E7 (mathematics)

Last updated

In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank  7. The designation E7 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E7 algebra is thus one of the five exceptional cases.

Contents

The fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z, 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 E7, corresponding to a complex group of complex dimension 133. The complex adjoint Lie group E7 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 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:

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

The compact real form of E7 is the isometry group of the 64-dimensional exceptional compact Riemannian symmetric space EVI (in Cartan's classification). It is known informally as the "quateroctonionic projective plane" because it can be built using an algebra that is the tensor product of the quaternions and the octonions, 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 , due to Hans Freudenthal and Jacques Tits.

The Tits–Koecher construction produces forms of the E7 Lie algebra from Albert algebras, 27-dimensional exceptional Jordan algebras.

E7 as an algebraic group

By means of a Chevalley basis 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 “7, which are classified in the general framework of Galois cohomology (over a perfect field k) by the set H1(k, Aut(E7)) which, because the Dynkin diagram of E7 (see below) has no automorphisms, coincides with H1(k, E7, ad). [1]

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 implies that H1(k, E7) = 0, meaning that E7 has no twisted forms: see below.

Algebra

Dynkin diagram

The Dynkin diagram for E7 is given by Dynkin diagram type E7.svg .

Root system

The 126 vertices of the 231 polytope represent the root vectors of E7, as shown in this Coxeter plane projection
Coxeter-Dynkin diagram: Gosset 2 31 polytope.svg
The 126 vertices of the 231 polytope represent the root vectors of E7, as shown in this Coxeter plane projection
Coxeter–Dynkin diagram: CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry:
u = (1, ph, 0, -1, ph, 0,0)
v = (ph, 0, 1, ph, 0, -1,0)
w = (0, 1, ph, 0, -1, ph,0)
The projected 231 polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls of each set of tallied norms. These show:
1) 2 points at the origin
2) 2 icosahedrons
3) 1 icosadodecahedron
4) 2 dodecahedrons
5) 1 icosadodecahedron
for a total of 126 vertices. 3 21 E7 to 3D H3 symmetry concentric hulls.png
Shown in 3D projection using the basis vectors [u,v,w] giving H3 symmetry:
u = (1, φ, 0, -1, φ, 0,0)
v = (φ, 0, 1, φ, 0, -1,0)
w = (0, 1, φ, 0, -1, φ,0)
The projected 231 polytope vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls of each set of tallied norms. These show:
1) 2 points at the origin
2) 2 icosahedrons
3) 1 icosadodecahedron
4) 2 dodecahedrons
5) 1 icosadodecahedron
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 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 E7 × SU(2) as a subgroup of E8, 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 E7 Cartan matrix (below) and a Dynkin diagram node ordering of: DynkinE7.svg

one choice of simple roots is given by the rows of the following matrix:

Weyl group

The Weyl group of E7 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 PSp6(2) or PSΩ7(2)). [2]

Cartan matrix

Hasse diagram of E7 root poset with edge labels identifying added simple root position E7HassePoset.svg
Hasse diagram of E7 root poset with edge labels identifying added simple root position

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 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. The dimensions of the smallest irreducible representations are (sequence A121736 in the OEIS ):

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 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

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 hermitian matrices. Then the first invariant is the symplectic invariant of Sp(56, R):

The second more complicated invariant is a symmetric quartic polynomial:

Where and the binary circle operator is defined by .

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

Chevalley groups of type E7

The points over a finite field 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. This is closely connected to the group written E7(q), however there is ambiguity in this notation, which can stand for several things:

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, 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 Fq unlike E7,sc(q) and E7,ad(q).

As mentioned above, E7(q) is simple for any q, [3] [4] 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 (sequence A008870 in the OEIS ):

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) (sequence A008869 in the OEIS ). 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 four dimensions, which is a dimensional reduction from 11 dimensional supergravity, admit an E7 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 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 E7 / SU(8).

In string theory, E7 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 E8 × E7 in six-dimensional compactifications of heterotic string theory, for instance on the four-dimensional surface K3.

See also

Notes

  1. Platonov, Vladimir; Rapinchuk, Andrei (1994) [1991], Algebraic groups and number theory, Pure and Applied Mathematics, 139, Boston, MA: Academic Press, ISBN   978-0-12-558180-6, MR   1278263 (original version: Платонов, Владимир П.; Рапинчук, Андрей С. (1991). Алгебраические группы и теория чисел. Наука. ISBN   5-02-014191-7.), §2.2.4
  2. Conway, John Horton; Curtis, Robert Turner; Norton, Simon Phillips; Parker, Richard A; Wilson, Robert Arnott (1985). Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford University Press. p. 46. ISBN   0-19-853199-0.
  3. Carter, Roger W. (1989). Simple Groups of Lie Type. Wiley Classics Library. John Wiley & Sons. ISBN   0-471-50683-4.
  4. Wilson, Robert A. (2009). The Finite Simple Groups. Graduate Texts in Mathematics. 251. Springer-Verlag. ISBN   1-84800-987-9.

Related Research Articles

Dynkin diagram Pictoral representation of symmetry

In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled. The multiple edges are, within certain constraints, directed.

Simple Lie group connected non-abelian Lie group lacking nontrivial connected normal subgroups

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 symmetric spaces.

G<sub>2</sub> (mathematics)

In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.

F<sub>4</sub> (mathematics)

In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimensional.

E<sub>6</sub> (mathematics) 78-dimensional exceptional simple Lie group

In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras , all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6. The designation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras (see Élie Cartan § Work). This classifies Lie algebras into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E6 algebra is thus one of the five exceptional cases.

Weyl group Subgroup of a root systems isometry group

In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.

E<sub>8</sub> (mathematics) 248-dimensional exceptional simple Lie group

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases.

In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defining module of a classical Lie group is a fundamental representation. Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed from the fundamental representations by a procedure due to Élie Cartan. Thus in a certain sense, the fundamental representations are the elementary building blocks for arbitrary finite-dimensional representations.

Triality

In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin(8), the double cover of 8-dimensional rotation group SO(8), arising because the group has an outer automorphism of order three. There is a geometrical version of triality, analogous to duality in projective geometry.

Killing form

In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.

Reductive group

In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group GL(n) of invertible matrices, the special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraic groups and semisimple algebraic groups are reductive.

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 Killing form is due to Cartan.

Cartan subalgebra Nilpotent subalgebra of a Lie algebra


In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising. They were introduced by Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra over a field of characteristic .

Semisimple Lie algebra Direct sum of simple Lie algebras

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras.

In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. It is a Kac–Moody algebra for which the generalized Cartan matrix is positive semi-definite and has corank 1. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities.

Symmetric space A (pseudo-)Riemannian manifold whose geodesics are reversible.

In mathematics, a symmetric space is 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, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.

Coxeter–Dynkin diagram Pictoral representation of symmetry

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction: each graph "node" represents a mirror and the label attached to a branch encodes the dihedral angle order between two mirrors, that is, the amount by which the angle between the reflective planes can be multiplied by to get 180 degrees. An unlabeled branch implicitly represents order-3.

Complexification (Lie group) universal construction of a complex Lie group from a real Lie group

In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original group into another complex Lie group extends compatibly to a complex analytic homomorphism between the complex Lie groups. The complexification, which always exists, is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic if the original group has a quotient by a discrete normal subgroup which is linear.

This is a glossary of representation theory in mathematics.

Glossary of Lie groups and Lie algebras Wikipedia glossary

This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.

References