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On the structure of quantum permutation groups

Authors: Teodor Banica and Sergiu Moroianu
Journal: Proc. Amer. Math. Soc. 135 (2007), 21-29
MSC (2000): Primary 16W30; Secondary 81R50
Published electronically: June 22, 2006
MathSciNet review: 2280170
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Abstract: The quantum permutation group of the set $ X_n=\{ 1,\ldots ,n\}$ corresponds to the Hopf algebra $ A_{aut}(X_n)$. This is an algebra constructed with generators and relations, known to be isomorphic to $ \mathbb{C} (S_n)$ for $ n\leq 3$, and to be infinite dimensional for $ n\geq 4$. In this paper we find an explicit representation of the algebra $ A_{aut}(X_n)$, related to Clifford algebras. For $ n=4$ the representation is faithful in the discrete quantum group sense.

References [Enhancements On Off] (What's this?)

  • 1. E. Abe, Hopf algebras, Cambridge Univ. Press (1977). MR 0594432 (83a:16010)
  • 2. T. Banica, Hopf algebras and subfactors associated to vertex models, J. Funct. Anal. 159 (1998), 243-266. MR 1654119 (2001a:46069)
  • 3. T. Banica, Symmetries of a generic coaction, Math. Ann. 314 (1999), 763-780. MR 1709109 (2001g:46146)
  • 4. T. Banica, Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2005), 243-280. MR 2146039
  • 5. J. Bichon, Quantum automorphism groups of finite graphs, Proc. Amer. Math. Soc. 131 (2003), 665-673. MR 1937403 (2003j:16049)
  • 6. J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003), 68-92. MR 1951446 (2003k:46081)
  • 7. S. Vaes, Strictly outer actions of groups and quantum groups, J. Reine Angew. Math. 578 (2005), 147-184. MR 2113893 (2005k:46167)
  • 8. D. Voiculescu, K. Dykema and A. Nica, Free random variables, CRM Monograph Series 1, AMS (1993). MR 1217253 (94c:46133)
  • 9. S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195-211. MR 1637425 (99h:58014)
  • 10. S. Wang, Simple compact quantum groups I, preprint.
  • 11. S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665. MR 0901157 (88m:46079)
  • 12. S.L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted $ SU(N)$ groups, Invent. Math. 93 (1988), 35-76. MR 0943923 (90e:22033)
  • 13. S.L. Woronowicz, Compact quantum groups, in Symétries Quantiques - Les Houches 1995, North-Holland (1998), 845-884. MR 1616348 (99m:46164)

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Additional Information

Teodor Banica
Affiliation: Laboratoire Emile Picard, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France

Sergiu Moroianu
Affiliation: Institutul de Matematică al Academiei Române, P.O. Box 1-764, RO-014700 Bucureşti, Romania

Keywords: Hopf algebra, magic biunitary matrix, inner faithful representation
Received by editor(s): November 24, 2004
Received by editor(s) in revised form: July 28, 2005
Published electronically: June 22, 2006
Additional Notes: The second author was partially supported by the Marie Curie grant MERG-CT-2004-006375 funded by the European Commission, and by a CERES contract (2004)
Communicated by: Martin Lorenz
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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