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{{Short description|133-dimensional exceptional simple Lie group}}
{{DISPLAYTITLE:E<sub>7</sub> (mathematics)}}
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{{Lie groups |Simple}}

In [[mathematics]], '''E<sub>7</sub>''' is the name of several closely related [[Lie group]]s, linear [[algebraic group]]s or their [[Lie algebra]]s '''e'''<sub>7</sub>, all of which have dimension 133; the same notation E<sub>7</sub> is used for the corresponding [[root lattice]], which has [[Rank of a Lie group|rank]]&nbsp;7. The designation E<sub>7</sub> comes from the [[Killing form|Cartan–Killing classification]] of the complex [[simple Lie algebra]]s, which fall into four infinite series labeled A<sub>''n''</sub>, B<sub>''n''</sub>, C<sub>''n''</sub>, D<sub>''n''</sub>, and [[Exceptional simple Lie group|five exceptional cases]] labeled [[E6 (mathematics)|E<sub>6</sub>]], E<sub>7</sub>, [[E8 (mathematics)|E<sub>8</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], and [[G2 (mathematics)|G<sub>2</sub>]]. The E<sub>7</sub> algebra is thus one of the five exceptional cases.

The fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E<sub>7</sub> is the [[cyclic group]] '''Z'''/2'''Z''', and its [[outer automorphism group]] is the [[trivial group]]. The dimension of its [[fundamental representation]] is 56.

==Real and complex forms==
There is a unique complex Lie algebra of type E<sub>7</sub>, corresponding to a complex group of complex dimension 133. The complex adjoint Lie group E<sub>7</sub> of [[complex dimension]] 133 can be considered as a simple real Lie group of real dimension 266. This has fundamental group '''Z'''/2'''Z''', has maximal [[Compact space|compact]] subgroup the compact form (see below) of E<sub>7</sub>, and has an outer automorphism group of order 2 generated by complex conjugation.

As well as the complex Lie group of type E<sub>7</sub>, there are four real forms of the Lie algebra, and correspondingly four real forms of the group with trivial center (all of which have an algebraic double cover, and three of which have further non-algebraic covers, giving further real forms), all of real dimension 133, as follows:
* The compact form (which is usually the one meant if no other information is given), which has fundamental group '''Z'''/2'''Z''' and has trivial outer automorphism group.
* The split form, EV (or E<sub>7(7)</sub>), which has maximal compact subgroup SU(8)/{±1}, fundamental group cyclic of order 4 and outer automorphism group of order 2.
* EVI (or E<sub>7(-5)</sub>), which has maximal compact subgroup SU(2)·SO(12)/(center), fundamental group non-cyclic of order 4 and trivial outer automorphism group.
* EVII (or E<sub>7(-25)</sub>), which has maximal compact subgroup SO(2)·E<sub>6</sub>/(center), infinite cyclic fundamental group and outer automorphism group of order 2.

For a complete list of real forms of simple Lie algebras, see the [[list of simple Lie groups]].

The compact real form of E<sub>7</sub> is the [[isometry group]] of the 64-dimensional exceptional compact [[Riemannian symmetric space]] EVI (in Cartan's [[Riemannian symmetric space#Classification of Riemannian symmetric spaces|classification]]). It is known informally as the "{{Not a typo|[[quateroctonionic projective plane]]}}<!-- this is not a typo: please don't change this to "qua*r*teroctonionic" -->" because it can be built using an algebra that is the tensor product of the [[quaternion]]s and the [[octonion]]s, and is also known as a [[Rosenfeld projective plane]], though it does not obey the usual axioms of a projective plane. This can be seen systematically using a construction known as the ''[[Freudenthal magic square|magic square]]'', due to [[Hans Freudenthal]] and [[Jacques Tits]].

The algebraic double cover of the complex form, or of the split real form, or of EVII can be described as the automorphism group of a Freudenthal triple system, which is a special kind of [[triple system]] defined on a 56-dimensional vector space.<ref>See Springer.</ref>

The [[Tits–Koecher construction]] produces forms of the E<sub>7</sub> Lie algebra from [[Albert algebra]]s, 27-dimensional exceptional [[Jordan algebra]]s.

