Vertex-IRF transformations, dynamical quantum groups and harmonic analysis

It is shown that a dynamical quantum group arising from a vertex-IRF transformation has a second realization with untwisted dynamical multiplication but nontrivial bigrading. Applied to the SL 2; ) dynamical quantum group, the second realization is naturally described in terms of Koornwinder's twisted primitive elements. This leads to an intrinsic explanation why harmonic analysis on the “classical” SL(2; ) quantum group with respect to twisted primitive elements, as initiated by Koornwinder, is the same as harmonic analysis on the SL(2; C) dynamical quantum group.