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E7 (mathematics)

In mathematics, E₇ is the name of a Lie group and also its Lie algebra $\mathfrak{e}_7$ . 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.

Contents

1 Algebra

1.1 Dynkin diagram
1.2 Root system
1.3 Cartan matrix

Algebra

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 $\begin{pmatrix}8\\4\end{pmatrix}$ 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.

$\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 2 \end{pmatrix}$

Categories: Lie groups

Last updated: 05-22-2005 00:51:24

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10-26-2009 08:16:03
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