Monoidal Rigidity for Free Wreath Products

TL;DR

This paper proves that certain monoidal equivalences between compact quantum groups involving free wreath products imply actual isomorphisms, clarifying the structure of these quantum groups.

Contribution

It establishes that monoidal equivalences of a specific form lead to isomorphisms between free wreath product quantum groups, revealing their rigidity.

Findings

Monoidal equivalence implies isomorphism for these quantum groups

Clarifies the structure of free wreath products with quantum automorphism groups

Provides a rigidity result for quantum group constructions

Abstract

In this note we observe that any compact quantum group monoidally equivalent, in a nice way, to a free wreath product of a compact quantum group $G$ by the quantum automorphism group of a finite dimensional C*-algebra with a $\delta$-form is actually isomorphic to a free wreath product of $G$ by the quantum automorphism group of another finite dimensional C*-algebra with a $\delta$-form.