- E₇
In

mathematics ,**E**is the name of several_{7}Lie group s and also theirLie algebra s $mathfrak\{e\}\_7$. It is one of the five exceptional compactsimple Lie group s as well as one of thesimply laced group s. "E"_{7}has rank 7 and dimension 133. The fundamental group of the compact form is thecyclic group **Z**_{2}, and itsouter automorphism group is thetrivial group . The dimension of itsfundamental representation is 56.The compact real form of E

_{7}is theisometry group of a 64-dimensionalRiemannian manifold known informally as the 'quateroctonionic projective plane' because it can be built using an algebra that is the tensor product of thequaternion s and theoctonion s. This can be seen systematically using a construction known as the "magic square", due toHans Freudenthal andJacques Tits . There are three other real forms, and one complex form.**Algebra**Dynkin diagram 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 $egin\{pmatrix\}8\backslash 4end\{pmatrix\}$ 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 root s 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.**An alternative description**An alternative (7-dimensional) description of the root system, which is useful in considering E

_{7 × SU(2) as a subgroup of E8, is the following:}All $4\; imesegin\{pmatrix\}6\backslash 2end\{pmatrix\}$ 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

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

and the two following roots:$left(0,0,0,0,0,0,pm\; sqrt\{2\}\; ight).$

Thus the generators comprise of a 66-dimensional so(12) subalgebra as well as 65 generators that transform as two self-conjugate

Weyl spinor s of spin(12) of opposite chirality and their chirality generator, and two other generators of chiralities $pm\; sqrt\{2\}$The

simple root s 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.Cartan matrix :$egin\{pmatrix\}\; 2\; -1\; 0\; 0\; 0\; 0\; 0\; \backslash -1\; 2\; -1\; 0\; 0\; 0\; 0\; \backslash \; 0\; -1\; 2\; -1\; 0\; 0\; -1\; \backslash \; 0\; 0\; -1\; 2\; -1\; 0\; 0\; \backslash \; 0\; 0\; 0\; -1\; 2\; -1\; 0\; \backslash \; 0\; 0\; 0\; 0\; -1\; 2\; 0\; \backslash \; 0\; 0\; -1\; 0\; 0\; 0\; 2end\{pmatrix\}$

=Important subalgebras and representations=E

_{7}has an 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 SU(8) (with the sameCartan subalgebra as in the E_{7}).In addition to the 133-dimensional adjoint representation, there is a 56-dimensional "vector" representation, to be found in the "E"

_{8}adjoint representation.**Importance in physics**"N" = 8

supergravity in four dimensions, which is adimensional reduction from 11 dimensional supergravity, admit an E_{7}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 E_{7}, and the scalars are in a representation of both (Gravitons aresinglet s with respect to both). Physical states are in representations of the coset E_{7}/ SU(8).In

string theory , E_{7}appears as a part of thegauge group of one the (unstable and non-supersymmetric ) versions of theheterotic string . It can also appear in the unbroken gauge group E_{8}× E_{7}in six-dimensional compactifications of heterotic string theory, for instance on the four-dimensional surface K3.**References***

John Baez , "The Octonions", Section 4.5: E_{7}, [*http://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html Bull. Amer. Math. Soc.*] . Online HTML version at http://math.ucr.edu/home/baez/octonions/node18.html.**39**(2002), 145-205* 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.

**ee also***

En (Lie algebra)

*ADE classification

*List of simple Lie groups

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