E8 (Gruppe)

Vorlage:Downsize 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. It was formulated between the years of 1888-1890 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 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, and is often the last case of various theorems to be proved.

Basic description

E₈ has rank 8 (the maximum number of mutually commutative degrees of freedom), 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 E_n for every integer n≥3, which is infinite dimensional if n is greater than 8.

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.

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.

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 {1}{4}}(\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 _{ac}^{[i}\gamma _{cb}^{j]}J_{ij}.

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

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)).

E₈ root system

Datei:E8 roots zome.jpg

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.

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 {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}\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.

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].

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.

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.

E₈ root 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.

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.

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.

In October 2007, physicist Garrett Lisi analyzed a non-compact real form of the E8 Lie algebra to provide a purported simple theory of everything.

References

J. F. Adams, Lectures on Exceptional Lie Groups (Chicago Lectures in Mathematics) ISBN 0226005275
Killing, Die Zusammensetzung der stetigen/endlichen Transformationsgruppen Mathematische Annalen, Volume 31, Number 2 June, 1888, Pages 252-290 doi:10.1007/BF01211904, Volume 33, Number 1 March, 1888, Pages 1-48 doi:10.1007/BF01444109, Volume 34, Number 1 March, 1889, Pages 57-122 doi:10.1007/BF01446792, Volume 36, Number 2 June, 1890,Pages 161-189 doi:10.1007/BF01207837
J.M. Landsberg, L. Manivel, (2001)), The projective geometry of Freudenthal's magic square, Journal of Algebra (2001)doi:10.1006/jabr.2000.8697
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External links

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

Home page of Jeffrey Adams
Kazhdan-Lusztig-Vogan Polynomials for E₈
D. Vogan, Narrative of the Project to compute Kazhdan-Lusztig Polynomials for E₈
D. Vogan, The Character Table for E₈, or How We Wrote Down a 453,060 x 453,060 Matrix and Found Happiness Slides for a popular talk on E₈.
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The n-Category Café — University of Texas blog posting by John Baez on E₈

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