E₆

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.

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 where is one of $left(frac\{2\}\{3\},\; -frac\{1\}\{3\},\; -frac\{1\}\{3\}\; ight),\; left(\; -frac\{1\}\{3\},\; frac\{2\}\{3\},\; -frac\{1\}\{3\}\; ight),\; left(\; -frac\{1\}\{3\},\; -frac\{1\}\{3\},\; frac\{2\}\{3\}\; ight)$

All 27 combinations of where is one of $left(-frac\{2\}\{3\},frac\{1\}\{3\},frac\{1\}\{3\}\; ight),\; left(frac\{1\}\{3\},\; -frac\{2\}\{3\},\; frac\{1\}\{3\}\; ight),\; left(\; frac\{1\}\{3\},\; frac\{1\}\{3\},\; -frac\{2\}\{3\}\; ight)$

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)

:$left(frac\{1\}\{3\},-frac\{2\}\{3\},frac\{1\}\{3\};-frac\{2\}\{3\},frac\{1\}\{3\},frac\{1\}\{3\};-frac\{2\}\{3\},frac\{1\}\{3\},frac\{1\}\{3\}\; ight)$

An alternative description

An alternative (6-dimensional) description of the root system, which is useful in considering $E\_6\; imes\; SU(3)$ as a subgroup of $E\_8$, is the following:

All 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

:$left(pm\{1over\; 2\},pm\{1over\; 2\},pm\{1over\; 2\},pm\{1over\; 2\},pm\{1over\; 2\},pm\{sqrt\{3\}over\; 2\}\; ight)$.

Thus the 78 generators comprise of the following subalgebras:: A 45-dimensional $operatorname\{SO\}(10)$ subalgebra, including the above 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,−√3 / 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.

Cartan matrix

:

=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) × SU(3) × 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) × U(1) and SU(6) × SU(2).

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

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.

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 $E\_6\; /\; SP(8)$.

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

:

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

References

*. Online HTML version at [http://math.ucr.edu/home/baez/octonions/node17.html] .
*. Online scanned version at [http://ccdb4fs.kek.jp/cgi-bin/img_index?7904075] .

ee also

*En (Lie algebra)
*ADE classification
*Freudenthal magic square