E7 (mathematics)
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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.
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Algebra
Dynkin diagram
Dynkin_diagram_E7.png
Dynkin diagram of E_7
Root system
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.
Cartan matrix
- <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>
| E6 | E7 | E8 | F4 | G2 |