==E<sub>7</sub> as an algebraic group==
By means of a [[Chevalley basis]] for the Lie algebra, one can define E<sub>7</sub> as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as "untwisted") adjoint form of E<sub>7</sub>. Over an algebraically closed field, this and its double cover are the only forms; however, over other fields, there are often many other forms, or "twists" of E<sub>7</sub>, which are classified in the general framework of [[Galois cohomology]] (over a [[perfect field]] ''k'') by the set ''H''<sup>1</sup>(''k'', Aut(E<sub>7</sub>)) which, because the Dynkin diagram of E<sub>7</sub> (see [[#Dynkin diagram|below]]) has no automorphisms, coincides with ''H''<sup>1</sup>(''k'', E<sub>7, ad</sub>).<ref>{{Citation | last1=Platonov | first1=Vladimir | last2=Rapinchuk | first2=Andrei | title=Algebraic groups and number theory | orig-year=1991 | url=https://books.google.com/books?isbn=0125581807 | publisher=[[Academic Press]] | location=Boston, MA | series=Pure and Applied Mathematics | isbn=978-0-12-558180-6 | mr=1278263 | year=1994 | volume=139}} (original version: {{cite book | last1=Платонов | first1=Владимир П. | last2=Рапинчук | first2=Андрей С. | title=Алгебраические группы и теория чисел | year=1991 | publisher=Наука | isbn=5-02-014191-7 }}), §2.2.4</ref>

Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E<sub>7</sub> coincide with the three real Lie groups mentioned [[#Real and complex forms|above]], but with a subtlety concerning the fundamental group: all adjoint forms of E<sub>7</sub> have fundamental group '''Z'''/2'''Z''' in the sense of algebraic geometry, meaning that they admit exactly one double cover; the further non-compact real Lie group forms of E<sub>7</sub> are therefore not algebraic and admit no faithful finite-dimensional representations.

Over finite fields, the [[Lang–Steinberg theorem]] implies that ''H''<sup>1</sup>(''k'', E<sub>7</sub>) = 0, meaning that E<sub>7</sub> has no twisted forms: see [[#Chevalley groups of type E7|below]].

==Algebra==

===Dynkin diagram===
The [[Dynkin diagram]] for E<sub>7</sub> is given by [[File:Dynkin diagram type E7.svg|120px]].

===Root system===
[[File:Gosset 2 31 polytope.svg|thumb|300px|The 126 vertices of the [[2 31 polytope|2<sub>31</sub> polytope]] represent the root vectors of E<sub>7</sub>, as shown in this [[Coxeter plane]] projection<br>[[Coxeter–Dynkin diagram]]: {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}]]

[[File:3_21_E7_to_3D_H3_symmetry_concentric_hulls.png|thumb|300px|Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry:<br>
u = (1, ''φ'', 0, -1, ''φ'', 0,0)<br>
v = (''φ'', 0, 1, ''φ'', 0, -1,0)<br>
w = (0, 1, ''φ'', 0, -1, ''φ'',0)<br>
The projected [[2 31 polytope|2<sub>31</sub> polytope]] vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls of each set of tallied norms. These show:<br>
1) 2 points at the origin<br>
2) 2 icosahedrons<br>
3) 1 icosadodecahedron<br>
4) 2 dodecahedrons<br>
5) 1 icosadodecahedron<br>
for a total of 126 vertices.]]

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 <math>\begin{pmatrix}8\\4\end{pmatrix}</math> permutations of ({{sfrac|1|2}},{{sfrac|1|2}},{{sfrac|1|2}},{{sfrac|1|2}},−{{sfrac|1|2}},−{{sfrac|1|2}},−{{sfrac|1|2}},−{{sfrac|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 (root system)|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)
:({{sfrac|1|2}},{{sfrac|1|2}},{{sfrac|1|2}},{{sfrac|1|2}},−{{sfrac|1|2}},−{{sfrac|1|2}},−{{sfrac|1|2}},−{{sfrac|1|2}})

They are listed 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 {{nowrap|E<sub>7</sub> × SU(2)}} as a [[E8 (mathematics)#Subgroups|subgroup of]] E<sub>8</sub>, is the following:

All <math>4\times\begin{pmatrix}6\\2\end{pmatrix}</math> permutations of (±1,±1,0,0,0,0,0) preserving the zero at the last entry, all of the following roots with an even number of +{{sfrac|1|2}}

:<math>\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)</math>

and the two following roots

:<math>\left(0,0,0,0,0,0,\pm \sqrt{2}\right).</math>

Thus the generators consist of a 66-dimensional '''so'''(12) subalgebra as well as 64 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 <math>\pm \sqrt{2}</math>.

