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\begin{document}

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\section*{motivic Galois group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{motivic_cohomology}{}\paragraph*{{Motivic cohomology}}\label{motivic_cohomology}

[[!include motivic cohomology - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{mixed_motivic_galois_groups}{(Mixed) Motivic Galois groups}\dotfill \pageref*{mixed_motivic_galois_groups} \linebreak
\noindent\hyperlink{noris_motivic_galois_group}{Nori's Motivic Galois group}\dotfill \pageref*{noris_motivic_galois_group} \linebreak
\noindent\hyperlink{ayoubs_motivic_galois_group}{Ayoub's Motivic Galois group}\dotfill \pageref*{ayoubs_motivic_galois_group} \linebreak
\noindent\hyperlink{comparison}{Comparison}\dotfill \pageref*{comparison} \linebreak
\noindent\hyperlink{noncommutative_motivic_galois_groups}{Noncommutative Motivic Galois groups}\dotfill \pageref*{noncommutative_motivic_galois_groups} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{relation_to_grothendieckteichmller_group}{Relation to Grothendieck-Teichm\"u{}ller group}\dotfill \pageref*{relation_to_grothendieckteichmller_group} \linebreak
\noindent\hyperlink{in_quantum_field_theory}{In quantum field theory}\dotfill \pageref*{in_quantum_field_theory} \linebreak
\noindent\hyperlink{grothendiecks_yoga_remarks}{Grothendieck's Yoga Remarks}\dotfill \pageref*{grothendiecks_yoga_remarks} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

A slight variant $\tilde Mot_{num}(k,\mathbb{Q})$ of the [[category of pure motives|category of pure]] [[numerical motives]] $Mot_{num}(k,\mathbb{Q})$ is a [[Tannakian category]] [[equivalence of categories|equivalent]] to a [[category of representations]] of some [[algebraic group]] $GMot_k$:

\begin{displaymath}
Mot_{num}(k,\mathbb{Q}) \simeq Rep(GMot_k)
  \,.
\end{displaymath}
In the sense of [[Galois theory]], that [[algebraic group]] is called the \emph{motivic Galois group} for pure motives. There is also a motivic Galois group of mixed motives.

That group is, or is closely related to, the group of algebraic \emph{[[periods]]}, and as such is related to expressions appearing in [[deformation quantization]] and in [[renormalization]] in [[quantum field theory]], whence it is also sometimes referred to as the \emph{[[cosmic Galois group]]}. See there for more on this.

\hypertarget{mixed_motivic_galois_groups}{}\subsection*{{(Mixed) Motivic Galois groups}}\label{mixed_motivic_galois_groups}

\hypertarget{noris_motivic_galois_group}{}\subsubsection*{{Nori's Motivic Galois group}}\label{noris_motivic_galois_group}

[[M. Levine]], \emph{Mixed motives.} (3.3 Motives by Tannakian formalism) In, Handbook of K-theory, Vol. 1, Friedlander and Grayson, eds., p. 429--521, Springer Verlag (2005).(\href{https://www.uni-due.de/~bm0032/publ/MixMotKHB.pdf}{pdf}).

\hypertarget{ayoubs_motivic_galois_group}{}\subsubsection*{{Ayoub's Motivic Galois group}}\label{ayoubs_motivic_galois_group}

\begin{itemize}%
\item [[Joseph Ayoub]], \emph{L'alg\`e{}bre de Hopf et le groupe de Galois motiviques d'un corps de caract\'e{}ristique nulle I}, (\href{http://user.math.uzh.ch/ayoub/PDF-Files/GaloisMotivic-1.pdf}{pdf}).


\item [[Joseph Ayoub]], \emph{L'alg\`e{}bre de Hopf et le groupe de Galois motiviques d'un corps de caract\'e{}ristique nulle II}, (\href{http://user.math.uzh.ch/ayoub/PDF-Files/GaloisMotivic-2.pdf}{pdf}).


\item [[Joseph Ayoub]], \emph{From motives to comodules over the motivic Hopf algebra}, (\href{http://user.math.uzh.ch/ayoub/PDF-Files/Motivic-Hopf-3.pdf}{pdf}).



\end{itemize}
[[Joseph Ayoub]], \href{https://www.youtube.com/watch?v=f4vgBfPkiWg}{Lecture 1},\href{https://www.youtube.com/watch?v=JHSdJ28iED0}{Lecture 2},\href{https://www.youtube.com/watch?v=j967ozw9nXU}{Lecture 3}, The lectures were held within the framework of the (Junior) Hausdorff Trimester Program Topology and the Workshop: Interactions between operads and motives.

\href{http://mathoverflow.net/questions/254341/derived-version-of-equivalence-between-motives-and-representations-of-motivic-ga}{Mathoverflow, Derived version of equivalence between motives and representations of Motivic galois groups?}.

\hypertarget{comparison}{}\subsubsection*{{Comparison}}\label{comparison}

\emph{Ayoub's weak Tannakian formalism applied to the Betti realization for Voevodsky motives yields a pro-algebraic group, a candidate for the motivic Galois group. In particular, each Voevodsky motive gives rise to a representation of this group. On the other hand Nori motives are just representations of Nori's motivic Galois group.} These groups are isomorphic.

\begin{itemize}%
\item [[Utsav Choudhury]], [[Martin Gallauer Alves de Souza]], \emph{An isomorphism of motivic galois groups}, (\href{https://arxiv.org/pdf/1410.6104v2.pdf}{arXiv:1410.6104}).

