E₇ (mathematics)

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In mathematics, E₇ is the name of several Lie groups and also their Lie algebras . 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.

Group theory

Group theory Basic notions Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product Finite groups and classification of finite simple groups Cyclic group Zn Symmetric group, Sn Dihedral group, Dn Alternating group An Mathieu groups M11, M12, M22, M23, M24 Conway groups Co1, Co2, Co3 Janko groups J1, J2, J3, J4 Fischer groups F22, F23, F24 Baby Monster group B Monster group M Discrete groups and lattices The integers, Z Lattice (group) Modular groups, PSL(2,Z) and SL(2,Z) Topological groups and lie groups Solenoid (mathematics) Circle 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) G2 F4 E6 E7 E8 Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group Infinite-dimensional Lie groups O(∞) SU(∞) Sp(∞)

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Lie groups and algebras

Lie group Lie algebra Classical groups 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) Simple Lie groups G2 F4 E6 E7 E8 Other Lie groups Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group Infinite-dimensional Lie groups O(∞) SU(∞) Sp(∞) Lie algebras Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form Classification of simple and semi-simple Lie groups Dynkin diagrams Cartan subalgebra Root system Real form Complexification Split Lie algebra Compact Lie algebra Representation theory Representation of a Lie group Representation of a Lie algebra

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Contents 1 Algebra 1.1 Dynkin diagram 1.2 Root system 1.2.1 An alternative description 1.3 Cartan matrix 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 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 E₇ × SU(2) as a subgroup of E₈, is the following:

All permutations of

preserving the zero at the last entry,

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

and the two following roots

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

The simple roots in this description are

(−1/2,−1/2,−1/2,−1/2,−1/2,−1/2,−1/√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

[edit] Important subalgebras and representations

E₇ has an SU(8) subalgebra, as is evident by noting that in the 8-dimensional description of the root system, the first group of roots are identical to the roots of 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 SU(8) bosonic local symmetry. The fermions are in representations of 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 E₇ / SU(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 E₈ × E₇ in six-dimensional compactifications of heterotic string theory, for instance on the four-dimensional surface K3.

[edit] References

• John Baez, The Octonions, Section 4.5: E₇, Bull. Amer. Math. Soc. 39 (2002), 145-205. Online HTML version at http://math.ucr.edu/home/baez/octonions/node18.html.
• E. Cremmer and B. Julia, The N = 8 Supergravity Theory. 1. The Lagrangian, Phys.Lett.B80:48,1978. Online scanned version at http://ccdb4fs.kek.jp/cgi-bin/img_index?7810033.

[edit] See also

• En (Lie algebra)
• ADE classification
• List of simple Lie groups

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