In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
,
E8 is the name given to an exceptional simple Lie group of
dimensionIn mathematics and physics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify each point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
248 (see below); the same notation is sometimes used for its root lattice,
which has rank 8.
The group E
8 was discovered between the years of 1888 and 1890 by
Wilhelm KillingWilhelm Karl Joseph Killing was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry....
, though he did not prove its existence, which was first shown by
Élie CartanÉlie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
.
The designation E
8 comes from Killing and Cartan's classification of the complex simple Lie algebras, which fall into four infinite families labeled A
n, B
n, C
n, D
n, and five exceptional cases labeled
E6In mathematics, E
6 is the name of some Lie groups and also their Lie algebras . It is one of the five exceptional compact simple Lie groups as well as one of the simply laced groups...
,
E7In mathematics, E
7 is the name of several Lie groups and also their Lie algebras . It is one of the five exceptional compact simple Lie groups as well as one of the simply laced groups. E
7 has rank 7 and dimension 133. The fundamental group of the compact form is the cyclic...
, E
8,
F4In mathematics, F
4 is the name of a Lie group and also its Lie algebra . It is one of the five exceptional simple Lie groups. F
4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group...
, and
G2In mathematics, G
2 is the name of three simple Lie groups and of their Lie algebras . They are the smallest of the five exceptional simple Lie groups. G
2 has rank 2 and dimension 14...
. The E
8 algebra is the largest and most complicated of these exceptional cases, and is often the last case of various theorems to be proved.
Basic description
E
8 has dimension 248 (as a complex
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....
). Its
rankIn mathematics, a Cartan subgroup of a Lie group or algebraic group G is one of the subgroups whose Lie algebrais a Cartan subalgebra. The dimension of a Cartan subgroup, and therefore of a Cartan subalgebra, is the rank of G.-Conventions:...
, which is the dimension of its maximal torus, is 8. Therefore the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article. The
Weyl groupIn 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. For example, the root system...
of E
8, which is the
group of symmetriesThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
of the maximal torus which are induced by
conjugationConjugation may refer to:*Grammatical conjugation, the modification of a verb from its basic form, including:**Latin conjugation**Spanish conjugation**French conjugation**English verb*Marriage, relationship between two or more individuals....
s in the whole group, has order 696729600.
E
8 is unique among simple Lie groups in that its non-
trivialIn mathematics, the adjective trivial is frequently used for objects that have a very simple structure...
representation of smallest dimension is the
adjoint representationIn mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebra...
(of dimension 248) acting on the Lie algebra E
8 itself; it is also the unique one which has the following three properties: trivial center, simply connected, and simply laced (all roots have the
same length).
There is a Lie algebra
EnIn mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a line of n-1 points with an extra point attached to the third point from the end.-Finite dimensional Lie algebras:...
for every integer
n ≥ 3, which is infinite dimensional if
n is greater than 8.
Real forms
The complex Lie group E
8 of
complex dimensionIn mathematics, complex dimension usually refers to the dimension of a complex manifold M, or complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d...
248 can be considered as a simple real Lie group of (real) dimension 496. This is simply connected, has maximal
compactIn mathematics, more specifically general topology and metric topology, a compact space is an abstract mathematical space in which, intuitively, whenever one takes an infinite number of "steps" in the space, eventually one must get arbitrarily close to some other point of the space...
subgroup the compact form (see below) of E
8, and has an outer
automorphismIn mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
group of order 2 generated by complex conjugation.
As well as the complex Lie group of type E
8, there are three real forms of the group, all of real dimension 248, as follows:
- A compact form (which is usually the one meant if no other information is given), which is simply connected and has trivial outer automorphism group.
- A split form, which has maximal compact subgroup Spin(16)/(Z/2Z), fundamental group of order 2, and a non-algebraic double cover and has trivial outer automorphism group.
- A third form, which has maximal compact subgroup E7×SU(2)/(−1×−1), fundamental group of order 2, and a non-algebraic double cover and has trivial outer automorphism group.
For a complete list of real forms of simple Lie algebras, see the
list of simple Lie groups.
Representation theory
The coefficients of the character formulas for infinite dimensional irreducible representations of E
8 depend on some large square matrices consisting of polynomials, the Lusztig–Vogan polynomials, an analogue of
Kazhdan–Lusztig polynomialIn representation theory, a Kazhdan–Lusztig polynomial Py,w is a member of a family of integral polynomials introduced in work of David Kazhdan and George Lusztig...
s introduced for
reductive groupIn mathematics, a reductive group is an algebraic group G such that the unipotent radical of the identity component of G is trivial. Any semisimple algebraic group and any algebraic torus is reductive, as is any general linear group....
s in general by
George LusztigGeorge Lusztig is a Romanian-born American mathematician. He is a Norbert Wiener Professor at the Department of Mathematics, MIT.Born in Timişoara, he did his undergraduate studies at the University of Bucharest...
and
David KazhdanDavid Kazhdan or Každan, Kajdan, formerly named Dmitri Aleksandrovich Kazhdan is an Israeli mathematician known for work in representation theory.-Life:...
