E7 (mathematics)

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The correct title of this article is E₇ (mathematics). It features superscript or subscript characters that are substituted or omitted because of technical limitations.

In mathematics, E₇ is the name of several Lie groups and also their Lie algebras $\mathfrak{e}_7$ . It is one of the five exceptional compact simple Lie groups as well as one of the simply laced groups. E₇ has rank 7 and dimension 133. The fundamental group of the compact form is the cyclic group Z₂, and its outer automorphism group is the trivial group. The dimension of its fundamental representation is 56.

The compact real form of E₇ is the isometry group of a 64-dimensional Riemannian manifold known informally as the 'quateroctonionic projective plane' because it can be built using an algebra that is the tensor product of the quaternions and the octonions. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits. There are three other real forms, and one complex form.

Groups

Group theory

Basic notions
Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product direct sum

Discrete groups
Classification of finite simple groups Cyclic group Z_n Alternating group A_n Sporadic groups Mathieu group M_11..12,M_22..24 Conway group Co_1..3 Janko group J_1..4 Fischer group F_22..24 Baby Monster group B Monster group M

Continuous groups
Lie group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) E₆ E₇ E₈ F₄ G₂ SL₂ Lorentz group Poincaré group

Infinite dimensional groups
conformal group diffeomorphism group Quantum group O(∞) SU(∞) Sp(∞)

This box: view • talk • edit

1 Algebra
2 Important subalgebras and representations
3 Importance in physics
4 References
5 See also

[edit] Algebra

[edit] Dynkin diagram

[edit] Root system

Even though the roots span a 7-dimensional space, it is more symmetric and convenient to represent them as vectors lying in a 7-dimensional subspace of an 8-dimensional vector space.

The roots are all the 8×7 permutations of (1,−1,0,0,0,0,0,0)

and all the $\begin{pmatrix}8\\4\end{pmatrix}$ permutations of (1/2,1/2,1/2,1/2,−1/2,−1/2,−1/2,−1/2)

Note that the 7-dimensional subspace is the subspace where the sum of all the eight coordinates is zero. There are 126 roots.

The simple roots are

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

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

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

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

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

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

(1/2,1/2,1/2,1/2,−1/2,−1/2,−1/2,−1/2)

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] An alternative description

An alternative (7-dimensional) description of the root system, which is useful in considering $E_7 \times SU(2)$ as a subgroup of $E 8$ , is the following:

All $4\times\begin{pmatrix}6\\2\end{pmatrix}$ permutations of

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

all of the following roots with an even number of +1/2

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

and the two following roots

$\left(0,0,0,0,0,0,\pm \sqrt{2}\right)$ .

Thus the generators comprise of a 66-dimensional $\operatorname{so}(12)$ subalgebra as well as 65 generators that transform as two self-conjugate Weyl spinors of $\operatorname{spin}(12)$ of opposite chirality and their chirality generator, and two other generators of chiralities $\pm \sqrt{2}$

The simple roots in this description are

(-1/2,-1/2,-1/2,-1/2,-1/2,-1/2, $-1/\sqrt{2}$ )

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

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

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

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

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

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

Again 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

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

[edit] Important subalgebras and representations

E₇ has an $\operatorname{SU}(8)$ subalgebra, as is evident by noting that the in 8-dimensional description of the root system, the first group of roots are identical to the roots of $\operatorname{SU}(8)$ (with the same Cartan subalgebra as in the E₇).

In addition to the 133-dimensional adjoint representation, there is a 56-dimensional "vector" representation, to be found in the E₈ adjoint representation.

[edit] Importance in physics

N=8 supergravity in four dimensions, which is a dimensional reduction from 11 dimensional supergravity, admit an E₇ bosonic global symmetry and an $\operatorname{SU}(8)$ bosonic local symmetry. The fermions are in representations of $\operatorname{SU}(8)$ , the gauge fields are in a representation of E₇, and the scalars are in a representation of both (Gravitons are singlets with respect to both). Physical states are in representations of the coset $E 7 / S U (8)$ .

In string theory, E₇ appears as a part of the gauge group of one the (unstable and non-supersymmetric) versions of the heterotic string. It can also appear in the unbroken gauge group $E_8\times E_7$ in six-dimensional compactifications of heterotic string theory, for instance on the four-dimensional surface K3.