In mathematics, E₆ is the name of a Lie group and also its Lie algebra <math>\mathfrak{e}_6<math>. It is one of the five exceptional simple Lie groups as well as one of the simply laced groups. E₆ has rank 6 and dimension 78. Its center is the cyclic group Z₃. Its outer automorphism group is the cyclic group Z₂. Its fundamental representation is 27-dimensional (complex) and its dual representation, which is inequivalent to it is also 27-dimensional.

In particle physics, E₆ plays a role in some grand unified theories.

Contents

1 Algebra 1.1 Dynkin diagram 1.2 Roots of E6 1.3 Weyl/Coxeter group 1.4 Cartan matrix

Algebra

Dynkin diagram

Roots of E₆

Although they span a six-dimensional space, it's much more symmetrical to consider them as vectors in a six-dimensional subspace of a nine-dimensional space.

(1,-1,0;0,0,0;0,0,0), (-1,1,0;0,0,0;0,0,0),

(-1,0,1;0,0,0;0,0,0), (1,0,-1;0,0,0;0,0,0),

(0,1,-1;0,0,0;0,0,0), (0,-1,1;0,0,0;0,0,0),

(0,0,0;1,-1,0;0,0,0), (0,0,0;-1,1,0;0,0,0),

(0,0,0;-1,0,1;0,0,0), (0,0,0;1,0,-1;0,0,0),

(0,0,0;0,1,-1;0,0,0), (0,0,0;0,-1,1;0,0,0),

(0,0,0;0,0,0;1,-1,0), (0,0,0;0,0,0;-1,1,0),

(0,0,0;0,0,0;-1,0,1), (0,0,0;0,0,0;1,0,-1),

(0,0,0;0,0,0;0,1,-1), (0,0,0;0,0,0;0,-1,1),

All 27 combinations of <math>(\bold{3};\bold{3};\bold{3})<math> where <math>\bold{3}<math> is one of <math>(\frac{2}{3},-\frac{1}{3},-\frac{1}{3})<math>, <math>(-\frac{1}{3},\frac{2}{3},-\frac{1}{3})<math>, <math>(-\frac{1}{3},-\frac{1}{3},\frac{2}{3})<math>

All 27 combinations of <math>(\bold{\bar{3}};\bold{\bar{3}};\bold{\bar{3}})<math> where <math>\bold{\bar{3}}<math> is one of <math>(-\frac{2}{3},\frac{1}{3},\frac{1}{3})<math>, <math>(\frac{1}{3},-\frac{2}{3},\frac{1}{3})<math>, <math>(\frac{1}{3},\frac{1}{3},-\frac{2}{3})<math>

Simple roots

(0,0,0;0,0,0;0,1,-1)

(0,0,0;0,0,0;1,-1,0)

(0,0,0;0,1,-1;0,0,0)

(0,0,0;1,-1,0;0,0,0)

(0,1,-1;0,0,0;0,0,0)

<math>(\frac{1}{3},-\frac{2}{3},\frac{1}{3};-\frac{2}{3},\frac{1}{3},\frac{1}{3};-\frac{2}{3},\frac{1}{3},\frac{1}{3})<math>

Weyl/Coxeter group

Its Weyl/Coxeter group is symmetry group of the E6 polytope.

Cartan matrix

<math>

\begin{pmatrix} 2&-1&0&0&0&0\\ -1&2&-1&0&0&0\\ 0&-1&2&-1&-1&0\\ 0&0&-1&2&0&0\\ 0&0&-1&0&2&-1\\ 0&0&0&0&-1&2 \end{pmatrix} <math>

Exceptional Lie groups

E6 | E7 | E8 | F4 | G2