Infinite quantum permutations

TL;DR

This paper introduces the concept of quantum permutations for infinite sets, leading to the development of infinite quantum groups that generalize finite quantum permutation groups and explore quantum symmetries of infinite structures.

Contribution

It extends quantum permutation groups to infinite sets, defining infinite quantum groups and analyzing their role as universal quantum symmetries and automorphisms of infinite graphs.

Findings

Defined infinite quantum permutation groups

Identified universal quantum symmetries of infinite sets

Discussed quantum automorphisms of infinite graphs

Abstract

We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups encode universal quantum symmetries of the underlying sets among all discrete quantum groups. We also discuss quantum automorphisms of infinite graphs, including some examples and open problems regarding both the existence and non-existence of quantum symmetries in this setting.