Geodesic
Quantum Symmetries of Graph C *-algebras
Canadian mathematical bulletin, Tome 61 (2018) no. 4, pp. 848-864

Voir la notice de l'article provenant de la source Cambridge University Press

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.
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 
Schmidt, Simon; Weber, Moritz. Quantum Symmetries of Graph C *-algebras. Canadian mathematical bulletin, Tome 61 (2018) no. 4, pp. 848-864. doi: 10.4153/CMB-2017-075-4
@article{10_4153_CMB_2017_075_4,
     author = {Schmidt, Simon and Weber, Moritz},
     title = {Quantum {Symmetries} of {Graph} {C} *-algebras},
     journal = {Canadian mathematical bulletin},
     pages = {848--864},
     year = {2018},
     volume = {61},
     number = {4},
     doi = {10.4153/CMB-2017-075-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-075-4/}
}
TY  - JOUR
AU  - Schmidt, Simon
AU  - Weber, Moritz
TI  - Quantum Symmetries of Graph C *-algebras
JO  - Canadian mathematical bulletin
PY  - 2018
SP  - 848
EP  - 864
VL  - 61
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-075-4/
DO  - 10.4153/CMB-2017-075-4
ID  - 10_4153_CMB_2017_075_4
ER  - 
%0 Journal Article
%A Schmidt, Simon
%A Weber, Moritz
%T Quantum Symmetries of Graph C *-algebras
%J Canadian mathematical bulletin
%D 2018
%P 848-864
%V 61
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-075-4/
%R 10.4153/CMB-2017-075-4
%F 10_4153_CMB_2017_075_4

[1] [1] Banica, T., Quantum automorphism groups of homogeneous graphs. J. Funct. Anal. 224 (2005), 243–280. http://dx.doi.org/!0.1016/j.jfa.2004.11.002 Google Scholar

[2] [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. http://dx.doi.org/10.1007/s10801-006-0049-9 Google Scholar

[3] [3] Banica, T., Bichon, J., and Chenevier, G., Graphs having no quantum symmetry. Ann. Inst. Fourier (Grenoble) 57 (2007), 955–971. http://dx.doi.org/10.5802/aif.2282 Google Scholar

[4] [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 Google Scholar

[5] [5] Bichon, J., Quantum automorphism groups of finite graphs. Proc. Amer. Math. Soc. 131 (2003), 665–673. http://dx.doi.Org/10.1090/S0002-9939-02-06798-9 Google Scholar

[6] [6] Bichon, J., Free wreath product by the quantum permutation group. Algebr. Represent. Theory 7 (2004), 343–362. http://dx.doi.Org/10.1023/B:ALCE.0000042148.97035.ca Google Scholar

[7] [7] Chassaniol, A., Quantum automorphism group of the lexicographic product of finite regular graphs. J. Algebra 456 (2016), 23–45. http://dx.doi.Org/10.1016/j.jalgebra.2O16.01.036 Google Scholar

[8] [8] Cuntz, J. and Krieger, W., A class of C*-algebras and topological Markov chains. Invent. Math. 56 (1980), no. 3, 251-268. http://dx.doi.org/10.1007/BF01390048 Google Scholar

[9] [9] Eilers, S., Restorff, G., Ruiz, E., and Sorensen, A., The complete classification of unital graph C*-algebras: Geometric and strong. arxiv:1611.07120 Google Scholar

[10] [10] Fulton, M. B., The quantum automorphism group and undirected trees. Ph.D. Thesis, Virginia Polytechnic Institute and State University, 2006. Google Scholar

[11] [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 Google Scholar

[12] [12] Joardar, S. and Mandai, A., Quantum symmetry of graph C* -algebras associated with connected graphs. 2017. arxiv:1 711.04253 Google Scholar

[13] [13] Neshveyev, S. and Tuset, L., Compact quantum groups and their representation categories. Cours Specialises, 20, Société Mathématique de France, Paris, 2013. Google Scholar

[14] [14] Podles, P., Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups. Comm. Math. Phys. 170 (1995), 1–20. http://dx.doi.Org/10.1007/BF02099436 Google Scholar

[15] [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 Google Scholar

[16] [16] Speicher, R. and Weber, M., Quantum groups with partial commutation relations. arxiv:1603.09192 Google Scholar

[17] [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 Google Scholar

[18] [18] Wang, S., Quantum symmetry groups of finite spaces. Comm. Math. Phys. 195 (1998), 195–211. http://dx.doi.Org/10.1OO7/sOO22OOO5O385 Google Scholar

[19] [19] Woronowicz, S. L., Compact matrix pseudogroups. Comm. Math. Phys. 111 (1987), 613–665. http://dx.doi.Org/10.1007/BF01219077 Google Scholar

[20] [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 Google Scholar

Cité par Sources :