Given the E<sub>7</sub> [[Cartan matrix]] (below) and a [[Dynkin diagram]] node ordering of: [[File:DynkinE7.svg|120px]]

:one choice of [[Simple root (root system)|simple root]]s is given by the rows of the following matrix:
:<math>\begin{bmatrix}
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 \\
-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&\frac{\sqrt{2}}{2}\\
0&0&0&0&1&-1&0 \\
\end{bmatrix}.</math>

[[File:E7HassePoset.svg|thumb|320px|[[Hasse diagram]] of E<sub>7</sub> [[Root system#The root poset|root poset]] with edge labels identifying added simple root position]]
===Weyl group===
The [[Weyl group]] of E<sub>7</sub> is of order 2903040: it is the direct product of the cyclic group of order 2 and the unique [[simple group]] of order 1451520 (which can be described as PSp<sub>6</sub>(2) or PSΩ<sub>7</sub>(2)).<ref>{{cite book |last1=Conway |first1=John Horton |author-link1=John Horton Conway |last2=Curtis |first2=Robert Turner |last3=Norton |first3=Simon Phillips |author-link3=Simon P. Norton |last4=Parker |first4=Richard A |author-link4=Richard A. Parker |last5=Wilson |first5=Robert Arnott |author-link5=Robert Arnott Wilson |title=[[ATLAS of Finite Groups|Atlas of Finite Groups]]: Maximal Subgroups and Ordinary Characters for Simple Groups |year=1985 |publisher=Oxford University Press |isbn=0-19-853199-0 |page=46 }}</ref>

===Cartan matrix===
:<math>\begin{bmatrix}
 2 & -1 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 & 0 & 0 \\
 0 & -1 & 2 & -1 & 0 & 0 & 0 \\
 0 & 0 & -1 & 2 & -1 & 0 & -1 \\
 0 & 0 & 0 & -1 & 2 & -1 & 0 \\
 0 & 0 & 0 & 0 & -1 & 2 & 0 \\
 0 & 0 & 0 & -1 & 0 & 0 & 2
\end{bmatrix}.</math>

==Important subalgebras and representations==
[[File:E7 Maximal Embeddings.svg|thumb|320px|Embeddings of the maximal subgroups of E<sub>7</sub> up to dimension 133 shown with associated projection matrix.]]
E<sub>7</sub> 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 E<sub>7</sub>).

In addition to the 133-dimensional adjoint representation, there is a [[E8 (mathematics)#Subgroups|56-dimensional "vector" representation]], to be found in the E<sub>8</sub> adjoint representation.

The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the [[Weyl character formula]]. The dimensions of the smallest irreducible representations are {{OEIS|id=A121736}}:

:<u>1</u>, 56, <u>133</u>, 912, <u>1463</u>, <u>1539</u>, 6480, <u>7371</u>, <u>8645</u>, 24320, 27664, <u>40755</u>, 51072, 86184, <u>150822</u>, <u>152152</u>, <u>238602</u>, <u>253935</u>, <u>293930</u>, 320112, 362880, <u>365750</u>, <u>573440</u>, <u>617253</u>, 861840, 885248, <u>915705</u>, <u>980343</u>, 2273920, 2282280, 2785552, <u>3424256</u>, 3635840...

The underlined terms in the sequence above are the dimensions of those irreducible representations possessed by the adjoint form of E<sub>7</sub> (equivalently, those whose weights belong to the root lattice of E<sub>7</sub>), whereas the full sequence gives the dimensions of the irreducible representations of the simply connected form of E<sub>7</sub>. There exist non-isomorphic irreducible representation of dimensions 1903725824, 16349520330, etc.

