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

Christian Kassel (9/12/99)

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

1. provides the right categorical setting for the theory of Vassiliev invariants of links,
2. 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.

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Steve Lack

Last modified: Tue Dec 7 10:13:29 EST 1999