Xen qabbalah dictionary

E8 lie group

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 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.

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 (2^14)(3^5)(5^2)7 = 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 Ek 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 E8(-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 fieldk) by the set H^1(k,Aut(E8)) which, because the Dynkin diagram of E8 (see below) has no automorphisms, coincides with H^1(k,E8).

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.