Abstract

I will explain what are associators (and why are they useful and natural) and what is the Grothendieck-Teichmüller group, and why it is completely obvious that the Grothendieck-Teichmuller group acts simply transitively on the set of all associators. Not enough will be said about how this can be used to show that “every bounded-degree associator extends”, that “rational associators exist”, and that “the pentagon implies the hexagon”.

In a nutshell: the filtered tower of braid groups (with bells and whistles attached) is isomorphic to its associated graded, but the isomorphism is neither canonical nor unique – such an isomorphism is precisely the thing called “an associator”. But the set of isomorphisms between two isomorphic objects always has two groups acting simply transitively on it – the group of automorphisms of the first object acting on the right, and the group of automorphisms of the second object acting on the left. In the case of associators, that first group is what Drinfel’d calls the Grothendieck-Teichmuller group GT, and the second group, isomorphic (though not canonically) to the first, is the “graded version” GRT of GT. All the references and material for this talk can be found there: http://www.math.toronto.edu/~drorbn/Talks/Newton-1301/ .