Idempotent states on compact quantum groups and their classification on ${\mathrm{U}}_q(2)$, ${\mathrm{SU}}_q(2)$, and ${\mathrm{SO}}_q(3)$

Abstract

Unlike for locally compact groups, idempotent states on locally compact quantum groups do not necessarily arise as Haar states of compact quantum subgroups. We give a simple characterisation of those idempotent states on compact quantum groups that do arise as Haar states on quantum subgroups. We also show that all idempotent states on the quantum groups ${\mathrm{U}}_q(2)$, ${\mathrm{SU}}_q(2)$, and ${\mathrm{SO}}_q(3)$ ($q\in (-1,0)\cup (0,1]$) arise in this manner and list the idempotent states on the compact quantum semigroups ${\mathrm{U}}_0(2)$, ${\mathrm{SU}}_0(2)$, and ${\mathrm{SO}}_0(3)$. In the Appendix we provide a short new proof of the coamenability of deformations of classical compact Lie groups based on their representation theory.