En (Lie Algebra) - Finite Dimensional Lie Algebras

Finite Dimensional Lie Algebras

The E_n group is similar to the A_n 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 E_n is 9-n.

E₃ is another name for the Lie algebra A₁A₂ of dimension 11, with Cartan determinant 6.

$\left [ \begin{smallmatrix} 2 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 2 \end{smallmatrix}\right ]$
E₄ is another name for the Lie algebra A₄ of dimension 24, with Cartan determinant 5.

$\left [ \begin{smallmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -1& 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{smallmatrix}\right ]$
E₅ is another name for the Lie algebra D₅ of dimension 45, with Cartan determinant 4.

$\left [ \begin{smallmatrix} 2 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 \\ 0 & -1 & 2 & -1 & -1 \\ 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 2 \end{smallmatrix}\right ]$
E₆ is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.

$\left [ \begin{smallmatrix} 2 & -1 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 2 \end{smallmatrix}\right ]$
E₇ is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.

$\left [ \begin{smallmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 2 \end{smallmatrix}\right ]$
E₈ is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.

$\left [ \begin{smallmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2 \end{smallmatrix}\right ]$

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