On two possible constructions of the quantum semigroup of all quantum permutations of an infinite countable set

TL;DR

This paper introduces two models for a quantum semigroup of permutations on an infinite set, using projective limits and universal properties, advancing the mathematical understanding of quantum symmetries.

Contribution

It proposes two novel constructions of the quantum semigroup of all permutations on an infinite countable set, expanding the framework of quantum symmetry structures.

Findings

Two models for the quantum permutation semigroup are developed

The models are based on projective limits and universal properties

The approaches provide new insights into quantum symmetries of infinite sets

Abstract

Two different models for a Hopf-von Neumann algebra of bounded functions on the quantum semigroup of all (quantum) permutations of infinitely many elements are proposed, one based on projective limits of enveloping von Neumann algebras related to finite quantum permutation groups, and the second on a universal property with respect to infinite magic unitaries.