(1983).
The values at 1 of the Lusztig-Vogan polynomials give the coefficients of the matrices relating the standard representations (whose characters are easy to describe) with the irreducible representations.
These matrices were computed after four years of collaboration by a
group of 18 mathematicians and computer scientistsThe Atlas of Lie Groups and Representations is a mathematical project to solve the problem of the unitary dual for real reductive Lie groups. The completion of E8in March 2007 received media attention....
, led by
Jeffrey AdamsJeffrey Adams is a mathematician at the University of Maryland who works on unitary representations of reductive Lie groups, and who led the project Atlas of Lie groups and representations that calculated the characters of the representations of E8. The project to calculate the...
, with much of the programming done by
Fokko du ClouxFokko du Cloux was a French mathematician and computer scientist. He worked on the Atlas of Lie groups and representations until his death. One of the founding members of the project, he was responsible for building the Atlas software which was instrumental in the mapping of the E8 Lie...
. The most difficult case (for exceptional groups) is the split real form of E
8 (see above), where the largest matrix is of size 453060×453060. The Lusztig-Vogan polynomials for all other exceptional simple groups have been known for some time; the calculation for the split form of
E8 is far longer than any other case. The announcement of the result in March 2007 by the
Atlas of Lie groups and representationsThe Atlas of Lie Groups and Representations is a mathematical project to solve the problem of the unitary dual for real reductive Lie groups. The completion of E8in March 2007 received media attention....
received extraordinary attention from the media (see the external links), to the surprise of the mathematicians working on it.
Constructions
One can construct the (compact form of the) E
8 group as the automorphism group of the corresponding
e8 Lie algebra . This algebra has a 120-dimensional subalgebra
so(16) generated by
Jij as well as 128 new generators
Qa that transform as a Weyl-Majorana spinor of
spin(16). These statements determine the commutators
as well as
while the remaining commutator (not anticommutator!) is defined as
It is then possible to check that the
Jacobi identityIn mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation...
is satisfied.
Geometry
The compact real form of E
8 is the
isometry groupIn mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...
of a 128-dimensional
Riemannian manifoldIn Riemannian geometry, a Riemannian manifold is a real differentiable manifold M in which each tangent space is equipped with an inner product g in a manner which varies smoothly from point to point. The metric g is a positive definite symmetric tensor: a metric tensor...
known informally as the 'octo-octonionic projective plane' because it can be built using an algebra that is the tensor product of the
octonionIn mathematics, the octonions are a nonassociative and noncommutative extension of the quaternions. Their 8-dimensional normed division algebra over the real numbers is the widest possible that can be obtained from the Cayley-Dickson construction...
s with themselves. This can be seen systematically using a construction known as the
magic squareIn mathematics, the Freudenthal magic square is a construction relating several Lie groups. It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie group to a pair of division algebras A, B...
, due to
Hans FreudenthalHans Freudenthal was a Dutch mathematician. He made substantial contributions to algebraic topology and also took an interest in literature, philosophy, history and mathematics education....
and
Jacques TitsJacques Tits is a Belgian and French mathematician. He has written and cowritten a large number of papers on a number of subjects, principally group theory....
(see J.M. Landsberg, L. Manivel, (2001)).
E8 root system
A
root systemIn 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...
of rank
r is a particular finite configuration of vectors, called
roots, which span an
r-dimensional
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions...
and satisfy certain geometrical properties. In particular, the root system must be invariant under
reflectionIn mathematics, a reflection is a map that transforms an object into its mirror image. For example, a reflection of the small English letter p in respect to a vertical line would look like q...
through the hyperplane perpendicular to any root.
The
E8 root system is a rank 8 root system containing 240 root vectors spanning
R8. It is irreducible in the sense that it cannot be built from root systems of smaller rank. All the root vectors in E
8 have the same length. It is convenient for many purposes to normalize them to have length √2.
Construction
In the so-called
even coordinate system E
8 is given as the set of all vectors in
R8 with length squared equal to 2 such that coordinates are either all
integerThe integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....
s or all
half-integerIn mathematics, a half-integer is a number of the form,where is an integer. For example,are all half-integers. Note that a half of an integer is not always a half-integer: half of an even integer is an integer but not a half-integer...
s and the sum of the coordinates is even.
