# 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.
${\displaystyle \left[{\begin{smallmatrix}2&-1&0\\-1&2&0\\0&0&2\end{smallmatrix}}\right]}$
• E4 is another name for the Lie algebra A4 of dimension 24, with Cartan determinant 5.
${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0\\-1&2&-1&0\\0&-1&2&-1\\0&0&-1&2\end{smallmatrix}}\right]}$
• E5 is another name for the Lie algebra D5 of dimension 45, with Cartan determinant 4.
${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0\\-1&2&-1&0&0\\0&-1&2&-1&-1\\0&0&-1&2&0\\0&0&-1&0&2\end{smallmatrix}}\right]}$
• E6 is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.
${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0&0\\-1&2&-1&0&0&0\\0&-1&2&-1&0&-1\\0&0&-1&2&-1&0\\0&0&0&-1&2&0\\0&0&-1&0&0&2\end{smallmatrix}}\right]}$
• E7 is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.
${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0&0&0\\-1&2&-1&0&0&0&0\\0&-1&2&-1&0&0&-1\\0&0&-1&2&-1&0&0\\0&0&0&-1&2&-1&0\\0&0&0&0&-1&2&0\\0&0&-1&0&0&0&2\end{smallmatrix}}\right]}$
• E8 is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.
${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&-1\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&0\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&0\\0&0&-1&0&0&0&0&2\end{smallmatrix}}\right]}$

## Infinite-dimensional Lie algebras

• E9 is another name for the infinite-dimensional affine Lie algebra ${\displaystyle {\tilde {E}}_{8}}$ (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.
${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0&0\\0&-1&2&-1&0&0&0&0&-1\\0&0&-1&2&-1&0&0&0&0\\0&0&0&-1&2&-1&0&0&0\\0&0&0&0&-1&2&-1&0&0\\0&0&0&0&0&-1&2&-1&0\\0&0&0&0&0&0&-1&2&0\\0&0&-1&0&0&0&0&0&2\end{smallmatrix}}\right]}$
• 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:
• ${\displaystyle \left[{\begin{smallmatrix}2&-1&0&0&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0&0&0\\0&-1&2&-1&0&0&0&0&0&-1\\0&0&-1&2&-1&0&0&0&0&0\\0&0&0&-1&2&-1&0&0&0&0\\0&0&0&0&-1&2&-1&0&0&0\\0&0&0&0&0&-1&2&-1&0&0\\0&0&0&0&0&0&-1&2&-1&0\\0&0&0&0&0&0&0&-1&2&0\\0&0&-1&0&0&0&0&0&0&2\end{smallmatrix}}\right]}$
• E11 (or E8+++ as a (three-node) very-extended E8) is a Lorentzian algebra, containing 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

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.