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In mathematics, E7 is the name of several closely related Lie groups, linear
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
s or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding
root lattice In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation ...
, which has
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
 7. The designation E7 comes from the Cartan–Killing classification of the complex
simple Lie algebra In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of s ...
s, which fall into four infinite series labeled A''n'', B''n'', C''n'', D''n'', and five exceptional cases labeled E6, E7, E8, F4, and G2. The E7 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 E7 is the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
Z/2Z, and its
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
is the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
. The dimension of its
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group or Lie algebra whose highest weig ...
is 56.


Real and complex forms

There is a unique complex Lie algebra of type E7, corresponding to a complex group of complex dimension 133. The complex adjoint Lie group E7 of
complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the case of a variety) are modeled on a ...
133 can be considered as a simple real Lie group of real dimension 266. This has fundamental group Z/2Z, has maximal
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
subgroup the compact form (see below) of E7, and has an outer automorphism group of order 2 generated by complex conjugation. As well as the complex Lie group of type E7, 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/2Z and has trivial outer automorphism group. * The split form, EV (or E7(7)), which has maximal compact subgroup SU(8)/, fundamental group cyclic of order 4 and outer automorphism group of order 2. * EVI (or E7(-5)), which has maximal compact subgroup SU(2)·SO(12)/(center), fundamental group non-cyclic of order 4 and trivial outer automorphism group. * EVII (or E7(-25)), which has maximal compact subgroup SO(2)·E6/(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 In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian sym ...
. The compact real form of E7 is the
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
of the 64-dimensional exceptional compact
Riemannian symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
EVI (in Cartan's classification). It is known informally as the "" because it can be built using an algebra that is the tensor product of the quaternions and the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s, and is also known as a
Rosenfeld projective plane In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea indep ...
, though it does not obey the usual axioms of a projective plane. This can be seen systematically using a construction known as the '' magic square'', due to
Hans Freudenthal Hans Freudenthal (17 September 1905 – 13 October 1990) was a Jewish-German-born Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education ...
and
Jacques Tits Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life an ...
. The Tits–Koecher construction produces forms of the E7 Lie algebra from Albert algebras, 27-dimensional exceptional
Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan alg ...
s.


E7 as an algebraic group

By means of a
Chevalley basis In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers. Chevalley used these bases to construct analogues of Lie groups over finite f ...
for the Lie algebra, one can define E7 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 E7. 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 E7, which are classified in the general framework of
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natur ...
(over a
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k' ...
''k'') by the set ''H''1(''k'', Aut(E7)) which, because the Dynkin diagram of E7 (see below) has no automorphisms, coincides with ''H''1(''k'', E7, ad). Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E7 coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all adjoint forms of E7 have fundamental group Z/2Z in the sense of algebraic geometry, meaning that they admit exactly one double cover; the further non-compact real Lie group forms of E7 are therefore not algebraic and admit no faithful finite-dimensional representations. Over finite fields, the
Lang–Steinberg theorem In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if ''G'' is a connected smooth algebraic group over a finite field \mathbf_q, then, writing \sigma: G \to G, \, x \mapsto x^q for the Frobenius, the morphism of varieties :G ...
implies that ''H''1(''k'', E7) = 0, meaning that E7 has no twisted forms: see below.


Algebra


Dynkin diagram

The
Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
for E7 is given by .


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 \begin8\\4\end permutations of (½,½,½,½,−½,−½,−½,−½) 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 Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
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) :(½,½,½,½,−½,−½,−½,−½) They are listed so that their corresponding nodes in the
Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
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 as a subgroup of E8, is the following: All 4\times\begin6\\2\end 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 +½ :\left(\pm,\pm,\pm,\pm,\pm,\pm,\pm\right) and the two following roots :\left(0,0,0,0,0,0,\pm \sqrt\right). Thus the generators consist of a 66-dimensional so(12) subalgebra as well as 64 generators that transform as two self-conjugate
Weyl spinor In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
s of spin(12) of opposite chirality, and their chirality generator, and two other generators of chiralities \pm \sqrt. Given the E7
Cartan matrix In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Ki ...
(below) and a
Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
node ordering of: :one choice of
simple root Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
s is given by the rows of the following matrix: :\begin 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&-\frac&-\frac&-\frac&-\frac&-\frac&\frac\\ 0&0&0&0&1&-1&0 \\ \end.


Weyl group

The
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
of E7 is of order 2903040: it is the direct product of the cyclic group of order 2 and the unique
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
of order 1451520 (which can be described as PSp6(2) or PSΩ7(2)).


Cartan matrix

:\begin 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.


