Projective limits of quantum symmetry groups and the doubling construction for Hopf algebras

TL;DR

This paper investigates the projective limits of quantum symmetry groups associated with inductive limits of C*-algebras, revealing connections to the doubling construction for Hopf algebras and providing explicit examples.

Contribution

It establishes that the quantum symmetry group of an inductive limit of C*-algebras is the projective limit of the individual quantum symmetry groups, and explores the link to the doubling construction for Hopf algebras.

Findings

Quantum symmetry groups form projective limits in inductive systems.

Explicit examples include quantum symmetry groups of duals of finite symmetric groups.

Deep connection identified between quantum symmetry groups and Hopf algebra doubling.

Abstract

The quantum symmetry group of the inductive limit of C*-algebras equipped with orthogonal filtrations is shown to be the projective limit of the quantum symmetry groups of the C*-algebras appearing in the sequence. Some explicit examples of such projective limits are studied, including the case of quantum symmetry groups of the duals of finite symmetric groups, which do not fit directly into the framework of the main theorem and require further specific study. The investigations reveal a deep connection between quantum symmetry groups of discrete group duals and the doubling construction for Hopf algebras.