THE FREE WREATH PRODUCT OF A COMPACT QUANTUM GROUP BY A QUANTUM AUTOMORPHISM GROUP

Abstract. Let G be a compact quantum group and Gaut(B,ψ) be the quantum automor-phism group of a finite dimensional C*-algebra (B,ψ). In this paper, we study the free wreath product G o∗Gaut(B,ψ). First of all, we describe its space of intertwiners and find its fusion semiring. Then, we prove some stability properties of the free wreath product opera-tion. In particular, we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings. We also establish some analytic and alge-braic properties of G o ∗ Gaut(B,ψ). As a last result, we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group