E₈ (mathematics)

Group theory

[show]Basic notions
Subgroup Normal subgroup Quotient group Group homomorphism (semi-)direct product

[show]Finite groups and classification of finite simple groups
Cyclic group Z_n Symmetric group, S_n Dihedral group, D_n Alternating group A_n Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄ Conway groups Co₁, Co₂, Co₃ Janko groups J₁, J₂, J₃, J₄ Fischer groups F₂₂, F₂₃, F₂₄ Baby Monster group B Monster group M

[show]Discrete groups and lattices
The integers, Z Lattice (group) Modular groups, PSL(2,Z) and SL(2,Z)

[show]Topological groups and Lie groups
Solenoid (mathematics) Circle group General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n) G₂ F₄ E₆ E₇ E₈ Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group Infinite-dimensional Lie groups O(∞) SU(∞) Sp(∞)

Lie groups

[show]Classical groups
General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n)

[show]Simple Lie groups
List of simple Lie groups Infinite simple Lie groups: A_n, B_n, C_n, D_n, Exceptional simple Lie groups: G₂ F₄ E₆ E₇ E₈

[show]Other Lie groups
Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group

[show]Lie algebras
Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form Lie point symmetry

[show]Structure of semi-simple Lie groups
Dynkin diagrams Cartan subalgebra Root system Real form Complexification Split Lie algebra Compact Lie algebra

[show]Representation theory
Representation of a Lie group Representation of a Lie algebra

[show]Lie groups in Physics
Particle physics and representation theory Representation theory of the Lorentz group Representation theory of the Poincaré group Representation theory of the Galilean group

In mathematics, E₈ is any of several closely related exceptional simple Lie groups, linear algebraic groups and 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 A_n, B_n, C_n, D_n, and five exceptional cases labeled E₆, E₇, E₈, F₄, and G₂. The E₈ algebra is the largest and most complicated of these exceptional cases.
Wilhelm Killing (1888, 1888, 1889, 1890) discovered the complex Lie algebra E₈ during his classification of simple compact Lie algebras, though he did not prove its existence, which was first shown by Élie Cartan. Cartan determined that a complex simple Lie algebra of type E₈ admits three real forms. Each of them gives rise to a simply-connected simple Lie group of dimension 248, exactly one of which is compact. Algebraic groups and Lie algebras of type E₈ over other fields have also been considered: for example, in the case of finite fields they lead to an infinite family of finite simple groups of Lie type.

[edit] Basic description

The compact Lie group E₈ has dimension 248. Its rank, 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 group of E₈, which is the group of symmetries of the maximal torus which are induced by conjugations in the whole group, has order 696729600.
The compact group E₈ is unique among simple compact Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra E₈ itself; it is also the unique one which has the following four properties: trivial center, compact, simply connected, and simply laced (all roots have the same length).
There is a Lie algebra E_n for every integer n ≥ 3, which is infinite dimensional if n is greater than 8.

[edit] Real and complex forms

There is a unique complex Lie algebra of type E₈, corresponding to a complex group of complex dimension 248. The complex Lie group E₈ of complex dimension 248 can be considered as a simple real Lie group of real dimension 496. This is simply connected, has maximal compact subgroup the compact form (see below) of E₈, and has an outer automorphism group of order 2 generated by complex conjugation.
As well as the complex Lie group of type E₈, there are three real forms of the Lie algebra, three real forms of the group with trivial center (two of which have non-algebraic double covers, giving two further real forms), 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.
There is also at least one algebraic group and Lie algebra of type E₈ over any field (or even commutative ring), called the split form. Over algebraically closed fields this is unique, but over other fields there are often many other forms of E₈.

