Quantum automorphism groups of vertex-transitive graphs of order \leq 11
Teodor Banica1
and Julien Bichon2
1Universite Paul Sabatier Departement of Mathematics 118 route de Narbonne 31062 Toulouse France
2Universite de Pau et des Pays de l'Adour, IPRA Laboratoire de Mathematiques Appliquees Avenue de l'Universite 64000 Pau France
2Universite de Pau et des Pays de l'Adour, IPRA Laboratoire de Mathematiques Appliquees Avenue de l'Universite 64000 Pau France
DOI: 10.1007/s10801-006-0049-9
Abstract
We study quantum automorphism groups of vertex-transitive graphs having less than 11 vertices. With one possible exception, these can be obtained from cyclic groups \mathbb Z n {\mathbb Z}_n, symmetric groups S n and quantum symmetric groups Q n \mathcal Q_n, by using various product operations. The exceptional case is that of the Petersen graph, and we present some questions about it.
Pages: 83–105
Keywords: keywords quantum permutation group; transitive graph
Full Text: PDF
References
1. T. Banica, “Symmetries of a generic coaction,” Math. Ann. 314 (1999), 763-780.
2. T. Banica, “Quantum automorphism groups of small metric spaces,” Pacific J. Math. 219 (2005), 27-51.
3. T. Banica, “Quantum automorphism groups of homogeneous graphs,” J. Funct. Anal. 224 (2005), 243-280.
4. T. Banica and J. Bichon, “Free product formulae for quantum permutation groups,” J. Inst. Math. Jussieu, to appear.
5. T. Banica and B. Collins, “Integration over compact quantum groups,” Publ. Res. Inst. Math. Sci., to appear.
6. T. Banica and S. Moroianu, “On the structure of quantum permutation groups,” Proc. Amer. Math. Soc., to appear.
7. J. Bichon, “Quantum automorphism groups of finite graphs,” Proc. Amer. Math. Soc. 131 (2003), 665-673.
8. J. Bichon, “Free wreath product by the quantum permutation group,” Alg. Rep. Theory 7 (2004), 343- 362.
9. J. Bichon, A. De Rijdt, and S. Vaes, “Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups,” Comm. Math. Phys. 262 (2006), 703-728. Springer
10. W. Fulton and J. Harris, Representation Theory: A First Course, GTM 129, Springer, 1991.
11. J. Kustermans and S. Vaes, “Locally compact quantum groups,” Ann. Sci. Éc. Norm. Supér. (4) 33(6) (2000), 837-934.
12. A. Maes and A. Van Daele, “Notes on compact quantum groups,” Nieuw Arch. Wiskd. (4) 16(1-2) (1998), 73-112.
13. S. Vaes, “Strictly outer actions of groups and quantum groups,” J. Reine Angew. Math. 578 (2005), 147-184.
14. S. Vaes and R. Vergnioux, “The boundary of universal discrete quantum groups, exactness and factoriality,” arxiv:math.OA/0509706.
15. A. Van Daele and S. Wang, “Universal quantum groups,” Internat. J. Math. 7 (1996), 255-264.
16. R. Vergnioux, “The property of rapid decay for discrete quantum groups,” preprint.
17. S. Wang, “Free products of compact quantum groups,” Comm. Math. Phys. 167 (1995), 671-692.
18. S. Wang, “Quantum symmetry groups of finite spaces,” Comm. Math. Phys. 195 (1998), 195-211.
19. S.L. Woronowicz, “Compact matrix pseudogroups,” Comm. Math. Phys. 111 (1987), 613-665.
2. T. Banica, “Quantum automorphism groups of small metric spaces,” Pacific J. Math. 219 (2005), 27-51.
3. T. Banica, “Quantum automorphism groups of homogeneous graphs,” J. Funct. Anal. 224 (2005), 243-280.
4. T. Banica and J. Bichon, “Free product formulae for quantum permutation groups,” J. Inst. Math. Jussieu, to appear.
5. T. Banica and B. Collins, “Integration over compact quantum groups,” Publ. Res. Inst. Math. Sci., to appear.
6. T. Banica and S. Moroianu, “On the structure of quantum permutation groups,” Proc. Amer. Math. Soc., to appear.
7. J. Bichon, “Quantum automorphism groups of finite graphs,” Proc. Amer. Math. Soc. 131 (2003), 665-673.
8. J. Bichon, “Free wreath product by the quantum permutation group,” Alg. Rep. Theory 7 (2004), 343- 362.
9. J. Bichon, A. De Rijdt, and S. Vaes, “Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups,” Comm. Math. Phys. 262 (2006), 703-728. Springer
10. W. Fulton and J. Harris, Representation Theory: A First Course, GTM 129, Springer, 1991.
11. J. Kustermans and S. Vaes, “Locally compact quantum groups,” Ann. Sci. Éc. Norm. Supér. (4) 33(6) (2000), 837-934.
12. A. Maes and A. Van Daele, “Notes on compact quantum groups,” Nieuw Arch. Wiskd. (4) 16(1-2) (1998), 73-112.
13. S. Vaes, “Strictly outer actions of groups and quantum groups,” J. Reine Angew. Math. 578 (2005), 147-184.
14. S. Vaes and R. Vergnioux, “The boundary of universal discrete quantum groups, exactness and factoriality,” arxiv:math.OA/0509706.
15. A. Van Daele and S. Wang, “Universal quantum groups,” Internat. J. Math. 7 (1996), 255-264.
16. R. Vergnioux, “The property of rapid decay for discrete quantum groups,” preprint.
17. S. Wang, “Free products of compact quantum groups,” Comm. Math. Phys. 167 (1995), 671-692.
18. S. Wang, “Quantum symmetry groups of finite spaces,” Comm. Math. Phys. 195 (1998), 195-211.
19. S.L. Woronowicz, “Compact matrix pseudogroups,” Comm. Math. Phys. 111 (1987), 613-665.