E8 (mathematics)

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In mathematics, E8 is the name given to a family of closely related structures. In particular, it is the name of an exceptional simple Lie algebra 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.

The designation E8 comes from Élie Cartan's classification of the 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.[1] The E8 algebra is the largest and most complicated of these exceptional cases.

While E8 and the other exceptional Lie groups were discovered around 1890, there is no definitive explanation of their role in mathematics. They have been taken as enigmatic possibilities. Up until recent times the case of E8 had generally been considered the most awkward, for example in completing the proof of the Weil conjecture on Tamagawa numbers. Mathematician Michael Freedman showed in 1982 that the E8 manifold was one of the strangest examples of a 4-manifold. Mathematical physicist John C. Baez wrote in 1996 how he had been puzzled for years about why such an 'exceptional' object existed, as a possibility for symmetry; and how Bertram Kostant had answered with another special phenomenon, triality.[2]

The E8 Lie group has important applications in theoretical physics. In particular, the group E8×E8 (the Cartesian product of two copies of E8) serves as the gauge group of one type of heterotic string. The group is one of two anomaly-free gauge groups that can be coupled to the N = 1 supergravity in 10 dimensions.

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[edit] Basic description

We say that E8 has rank 8 (the maximum number of mutually commutative degrees of freedom), and dimension 248 (as a manifold). This means that a maximal torus of the compact Lie group E8 has dimension 8. 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.

The compact Lie group E8 is simply connected and its center is the trivial subgroup. Its outer automorphism group is the trivial group, meaning that all its automorphisms are inner automorphisms. Its fundamental representation is the 248-dimensional adjoint representation (in other words conjugation acting on tangent vectors at the identity element).

[edit] Computation of its representation theory

In terms of its entire representation theory, E8 was computationally tamed by a group of mathematicians in early 2007, with an announcement on March 19, 2007 at the American Institute of Mathematics. A representation of a Lie group (as for any group) is a mapping of that group into the symmetries of Euclidean space (to be more precise, we admit the complex numbers also as scalars). E8 is unique among Lie groups in that its smallest (i.e., lowest-dimension) representation is the so-called adjoint representation, in which the space whose symmetries provide the representation is the Lie algebra E8 itself. In a manner of speaking, the easiest way to understand E8 is as (some of) its own symmetries.

The effort was to compute a large square matrix consisting of polynomials, the Kazhdan–Lusztig polynomials[3] introduced for reductive groups in general[4] by David Kazhdan and George Lusztig during the 1980s. The particularly difficult case for E8, is the split real form (in other words the realisation that is least 'compact', see below[5]) where the required data was in a table of size 453060×453060. The resulting data required over 60 gigabytes to store. Modular arithmetic for several moduli was required, and the results reconstituted by means of the Chinese remainder theorem.[6] For groups of lower rank, these data have been known for some time.[7]

This computation comes after four years of collaboration by a group of 18 mathematicians and computer scientists. Jeffrey Adams[8], project leader and mathematics professor at the University of Maryland, College Park said "This groundbreaking achievement is significant both as an advance in basic knowledge, as well as a major advance in the use of large scale computing to solve complicated mathematical problems." The late Fokko du Cloux started developing the program to compute the polynomials in 2003. The final calculation was carried out on an 8-core AMD machine with 64 gigabytes of RAM, run by William Stein at the University of Washington.[7] The group is working on the larger task of producing an atlas of Lie groups and representations.

[edit] Real forms

As well as the complex Lie group E8, of complex dimension 248 or real dimension 496, there are three real forms of the group, all of real dimension 248. There is one compact one (which is usually the one meant if no other information is given), one split one, and a third one.

[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 'octooctonionic 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.[9]

[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] 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[10], which may be labelled as a pair of dimensions as

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

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] Importance in physics

The group E8 frequently appears in string theory and supergravity. For example, it is the U-duality group of supergravity on an eight-torus (in its split form). The gauge group of one of the two supersymmetric versions of the heterotic string is E8×E8.

One way to incorporate the standard model of particle physics in the heterotic string includes the symmetry breaking of E8 to its maximal subgroup

(\operatorname{SU}(3) \times \operatorname{E}_6)/(\mathbb{Z}/3\mathbb{Z}) .

[edit] See also

[edit] Notes

  1. ^ The subscripts here refer to the rank of the Lie algebra which is the dimension of a maximal abelian subalgebra (cf. Cartan subalgebra), or, equivalently, the dimension of the maximal torus in the associated compact Lie group.
  2. ^ Baez, John (1996-09-30). This Week's Finds in Mathematical Physics (Week 90). Retrieved on 2007-03-23.
  3. ^ Strictly speaking these are Kazhdan–Lusztig–Vogan polynomials, crediting David Vogan, who was himself involved in the work.
  4. ^ For Coxeter groups, which include the Weyl groups of simple Lie groups.
  5. ^ The maximal compact subgroup of the split form of E8 is described by this PDF, p.17. It is a quotient of the spin group Spin(16) by a subgroup of order 2, but is not isomorphic with the special orthogonal group SO(16) (though necessarily sharing its Lie algebra).
  6. ^ [1],[2]
  7. ^ a b [3] gives an abstract, topological definition of Kazhdan–Lusztig polynomial as a type of Poincaré polynomial, and reports on the state of the art in the mid-1990s.
  8. ^ Official home page.[4]
  9. ^ J.M. Landsberg, L. Manivel, The projective geometry of Freudenthal's magic square, Journal of Algebra (2001)
  10. ^ Since there is a quotient in the product, these notations may strictly be taken as indicating the infinitesimal (Lie algebra) representations.

[edit] References

[edit] External links

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