On the structure of quantum permutation groups

Banica, Teodor; Moroianu, Sergiu

doi:10.1090/S0002-9939-06-08464-4

On the structure of quantum permutation groups
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by Teodor Banica and Sergiu Moroianu PDF

Proc. Amer. Math. Soc. 135 (2007), 21-29 Request permission

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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