E6 (mathematics)

Encyclopedia : E : E6 : E6M : E6 (mathematics)

In mathematics, E₆ is the name of some Lie groups and also their Lie algebras [\mathfrak_6]. It is one of the five exceptional compact simple Lie groups as well as one of the simply laced groups. E₆ has rank 6 and dimension 78. The fundamental group of the comapct form is the cyclic group Z₃ and its outer automorphism group is the cyclic group Z₂. Its fundamental representation is 27-dimensional (complex). The dual representation, which is inequivalent, is also 27-dimensional.

A certain noncompact real form of E₆ is the group of collineations (line-preserving transformations) of the octonionic projective plane OP². It is also the group of determinant-preserving linear transformations of the exceptional Jordan algebra. The exceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E₆ has a 27-dimensional complex representation. The compact real form of E₆ is the isometry group of a 32-dimensional Riemannian manifold known as the 'bioctonionic projective plane'. Altogether there are 5 real forms and one complex form.

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

Contents

1 Algebra

1.1
1.2 Roots of E₆
1.3

2 E6 polytope
3 References
4 See also

Algebra

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 [(\bold;\bold;\bold)] where [\bold] is one of [\left(\frac,-\frac,-\frac\right)], [\left(-\frac,\frac,-\frac\right)], [\left(-\frac,-\frac,\frac\right)]

All 27 combinations of [(\bold};\bold};\bold})] where [\bold}] is one of [(-\frac,\frac,\frac)], [(\frac,-\frac,\frac)], [(\frac,\frac,-\frac)]

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)

[\left(\frac,-\frac,\frac;-\frac,\frac,\frac;-\frac,\frac,\frac\right)]

[\begin2&-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]

E6 polytope

The E₆ polytope is the convex hull of the roots of E₆. It therefore exists in 6 dimensions; its symmetry group contains the Coxeter group for E₆ as an index 2 subgroup.

References

John Baez, The Octonions, Section 4.4: E₆, [Bull. Amer. Math. Soc. 39 (2002), 145-205]. Online HTML version at [link].