Ergodic actions of compact quantum groups from solutions of the conjugate equations

TL;DR

This paper constructs ergodic actions of specific compact quantum groups on C*-algebras using solutions to conjugate equations within tensor C*-categories, advancing the understanding of quantum group actions.

Contribution

It introduces a functorial method to build C*-algebras with ergodic quantum group actions from solutions of conjugate equations, linking tensor categories and quantum symmetries.

Findings

Constructed C*-algebras with ergodic actions of A_u(Q) and B_u(Q)

Established a functorial framework connecting conjugate equations to quantum group actions

Demonstrated the maximal C*-norm completion for tensorial functors

Abstract

We use a tensor C*-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a *-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal C*-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the C*-algebra in question. The quantum groups involved are A_u(Q) and B_u(Q).