The [[fundamental representation]]s are those with dimensions 133, 8645, 365750, 27664, 1539, 56 and 912 (corresponding to the seven nodes in the [[#Dynkin diagram|Dynkin diagram]] in the order chosen for the [[#Cartan matrix|Cartan matrix]] above, i.e., the nodes are read in the six-node chain first, with the last node being connected to the third).

The embeddings of the maximal subgroups of E<sub>7</sub> up to dimension 133 are shown to the right.

===E<sub>7</sub> Polynomial Invariants===
E<sub>7</sub> is the automorphism group of the following pair of polynomials in 56 non-commutative variables. We divide the variables into two groups of 28, (''p'', ''P'') and (''q'', ''Q'') where ''p'' and ''q'' are real variables and ''P'' and ''Q'' are 3×3 [[octonion]] hermitian matrices. Then the first invariant is the symplectic invariant of Sp(56, '''R'''):

:<math>C_1 = pq - qp + Tr[PQ] - Tr[QP]</math>

The second more complicated invariant is a '''symmetric''' quartic polynomial:

:<math>C_2 = (pq + Tr[P\circ Q])^2 + p Tr[Q\circ \tilde{Q}]+q Tr[P\circ \tilde{P}]+Tr[\tilde{P}\circ \tilde{Q}] </math>

Where <math>\tilde{P} \equiv \det(P) P^{-1}</math> and the binary circle operator is defined by <math>A\circ B = (AB+BA)/2</math>.

An alternative quartic polynomial invariant constructed by Cartan uses two anti-symmetric 8x8 matrices each with 28 components.

:<math> C_2 = Tr[(XY)^2] - \dfrac{1}{4} Tr[XY]^2 +\frac{1}{96}\epsilon_{ijklmnop}\left( X^{ij}X^{kl}X^{mn}X^{op} + Y^{ij}Y^{kl}Y^{mn}Y^{op} \right)</math>

==Chevalley groups of type E<sub>7</sub>==
The points over a [[finite field]] with ''q'' elements of the (split) algebraic group E<sub>7</sub> (see [[#E7 as an algebraic group|above]]), whether of the adjoint (centerless) or simply connected form (its algebraic universal cover), give a finite [[Group of Lie type|Chevalley group]]. This is closely connected to the group written E<sub>7</sub>(''q''), however there is ambiguity in this notation, which can stand for several things:

* the finite group consisting of the points over '''F'''<sub>''q''</sub> of the simply connected form of E<sub>7</sub> (for clarity, this can be written E<sub>7,sc</sub>(''q'') and is known as the "universal" Chevalley group of type E<sub>7</sub> over '''F'''<sub>''q''</sub>),
* (rarely) the finite group consisting of the points over '''F'''<sub>''q''</sub> of the adjoint form of E<sub>7</sub> (for clarity, this can be written E<sub>7,ad</sub>(''q''), and is known as the "adjoint" Chevalley group of type E<sub>7</sub> over '''F'''<sub>''q''</sub>), or
* the finite group which is the image of the natural map from the former to the latter: this is what will be denoted by E<sub>7</sub>(''q'') in the following, as is most common in texts dealing with finite groups.

From the finite group perspective, the relation between these three groups, which is quite analogous to that between SL(''n'', ''q''), PGL(''n'', ''q'') and PSL(''n'', ''q''), can be summarized as follows: E<sub>7</sub>(''q'') is simple for any ''q'', E<sub>7,sc</sub>(''q'') is its [[Schur multiplier|Schur cover]], and the E<sub>7,ad</sub>(''q'') lies in its automorphism group; furthermore, when ''q'' is a power of 2, all three coincide, and otherwise (when ''q'' is odd), the Schur multiplier of E<sub>7</sub>(''q'') is 2 and E<sub>7</sub>(''q'') is of index 2 in E<sub>7,ad</sub>(''q''), which explains why E<sub>7,sc</sub>(''q'') and E<sub>7,ad</sub>(''q'') are often written as 2·E<sub>7</sub>(''q'') and E<sub>7</sub>(''q'')·2. From the algebraic group perspective, it is less common for E<sub>7</sub>(''q'') to refer to the finite simple group, because the latter is not in a natural way the set of points of an algebraic group over '''F'''<sub>''q''</sub> unlike E<sub>7,sc</sub>(''q'') and E<sub>7,ad</sub>(''q'').

