In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, E
7 is the name of several closely related
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
s, linear
algebraic groups or their
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
s e
7, all of which have dimension 133; the same notation E
7 is used for the corresponding
root lattice, 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 algebras, 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 ...
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 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 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 Britis ...
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.
The compact real form of E
7 is the
isometry group of the 64-dimensional exceptional compact
Riemannian symmetric space EVI (in Cartan's
classification Classification is a process related to categorization, the process in which ideas and objects are recognized, differentiated and understood.
Classification is the grouping of related facts into classes.
It may also refer to:
Business, organizat ...
). It is known informally as the "" because it can be built using an algebra that is the tensor product of the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
s 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, 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
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...
'', due to
Hans Freudenthal and
Jacques Tits.
The
Tits–Koecher construction produces forms of the E
7 Lie algebra from
Albert algebra In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there ...
s, 27-dimensional exceptional
Jordan algebras.
E7 as an algebraic group
By means of a
Chevalley basis 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 (over a
perfect field ''k'') by the set ''H''
1(''k'', Aut(E
7)) which, because the Dynkin diagram of E
7 (see
below
Below may refer to:
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* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred 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 implies that ''H''
1(''k'', E
7) = 0, meaning that E
7 has no twisted forms: see
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred Below ...
.
Algebra
Dynkin diagram
The
Dynkin diagram 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 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)
:(½,½,½,½,−½,−½,−½,−½)
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 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 spinors of spin(12) of opposite chirality, and their chirality generator, and two other generators of chiralities
.
Given the E
7 Cartan matrix (below) and a
Dynkin diagram node ordering of:
:one choice of
simple roots 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 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 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. 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 representations are those with dimensions 133, 8645, 365750, 27664, 1539, 56 and 912 (corresponding to the seven nodes in the
Dynkin diagram in the order chosen for the
Cartan matrix 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):
: