# En (Lie algebra)

Dynkin diagrams :
* Finite |
* E$_3$=A$_2$A$_1$
* E$_4$=A$_4$
* E$_5$=D$_5$
* E$_6$
* E$_7$
* E$_8$
* Affine (Extended) |
* E$_9$ or E$^{(1)}$ $_8$ or E$^+$ $_8$
* Hyperbolic (Over-extended) |
* E$_{10}$ or E$^{(1)^}$ $_8$ or E$^{++}$ $_8$
* Lorentzian (Very-extended) |
* E$_{11}$ or E$^{+++}$ $_8$
* Kac–Moody |
* E$_{12}$ or E$^{++++}$ $_8$
In mathematics, especially in Lie theory, E$_n$ 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, E$_2$ and E$_4$ are used as names for G$_2$ and F$_4$.

## 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.
* E$_3$ is another name for the Lie algebra A$_1$A$_2$ of dimension 11, with Cartan determinant 6.
    $\left[{\begin{matrix}2&-1&0\\-1&2&0\\0&0&2\end{matrix}}\right]$
* E$_4$ is another name for the Lie algebra A$_4$ of dimension 24, with Cartan determinant 5.
    $\left[{\begin{matrix}2&-1&0&0\\-1&2&-1&0\\0&-1&2&-1\\0&0&-1&2\end{matrix}}\right]$
* E$_5$ is another name for the Lie algebra D$_5$ of dimension 45, with Cartan determinant 4.
    $\left[{\begin{matrix}2&-1&0&0&0\\-1&2&-1&0&0\\0&-1&2&-1&-1\\0&0&-1&2&0\\0&0&-1&0&2\end{matrix}}\right]$
* E$_6$ is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.
    $\left[{\begin{matrix}2&-1&0&0&0&0\\-1&2&-1&0&0&0\\0&-1&2&-1&0&-1\\0&0&-1&2&-1&0\\0&0&0&-1&2&0\\0&0&-1&0&0&2\end{matrix}}\right]$
* E$_7$ is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.
    $\left[{\begin{matrix}2&-1&0&0&0&0&0\\-1&2&-1&0&0&0&0\\0&-1&2&-1&0&0&-1\\0&0&-1&2&-1&0&0\\0&0&0&-1&2&-1&0\\0&0&0&0&-1&2&0\\0&0&-1&0&0&0&2\end{matrix}}\right]$
* E$_8$ is the exceptional Lie algebra of dimension 248, with Cartan determinant 1.
    $\left[{\begin{matrix}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&-1\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&0\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&0\\0&0&-1&0&0&0&0&2\end{matrix}}\right]$

## Infinite-dimensional Lie algebras

* E$_9$ is another name for the infinite-dimensional affine Lie algebra Ẽ$_8$ (also as E$^+$ $_8$ or E$^{(1)}$ $_8$ as a (one-node) extended E$_8$) (or E$_8$ lattice) corresponding to the Lie algebra of type E$_8$. E$_9$ has a Cartan matrix with determinant 0.
    $\left[{\begin{matrix}2&-1&0&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0&0\\0&-1&2&-1&0&0&0&0&-1\\0&0&-1&2&-1&0&0&0&0\\0&0&0&-1&2&-1&0&0&0\\0&0&0&0&-1&2&-1&0&0\\0&0&0&0&0&-1&2&-1&0\\0&0&0&0&0&0&-1&2&0\\0&0&-1&0&0&0&0&0&2\end{matrix}}\right]$
* E$_{10}$ (or E$^{++}$ $_8$ or E$^{(1)^}$ $_8$ as a (two-node) over-extended E$_8$) is an infinite-dimensional Kac–Moody algebra whose root lattice is the even Lorentzian unimodular lattice II$_{9,1}$ 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. E$_{10}$ has a Cartan matrix with determinant −1:
    $\left[{\begin{matrix}2&-1&0&0&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0&0&0\\0&-1&2&-1&0&0&0&0&0&-1\\0&0&-1&2&-1&0&0&0&0&0\\0&0&0&-1&2&-1&0&0&0&0\\0&0&0&0&-1&2&-1&0&0&0\\0&0&0&0&0&-1&2&-1&0&0\\0&0&0&0&0&0&-1&2&-1&0\\0&0&0&0&0&0&0&-1&2&0\\0&0&-1&0&0&0&0&0&0&2\end{matrix}}\right]$
* E$_{11}$ (or E$^{+++}$ $_8$ as a (three-node) very-extended E$_8$) is a Lorentzian algebra, containing one time-like imaginary dimension, that has been conjectured to generate the symmetry "group" of M-theory.
* E$_n$ for n ≥ 12 is a family of infinite-dimensional Kac–Moody algebras that are not well studied.

## Root lattice

The root lattice of E$_n$ has determinant 9 − n, and can be constructed as the lattice of vectors in the unimodular Lorentzian lattice Z$_{n,1}$ that are orthogonal to the vector (1,1,1,1,...,1|3) of norm n × 1$^2$ − 3$^2$ = n − 9.

## E$_{7+1⁄2}$

Landsberg and Manivel extended the definition of E$_n$ for integer n to include the case n = 7+1⁄2. They did this in order to fill the "hole" in dimension formulae for representations of the E$_n$ series which was observed by Cvitanovic, Deligne, Cohen and de Man. E$_{7+1⁄2}$ has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.