# Operator algebra seminar

Speaker: Olivier Gabriel, Glasgow University

Title: Representations of compact quantum groups and classification theory

Abstract: We will start our talk by a brief discussion of compact quantum groups. Any dimension d representation of a compact quantum group G induces an action on the Cuntz algebra Od. We study the fixed point algebra for this action and prove that under certain conditions, it is a Kirchberg algebra in the bootstrap class, hence classified by its K-theory. As a consequence, the fixed point algebra has a "stability property": it only depends on G through its representation category and its fusion rules.

In particular, for actions of G := SUq(2), the fixed points are the Cuntz algebra O and we therefore get a family of inclusions of O inside O2. As a basis for future developments, we sketch how von Neumann algebra techniques could help us extract information from these inclusions, beyond classification techniques