The Grothendieck-Teichmüller tower construction involves moduli spaces outlined in Grothendieck’s Esquisse d'un programme and then developed by Vladimir Drinfel'd and others. It involves the Teichmüller groupoids which are the fundamental groupoids of moduli stacks of genus $g$ curves with $n$ points removed. It is a basis for the definition of the Grothendieck-Teichmüller group which is by the definition inertia-preserving automorphism group of the Grothendieck-Teichmüller tower.
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GT Lie algebra appears as the tangent Lie algebra to the GT group.
Drinfel’d defined it more explicitly as follows.
Let $\mathfrak{lie}_n$ be the degree completion of the free Lie algebra on $n$-generators over a fixed ground field $K$ of characteristic $0$. Let $\mathfrak{der}_n$ be the space of $K$-linear derivations $\mathfrak{lie}_n\to \mathfrak{lie}_n$. A derivation $u\in\mathfrak{der}_n$ is tangential if there exist $a_i\in\mathfrak{lie}_n$ for $i=1,\ldots,n$, such that $u(x_i)=[x_i,a_i]$. In particular, $u$ is determined by elements $a_1,\ldots,a_n$ and is below denoted by $(a_1,\ldots,a_n)$. The tangential derivations form a Lie subalgebra $\mathfrak{tder}_n\subset\mathfrak{der}_n$. The GT Lie algebra is the subspace $\mathfrak{grt}\subset\mathfrak{tder}_2$ consisting of the tangential derivations of the form $(0,\psi)$, where the identities
and the pentagon-type identity
hold, and which is equipped with the Ihara Lie bracket
The GT group acts freely on the set of Drinfeld associators.
(Drinfeld, Ihara, Deligne)
There is an inclusion of the absolute Galois group of the rational numbers into the Grothendieck-Teichmüller group (recalled e.g. Stix 04, theorem 6).
The Grothendieck-Teichmüller group is supposed to be a quotient of the motivic Galois group. This is a conjecture due to (Drinfeld 91).
The Grothendieck-Teichmüller Lie algebra is isomorphic to the 0th cohomology of Kontsevich’s graph complex (Willwacher 10).
Grothendieck predicted that the GT group is closely related to the absolute Galois group. Maxim Kontsevich later conjectured its action on certain space of quantum field theories and outlined its motivic aspects.
This was later proven by Vasily Dolgushev, see at formal deformation quantization – Motivic Galois group action on the space of quantizations for details and pointers.
For more see also at cosmic Galois group for more on this.
The Grothendieck-Teichmüller group $GRT$ was originally introduced in
inspired by
See also
Leila Schneps, Pierre Lotchak, Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser. 242 (doi)– it contains a reprint of Groethendieck’s Esquisse and some surveys by contributors, including Leila Schneps, The Grothendieck-Teichmueller group GT: a survey, pdf
Maxim Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), 35-72, math.QA/9904055
The following monograph is in progress (and as of early 2016, recently completed):
The second purpose of the book is to explain, from a homotopical viewpoint, a deep relationship between operads and Grothendieck-Teichmüller groups. This connection, which has been foreseen by M. Kontsevich (from researches on the deformation quantization process in mathematical physics), gives a new approach to understanding internal symmetries of structures occurring in various constructions of algebra and topology. In the book, we set up the background required by an in-depth study of this subject, and we make precise the interpretation of the Grothendieck-Teichmüller group in terms of the homotopy of operads. The book is actually organized for this ultimate objective, which readers can take either as a main motivation or as a leading example to learn about general theories.
Videos from a seminar at the Newton Institute:
The Drinfeld conjeture is stated in
Relation to the graph complex
Jakob Stix, The Grothendieck-Teichmüller group and Galois theory of the rational numbers, 2004 (pdf)
Anton Alekseev, Charles Torossian, The Kashiwara-Vergne conjecture and Drinfeld’s associators, arxiv/0802.4300