\end{itemize}
\emph{the existence of a motivic t-structure (which renders the Betti realization t-exact) would imply the isomorphism of motivic Galois groups.}

\begin{itemize}%
\item [[J.P.Pridham]], \emph{Tannaka duality for enhanced triangulated categories}, (\href{https://arxiv.org/pdf/1309.0637v4.pdf}{arXiv:1309.0637}).

\end{itemize}
\hypertarget{noncommutative_motivic_galois_groups}{}\subsection*{{Noncommutative Motivic Galois groups}}\label{noncommutative_motivic_galois_groups}

\begin{itemize}%
\item [[Matilde Marcolli]], [[Gonçalo Tabuada]] \emph{Unconditional noncommutative motivic galois groups},(\href{https://arxiv.org/pdf/1112.5422.pdf}{arXiv:1112.5422}).


\item [[Matilde Marcolli]], [[Gonçalo Tabuada]] \emph{Noncommutative numerical motives, Tannakian structures, and motivic Galois groups.} J. Eur. Math. Soc. 18 (2016), 623-655. doi: 10.4171/JEMS/598



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

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

A quick survey is in

\begin{itemize}%
\item [[Bruno Kahn]], \emph{Motivic Galois groups} (\href{http://www.math.jussieu.fr/~kahn/preprints/PaloAlto2.pdf}{pdf})

\end{itemize}
More is in

\begin{itemize}%
\item [[Jean-Pierre Serre]], \emph{Propri\'e{}t\'e{}s conjecturales des groups de Galois motiviques et des repr\'e{}sentations l-adiques}, in U. Jannsen, S. Kleiman, J.-P. Serre (eds) \emph{Motives}, Proceedings of Symposia in Pure Mathematics, AMS, vol. 55, part 1, 377 - 400 (1994).


\item [[Yves André]], \emph{Groupes de Galois motiviques et p\'e{}riodes}, (\href{http://www.bourbaki.ens.fr/TEXTES/1104.pdf}{pdf}), S\'e{}minaire Bourbaki, 68\`e{}me ann\'e{}e, 2015-2016 n\textdegree{}1104.


\item AB Goncharov, \emph{Polylogarithms and Motivic galois groups}, (\href{http://users.math.yale.edu/users/ag727/polylog.pdf}{pdf}).


\item R Sujatha, \emph{Motives from a categorical point of view}, (\href{http://www.math.tifr.res.in/~sujatha/ihes.pdf}{pdf}).


\item [[Minhyong Kim]], \emph{(Videos) Motivic fundamental groups and Diophantine geometry}, \href{https://www.youtube.com/watch?v=qiR8un1mwIA}{I}, \href{https://www.youtube.com/watch?v=SZLE_dqf5lk}{II}.


\item Annette Huber, Stefan M\"u{}ller-Stach, \emph{On the relation between Nori motives and Kontsevich periods}, \href{http://arxiv.org/abs/1105.0865}{arxiv/1105.0865}



\end{itemize}
\begin{quote}%
We show that the spectrum of Kontsevich's algebra of formal periods is a torsor under the motivic Galois group for mixed motives over the rational numbers. This assertion is stated without proof by Kontsevich and originally due to Nori. In a series of appendices, we also provide the necessary details on Nori's category of motives.


\end{quote}
\href{http://mathoverflow.net/questions/127633/what-are-the-possible-motivic-galois-groups-over-mathbb-q?rq=1}{Mathoverflow, What are the possible motivic Galois groups over Q ?}.

\href{http://mathoverflow.net/questions/71660/why-would-the-category-of-motives-be-tannakian}{Mathoverflow, Why would the category of Motives be Tannakian?}.

\hypertarget{relation_to_grothendieckteichmller_group}{}\subsubsection*{{Relation to Grothendieck-Teichm\"u{}ller group}}\label{relation_to_grothendieckteichmller_group}

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

\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}
\hypertarget{in_quantum_field_theory}{}\subsubsection*{{In quantum field theory}}\label{in_quantum_field_theory}

A conjecture that the motivic Galois group naturally acts on the space of [[deformation quantizations]] of [[free field theories]] is in section 5 of

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

\end{itemize}
Regarding that this approach is related to Hodge theory periods, and that QFT can be attacked using the Batalin-Vilkovsky formalism, Barannikov has developed further the Batalin-Vilkovisky geometry and related Hodge theory in a direction which is probably a ground for better understanding of motivic Galois group as well.

An analogous statement, that a motivic Galois group naturally acts on structures in [[renormalization]] is in

\begin{itemize}%
\item [[Alain Connes]], [[Matilde Marcolli]], \emph{Renormalization and motivic Galois theory}, \href{http://arxiv.org/abs/math/0409306}{arXiv:math/0409306}

\end{itemize}
\begin{itemize}%
\item [[Alain Connes]], \emph{Symetries Galoisiennes et Renormalisation}, \href{https://arxiv.org/abs/math/0211199}{arXiv:math/0211199}

\end{itemize}
For more along these lines see at \emph{[[cosmic Galois group]]}.

\hypertarget{grothendiecks_yoga_remarks}{}\subsubsection*{{Grothendieck's Yoga Remarks}}\label{grothendiecks_yoga_remarks}

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

\begin{itemize}%
\item [[Galois theory]]


\item [[motive]]


\item [[cosmic Galois group]], [[Grothendieck-Teichmüller group]]



\end{itemize}
[[!redirects motivic Galois groups]]



\end{document}