En (Lie algebra)

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.

\begin{matrix} 2 & -1 & 0 \\ -1 &  2 &  0 \\ 0 &  0 &  2 \end{matrix}\right ]$$ \begin{matrix} 2 & -1 & 0 &  0 \\ -1 &  2 & -1&  0  \\ 0 & -1 &  2 & -1 \\ 0 &  0 & -1 &  2 \end{matrix}\right ]$$ \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 ]$$ \begin{matrix} 2 & -1 & 0 &  0 &  0 &  0 \\ -1 &  2 & -1&  0 &  0 &  0 \\ 0 & -1 &  2 & -1 &  0 &  -1 \\ 0 &  0 & -1 &  2 & -1 &  0 \\ 0 &  0 &  0 & -1 &  2 & 0 \\ 0 &  0 &  -1 &  0 & 0 &  2 \end{matrix}\right ]$$ \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 ]$$ \begin{matrix} 2 & -1 & 0 &  0 &  0 &  0 &  0 & 0 \\ -1 &  2 & -1&  0 &  0 &  0 &  0 & 0 \\ 0 & -1 &  2 & -1 &  0 &  0 &  0 & -1 \\ 0 &  0 & -1 &  2 & -1 &  0 &  0 & 0 \\ 0 &  0 &  0 & -1 &  2 & -1 &  0 & 0 \\ 0 &  0 &  0 &  0 & -1 &  2 & -1 & 0 \\ 0 &  0 &  0 &  0 &  0 & -1 &  2 & 0 \\ 0 &  0 & -1 &  0 &  0 &  0 &  0 & 2 \end{matrix}\right ]$$
 * E3 is another name for the Lie algebra A1A2 of dimension 11, with Cartan determinant 6.
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 * E4 is another name for the Lie algebra A4 of dimension 24, with Cartan determinant 5.
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 * E5 is another name for the Lie algebra D5 of dimension 45, with Cartan determinant 4.
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 * E6 is the exceptional Lie algebra of dimension 78, with Cartan determinant 3.
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 * E7 is the exceptional Lie algebra of dimension 133, with Cartan determinant 2.
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 * E8 is the exceptional Lie algebra  of dimension 248, with Cartan determinant 1.
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Infinite-dimensional Lie algebras
\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 ]$$ \begin{matrix} 2 & -1 & 0 &  0 &  0 &  0 &  0 & 0 & 0  & 0 \\ -1 &  2 & -1&  0 &  0 &  0 &  0 & 0 & 0 & 0 \\ 0 & -1 &  2 & -1 &  0 &  0 &  0 & 0 & 0 & -1 \\ 0 &  0 & -1 &  2 & -1 &  0 &  0 & 0 & 0 & 0  \\ 0 &  0 &  0 & -1 &  2 & -1 &  0 & 0 & 0 & 0  \\ 0 &  0 &  0 &  0 & -1 &  2 & -1 & 0 & 0 & 0  \\ 0 &  0 &  0 &  0 &  0 & -1 &  2 & -1 & 0 & 0  \\ 0 &  0 &  0 &  0 &  0 & 0 &  -1 & 2 & -1 & 0  \\ 0 &  0 &  0 &  0 &  0 & 0 &  0 & -1 & 2 & 0  \\ 0 & 0 &  -1 &  0 &  0 &  0 &  0 &  0 &  0 & 2 \end{matrix}\right ]$$
 * E9 is another name for the infinite-dimensional affine Lie algebra Ẽ8 (also as E$(1) 8$ or E$+ 8$ 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.
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 * E10 (or E$(1)^ 8$ or E$++ 8$ 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:
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 * E11 (or E$+++ 8$ 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 a family of infinite-dimensional Kac–Moody algebras that are not well studied.

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 of norm n × 12 − 32 = n − 9.

E$++++ 8$
Landsberg and Manivel extended the definition of En for integer n to include the case n = $+ 8$. 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$(1) 8$ has dimension 190, but is not a simple Lie algebra: it contains a 57 dimensional Heisenberg algebra as its nilradical.