E8 LIE

In mathematics, E₈ is the name given to a family of closely related structures. In particular, it is the name of four exceptional simple Lie algebras as well as that of the six 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. E₈ was discovered between the years of 1888 and 1890 by Wilhelm Killing, though he did not prove its existence, which was first shown by E. Cartan..

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 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.

Source: Wikipedia