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 In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
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. The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the
Weyl character formula In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...
. The dimensions of the smallest irreducible representations are : :1, 56, 133, 912, 1463, 1539, 6480, 7371, 8645, 24320, 27664, 40755, 51072, 86184, 150822, 152152, 238602, 253935, 293930, 320112, 362880, 365750, 573440, 617253, 861840, 885248, 915705, 980343, 2273920, 2282280, 2785552, 3424256, 3635840... The underlined terms in the sequence above are the dimensions of those irreducible representations possessed by the adjoint form of E7 (equivalently, those whose weights belong to the root lattice of E7), whereas the full sequence gives the dimensions of the irreducible representations of the simply connected form of E7. There exist non-isomorphic irreducible representation of dimensions 1903725824, 16349520330, etc. The
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible representation, irreducible finite-dimensional representation of a semisimple Lie algebra, semisimple Lie group or Lie algebra whose highest weig ...
s are those with dimensions 133, 8645, 365750, 27664, 1539, 56 and 912 (corresponding to the seven nodes in the
Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
in the order chosen for the
Cartan matrix In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Ki ...
above, i.e., the nodes are read in the six-node chain first, with the last node being connected to the third).


E7 Polynomial Invariants

E7 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 In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
hermitian matrices. Then the first invariant is the symplectic invariant of Sp(56, R): :C_1 = pq - qp + Tr Q- Tr P/math> The second more complicated invariant is a symmetric quartic polynomial: :C_2 = (pq + Tr \circ Q^2 + p Tr \circ \tildeq Tr \circ \tildeTr tilde\circ \tilde Where \tilde \equiv \det(P) P^ and the binary circle operator is defined by A\circ B = (AB+BA)/2. An alternative quartic polynomial invariant constructed by Cartan uses two anti-symmetric 8x8 matrices each with 28 components. : C_2 = Tr XY)^2- \dfrac Tr Y2 +\frac\epsilon_\left( X^X^X^X^ + Y^Y^Y^Y^ \right)


Chevalley groups of type E7

The points over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
with ''q'' elements of the (split) algebraic group E7 (see above), whether of the adjoint (centerless) or simply connected form (its algebraic universal cover), give a finite
Chevalley group In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phra ...
. This is closely connected to the group written E7(''q''), however there is ambiguity in this notation, which can stand for several things: * the finite group consisting of the points over F''q'' of the simply connected form of E7 (for clarity, this can be written E7,sc(''q'') and is known as the “universal” Chevalley group of type E7 over F''q''), * (rarely) the finite group consisting of the points over F''q'' of the adjoint form of E7 (for clarity, this can be written E7,ad(''q''), and is known as the “adjoint” Chevalley group of type E7 over F''q''), 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 E7(''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: E7(''q'') is simple for any ''q'', E7,sc(''q'') is its
Schur cover In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \oper ...
, and the E7,ad(''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 E7(''q'') is 2 and E7(''q'') is of index 2 in E7,ad(''q''), which explains why E7,sc(''q'') and E7,ad(''q'') are often written as 2·E7(''q'') and E7(''q'')·2. From the algebraic group perspective, it is less common for E7(''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''q'' unlike E7,sc(''q'') and E7,ad(''q''). As mentioned above, E7(''q'') is simple for any ''q'', and it constitutes one of the infinite families addressed by the
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else i ...
. Its number of elements is given by the formula : :\fracq^(q^-1)(q^-1)(q^-1)(q^-1)(q^8-1)(q^6-1)(q^2-1) The order of E7,sc(''q'') or E7,ad(''q'') (both are equal) can be obtained by removing the dividing factor gcd(2, ''q''−1) . The Schur multiplier of E7(''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 E7,ad(''q'')) and the group of field automorphisms (i.e., cyclic of order ''f'' if ''q'' = ''pf'' where ''p'' is prime).


Importance in physics

''N'' = 8
supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
in four dimensions, which is a
dimensional reduction Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. In physics, a theory in ''D'' spacetime dimensions can be redefined in a lower number of dimensions ''d'', by taking all the fields ...
from 11 dimensional supergravity, admit an E7 bosonic global symmetry and an SU(8) bosonic
local symmetry In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be ''continuo ...
. 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 . In string theory, E7 appears as a part of the
gauge group In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
of one of the (unstable and non-
supersymmetric In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories ...
) versions of the heterotic string. It can also appear in the unbroken gauge group in six-dimensional compactifications of heterotic string theory, for instance on the four-dimensional surface K3.


See also

*
En (Lie algebra) In mathematics, especially in Lie theory, E''n'' is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and ''k'', with ''k'' = ''n'' − 4. In some older books and papers, ''E''2 and ''E''4 a ...
*
ADE classification In mathematics, the ADE classification (originally ''A-D-E'' classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, r ...
*
List of simple Lie groups In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian sym ...


Notes


References

* *
John Baez John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
, ''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 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. {{String theory topics , state=collapsed Algebraic groups Lie groups Exceptional Lie algebras