Idempotent states on compact quantum groups and their classification on U_q(2), SU_q(2), and 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 which 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 in (-1,0) \cup (0,1]) arise in this manner and list the idempotent states on the compact quantum semigroups U_0(2), SU_0(2), and SO_0(3). In the Appendix we provide a short new proof of coamenability of the deformations of classical compact Lie groups based on their representation theory.

32 pages; version 2 revises the terminology, adds a few new results in Section 3 and introduces several minor corrections. The paper will appear in the Journal of the Noncommutative Geometry