E8 (mathematics

E₈ (mathematics)
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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 E₆, E₇, E₈, F₄, and G₂. The E₈ algebra is the largest and most complicated of these exceptional cases.
Wilhelm Killing (1888a, 1888b, 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 simple Lie group of dimension 248, exactly one of which is compact. Chevalley (1955) introduced algebraic groups and Lie algebras of type E₈ over other fields: for example, in the case of finite fields they lead to an infinite family of finite simple groups of Lie type.

Basic description

The 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 2¹⁴3⁵5²7 = 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_k for every integer k ≥ 3, which is infinite dimensional if k is greater than 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 (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:

• The compact form (which is usually the one meant if no other information is given), which is simply connected and has trivial outer automorphism group.
• The split form, EVIII (or E₈₍₈₎), which has maximal compact subgroup Spin(16)/(Z/2Z), fundamental group of order 2 (implying that it has a double cover, which is a simply connected Lie real group but is not algebraic, see below) and has trivial outer automorphism group.
• EIX (or E_8(-24)), which has maximal compact subgroup E₇×SU(2)/(−1,−1), fundamental group of order 2 (again implying a double cover, which is not algebraic) 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 (sometimes also known as “untwisted”) 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 (over a perfect field k) by the set H¹(k,Aut(E₈)) which, because the Dynkin diagram of E₈ (see below) has no automorphisms, coincides with H¹(k,E₈).^[1]
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(k,E₈)=0, meaning that E₈ has no twisted forms: see below.

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 the 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 (twice), 12692520960…

The 248-dimensional representation is the adjoint representation. There are two non-isomorphic irreducible representations of dimension 8634368000 (it is not unique; however, the next integer with this property is 175898504162692612600853299200000 (sequence A181746 in the OEIS)). 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.

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

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

as well as

${\displaystyle [J_{ij},Q_{a}]={\frac {1}{4}}(\gamma _{i}\gamma _{j}-\gamma _{j}\gamma _{i})_{ab}Q_{b},}$

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

${\displaystyle [Q_{a},Q_{b}]=\gamma _{ac}^{[i}\gamma _{cb}^{j]}J_{ij}.}$

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



 
Applications
The E₈ Lie group has applications in theoretical physics and especially in string theory and supergravity. E₈×E₈ is 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 ten 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.
Antony Garrett Lisi's incomplete "An Exceptionally Simple Theory of Everything" attempts to describe all known fundamental interactions in physics as part of the E₈ Lie algebra.^[6][7]
R. Coldea, D. A. Tennant, and E. M. Wheeler et al. (2010) reported an experiment where the electron spins of a cobalt-niobium crystal exhibited, under certain conditions, two of the eight peaks related to E₈ that were predicted by Zamolodchikov (1989).^[8][9]