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
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.


">
" size=20>

 
 

Browse articles alphabetically:
#0">0 | #1">1 | #2">2 | #3">3 | #4">4 | #5">5 | #6">6 | #7">7 | #8">8 | #9">9 | #_">_ | #A">A | #B">B | #C">C | #D">D | #E">E | #F">F | #G">G | #H">H | #I">I | #J">J | #K">K | #L">L | #M">M | #N">N | #O">O | #P">P | #Q">Q | #R">R | #S">S | #T">T | #U">U | #V">V | #W">W | #X">X | #Y">Y | #Z">Z