Free wreath product quantum groups and standard invariants of subfactors

By a construction of Vaughan Jones, the bipartite graph Γ(A) associated with the natural inclusion of C inside a finite-dimensional C⁎-algebra A gives rise to a planar algebra PΓ(A). We prove that every subfactor planar subalgebra of PΓ(A) is the fixed point planar algebra of a uniquely determined action of a compact quantum group G on A. We use this result to introduce a conceptual framework for the free wreath product operation on compact quantum groups in the language of planar algebras/standard invariants of subfactors. Our approach will unify both previous definitions of the free wreath product due to Bichon and Fima-Pittau and extend them to a considerably larger class of compact quantum groups. In addition, we observe that the central Haagerup property for discrete quantum groups is stable under the free wreath product operation (on their duals).