Finite Dimensional Lie Algebras
The En group is similar to the An group, except the nth node is connected to the 3rd node. So the Cartan matrix appears similar, -1 above and below the diagonal, except for the last row and column, have -1 in the third row and column. The determinant of the Cartan matrix for En is 9-n.
- E3 is another name for the Lie algebra A1A2 of dimension 11, with Cartan determinant 6.
- E4 is another name for the Lie algebra A4 of dimension 24, with Cartan determinant 5.
- E5 is another name for the Lie algebra D5 of dimension 45, with Cartan determinant 4.
- E6 is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.
- E7 is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.
- E8 is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.
![\left [
\begin{smallmatrix} 2 & -1 & 0 \\
-1 & 2 & 0 \\ 0 & 0 & 2
\end{smallmatrix}\right ]](http://upload.wikimedia.org/math/e/8/2/e82cdff162638a17f2b51a75c3af820f.png)
![\left [
\begin{smallmatrix} 2 & -1 & 0 & 0 \\
-1 & 2 & -1& 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2
\end{smallmatrix}\right ]](http://upload.wikimedia.org/math/3/4/3/3432219ae2be2fde6a1f8581dbae3655.png)
![\left [
\begin{smallmatrix} 2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 \\ 0 & -1 & 2 & -1 & -1 \\ 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 2
\end{smallmatrix}\right ]](http://upload.wikimedia.org/math/d/4/a/d4afcf779bf922493b29d5abec6f6161.png)
![\left [
\begin{smallmatrix} 2 & -1 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 2
\end{smallmatrix}\right ]](http://upload.wikimedia.org/math/6/3/f/63f3379096cb1649d4ceb8f0edce558c.png)
![\left [
\begin{smallmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 2
\end{smallmatrix}\right ]](http://upload.wikimedia.org/math/f/1/3/f13798387144d63eed593916b120a1e1.png)
![\left [
\begin{smallmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\ 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 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2
\end{smallmatrix}\right ]](http://upload.wikimedia.org/math/4/0/5/405fbe105e0d0bab564fc3c54a6294bd.png)
Read more about this topic: En (Lie Algebra)
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