Free wreath product quantum groups and standard invariants of subfactors

TL;DR

This paper introduces a new framework for free wreath product quantum groups using subfactor planar algebras, unifying previous definitions and extending to broader classes, with implications for quantum group properties.

Contribution

It develops a conceptual approach to free wreath products via subfactor invariants, unifies existing definitions, and extends them to larger classes of quantum groups.

Findings

Every subfactor planar subalgebra is a fixed point algebra of a quantum group action.

The free wreath product operation preserves the central Haagerup property.

Confirmed a conjecture on the distribution of characters in free wreath products.

Abstract

By a construction of Vaughan Jones, the bipartite graph $\Gamma(A)$ associated with the natural inclusion of $\mathbb C$ inside a finite-dimensional $C^*$-algebra $A$ gives rise to a planar algebra $\mathcal P^{\Gamma(A)}$. We prove that every subfactor planar subalgebra of $\mathcal P^{\Gamma(A)}$ is the fixed point planar algebra of a uniquely determined action of a compact quantum group $\mathbb G$ 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 unifies both previous definitions of the free wreath product due to Bichon and Fima-Pittau and extends 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) and we deduce a positive answer to a conjecture of Banica and Bichon on the distribution of the character of a free wreath product.