[edit] Representation theory

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 A121732 in OEIS):

1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860, 2275896000, 2642777280, 2903770000, 3929713760, 4076399250, 4825673125, 6899079264, 8634368000, 8634368000, 12692520960…

The 248-dimensional representation is the adjoint representation. There are two non-isomorphic irreductible representations of dimension 8634368000. The fundamental representations are those with dimensions 3875, 6696000, 6899079264, 146325270, 2450240, 30380, 248 and 147250 (corresponding to the eight nodes in the Dynkin diagram in the order chosen for the Cartan matrix below, i.e., the nodes are read in the seven-node chain first, with the last node being connected to the third).
The coefficients of the character formulas for infinite dimensional irreducible representations of E₈ depend on some large square matrices consisting of polynomials, the Lusztig–Vogan polynomials, an analogue of Kazhdan–Lusztig polynomials introduced for reductive groups in general by George Lusztig and David Kazhdan (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 scientists, led by Jeffrey Adams, with much of the programming done by Fokko du Cloux. The most difficult case (for exceptional groups) is the split real form of E₈ (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 received extraordinary attention from the media (see the external links), to the surprise of the mathematicians working on it.
The representations of the E₈ groups over finite fields are given by Deligne–Lusztig theory.

[edit] Constructions

One can construct the (compact form of the) E₈ group as the automorphism group of the corresponding e₈ Lie algebra. This algebra has a 120-dimensional subalgebra so(16) generated by J_ij as well as 128 new generators Q_a that transform as a Weyl–Majorana spinor of spin(16). These statements determine the commutators

$[J_{ij},J_{k\ell}]=\delta_{jk}J_{i\ell}-\delta_{j\ell}J_{ik}-\delta_{ik}J_{j\ell}+\delta_{i\ell}J_{jk}$

as well as

$[J_{ij},Q_a] = \frac 14 (\gamma_i\gamma_j-\gamma_j\gamma_i)_{ab} Q_b,$

while the remaining commutator (not anticommutator!) is defined as

$[Q_a,Q_b]=\gamma^{[i}_{ac}\gamma^{j]}_{cb} J_{ij}.$

It is then possible to check that the Jacobi identity is satisfied.

[edit] Geometry

The compact real form of E₈ is the isometry group of a 128-dimensional Riemannian manifold known informally as the 'octo-octonionic projective plane' because it can be built using an algebra that is the tensor product of the octonions with themselves. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits (Landsberg & Manivel 2001).

[edit] E₈ root system

Zome model of the E₈ root system.

A root system of rank r is a particular finite configuration of vectors, called roots, which span an r-dimensional Euclidean space and satisfy certain geometrical properties. In particular, the root system must be invariant under reflection 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 E₈ have the same length. It is convenient for many purposes to normalize them to have length √2.

[edit] Construction

Graph of E8 projected into the Coxeter plane

In the so-called even coordinate system E₈ is given as the set of all vectors in R⁸ with length squared equal to 2 such that coordinates are either all integers or all half-integers and the sum of the coordinates is even.
Explicitly, there are 112 roots with integer entries obtained from

$(\pm 1,\pm 1,0,0,0,0,0,0)\,$

by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots with half-integer entries obtained from

$\left(\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12,\pm\tfrac12\right) \,$

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 D₈ root system. The E₈ root system also contains a copy of A₈ (which has 72 roots) as well as E₆ and E₇ (in fact, the latter two are usually defined as subsets of E₈).
In the odd coordinate system E₈ 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.

[edit] Simple roots

Graph of E8 Hasse diagram

A set of simple roots for a root system Φ is a set of roots that form a basis 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 E₈ is given by the rows of the following matrix:

$\left [\begin{smallmatrix} \frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}\\ -1&1&0&0&0&0&0&0 \\ 0&-1&1&0&0&0&0&0 \\ 0&0&-1&1&0&0&0&0 \\ 0&0&0&-1&1&0&0&0 \\ 0&0&0&0&-1&1&0&0 \\ 0&0&0&0&0&-1&1&0 \\ 1&1&0&0&0&0&0&0 \\ \end{smallmatrix}\right ].$

The set of simple roots is by no means unique (the number of possible choices of positive roots is the order of the Weyl group); however, the particular choice displayed above has the unique property that the positive roots are then exactly those whose rightmost nonzero coordinate is positive.

[edit] Dynkin diagram

The Dynkin diagram for E₈ 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.

