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 Uq (2), SUq (2), and SOq (3) (q ∈(- 1; 0) ∪ (0; 1)) arise in this manner and list the idempotent states on the compact quantum semigroups U0(2), SU0(2), and SO0(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.
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