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\begin{document}

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\section{Algèbres de Lie complexes semi-simples}

\subsection{Décomposition de Cartan}

Si $L$ est semi-simple finie sur un corps de caractéristique nulle, alors toute algèbre de Cartan $H$ de $L$ est une algèbre \emph{torique} maximale, c'est à dire abélienne et constituée d'éléments semi-simples, et réciproquement. Lorsque le corps est algébriquement clos, les algèbres de Cartan sont celles maximales constituées d'éléments diagonalisables dans la représentation adjointe.

On obtient une décomposition de $L$ suivant les valeurs propres des éléments de $H$ appelées alors \emph{racines} de $L$.

On note $\Phi\subset H^{*}$ l'ensemble des racines non nulles de la décomposition de Cartan $L$ suivant $H$. C'est l'ensemble des formes linéaires $\alpha$ sur $H$ correspondant aux valeurs propres non nulles de la représentation adjointe de $H$ sur $L$. On a toujours $H^*=\C \Phi$. L'espace propre correspondant est noté 
$$L_{\alpha}=\set{x\in L \mid \forall h\in H\ \lie{h}{x}=\alpha(h)x}$$
On a alors la \emph{décomposition de Cartan}
$$L=H \oplus_{\alpha\in \Phi}L_{\alpha}$$
Pour $h\in H$, on a toujours 
$$\lie{h}{e_{\alpha}}=\alpha(h)e_{\alpha}$$
En particulier, les matrices des éléments de $H$ dans la représentation adjointe sont diagonales de diagonale $(\alpha(h))_{\alpha\in \Phi\cup \set{0}}$ dans la base constituée d'une base de $H$ et des générateurs des $L_{\alpha}$ avec $\alpha\in \Phi$

\end{document}