# Inductive limits of compact quantum groups and their unitary representations

## TL;DR

This paper introduces inductive limits of compact quantum groups as $W^*$-bialgebras, develops their unitary representation theory, and connects these concepts to quantized characters and $q$-central probability measures.

## Contribution

It formalizes inductive limits of compact quantum groups as $W^*$-bialgebras and explores their unitary representations, linking to previous work on quantized characters and probability measures.

## Key findings

- Defined inductive limits as $W^*$-bialgebras with additional structures
- Provided a representation-theoretic interpretation of quantized characters
- Connected transformations to analysis of $q$-central probability measures

## Abstract

We will introduce the notion of inductive limits of compact quantum groups as $W^*$-bialgebras equipped with some additional structures. We also formulate their unitary representation theories. Those give a more explicit representation-theoretic meaning to our previous study of quantized characters associated with a given inductive system of compact quantum groups. As a byproduct, we will give an explicit representation-theoretic interpretation to some transformations that play an important role in the analysis of $q$-central probability measures on the paths in the Gelfand-Tsetlin graph.