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

TL;DR

This paper characterizes idempotent states on compact quantum groups, showing they correspond to Haar states on quantum subgroups, and classifies these states for specific quantum groups like U_q(2), SU_q(2), and SO_q(3).

Contribution

It provides a simple characterization of idempotent states that are Haar states on quantum subgroups and classifies all such states for certain quantum groups.

Findings

All idempotent states on U_q(2), SU_q(2), and SO_q(3) arise as Haar states on quantum subgroups.

List of idempotent states on U_0(2), SU_0(2), and SO_0(3).

New proof of coamenability for deformations of classical compact Lie groups.

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.