# Associators, Grothendieck-Teichmüller group and monoidal categories

The standard definition of an associator seems to be that it a a grouplike power series in two variables $x$ and $y$ satisfying some pentagon and hexagon relations.
In other words, denoting by $\mathfrak{lie}_2$ the free Lie algebra on two variables, an associator is a grouplike element in the completed hopf algebra $\widehat{U}(\mathfrak{lie_2})$ satisfying the above mentionned relations.
In more abstract terms, there is a bijection between the set of associators and a certain morphism of operads. Using Bar-Natan's notations it is a morphism of operads between $\widehat{PaB_K}$ and $PaCD$.

In this context, the Grothendieck-Teichmüller group $\widehat{GT}$ is the group of automorphism of $\widehat{PaB_K}$. It can also be described as pairs $(f,\lambda) \in \widehat{F_2(K)} \times K^*$ again satisfying some relations.

These seem to be the "textbook" definitions, however I don't understand their motivation and what these objects do. My questions are the following. (The lack of precision comes from my lack of knowledge of the subject)

0) What is the purpose of associators? What is the purpose of the G-T group in this context (not in algebraic geometry).

The following questions are possible answers to the previous one.

1) Let $(C, \beta, \gamma)$ be a braided monoidal category with associativity constraint $\gamma$ and braiding $\beta$. Suppose we wish to change $\beta$ and $\gamma$ into $\beta'$ and $\gamma'$ such that $(C, \beta', \gamma')$ is again a braided monoidal category. Does this procedure have a name? Do elements of the Grothendieck-Teichmüller control this?

2) Let $(C, \sigma, \gamma)$ be a symmetric monoidal category. Do the associators control how to transform $C$ into a braided monoidal category?

3) Does the term infinitesimally braided monoidal category exist?

4) Does the Kontsevich integral somehow fit into this story?

As I mentioned above, I know little about the subject: I have skimmed through Bar-Natan's article "On associators and the Grothendieck-Teichmüller group group" and the paper arXiv:0903.4067, therefore any reference is welcome.

0) This is a wide question. Probably the best answer/definition is precisely Bar Natan's one, which of course is implicit in Drinfeld's work: an associator is a filtered isomorphism between the completion of a naturally filtered, complicated, topological category, and a much more manageable, explicit, naturally graded, combinatorial one. In addition to the obvious interest in braid/knot theory, it turns out that the former is closely related to "quantum" objects, while the latter is related to "classical" objects. Hence associators shows up everywhere in deformation-quantization. Another "conceptual" explanation of this is that they are also responsible for the formality of the little disc operad, which is a close relative of the above isomorphism.

1) yes, if your category is $k[[\hbar]]$-linear for $k$ of char 0, this is basically the motivation behind the definition of GT. This is essentially the same things as the definition you gave, as the automorphism group of the completion of the braid category (as opposed to the automorphism group of the braid category, which is almost trivial).

2),3) yes, if the symmetric category is infinitesimal braided (i.e. if it's a representation of PaCD) then an associator turns it into a braided monoidal category (and even ribbon if it has duals).

4) Yes, it is exactly this: the Kontsevich integral for braids is exactly the above mentionned isomorphism, which extends to a functor from the category of tangles which is again an isomorphism after completion, and whose restriction to links is equal to the Kontsevich integral for any choice of associator.

edit: More precisely: as you say, associators are in bijection with isomorphisms $\widehat{PaB}\rightarrow PaCD$ compatible with the operadic structure. The Kontsevich integral provides such an isomorphism, hence produces a particular associator, namely the image of the trivial braid from $(\cdot \cdot) \cdot$ to $\cdot (\cdot \cdot)$ as explained in Bar-Natan's paper. This is the so-called KZ associator, which is the first (and for some time only) known associator. This is how Drinfeld showed that the set of associators in non-empty (and then deduced that there exists rational associators as well). Then this isomorphism extends to tangles, and by a result of Le-Murakami (implicit in Drinfeld's paper) its restriction to links doesn't depends on the choice of the associator (which proves the rationality of the Kontsevich integral of links).

All of this is explained in a nice, clear, pictorial way in many of Bar Natan's paper, especially of course those on Vassiliev invariants, and in Kassel--Turaev "Chord diagramms invariants of tangles and graphs".

• Thank you, I will try to understand your answer and then perhaps ask some follow-up questions. – Anthony Conway Jan 3 '14 at 12:54
• I am still having difficulties with 4). I understand that the Kontsevich integral sends a braid to chord diagrams. Hence it must have something to do with $\widehat{PaB}$ and $PaCD$. However I fail to find any precise statement linking it with associators. – Anthony Conway Jan 3 '14 at 21:45
• I've added an explanation, does it clarify things ? – Adrien Jan 4 '14 at 13:45