Explicitly, there are 112 roots with integer entries obtained from
by taking an arbitrary combination of signs and an arbitrary
permutationIn several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the elements of a set to other elements of the same set, i.e., exchanging elements of a set.- Definitions :The general concept of permutation can be...
of coordinates, and 128 roots with half-integer entries obtained from
by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even). There are 240 roots in all.
The 112 roots with integer entries form a D
8 root system. The E
8 root system also contains a copy of A
8 (which has 72 roots) as well as
E6In mathematics, E
6 is the name of some Lie groups and also their Lie algebras . It is one of the five exceptional compact simple Lie groups as well as one of the simply laced groups...
and
E7In mathematics, E
7 is the name of several Lie groups and also their Lie algebras . It is one of the five exceptional compact simple Lie groups as well as one of the simply laced groups. E
7 has rank 7 and dimension 133. The fundamental group of the compact form is the cyclic...
(in fact, the latter two are usually
defined as subsets of E
8).
In the
odd coordinate system E
8 is given by taking the roots in the even coordinate system and changing the sign of any one coordinate. The roots with integer entries are the same while those with half-integer entries have an odd number of minus signs rather than an even number.
Simple roots
A set of simple roots for a root system Φ is a set of roots that form a
basisIn linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others...
for the Euclidean space spanned by Φ with the special property that each root has components with respect to this basis that are either all nonnegative or all nonpositive.
One choice of simple roots for E
8 (by no means unique) is given by the rows of the following matrix:
Dynkin diagram
The Dynkin diagram for E
8 is given by
This diagram gives a concise visual summary of the root structure. Each node of this diagram represents a simple root. A line joining two simple roots indicates that they are at an angle of 120° to each other. Two simple roots which are not joined by a line are orthogonal.
Cartan matrix
The
Cartan matrixIn mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. In fact, Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan.- Lie algebras :A generalized...
of a rank
r root system is an
r ×
r matrixIn mathematics, a matrix is a rectangular array of numbers, such asEntries of a matrix are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied...
whose entries are derived from the simple roots. Specifically, the entries of the Cartan matrix are given by
where (-,-) is the Euclidean inner product and
αi are the simple roots. The entries are independent of the choice of simple roots (up to ordering).
The Cartan matrix for E
8 is given by
The
determinantIn algebra, the determinant is a special number associated to any square matrix, that is to say, a rectangular array of numbers where the number of rows and columns are equal. The fundamental geometric meaning of a determinant is a scale factor for measure when the matrix is regarded as a linear...
of this matrix is equal to 1.
E8 root lattice
The integral span of the E
8 root system forms a
latticeIn mathematics, especially in geometry and group theory, a lattice in R
n is a discrete subgroup of R
n which spans the real vector space R
n. Every lattice in R
n can be generated from a basis for the vector space by forming all linear combinations with...
in
R8 naturally called the
E8 root latticeIn mathematics, the E
8 lattice is a special lattice in R
8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8...
. This lattice is rather remarkable in that it is the only (nontrivial) even,
unimodular latticeIn mathematics, a unimodular lattice is a lattice of discriminant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
with rank less than 16.
Simple subalgebras of E8
The Lie algebra E8 contains as subalgebras all the exceptional Lie algebras as well as many other important Lie algebras in mathematics and physics. The height of the Lie algebra on the diagram approximately corresponds to the rank of the algebra. A line from an algebra down to a lower algebra indicates that the lower algebra is a subalgebra of the higher algebra. Some algebras are more obvious such as SU(n) is a subalgebra of O(2n) and some are less obvious especially the exceptional algebras G2, F4, E6 & E7. The orthogonal and
unitaryUnitary may refer to:* In automotive design, unitary construction is another common term for a unibody or monocoque construction* In Christian doctrine, unitarianism is the belief in a "unitary God" as opposed to the concept of the Trinity....
subalgebras are particularly important in physics as they are used to represent space-time and bosonic symmetries respectively. Some of the smaller algebras are equivalent e.g. O(3)~SU(2).
Subgroups
The smaller exceptional groups
E7In mathematics, E
7 is the name of several Lie groups and also their Lie algebras . It is one of the five exceptional compact simple Lie groups as well as one of the simply laced groups. E
7 has rank 7 and dimension 133. The fundamental group of the compact form is the cyclic...
and
E6In mathematics, E
6 is the name of some Lie groups and also their Lie algebras . It is one of the five exceptional compact simple Lie groups as well as one of the simply laced groups...
sit inside E
8. In the compact group, both (E
7×SU(2)) / (
Z/2
Z) and (E
6×SU(3)) / (
Z/3
Z) are
maximal subgroupIn mathematics, the term maximal subgroup is used to mean slightly different things in different areas of algebra.In group theory, a maximal subgroup H of a group G is a proper subgroup, such that no proper subgroup K contains H strictly. In other words H is a maximal element of the partially...
s of
E
8.
