Free Wreath Product by the Quantum Permutation Group

Abstract

Let A be a compact quantum group, let n ∈ N * and let A aut (X n ) be the quantum permutation group on n letters. A free wreath product construction A * w A aut (X n ) is done. This construction provides new examples of quantum groups, and is useful to describe the quantum automorphism group of the n-times disjoint union of a finite connected graph.

The Haagerup property for locally compact quantum groups

“…Example 7.9 ([14,15,56]). The duals of the free orthogonal quantum groups, of the free unitary quantum groups, of the quantum automorphism groups of certain finite-dimensional C -algebras equipped with canonical traces, and of the quantum reflection groups H sC n (the free wreath products Z s o S C n , see [9]) for n 4 and 1 Ä s < 1 have the Haagerup property.…”

“…O G//˝C.Z 2 /. Moreover it follows from [9] that the Haar measure of O H is given with respect to this decomposition by the formula h D h 1˝h2 , where h 1 is the free product of Haar states of O G and h 2 is induced by the Haar measure of Z 2 . It follows that Remark 7.12.…”

“…We finish the subsection by exhibiting another corollary, related to the wreath product of compact quantum groups [9] and also to the considerations which will follow in Section 8.…”

“…The free wreath product of a coamenable compact quantum group by S C 2 is not coamenable in general. For example, taking G D Z, the free wreath product of O G D T by S C 2 is a non-coamenable compact quantum group (whose fusion rules are even non-commutative, see [9]), and yet the above corollary shows that its dual has the Haagerup property.…”

Free Product Formulae for Quantum Permutation Groups

“…This is pointed out in [5], where the case of G = Z 2 is worked out explicitely, with a complete discussion of corepresentations of the free wreath product. The fusion rules found there allow one to compute the spectral measure of the free wreath product.…”