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 , , and () arise in this manner and list the idempotent states on the compact quantum semigroups , , and . 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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Uwe Franz, Adam Skalski, Reiji Tomatsu, Idempotent states on compact quantum groups and their classification on , , and . J. Noncommut. Geom. 7 (2013), no. 1, pp. 221–254