Caractères de rigidité du groupe de Grothendieck–Teichmüller

Abstract

— Let k be a (topological) field of characteristic 0. Using a Drinfeld associator, a representation b Φ(ρ) of the braid group over the field k((h)) of Laurent series can be associated to any representation of a certain Hopf algebra Bn(k). We investigate the dependance in Φ of b Φ(ρ) for a certain class of representations — so-called GT-rigid representations — and deduce from it (continuous) projective representations of the Grothendieck-Teichmuller group GT1(k), hence for k = Ql representations of the absolute Galois group of Q(μl∞). In most situations, these projective representations can be decomposed into linear characters, which we do for the representations of the Iwahori-Hecke algebra of type A. In this case, we moreover express b Φ(ρ) when Φ is even, and get unitary matrix models for the representations of the Iwahori-Hecke algebra. With respect to the action of GT1(k), the representations of this algebra corresponding to hook diagrams have noticeable properties.