Examples of quantum commutants

TL;DR

This paper explores quantum commutants, describing their properties, providing examples on classical and quantum spaces, and highlighting that some are not compact quantum groups, thus contributing to the understanding of quantum semigroups.

Contribution

It introduces the concept of quantum commutants, provides explicit examples, and discusses their properties, including cases that are not compact quantum groups.

Findings

Quantum commutants are quantum semigroups with a universal property.

Some quantum semigroups acting on classical and quantum spaces are not compact quantum groups.

The paper addresses an open problem related to compact quantum groups.

Abstract

We describe the notion of a quantum family of maps of a quantum space and that of a quantum commutant of such a family. Quantum commutants are quantum semigroups defined by a certain universal property. We give a few examples of these objects acting on a classical $n$-point space and on the quantum space underlying the algebra of two by two matrices. We show that some of the resulting quantum semigroups are not compact quantum groups. The proof of one result touches on an interesting problem of the theory of compact quantum groups.