The 248-dimensional adjoint representation of E
8 may be considered in terms of its
restricted representationIn mathematics, restriction is a fundamental construction in representation theory of groups. Restriction forms a representation of a subgroup from a representation of the whole group. Often the restricted representation is simpler to understand...
to the first of these subgroups. It transforms under SU(2)×E
7 as a sum of tensor product representations, which may be labelled as a pair of dimensions as
(Since there is a quotient in the product, these notations may strictly be taken as indicating the infinitesimal (Lie algebra) representations.)
Since the adjoint representation can be described by the roots together with the generators in the
Cartan subalgebraIn 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....
, we may see that decomposition by looking at these. In this description:
- The (3,1) consists of the roots (0,0,0,0,0,0,1,−1), (0,0,0,0,0,0,−1,1) and the Cartan generator corresponding to the last dimension.
- The (1,133) consists of all roots with (1,1), (−1,−1), (0,0), (−1/2,−1/2) or (1/2,1/2) in the last two dimensions, together with the Cartan generators corresponding to the first 7 dimensions.
- The (2,56) consists of all roots with permutations of (1,0), (−1,0) or (1/2,−1/2) in the last two dimensions.
The 248-dimensional adjoint representation of E
8, when similarly restricted, transforms under SU(3)×E
6 as:
We may again see the decomposition by looking at the roots together with the generators in the
Cartan subalgebraIn 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....
. In this description:
- The (8,1) consists of the roots with permutations of (1,−1,0) in the last three dimensions, together with the Cartan generator corresponding to the last two dimensions.
- The (1,78) consists of all roots with (0,0,0), (−1/2,−1/2,−1/2) or (1/2,1/2,1/2) in the last three dimensions, together with the Cartan generators corresponding to the first 6 dimensions.
- The (3,27) consists of all roots with permutations of (1,0,0), (1,1,0) or (−1/2,1/2,1/2) in the last three dimensions.
- The (3,27) consists of all roots with permutations of (−1,0,0), (−1,−1,0) or (1/2,−1/2,−1/2) in the last three dimensions.
The finite quasisimple groups that can embed in (the compact form of) E
8 were found by
Applications
The E
8 Lie group has applications in
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics in an attempt to explain natural phenomena. Its central core is mathematical physics,[Sometimes mathematical physics and theoretical physics are used synonymously to refer to the...]
, in particular in
string theoryString theory is a developing branch of theoretical physics that combines quantum mechanics and general relativity into a quantum theory of gravity...
and
supergravityIn theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity...
. The group E
8×E
8 (the
Cartesian productIn mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept....
of two copies of E
8) serves as the gauge group of one of the two types of
heterotic stringIn physics, a heterotic string is a peculiar mixture of the bosonic string and the superstring...
and is one of two
anomaly-freeIn quantum physics an anomaly or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical physics an anomaly is the failure of a symmetry to be restored in the limit in which the symmetry-breaking...
gauge groups that can be coupled to the
N = 1
supergravityIn theoretical physics, supergravity is a field theory that combines the principles of supersymmetry and general relativity...
in 10 dimensions.
E
8 is the
U-dualityU-duality is a symmetry of string theory or M-theory combining S-duality and T-duality transformations. The term is most often met in the context of the "U-duality group" of M-theory as defined on a particular background space . This is the union of all the S- and T-dualities available in that...
group of supergravity on an eight-torus (in its split form).
One way to incorporate the
standard modelThe Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions. These particles make up all visible matter in the universe...
of particle physics into heterotic string theory is the
symmetry breakingIn physics, spontaneous symmetry breaking occurs when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. When that happens, the system no longer appears to behave in a symmetric manner...
of E
8 to its maximal subalgebra SU(3)×E
6.
In 1982,
Michael FreedmanMichael Hartley Freedman is a mathematician at Microsoft Station Q. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born into a Jewish family in Los Angeles...
used the
E8 latticeIn mathematics, the E
8 lattice is a special lattice in R
8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8...
to construct an example of a
topologicalIn mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
4-manifoldIn mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...
, the
E8 manifoldIn mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice.The E8 manifold was discovered by Michael Freedman in 1982...
, which has no
smooth structureIn mathematics, an n-dimensional differential structure on a set M makes it into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows us to do differential calculus on the manifold...
.
External links
Links related to the calculation of the Lusztig-Vogan polynomials.
Other external links: