E6 (mathematics)

In mathematics, E₆ is the name of a Lie group (and also sometimes of its Lie algebra). It is one of the exceptional simple Lie groups.

Table of contents

1 Roots of E6 1.1 Simple roots 2 Weyl/Coxeter group 3 Cartan matrix

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 where is one of , ,

All 27 combinations of where is one of , ,

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)

Weyl/Coxeter group

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

Cartan matrix

See also Simple Lie group, Lie group, Weyl group, Dynkin diagram.