Frédéric Wang Subscribe   About Me   Blog Archive   Mathematics   Computer Science

A Théorie Classique

A.1 Algèbres de Lie simples de dimension finie

Type Matrice de Symétrisation did_{i} Matrice de Cartan aija_{{ij}} Nombres de Coxeter
AnA_{n} In=(11)I_{n}=\begin{pmatrix}1&&\\ &\ddots&\\ &&1\end{pmatrix} (2-1-12-1-12)\begin{pmatrix}2&-1&&\\ -1&2&\ddots&\\ &\ddots&\ddots&-1\\ &&-1&2\end{pmatrix} h=h=n+1h=h^{\vee}=n+1
BnB_{n} (112)\begin{pmatrix}1&&&\\ &\ddots&&\\ &&1&\\ &&&2\\ \end{pmatrix} (2-1-12-1-12-2-12)\begin{pmatrix}2&-1&&&\\ -1&2&\ddots&&\\ &\ddots&\ddots&-1&\\ &&-1&2&-2\\ &&&-1&2\end{pmatrix} h=2n,h=2n-1h=2n,h^{\vee}=2n-1
CnC_{n} (221)\begin{pmatrix}2&&&\\ &\ddots&&\\ &&2&\\ &&&1\\ \end{pmatrix} (2-1-12-1-12-1-22)\begin{pmatrix}2&-1&&&\\ -1&2&\ddots&&\\ &\ddots&\ddots&-1&\\ &&-1&2&-1\\ &&&-2&2\end{pmatrix} h=2n,h=n+1h=2n,h^{\vee}=n+1
DnD_{n} In=(11)I_{n}=\begin{pmatrix}1&&\\ &\ddots&\\ &&1\end{pmatrix} (2-1-12-1-1-120-102)\begin{pmatrix}2&-1&&&\\ -1&2&\ddots&&\\ &\ddots&\ddots&-1&-1\\ &&-1&2&0\\ &&-1&0&2\end{pmatrix} h=h=2n-2h=h^{\vee}=2n-2
E6E_{6} I6=(11)I_{6}=\begin{pmatrix}1&&\\ &\ddots&\\ &&1\end{pmatrix} (20-1000020-100-102-1000-1-12-10000-12-10000-12)\begin{pmatrix}2&0&-1&0&0&0\\ 0&2&0&-1&0&0\\ -1&0&2&-1&0&0\\ 0&-1&-1&2&-1&0\\ 0&0&0&-1&2&-1\\ 0&0&0&0&-1&2\end{pmatrix} h=h=12h=h^{\vee}=12
E7E_{7} I7=(11)I_{7}=\begin{pmatrix}1&&\\ &\ddots&\\ &&1\end{pmatrix} (20-10000020-1000-102-10000-1-12-100000-12-100000-12-100000-12)\begin{pmatrix}2&0&-1&0&0&0&0\\ 0&2&0&-1&0&0&0\\ -1&0&2&-1&0&0&0\\ 0&-1&-1&2&-1&0&0\\ 0&0&0&-1&2&-1&0\\ 0&0&0&0&-1&2&-1\\ 0&0&0&0&0&-1&2\end{pmatrix} h=h=18h=h^{\vee}=18
E8E_{8} I8=(11)I_{8}=\begin{pmatrix}1&&\\ &\ddots&\\ &&1\end{pmatrix} (20-100000020-10000-102-100000-1-12-1000000-12-1000000-12-1000000-12-1000000-12)\begin{pmatrix}2&0&-1&0&0&0&0&0\\ 0&2&0&-1&0&0&0&0\\ -1&0&2&-1&0&0&0&0\\ 0&-1&-1&2&-1&0&0&0\\ 0&0&0&-1&2&-1&0&0\\ 0&0&0&0&-1&2&-1&0\\ 0&0&0&0&0&-1&2&-1\\ 0&0&0&0&0&0&-1&2\end{pmatrix} h=h=30h=h^{\vee}=30
F4F_{4} (1000010000200002)\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&2&0\\ 0&0&0&2\end{pmatrix} (2-100-12-200-12-100-12)\begin{pmatrix}2&-1&0&0\\ -1&2&-2&0\\ 0&-1&2&-1\\ 0&0&-1&2\end{pmatrix} h=12,h=9h=12,h^{\vee}=9
G2G_{2} (3001)\begin{pmatrix}3&0\\ 0&1\end{pmatrix} (2-1-32)\begin{pmatrix}2&-1\\ -3&2\end{pmatrix} h=6,h=4h=6,h^{\vee}=4