E8 (mathematics)

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Graph of E8 Gosset polytope, 42,1Coxeter-Dynkin diagram:
Graph of E8 Gosset polytope, 42,1

Coxeter-Dynkin diagram:Image:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD downbranch-00.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD dot.pngImage:CD 3b.pngImage:CD ring.png

The correct title of this article is E8 (mathematics). It features superscript or subscript characters that are substituted or omitted because of technical limitations.

In mathematics, E8 is the name given to a family of closely related structures. In particular, it is the name of some exceptional simple Lie algebras as well as that of the associated simple Lie groups. It is also the name given to the corresponding root system, root lattice, and Weyl/Coxeter group, and to some finite simple Chevalley groups. E8 was formulated between the years of 1888 and 1890 by Wilhelm Killing.

The designation E8 comes from Wilhelm Killing and Élie Cartan's classification of the complex simple Lie algebras, which fall into four infinite families labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E8 algebra is the largest and most complicated of these exceptional cases, and is often the last case of various theorems to be proved.


Groups
Group theory
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[edit] Basic description

E8 has rank 8 and dimension 248 (as a manifold). The vectors of the root system are in eight dimensions and are specified later in this article. The Weyl group of E8, which acts as a symmetry group of the maximal torus by means of the conjugation operation from the whole group, is of order 696729600.

E8 is unique among simple 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.

There is a Lie algebra En for every integer n≥3, which is infinite dimensional if n is greater than 8.

[edit] Real forms

The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of (real) dimension 496, which is simply connected, has maximal compact subgroup the compact form 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 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.

[edit] Representation theory

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 Vogan (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.

[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 Jij as well as 128 new generators Qa 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 (see J.M. Landsberg, L. Manivel, (2001)).

[edit] E8 root system

Zome Model of the 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. Each of the root vectors in E8 have equal length. It is convenient for many purposes to normalize them to have length √2.

[edit] Construction

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

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 (by no means unique) 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 ].

[edit] Dynkin diagram

The Dynkin diagram for E8 is given by

Dynkin diagram of E8

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] 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 αi 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

Simple subalgebra tree of E8
Simple subalgebra 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 and E8. The orthogonal and unitary 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] Significant maximal 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.

[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.

[edit] See also

[edit] References

[edit] External links

Links related to the calculation of the Lusztig-Vogan polynomials in 2007 with mathematical content:

Links related to the calculation of the Lusztig-Vogan polynomials in 2007 without mathematical content:

Other external links:

Exceptional Lie groups
E6 | E7 | E8 | F4 | G2
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