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In mathematics, E7 is the name of a Lie group and also its Lie algebra <math>\mathfrak{e}_7<math>. It is one of the five exceptional simple Lie groups as well as one of the simply laced groups. E7 has rank 7 and dimension 133. Its center is the cyclic group Z2. Its outer automorphism group is the trivial group. The dimensional of its fundamental representation is 56.
Algebra
Even though the roots span a 7 dimensional space, it is more symmetric and convenient to represent them as vectors lying in a 7 dimensional subspace of an eight dimensional vector space.
The roots are all the 8×7 permutations of
(1,-1,0,0,0,0,0,0)
and all the <math>\begin{pmatrix}8\\4\end{pmatrix}<math> permutations of
(1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2,-1/2)
Note that the 7 dimensional subspace is the subspace where the sum of all the eight coordinates is zero. There are 126 roots.
- <math>
\begin{pmatrix}
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{pmatrix}<math>
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