  We develop theory of multiplicity maps for compact quantum groups, as an
application, we obtain a complete classification of right coideal
$C^*$-algebras of $C(SU_q(2))$ for $q\in [-1,1]\setminus \{0\}$. They are
labeled with Dynkin diagrams, but classification results for positive and
negative cases of $q$ are different. Many of the coideals are quantum spheres
or quotient spaces by quantum subgroups, but we do have other ones in our
classification list.
