E6 (mathematics)

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 $({\mathbf {3}};{\mathbf {3}};{\mathbf {3}})$ where ${\mathbf {3}}$ 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 $({\mathbf {\bar {3}} };{\mathbf {\bar {3}} };{\mathbf {\bar {3}} })$ where ${\mathbf {\bar {3}}}$ 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)

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 $E_{8}$ , 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 generators comprise of a 45-dimensional $\operatorname {so} (10)$ subalgebra as well as 32 generators that transform as a Majorana spinor of $\operatorname {spin} (10)$ and its chirality 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)

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

(-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

{\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}}

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.

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 $SU(3)\times SU(2)\times U(1)$ gauge group of the standard model (also see Importance in physics of E8).

References

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