A remark on the motivic Galois group and the quantum coadjoint action

TL;DR

This paper explores the connection between a quotient of the motivic Galois group and the quantum coadjoint action in quantum groups, suggesting a nontrivial realization that could extend to broader contexts and relate to physics.

Contribution

It demonstrates a link between a solvable quotient of the motivic Galois group and quantum coadjoint action in quantum groups at roots of unity, proposing a broader relation to quantum Galois groups.

Findings

A nontrivial realization of a quotient of the motivic Galois group in quantum groups.

Relation between motivic Galois groups and quantum coadjoint actions at roots of unity.

Potential extension to more general cases and connections to physics.

Abstract

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.