A remark on the motivic Galois group and the quantum coadjoint action
Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, 1090, Vienna, Austria
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Published online: 13 March 2006
It has been suggested that the Grothendieck–Teichmüller 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 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 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.
© Springer-Verlag, 2006