In
mathematics, E
7 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 e
7, all of which have dimension 133; the same notation E
7 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 E
7 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, E
7,
E8,
F4, and
G2. The E
7 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
7 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 E
7, corresponding to a complex group of complex dimension 133. The complex adjoint Lie group E
7 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 E
7, and has an outer automorphism group of order 2 generated by complex conjugation.
As well as the complex Lie group of type E
7, 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 E
7(7)), which has maximal compact subgroup SU(8)/, fundamental group cyclic of order 4 and outer automorphism group of order 2.
* EVI (or E
7(-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 E
7(-25)), which has maximal compact subgroup SO(2)·E
6/(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 E
7 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 E
7 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 E
7 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
7. 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
7, 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(E
7)) which, because the Dynkin diagram of E
7 (see
below) has no automorphisms, coincides with ''H''
1(''k'', E
7, ad).
Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E
7 coincide with the three real Lie groups mentioned
above, but with a subtlety concerning the fundamental group: all adjoint forms of E
7 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 E
7 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'', E
7) = 0, meaning that E
7 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 E
7 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
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 E
8, is the following:
All
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
:
and the two following roots
:
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
.
Given the E
7 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:
:
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 E
7 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 PSp
6(2) or PSΩ
7(2)).
Cartan matrix
:
Important subalgebras and representations
E
7 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 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.
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 E
7 (equivalently, those whose weights belong to the root lattice of E
7), whereas the full sequence gives the dimensions of the irreducible representations of the simply connected form of E
7. 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
E
7 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):
: