The free wreath product of a compact quantum group by a quantum automorphism group

TL;DR

This paper investigates the structure and properties of the free wreath product of a compact quantum group with a quantum automorphism group, revealing its intertwiners, fusion semiring, stability, and algebraic features.

Contribution

It provides a detailed description of the free wreath product's intertwiners, fusion semiring, and stability properties, and relates it to quantum automorphism groups.

Findings

Describes the space of intertwiners for the free wreath product.

Identifies conditions for monoidal equivalence and isomorphic fusion semirings.

Shows the free wreath product of two quantum automorphism groups as a quotient of a quantum automorphism group.

Abstract

Let $\mathbb{G}$ be a compact quantum group and $\mathbb{G}^{aut}(B,\psi)$ be the quantum automorphism group of a finite dimensional C*-algebra $(B,\psi)$. In this paper, we study the free wreath product $\mathbb{G}\wr_{*} \mathbb{G}^{aut}(B,\psi)$. 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 operation. 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 algebraic properties of $\mathbb{G}\wr_{*} \mathbb{G}^{aut}(B,\psi)$. 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.