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.
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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.
