# Cuntz semigroups of compact-type Hopf C*-algebras

## TL;DR

This paper explores the structure of Cuntz semigroups in compact-type Hopf C*-algebras, extending classification tools to quantum groups with Hopf algebra structures.

## Contribution

It generalizes the equivariant Cuntz semigroup to Hopf C*-algebras, developing new theoretical aspects for classification of quantum groups.

## Key findings

- Identification of additional structure in Cuntz semigroups of Hopf C*-algebras
- Development of theory enabling classification of quantum groups
- Extension of Elliott classification program results

## Abstract

The classical Cuntz semigroup has an important role in the study of C*-algebras, being one of the main invariants used to classify recalcitrant C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra structure, and find additional structure in their Cuntz semigroups, thus generalizing the equivariant Cuntz semigroup. We develop various aspects of the theory of such semigroups, and in particular, we give general results allowing classification results of the Elliott classification program to be extended to classification results for C*-algebraic quantum groups.