E7 (mathematics)
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In mathematics, E7 is the name of several Lie groups and also their Lie algebras <math>\mathfrak{e}_7</math>. It is one of the five exceptional compact 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.
The compact real form of E7 is the isometry group of a 64-dimensional Riemannian manifold known informally as the 'quateroctonionic projective plane' because it can be built using an algebra that is the tensor product of the quaternions and the octonions. This can be seen systematically using a construction known as the 'magic square', due to Hans Freudenthal and Jacques Tits. There are three other real forms, and one complex form.
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Algebra
Dynkin diagram
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 8-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>
References
- John Baez, The Octonions, Section 4.5: E7, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTML version at
http://math.ucr.edu/home/baez/octonions/node18.html.
See also
| E6 | E7 | E8 | F4 | G2 |