO

Quantum Symmetries of Graph C *-algebras

  • Simon Schmidt (a1) and Moritz Weber (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.