Self-similar quantum groups

TL;DR

This paper introduces the concept of self-similarity in compact quantum groups, focusing on quantum automorphism groups of infinite rooted trees and exploring finitely-constrained examples related to quantum wreath products.

Contribution

It defines self-similar quantum groups, constructs a new $C^*$-algebra for automorphisms of infinite trees, and identifies a class of finitely-constrained examples as quantum wreath products.

Findings

Defined self-similar quantum groups.

Constructed the quantum automorphism group $ ext{A}_X$ for infinite trees.

Identified finitely-constrained examples as quantum wreath products.

Abstract

We introduce the notion of self-similarity for compact quantum groups. For a finite set $X$, we introduce a $C^*$-algebra $\mathbb{A}_X$, which is the quantum automorphism group of the infinite homogeneous rooted tree $X^*$. Self-similar quantum groups are then certain quantum subgroups of $\mathbb{A}_X$. Our main class of examples are called finitely-constrained self-similar quantum groups, and we find a class of these examples that can be described as quantum wreath products by subgroups of the quantum permutation group.