On the structure of quantum permutation groups

Abstract

Abstract. The quantum permutation group of the set Xn = {1, . . . , n} corresponds to the Hopf algebra Aaut(Xn). This is an algebra constructed with generators and relations, known to be isomorphic to C(Sn) for n ≤ 3, and to be infinite dimensional for n ≥ 4. In this paper we find an explicit representation of the algebra Aaut(Xn), related to Clifford algebras. For n = 4 the representation is faithful in the discrete quantum group sense.

Kirchberg's factorization property for discrete quantum groups

“…Problem 4.7. For other quantum groups of Kac type such as the quantum automorphism groups in [4,21], it would be interesting to determine if they have the factorization property, though we have the following obvious result because C(S + 4 ) is nuclear [5], C(S + 4 ) being the notation of Banica et al for A aut (X 4 ) in [21]. In forthcoming work, we will generalize Kirchberg's factorization property to non-Kac-type discrete quantum groups.…”

Quantum permutation groups: a survey

“…This representation is introduced in [13]. In [12] we use integration techniques for proving that π is faithful.…”

Free Product Formulae for Quantum Permutation Groups

“…In order to get a more enlightening explanation here, some kind of matrix model for A(X 4+ ) seems to be needed. So far, the only result in this sense is the one in [3], where an explicit realisation of A(X 4 ) is found. The model there uses a 4 × 4 matrix constructed using quaternions, which shows that η 4 is indeed the law of the square of a semicircle, appearing as a meridian on the real sphere S 3 .…”

“…Summarizing, a quantum permutation group should be regarded as a mixture of finite, compact and discrete groups, with a flavour of statistical mechanics, knot invariants and planar algebras. Several results are obtained along these lines in [1][2][3][4][5].…”