# En (Lie algebra)

Dynkin diagrams
Finite
E3=A2A1
E4=A4
E5=D5
E6
E7
E8
Affine (Extended)
E9 or E8(1) or E8+
Hyperbolic (Over-extended)
E10 or E8(1)^ or E8++
Lorentzian (Very-extended)
E11 or E8+++
Kac–Moody
E12 or E8++++
...

In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1,2, and k, with k=n-4.

In some older books and papers, E2 and E4 are used as names for G2 and F4.

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

## Infinite-dimensional Lie algebras

• E9 is another name for the infinite-dimensional affine Lie algebra ${\tilde{E}}_8$ (also as E8+ or E8(1) as a (one-node) extended E8) (or E8 lattice) corresponding to the Lie algebra of type E8. E9 has a Cartan matrix with determinant 0.
$\left [ \begin{smallmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 2 \end{smallmatrix}\right ]$
• E10 (or E8++ or E8(1)^ as a (two-node) over-extended E8) is an infinite-dimensional Kac–Moody algebra whose root lattice is the even Lorentzian unimodular lattice II9,1 of dimension 10. Some of its root multiplicities have been calculated; for small roots the multiplicities seem to be well behaved, but for larger roots the observed patterns break down. E10 has a Cartan matrix with determinant -1:
• $\left [ \begin{smallmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 2 \end{smallmatrix}\right ]$
• E11 (or E8+++ as a (three-node) very-extended E8) is a Lorentzian algebra, containining one time-like imaginary dimension, that has been conjectured to generate the symmetry "group" of M-theory.
• En for n≥12 is an infinite-dimensional Kac–Moody algebra that has not been studied much.

## Root lattice

The root lattice of En has determinant 9−n, and can be constructed as the lattice of vectors in the unimodular Lorentzian lattice Zn,1 that are orthogonal to the vector (1,1,1,1,....,1|3) of norm n×12 − 32 = n − 9.

## E7½

Main article: E7½

Landsberg and Manivel extended the definition of En for integer n to include the case n = 7½. They did this in order to fill the "hole" in dimension formulae for representations of the En series which was observed by Cvitanovic, Deligne, Cohen and de Man. E has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.

• k21, 2k1, 1k2 polytopes based on En Lie algebras.

## References

• Kac, Victor G; Moody, R. V.; Wakimoto, M. (1988). "On E10". Differential geometrical methods in theoretical physics (Como, 1987). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 250. Dordrecht: Kluwer Acad. Publ. pp. 109–128. MR 981374.