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\begin{document}
\title[Groupe de Galois et équation de Drinfeld]{Groupes de Galois d'équations différentielles non-commutatives et équation de Drinfel'd}
\author[M. Deneufchâtel]{\vspace{2cm}{\color{blue}Matthieu Deneufchâtel}}
\date[18/10/2011]{{ \tiny Séminaire CIP, 18 octobre 2011} }
%\institute[LIPN - P13]{\vspace{3cm}}
\institute[LIPN - P13]{\textit{Laboratoire d'Informatique de Paris Nord,} \\ Université Paris 13 }
\frame[plain]{\titlepage}

\AtBeginSection[]{\frame{\frametitle{Outline}
\tableofcontents[current]}}

\begin{frame}
  \frametitle{Outline}
  \tableofcontents[currentsubsection]
\end{frame}

\begin{frame}
\,{\bf Rappels : }$X = \left\{ x_1 , \dots , x_n \right\}$.
\begin{equation*}
   P_w = \left\{
\begin{aligned}
   x_i & \text{ if }w=x_i ; \\
   \left[ l_1 , l_2 \right] & \text{ if }w \text{ is a Lyndon word with standard factorization } l_1l_2 ; \\
   P_{l_{i_1}}^{\alpha_1} \dots P_{l_{i_k}}^{\alpha_k} & \text{ if } w = l_{i_1}^{\alpha_1} \dots l_{i_k}^{\alpha_k} , \, l_{i_1} > \dots > l_{i_k}.
\end{aligned}
\right.
\end{equation*}
\pause
 \begin{equation*}
   S_w = \left\{
\begin{aligned}
   w \quad \quad \quad \quad \, & \, \text{if }|w|=1 ; \\
   x S_u \quad \quad \quad \, \, \, \, \, & \, \text{if }w=xu \text{ and }w \text{ is a Lyndon word} ; \\
   \frac{S_{l_{i_1}}^{\shuffle \alpha_1} \dots S_{l_{i_k}}^{\shuffle \alpha_k}}{\alpha_1 ! \dots \alpha_k !} \quad \, & \, \text{otherwise, if } w = l_{i_1}^{\alpha_1} \dots l_{i_k}^{\alpha_k}.
\end{aligned}
\right.
\end{equation*}
\pause
{\bf Minh : }
\begin{equation*}
 \sum_{w \in X^*} w \otimes w = \prod_{l \in {\rm Lyn}(X)}^{\searrow} \exp(S_l \otimes P_l).
\end{equation*}
(produit : $\shuffle$ à gauche et concaténation à droite).
\end{frame}


\begin{frame}
 Objectif : Donner un cadre général dans lequel l'égalité
\begin{equation*}
 \sum_{w \in X^*} w \otimes w = \prod_{i \in I}^{\rightarrow} \exp(S_i \otimes P_i)
\end{equation*}
où $(P_i)_{i \in I}$ et $(S_i)_{i \in I}$ sont deux bases en dualité.
\end{frame}





\end{document}