E8 (mathematics)

In mathematics, E₈ is any of several closely related exceptional simple Lie groups, linear algebraic groups or 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 G₂, F₄, E₆, E₇, and E₈. The E₈ algebra is the largest and most complicated of these exceptional cases.

Basic description

The Lie group E₈ has dimension 248. Its rank, which is the dimension of its maximal torus, is eight.
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 that are induced by conjugations in the whole group, has order 2357 =.
The compact group E₈ is unique among simple compact Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation acting on the Lie algebra E₈ itself; it is also the unique one that has the following four properties: trivial center, compact, simply connected, and simply laced.
There is a Lie algebra E_k for every integer k ≥ 3. The largest value of k for which E_k is finite-dimensional is k = 8, that is, E_k is infinite-dimensional for any k > 8.

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 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, all of real dimension 248, as follows:

The compact form, which is simply connected and has trivial outer automorphism group.
The split form, EVIII, which has maximal compact subgroup Spin/, fundamental group of order 2 and has trivial outer automorphism group.
EIX, which has maximal compact subgroup E₇ × SU/, fundamental group of order 2 and has trivial outer automorphism group.

For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups.

E₈ as an algebraic group

By means of a Chevalley basis for the Lie algebra, one can define E₈ as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split form of E₈. Over an algebraically closed field, this is the only form; however, over other fields, there are often many other forms, or "twists" of E₈, which are classified in the general framework of Galois cohomology by the set H¹, which, because the Dynkin diagram of E₈ has no automorphisms, coincides with H¹.
Over R, the real connected component of the identity of these algebraically twisted forms of E₈ coincide with the three real Lie groups mentioned [|above], but with a subtlety concerning the fundamental group: all forms of E₈ are simply connected in the sense of algebraic geometry, meaning that they admit no non-trivial algebraic coverings; the non-compact and simply connected real Lie group forms of E₈ are therefore not algebraic and admit no faithful finite-dimensional representations.
Over finite fields, the Lang–Steinberg theorem implies that H¹ = 0, meaning that E₈ has no twisted forms: see [|below].
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 :
The 248-dimensional representation is the adjoint representation. There are two non-isomorphic irreducible representations of dimension 8634368000. The fundamental representations are those with dimensions 3875, 6696000, 6899079264, 146325270, 2450240, 30380, 248 and 147250.
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. The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations 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 is the split real form of E₈, 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, to the surprise of the mathematicians working on it.
The representations of the E₈ groups over finite fields are given by Deligne–Lusztig theory.

Constructions

One can construct the E₈ group as the automorphism group of the corresponding e₈ Lie algebra. This algebra has a 120-dimensional subalgebra so generated by J_ij as well as 128 new generators Q_a that transform as a Weyl–Majorana spinor of spin. These statements determine the commutators
as well as
while the remaining commutators between the spinor generators are defined as
It is then possible to check that the Jacobi identity is satisfied.

Geometry

The compact real form of E₈ is the isometry group of the 128-dimensional exceptional compact Riemannian symmetric space EVIII. It is 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, and is also known as a Rosenfeld projective plane, though it does not obey the usual axioms of a projective plane. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.

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 a number of purposes to normalize them to have length. These 240 vectors are the vertices of a semi-regular polytope discovered by Thorold Gosset in 1900, sometimes known as the 4₂₁ polytope.

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
by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots with half-integer entries obtained from
by taking an even number of minus signs. 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 as well as E₆ and E₇.
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.

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 that are not joined by a line are orthogonal.

[|Cartan matrix]

The Cartan matrix of a rank root system is an matrix 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 are the simple roots. The entries are independent of the choice of simple roots.
The Cartan matrix for E₈ is given by
The determinant of this matrix is equal to 1.

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.
Given the E₈ Cartan matrix and a Dynkin diagram node ordering of:
One choice of simple roots is given by the rows of the following matrix:
With this numbering of nodes in the Dynkin diagram, the highest root in the root system has Coxeter labels. Using this representation of the simple roots, the lowest root is given by
The only simple root that can be added to the lowest root to obtain another root is the one corresponding to node 1 in this labeling of the Dynkin diagram — as is to be expected from the affine Dynkin diagram for. The Hasse diagram to the right enumerates the 120 roots of positive height relative to any particular choice of simple roots consistent with this node numbering.
Note that the Hasse diagram does not represent the full Lie algebra, or even the full root system. The 120 roots of negative height relative to the same set of simple roots can be adequately represented by a second copy of the Hasse diagram with the arrows reversed; but it is less straightforward to connect these two diagrams via a basis for the eight-dimensional Cartan subalgebra. In the notation of the exposition of Chevalley generators and Serre relations: Insofar as an arrow represents the Lie bracket by the generator associated with a simple root, each root in the height -1 layer of the reversed Hasse diagram must correspond to some and can have only one upward arrow, connected to a node in the height 0 layer representing the element of the Cartan subalgebra given by. But the upward arrows from the height 0 layer must then represent, where is the Cartan matrix. One could draw multiple upward arrows from each associated with all for which is nonzero; but this neither captures the numerical entries in the Cartan matrix nor reflects the fact that each only has nonzero Lie bracket with one degree of freedom in the Cartan subalgebra.
More fundamentally, this organization implies that the span of the generators designated as "the" Cartan subalgebra is somehow inherently special, when in most applications, any mutually commuting set of eight of the 248 Lie algebra generators — or any eight linearly independent, mutually commuting Lie derivations on any manifold with E₈ structure — would have served just as well. Once a Cartan subalgebra has been selected, a basis of "Cartan generators" and a root system are a useful way to describe structure relative to this subalgebra. But the root system map is not the Lie algebra territory. Given a set of Chevalley generators, most degrees of freedom in a Lie algebra and their sparse Lie brackets with can be represented schematically as circles and arrows, but this simply breaks down on the chosen Cartan subalgebra. Such are the hazards of schematic visual representations of mathematical structures.