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\section*{Grothendieck-Teichmüller tower}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{grothendieckteichmller_lie_algebra}{Grothendieck-Teichm\"u{}ller Lie algebra}\dotfill \pageref*{grothendieckteichmller_lie_algebra} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_drinfeld_associators}{Relation to Drinfeld associators}\dotfill \pageref*{relation_to_drinfeld_associators} \linebreak
\noindent\hyperlink{relation_to_the_absolute_galois_group_of_the_rational_numbers}{Relation to the absolute Galois group of the rational numbers}\dotfill \pageref*{relation_to_the_absolute_galois_group_of_the_rational_numbers} \linebreak
\noindent\hyperlink{relation_to_the_motivic_galois_group}{Relation to the motivic Galois group}\dotfill \pageref*{relation_to_the_motivic_galois_group} \linebreak
\noindent\hyperlink{RelationToTheGraphComplex}{Relation to the graph complex}\dotfill \pageref*{RelationToTheGraphComplex} \linebreak
\noindent\hyperlink{relation_to_deformation_quantization_and_the_cosmic_galois_group}{Relation to deformation quantization and the cosmic Galois group}\dotfill \pageref*{relation_to_deformation_quantization_and_the_cosmic_galois_group} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{references_2}{References}\dotfill \pageref*{references_2} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Grothendieck-Teichm\"u{}ller tower} construction involves [[moduli spaces]] outlined in [[Grothendieck]]`s \emph{[[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 \emph{Grothendieck-Teichm\"u{}ller group} which is by the definition inertia-preserving [[automorphism group]] of the Grothendieck-Teichm\"u{}ller tower.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\ldots{}

\hypertarget{grothendieckteichmller_lie_algebra}{}\subsubsection*{{Grothendieck-Teichm\"u{}ller Lie algebra}}\label{grothendieckteichmller_lie_algebra}

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 \textbf{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 \textbf{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

\begin{displaymath}
\psi(x,y)=-\psi(y,x),\,\,\,for\,\,\,all\,\,\,x,y,
\end{displaymath}
\begin{displaymath}
\psi(x,y)+\psi(y,z)+\psi(z,x) = 0,\,\,\,whenever\,\,\,\,x+y+z=0,
\end{displaymath}
and the pentagon-type identity

\begin{displaymath}
\psi(t^{1,2},t^{2,34})+\psi(t^{12,3},t^{3,4})=
\psi(t^{2,3},t^{3,4})+\psi(t^{1,23},t^{23,4})+\psi(t^{1,2},t^{2,3}),
\end{displaymath}
hold, and which is equipped with the \textbf{Ihara Lie bracket}

\begin{displaymath}
[\psi_1,\psi_2]_{Ihara} = (0,\psi_1)(\psi_2)-(0,\psi_2)(\psi_1)+[\psi_1,\psi_2].
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_drinfeld_associators}{}\subsubsection*{{Relation to Drinfeld associators}}\label{relation_to_drinfeld_associators}

The GT group acts freely on the set of [[Drinfeld associators]].

\hypertarget{relation_to_the_absolute_galois_group_of_the_rational_numbers}{}\subsubsection*{{Relation to the absolute Galois group of the rational numbers}}\label{relation_to_the_absolute_galois_group_of_the_rational_numbers}

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Drinfeld, Ihara, Deligne)}

There is an inclusion of the [[absolute Galois group]] of the [[rational numbers]] into the Grothendieck-Teichm\"u{}ller group (recalled e.g. \hyperlink{Stix04}{Stix 04, theorem 6}).

\end{theorem}
\hypertarget{relation_to_the_motivic_galois_group}{}\subsubsection*{{Relation to the motivic Galois group}}\label{relation_to_the_motivic_galois_group}

The Grothendieck-Teichm\"u{}ller group is supposed to be a [[quotient]] of the [[motivic Galois group]]. This is a conjecture due to (\hyperlink{Drinfeld91}{Drinfeld 91}).

\hypertarget{RelationToTheGraphComplex}{}\subsubsection*{{Relation to the graph complex}}\label{RelationToTheGraphComplex}

The Grothendieck-Teichm\"u{}ller Lie algebra is isomorphic to the 0th cohomology of [[Kontsevich]]`s [[graph complex]] (\hyperlink{Willwacher10}{Willwacher 10}).

\hypertarget{relation_to_deformation_quantization_and_the_cosmic_galois_group}{}\subsubsection*{{Relation to deformation quantization and the cosmic Galois group}}\label{relation_to_deformation_quantization_and_the_cosmic_galois_group}

[[Grothendieck]] predicted that the GT group is closely related to the absolute Galois group. [[Maxim Kontsevich]] later conjectured its [[action]] on certain spaces of [[quantum field theories]] and outlined its [[motivic cohomology|motivic]] aspects.

