E6 (mathematics)

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In mathematics, E6 is the name of some Lie groups and also their Lie algebras \mathfrak{e}_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 compact 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

[edit] Algebra

Graph of E6 Gosset polytope, 221Coxeter-Dynkin diagram:
Graph of E6 Gosset polytope, 221

Coxeter-Dynkin diagram:Image:CD ring.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD downbranch-00.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.png

[edit] Dynkin diagram

Dynkin diagram of E 6

[edit] 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{3};\bold{3};\bold{3}) where \bold{3} is one of \left(\frac{2}{3},-\frac{1}{3},-\frac{1}{3}\right), \left(-\frac{1}{3},\frac{2}{3},-\frac{1}{3}\right), \left(-\frac{1}{3},-\frac{1}{3},\frac{2}{3}\right)

All 27 combinations of (\bold{\bar{3}};\bold{\bar{3}};\bold{\bar{3}}) where \bold{\bar{3}} is one of (-\frac{2}{3},\frac{1}{3},\frac{1}{3}), (\frac{1}{3},-\frac{2}{3},\frac{1}{3}), (\frac{1}{3},\frac{1}{3},-\frac{2}{3})

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{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)

[edit] An alternative description

An alternative (6-dimensional) description of the root system, which is useful in considering E_6 \times SU(3) as a subgroup of E8, is the following:

All 4\times\begin{pmatrix}5\\2\end{pmatrix} permutations of

(\pm 1,\pm 1,0,0,0,0) preserving the zero at the last entry,

and all of the following roots with an even number of plus signs

\left(\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{1\over 2},\pm{\sqrt{3}\over 2}\right).

Thus the 78 generators comprise of the following subalgebras:

A 45-dimensional \operatorname{SO}(10) subalgebra, including the above 4\times\begin{pmatrix}5\\2\end{pmatrix} generators plus the five Cartan generators corresponding to the first five entries.
Two 16-dimensional subalgebras that transform as a Weyl spinor of \operatorname{spin}(10) and its complex conjugate. These have a non-zero last entry.
1 generator which is their chirality generator, and is the sixth Cartan generator.

The simple roots in this description are

(-1/2,-1/2,-1/2,-1/2,-1/2,-{\sqrt{3}\over 2})

(1,1,0,0,0,0)

(0,-1,1,0,0,0)

(0,0,-1,1,0,0)

(0,0,0,-1,1,0)

(-1,1,0,0,0,0)

we have ordered them so that their corresponding nodes in the dynkin diagram are ordered from left to right (in the diagram depicted above) with the side node last.

[edit] Cartan matrix

\begin{pmatrix} 2&-1&0&0&0&0\\ -1&2&-1&0&0&0\\ 0&-1&2&-1&0&-1\\ 0&0&-1&2&-1&0\\ 0&0&0&-1&2&0\\ 0&0&-1&0&0&2 \end{pmatrix}

[edit] Important subalgebras and representations

The Lie algebra E6 has an F4 subalgebra, which is the fixed subalgebra of an outer automorphism, and an SU(3)\times SU(3)\times SU(3) subalgebra.

Other maximal subalgebras which have an importance in physics (see below) and can be read off the Dynkin diagram, are the algebras of SO(10) \times U(1) and SU(6) \times SU(2).

In addition to the 78-dimensional adjoint representation, there are two dual 27-dimensional "vector" representations.

[edit] 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.

[edit] Importance in physics

N=8 supergravity in five dimensions, which is a dimensional reduction from 11 dimensional supergravity, admits an E6 bosonic global symmetry and an \operatorname{SP}(8) bosonic local symmetry. The fermions are in representations of \operatorname{SP}(8), the gauge fields are in a representation of E6, and the scalars are in a representation of both (Gravitons are singlets with respect to both). Physical states are in representations of the coset E6 / SP(8).

In grand unification theories, E6 appears as a possible gauge group which, after its breaking, gives rise to the SU(3)\times SU(2) \times U(1) gauge group of the standard model (also see Importance in physics of E8). One way of achieving this is through breaking to \operatorname{SO}(10) \times \operatorname{U}(1). The adjoint 78 representation breaks, as explained above, into an adjoint 45, spinor 16 and \bar{16} as well as a singlet of the \operatorname{SO}(10) subalgebra. Including the \operatorname{U}(1) charge we have

78 \rightarrow 45_0 \oplus 16_{-3} \oplus \bar{16}_3 + 1_0.

Where the subscript denotes the \operatorname{U}(1) charge.

[edit] References

[edit] See also

Exceptional Lie groups
E6 | E7 | E8 | F4 | G2
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