E6 (mathematics)

From Free net encyclopedia

In mathematics, E₆ is the name of some Lie groups and also their Lie algebras <math>\mathfrak{e}_6</math>. 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. Its center is the cyclic group Z₃. 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 Dynkin diagram 1.2 Roots of E6 1.3 Weyl/Coxeter group 1.4 Cartan matrix 2 E6 polytope 3 References 4 See also

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>\left(\frac{2}{3},-\frac{1}{3},-\frac{1}{3}\right)</math>, <math>\left(-\frac{1}{3},\frac{2}{3},-\frac{1}{3}\right)</math>, <math>\left(-\frac{1}{3},-\frac{1}{3},\frac{2}{3}\right)</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>\left(\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}\right)</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>

E6 polytope

The E₆ polytope is the convex hull of the roots of E₆. It therefore exists in 6 dimensions; its symmetry group is the Coxeter group for E₆.

References

• John Baez, The Octonions, Section 4.4: E₆, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTML version at [1].

See also

• ADE classification

Exceptional Lie groups

E6 | E7 | E8 | F4 | G2

it:E6