This was later proven by [[Vasily Dolgushev]], see at \emph{\href{deformation+quantization#MotivicGaloisGroup}{formal deformation quantization -- Motivic Galois group action on the space of quantizations}} for details and pointers.

For more see also at \emph{[[cosmic Galois group]]} for more on this.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[stable symplectic category]],


\item [[motivic Galois group]], [[cosmic Galois group]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The Grothendieck-Teichm\"u{}ller group $GRT$ was originally introduced in

\begin{itemize}%
\item [[Vladimir Drinfel'd]], \emph{On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal(\overline{\mathbb{Q}/\mathbb{Q}})$}, \href{http://mi.mathnet.ru/aa199}{abs}, Rossiskaya Akademiya Nauk. Algebra i Analiz (in Russian) 2 (4): 149--181, ISSN 0234-0852, MR1080203 translation in Leningrad Math. J. 2 (1991), no. 4, 829--860

\end{itemize}
inspired by

\begin{itemize}%
\item [[Alexander Grothendieck]], \emph{Sketch of a program}, London Math. Soc. Lect. Note Ser., 242, Geometric Galois actions, 1, 5--48, Cambridge Univ. Press, Cambridge, 1997.

\end{itemize}
Review:

\begin{itemize}%
\item [[Benoit Fresse]], \emph{Little discs operads, graph complexes and Grothendieck--Teichmüller groups}, in [[Haynes Miller]] (ed.) \emph{[[Handbook of Homotopy Theory]]} (\href{https://arxiv.org/abs/1811.12536}{arXiv:1811.12536})

\end{itemize}
See also

\begin{itemize}%
\item Leila Schneps, Pierre Lotchak, \emph{Geometric Galois actions}, 1, London Math. Soc. Lecture Note Ser. 242 (\href{http://dx.doi.org/10.1017%2FCBO9780511666124}{doi})-- it contains a reprint of Groethendieck's Esquisse and some surveys by contributors, including Leila Schneps, \emph{The Grothendieck-Teichmueller group GT: a survey}, \href{http://www.math.jussieu.fr/~leila/SchnepsGT.pdf}{pdf}


\item [[Maxim Kontsevich]], \emph{Operads and motives in deformation quantization}, Lett. Math. Phys. 48 (1999), 35-72, \href{http://arxiv.org/abs/math/9904055}{math.QA/9904055}



\end{itemize}
The following monograph is in progress (and as of early 2016, recently completed):

\begin{itemize}%
\item Benoit Fresse, \emph{Homotopy of Operads and Grothendieck-Teichm\"u{}ller groups}, \href{http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/}{website}

\end{itemize}
\begin{quote}%
The second purpose of the book is to explain, from a homotopical viewpoint, a deep relationship between operads and Grothendieck-Teichm\"u{}ller 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-Teichm\"u{}ller 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.


\end{quote}
Videos from a seminar at the Newton Institute:

\begin{itemize}%
\item Newton Instititute Programme Jan-April 2013: Grothendieck-Teichm\"u{}ller Groups, Deformation and Operads, \href{https://www.newton.ac.uk/event/gdo/seminars}{seminars} (some with videos)

\end{itemize}
\hypertarget{references_2}{}\subsection*{{References}}\label{references_2}

The Drinfeld conjeture is stated in

\begin{itemize}%
\item [[Vladimir Drinfel'd]], \emph{On quasi-triangular Quasi-Hopf algebras and a group closely related with $Gal(\overline{\mathbb{Q}/\mathbb{Q}})$, Leningrad Math. J., 2 (1991), 829 - 860.}

\end{itemize}
Relation to the graph complex

\begin{itemize}%
\item [[Thomas Willwacher]], \emph{M. Kontsevich's graph complex and the Grothendieck-Teichmueller Lie algebra} (\href{http://arxiv.org/abs/1009.1654}{arXiv:1009.1654})

\end{itemize}
\begin{itemize}%
\item [[Jakob Stix]], \emph{The Grothendieck-Teichm\"u{}ller group and Galois theory of the rational numbers}, 2004 ([[StiXGaloisAndGT.pdf:file]])


\item [[Anton Alekseev]], Charles Torossian, \emph{The Kashiwara-Vergne conjecture and Drinfeld's associators}, \href{http://arxiv.org/abs/0802.4300}{arxiv/0802.4300}



\end{itemize}
[[!redirects Grothendieck-Teichmueller tower]] [[!redirects Grothendieck-Teichmüller group]] [[!redirects Grothendieck-Teichmuller tower]] [[!redirects Grothendieck-Teichmuller group]] [[!redirects Grothendieck-Teichmüller groups]]

[[!redirects Grothendieck-Teichmueller group]]

[[!redirects groupe de Grothendieck-Teichmüller]]



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