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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.

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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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