[edit] Weyl group

The Weyl group of E₈ is of order 696729600, and can be described as O₈⁺(2): it is of the form 2.G.2 (that is, a stem extension by the cyclic group of order 2 of an extension of the cyclic group of order 2 by a group G) where G is the unique simple group of order 174182400 (which can be described as PSΩ₈⁺(2)).^[1]

[edit] Cartan matrix

The Cartan matrix of a rank r root system is an r × r matrix whose entries are derived from the simple roots. Specifically, the entries of the Cartan matrix are given by

$A_{ij} = 2\frac{(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}$

where (−,−) is the Euclidean inner product and α_i are the simple roots. The entries are independent of the choice of simple roots (up to ordering).
The Cartan matrix for E₈ is given by

$\left [ \begin{smallmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2 \end{smallmatrix}\right ].$

The determinant of this matrix is equal to 1.

[edit] E₈ root lattice

Main article: E₈ lattice

The integral span of the E₈ root system forms a lattice in R⁸ naturally called the E₈ root lattice. This lattice is rather remarkable in that it is the only (nontrivial) even, unimodular lattice with rank less than 16.

[edit] Simple subalgebras of E₈

An incomplete simple subgroup tree 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 unitary^{[disambiguation needed]} 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).

[edit] Chevalley groups of type E₈

As an algebraic group, E₈ can be defined over the integers (by means of a Chevalley basis for the Lie algebra) hence, in particular, over any commutative ring. Its points over a finite field with q elements form a finite Chevalley group, generally written E₈(q), which is simple for any q.^[2]^[3] Its number of elements is given by the formula (sequence A008868 in OEIS):

q 120 (q 30 - 1)(q 24 - 1)(q 20 - 1)(q 18 - 1)(q 14 - 1)(q 12 - 1)(q 8 - 1)(q 2 - 1)

The first term in this sequence, the order of E₈(2), namely 337804753143634806261388190614085595079991692242467651576160959909068800000 ≈ 3.38×10⁷⁴, is already larger than the size of the Monster group; Coxeter and Moser^[4] remark that it is comparable (actually about 5×10⁵ times smaller) to Eddington's historical estimation of the number of protons in the Universe. This group E₈(2) is the last one described (but without its character table) in the ATLAS of Finite Groups^[5].

[edit] Subgroups

The smaller exceptional groups E₇ and E₆ sit inside E₈. In the compact group, both (E₇×SU(2)) / (Z/2Z) and (E₆×SU(3)) / (Z/3Z) are maximal subgroups of E₈.
The 248-dimensional adjoint representation of E₈ may be considered in terms of its restricted representation to the first of these subgroups. It transforms under SU(2)×E₇ as a sum of tensor product representations, which may be labelled as a pair of dimensions as

$(3,1) + (1,133) + (2,56). \,\!$

(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 subalgebra, 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 E₈, when similarly restricted, transforms under SU(3)×E₆ as:

$(8,1) + (1,78) + (3,27) + (\overline{3},\overline{27}).$

We may again see the decomposition by looking at the roots together with the generators in the Cartan subalgebra. 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) E₈ were found by Griess & Ryba (1999)

[edit] Invariant polynomial

E8 is the automorphism group of an octic polynomial invariant, thought to be the lowest order symmetric polynomial invariant of E8.^[6]

[edit] Applications

The E₈ Lie group has applications in theoretical physics, in particular in string theory and supergravity. The group E₈×E₈ (the Cartesian product of two copies of E₈) serves as the gauge group of one of the two types of heterotic string and is one of two anomaly-free gauge groups that can be coupled to the N = 1 supergravity in 10 dimensions. E₈ is the U-duality group of supergravity on an eight-torus (in its split form).
One way to incorporate the standard model of particle physics into heterotic string theory is the symmetry breaking of E₈ to its maximal subalgebra SU(3)×E₆.
In 1982, Michael Freedman used the E₈ lattice to construct an example of a topological 4-manifold, the E₈ manifold, which has no smooth structure.
R. Coldea, D. A. Tennant, and E. M. Wheeler et al. (2010) reported that in an experiment with a cobalt-niobium crystal, under certain physical conditions the electron spins in it exhibited two of the 8 peaks related to E₈ predicted by Zamolodchikov (1989) .

In your mind

Senin, 08 November 2010