Idempotent states on compact quantum groups and their classification on U_q(2), SU_q(2), and SO_q(3)

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 U_q (2), SU_q (2), and SO_q (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.