Résume | I will demonstrate that the free product construction for subfactors due to Bisch and Jones and its formulation for planar algebras can be used to give a systematic description of the free wreath product operation for compact quantum groups. In order to do, I will recall Jones's notion of graph planar algebra and I will show that any subfactor planar subalgebra of the graph planar algebra of a graph with one even vertex arises as the fixed point algebra of an action of a compact quantum group on the algebra of loops of length 2 on the graph. In addition, an application to (central) approximation properties of free wreath products is given. This is joint work with Pierre Tarrago. |