The Grothendieck-Teichmüller Lie Algebra

I'd like to talk about some recent work of Willwacher and coauthors related to the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}_1$ and graph homology. The goal of this first post is modest, just to define $\mathfrak{grt}_1$ and discuss some of what's known about it. I will be relying heavily on Willwacher's paper for my exposition.

First, we define the Drinfeld-Kohno Lie algebra $\mathfrak t_n$. It is freely generated as a Lie algebra by elements $t_{ij}$ which are symmetric in the indices $t_{ij}=t_{ji}$ where $i\neq j$ and $1\leq i,j\leq n$, subject to relations that $[t_{ij},t_{kl}]=0$ if the index sets are disjoint and
$$[t_{ij},t_{ik}]=-[t_{ij},t_{jk}], \text{ if } i,j,k \text{ are distinct indices}.$$ I encountered a similar Lie algebra in a paper of mine, where I noticed that you could interpret iterated brackets as labeled trees. Surely one can do the same here,  but with a different sign convention. For example $t_{ij}$ would be a line segment with ends labeled by $i$ and $j$. The bracket $[t_{ij},t_{ik}]$ would be a tree with one trivalent vertex and three univalent vertices labeled by $i,j,k$. I am not sure what the sign conventions should be.

Now consider the free Lie algebra on two generators, $X$ and $Y$, $\mathsf{L}(X,Y)$, and complete it to $\hat{\mathsf{L}}$. Let $\hat{\mathsf{L}}^+$ be spanned by elements of degree $3$ or higher. Consider $\varphi\in\hat{\mathsf{L}}^+$ satisfying
$$\varphi(t_{12},t_{23}+t_{24})+\varphi(t_{13}+t_{23},t_{34})=\varphi(t_{23},t_{24})+ \varphi(t_{12}+t_{13},t_{24}+t_{34})+\varphi(t_{12},t_{23}),$$ where we plug in elements of $\mathfrak{t}_4$.
The set of such $\varphi$ is the Grothendieck-Teichmüller Lie algebra $\mathfrak{grt}_1$.

There are actually two auxiliary conditions, such as the fact that $\varphi$ is antisymmetric in $X$ and $Y$, which are normally given, but they follow from a result of Furusho.

The simplest example of an element of $\mathfrak{grt}_1$ is the element $[X,[X,Y]]-[Y,[Y,X]]$.

So now that we know what $\mathfrak{grt}_1$ is, the next question is why do we care? It actually comes from a group $\mathsf{GRT}_1$, which is related to the space of Drinfeld associators with no quadratic term. It also arises in the algebraic geometry of the mapping class group and Teichmüller space, although I don't understand the story there very well. I believe that it is known that $\mathfrak{grt}_1$ embeds into the cokernel of the Johnson homomorphism as the set of so called Galois obstructions and this follows from work of Matsumoto and Nakamura.

An extremely important conjecture about $\mathfrak{grt}_1$ is the following.

Conjecture (Deligne-Mumford-Ihara): $\mathfrak{grt}_1$ is a free graded Lie algebra with generators $\sigma_3,\sigma_5,\ldots$.

F. Brown has proven half of this conjecture:

Theorem (F. Brown): The free graded Lie algebra with generators $\sigma_3,\sigma_5,\ldots$ embeds in $\mathfrak{grt}_1$.

Another remarkable result is the main theorem of Willwacher's paper. Namely, if you take Kontsevich's (commutative) even graph complex $\mathsf{GC}_2$, then

Theorem (Willwacher): There is an isomorphism $H^0(\mathsf{GC}_2)\cong \mathfrak{grt}_1$.

I am adopting Willwacher's notation here. Cohomological degree $0$ here picks up connected graphs with $E=2(V-1)$. The elements $\sigma_{2i+1}$ have a beautiful (conjectural) description as wheels with $2i+1$ spokes, which are indeed cocycles in the graph complex.

So now we have a definition of the Grothendieck-Teichmüller Lie algebra, and some idea of what is known about it and its importance. In my next posts, I intend to explore some of the ideas touched on here. For example I want to talk about $\mathsf{GC}_2$ in a lot more detail, and explore the connections to the Johnson homomorphism. I'd also like to delve into the details of Willwacher's paper.