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 An, Bn, Cn, Dn, 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 214 35 52 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 that 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. The largest value of k for which Ek is finite-dimensional is k = 8, that is, Ek is infinite-dimensional for any k > 8.