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

Dynkin diagram

Missing image
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 \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.

Cartan matrix

\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}

Exceptional Lie groups

E6 | E7 | E8 | F4 | G2

See also: E7 (mathematics), Cartan matrix, Center of a group, Cyclic group, Dynkin diagram, E6 (mathematics), E8 (mathematics), F4 (mathematics), Fundamental representation