As mentioned above, E<sub>7</sub>(''q'') is simple for any ''q'',<ref>{{cite book | first=Roger W. | last=Carter | title=Simple Groups of Lie Type | author-link=Roger Carter (mathematician) | publisher=John Wiley & Sons | series=Wiley Classics Library | isbn=0-471-50683-4 | year=1989 }}</ref><ref>{{cite book | first=Robert A. | last=Wilson | title=The Finite Simple Groups | author-link=Robert Arnott Wilson | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics]] | volume=251 | isbn=978-1-84800-987-5 | year=2009 }}</ref> and it constitutes one of the infinite families addressed by the [[classification of finite simple groups]]. Its number of elements is given by the formula {{OEIS|id=A008870}}:

:<math>\frac{1}{\mathrm{gcd}(2,q-1)}q^{63}(q^{18}-1)(q^{14}-1)(q^{12}-1)(q^{10}-1)(q^8-1)(q^6-1)(q^2-1)</math>

The order of E<sub>7,sc</sub>(''q'') or E<sub>7,ad</sub>(''q'') (both are equal) can be obtained by removing the dividing factor gcd(2, ''q''−1) {{OEIS|id=A008869}}. The Schur multiplier of E<sub>7</sub>(''q'') is gcd(2, ''q''−1), and its outer automorphism group is the product of the diagonal automorphism group '''Z'''/gcd(2, ''q''−1)'''Z''' (given by the action of E<sub>7,ad</sub>(''q'')) and the group of field automorphisms (i.e., cyclic of order ''f'' if ''q'' = ''p<sup>f</sup>'' where ''p'' is prime).

==Importance in physics==
''N'' = 8 [[supergravity]] in four dimensions, which is a [[dimensional reduction]] from [[eleven-dimensional supergravity]], admit an E<sub>7</sub> bosonic global symmetry and an SU(8) bosonic [[gauge symmetry|local symmetry]]. The fermions are in representations of SU(8), the gauge fields are in a representation of E<sub>7</sub>, and the scalars are in a representation of both (Gravitons are [[Singlet state|singlets]] with respect to both). Physical states are in representations of the coset {{nowrap|E<sub>7</sub> / SU(8)}}.

In [[string theory]], E<sub>7</sub> appears as a part of the [[gauge group]] of one of the (unstable and non-[[supersymmetric]]) versions of the [[heterotic string]]. It can also appear in the unbroken gauge group {{nowrap|E<sub>8</sub> × E<sub>7</sub>}} in six-dimensional compactifications of heterotic string theory, for instance on the four-dimensional surface [[K3 (surface)|K3]].

==See also==
* [[En (Lie algebra)]]
* [[ADE classification]]
* [[List of simple Lie groups]]

==Notes==
{{Reflist}}

==References==
* {{Citation | last1=Adams | first1=J. Frank | title=Lectures on exceptional Lie groups | url=https://books.google.com/books?isbn=0226005275 | publisher=[[University of Chicago Press]] | series=Chicago Lectures in Mathematics | isbn=978-0-226-00526-3 | mr=1428422 | year=1996}}
* [[John Baez]], ''The Octonions'', Section 4.5: E<sub>7</sub>, [https://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html 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 {{nowrap|N {{=}} 8}} Supergravity Theory. 1. The Lagrangian'', Phys.Lett.B80:48,1978. Online scanned version at http://ac.els-cdn.com/0370269378903039/1-s2.0-0370269378903039-main.pdf?_tid=79273f80-539d-11e4-a133-00000aab0f6c&acdnat=1413289833_5f3539a6365149b108ddcec889200964.
* {{cite journal
|first=T.A.
|last=Springer
|author-link=T._A._Springer
|title=Some groups of type E<sub>7</sub>
|journal=Nagoya Mathematical Journal
|year=2006
|volume=182
|pages=259-284
|doi=10.1017/S0027763000026891}}


{{Exceptional Lie groups}}
{{String theory topics |state=collapsed}}

[[Category:Algebraic groups]]
[[Category:Lie groups]]
[[Category:Exceptional Lie algebras]]