Completion of braided monoidal categories and action of the Grothendieck-Teichmüller group

Christian Kassel – 8 December 1999

I report on joint work with V. Turaev, published in the Duke Math. J. 92 (1998), 497-552. All braided monoidal categories I'll consider in this talk are R-linear, where R is a fixed commutative ring. If C is such a braided monoidal category, we can take its completion C^ with respect to the ideal of morphisms generated by c - c^{-1}, where c is the braiding in C and c^{-1} is its inverse. I'll show that such a completion provides the right categorical setting for the theory of Vassiliev invariants of links, is the right one for an action of Drinfeld's Grothendieck-Teichm\"uller group (whose definition I'll recall) on braided monoidal categories to exist. As a consequence of (1) and (2), we can construct an action of the absolute Galois group of the rationals on the Vassiliev invariants of framed links and tangles.