On actions of compact quantum groups

TL;DR

This paper investigates the algebraic structures of compact quantum group actions on C*-algebras, identifying differences caused by kernels and proposing a minimal kernel removal procedure, illustrated with a relevant example.

Contribution

It introduces a method to remove kernels from quantum group actions without altering the core algebraic structure, enhancing understanding of these actions.

Findings

Identifies the kernel difference in algebraic objects related to quantum group actions.

Proposes a minimal kernel removal procedure.

Provides an example of a reduced quantum group action with a non-trivial kernel.

Abstract

We compare algebraic objects related to a compact quantum group action on a unital C*-algebra in the sense of Podle\'s and Baum et al. and show that they differ by the kernel of the morphism describing the action. Then we address ways to remove the kernel without changing the Podle\'s algebraic core. A minimal such procedure is described. We end the paper with a natural example of an action of a reduced compact quantum group with non-trivial kernel.