Hostname: page-component-5cf477f64f-h6p2m Total loading time: 0 Render date: 2025-03-28T11:40:45.289Z Has data issue: false hasContentIssue false
  • English
  • Français

The Frucht property in the quantum group setting

Part of: Permutation groups Graph theory

Published online by Cambridge University Press:  29 November 2021

T. Banica Open the ORCID record for T. Banica [Opens in a new window]  and
J.P. McCarthy Open the ORCID record for J.P. McCarthy [Opens in a new window]
T. Banica
Affiliation:
Department of Mathematics, University of Cergy-Pontoise, Cergy-Pontoise, France. e-mail: teo.banica@gmail.com
J.P. McCarthy
Affiliation:
Department of Mathematics, Munster Technological University, Cork, Ireland. e-mail: jeremiah.mccarthy@mtu.ie
Get access Rights & Permissions [Opens in a new window]

Abstract

A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting, the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results.

Keywords

Quantum permutationQuantum automorphism

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 3 , September 2022 , pp. 603 - 633
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Banica, T., Quantum automorphism groups of small metric spaces, Pacific J. Math. 219 (2005), 27–51.CrossRefGoogle Scholar
Banica, T., Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2005), 243–280.CrossRefGoogle Scholar
Banica, T. and Bichon, J., Free product formulae for quantum permutation groups, J. Inst. Math. Jussieu 6 (2007), 381–414.CrossRefGoogle Scholar
Banica, T. and Bichon, J., Quantum automorphism groups of vertex-transitive graphs of order $\leq$ 11, J. Algebraic Combin. 26 (2007), 83–105.CrossRefGoogle Scholar
Banica, T. and Bichon, J., Quantum groups acting on 4 points, J. Reine Angew. Math. 626 (2009) 74–114.Google Scholar
Banica, T., Bichon, J. and Collins, B., The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), 345–384.Google Scholar
Banica, T., Bichon, J. and Chenevier, G., Graphs having no quantum symmetry, Ann. Inst. Fourier 57 (2007), 955–971.CrossRefGoogle Scholar
Banica, T., Bichon, J. and Natale, S., Finite quantum groups and quantum permutation groups, Adv. Math. 229 (2012), 3320–3338.CrossRefGoogle Scholar
Bichon, J., Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc. 131 (2003), 665–673.CrossRefGoogle Scholar
Bichon, J., Free wreath product by the quantum permutation group, Alg. Rep. Theory 7 (2004), 343––362.CrossRefGoogle Scholar
Bichon, J., Algebraic quantum permutation groups, Asian-Eur. J. Math. 1 (2008), 1–13.CrossRefGoogle Scholar
Brannan, M., Chirvasitu, A., Eifler, K., Harris, S., Paulsen, V., Su, X. and Wasilewski, M., Bigalois extensions and the graph isomorphism game, Comm. Math. Phys. 375 (2020), 1777–1809.CrossRefGoogle Scholar
Chassaniol, A., Quantum automorphism group of the lexicographic product of finite regular graphs, J. Algebra 456 (2016), 23–45.CrossRefGoogle Scholar
Chassaniol, A., Study of quantum symmetries for vertex-transitive graphs using intertwiner spaces (2019).Google Scholar
de Groot, J., Groups represented by homeomorphism groups, Math. Ann., 138 (1959), 80–102.CrossRefGoogle Scholar
Eder, C., Levandovskyy, V., Schanz, J., Schmidt, S., Steenpass, S. and Weber, M., Existence of quantum symmetries for graphs on up to seven vertices: A computer based approach (2019).Google Scholar
Franz, U. and Gohm, R., Random walks on finite quantum groups, quantum independent increment processes II, vol. 1866. Lecture Notes in Mathematics, (Springer, Berlin, 2006), 1–32.Google Scholar
Frucht, R., Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compos. Math. 6 (1939), 239–250.Google Scholar
Gromada, D., Free quantum analogue of Coxeter group $D_4$ (2020).Google Scholar
Kac, GI and Paljutkin, V.G., Finite group rings, Trudy Moskov. Mat. Obšč. 15 (1966), 224–261. Translated in Trans. Moscow Math. Soc. (1967), 251–284, 1966.Google Scholar
Levin, F., Factor groups of the modular group, J. London Math. Soc. 1(43) (1968), 195–203.CrossRefGoogle Scholar
Lupini, M., Mančinska, L. and Roberson, D.E., Nonlocal games and quantum permutation groups, J. Funct. Anal. 279 (2020), 1–39.CrossRefGoogle Scholar
Mančinska, L. and Roberson, D.E., Quantum isomorphism is equivalent to equality of homomorphism counts from planar graphs (2019).CrossRefGoogle Scholar
McCarthy, J.P., A state-space approach to quantum permutations (2021).CrossRefGoogle Scholar
Musto, B., Reutter, D.J. and Verdon, D., A compositional approach to quantum functions, J. Math. Phys. 59 (2018), 1–57.CrossRefGoogle Scholar
Neumann, B. H., Some remarks on infinite groups, J. Lond. Math. Soc. 12 (1937) 120–127.CrossRefGoogle Scholar
Neumann, B. H., An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. Lon. A 246, (1954), 503–554.CrossRefGoogle Scholar
Sabidussi, G., Graphs with given group and given graph-theoretical properties, Canad. J. Math. 9 (1960), 515–525.CrossRefGoogle Scholar
Schmidt, S., The Petersen graph has no quantum symmetry, Bull. Lond. Math. Soc. 50 (2018), 395–400.CrossRefGoogle Scholar
Schmidt, S., Quantum automorphisms of folded cube graphs, Ann. Inst. Fourier 70 (2020), 949–970.CrossRefGoogle Scholar
Schmidt, S., On the quantum symmetry groups of distance-transitive graphs, Adv. Math. 368 (2020), 1–43.CrossRefGoogle Scholar
Schmidt, S., Quantum automorphism groups of finite graphs, Saarland University, PhD Thesis (2020).Google Scholar
Sekine, Y., An example of finite-dimensional Kac algebras of Kac–Paljutkin type, Proc. Amer. Math. Soc. 124(4) (1996) 1139–1147.CrossRefGoogle Scholar
Stefan, D., The set of types of n-dimensional semisimple and cosemisimple Hopf algebras is finite, J. Algebra 193 (1997), 571–580.CrossRefGoogle Scholar
Wang, S., Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), 671–692.CrossRefGoogle Scholar
Wang, S., Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195–211.CrossRefGoogle Scholar
Woronowicz, S.L., Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613–665.CrossRefGoogle Scholar
Woronowicz, S.L., Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988), 35–76.CrossRefGoogle Scholar
Woronowicz, S.L., Compact quantum groups, in Symétries quantiques (Les Houches, 1995), (North-Holland, Amsterdam, 1998), 845–884.Google Scholar