A characteristic map for compact quantum groups

We show that if G is a compact Lie group and \(\mathfrak {g}\) is its Lie algebra, then there is a map from the Hopf-cyclic cohomology of the quantum enveloping algebra \(U_q(\mathfrak {g})\) to the twisted cyclic cohomology of quantum group algebra \({\mathcal O}(G_q)\). We also show that the Schmüdgen-Wagner index cocycle associated with the volume form of the differential calculus on the standard Podle? sphere \({\mathcal O}(S^2_q)\) is in the image of this map.