  It was suggested by Kontsevich that the Grothendieck-Teichmueller group GT
should act on the Duflo isomorphism of su(2) but the corresponding realization
of GT turned out to be trivial. We show that a solvable quotient of the motivic
Galois group - which is supposed to agree with GT - is closely related to the
quantum coadjoint action on U_q(sl_2) for q a root of unity, i.e. in the
quantum group case one has a nontrivial realization of a quotient of the
motivic Galois group. From a discussion of the algebraic properties of this
realization we conclude that in more general cases than U_q(sl_2) it should be
related to a quantum version of the motivic Galois group. Finally, we discuss
the relation of our construction to quantum field and string theory and explain
what we believe to be the "physical reason" behind this relation between the
motivic Galois group and the quantum coadjoint action. This might be a starting
point for the generalization of our construction to more involved examples.
