Fixed points of compact quantum groups actions on Cuntz algebras

TL;DR

This paper investigates fixed point algebras arising from actions of compact quantum groups on Cuntz algebras, establishing their structure, K-theory, and dependence on fusion rules, with explicit examples for quantum groups like SU_q(N).

Contribution

It introduces a method to analyze fixed point algebras of CQG actions on Cuntz algebras, showing they are purely infinite, nuclear, and identified as Pimsner algebras, with a focus on fusion rule dependence.

Findings

Fixed point algebras are purely infinite and nuclear.

Identified as Pimsner algebras and computed their K-theory.

Fixed points depend only on CQG fusion rules.

Abstract

Given an action of a Compact Quantum Group (CQG) on a finite dimensional Hilbert space, we can construct an action on the associated Cuntz algebra. We study the fixed point algebra of this action, using Kirchberg classification results. Under certain conditions, we prove that the fixed point algebra is purely infinite and nuclear. We further identify it as a Pimsner algebra, compute its $K$-theory and prove a "stability property": the fixed points only depend on the CQG via its fusion rules. We apply the theory to $SU_q(N)$ and illustrate by explicit computations for $SU_q(2)$ and $SU_q(3)$. This construction provides examples of free actions of CQG (or "principal noncommutative bundles").