The Classification of Root Systems

Abstract

In §11.4, we saw how to define the Dynkin diagram of a root system. By Theorem 11.16, which states that any two bases of a root system are conjugate by an element of the Weyl group, this diagram is unique up to the labelling of the vertices. (The labels merely indicate our notation for elements of the base; and so have no essential importance.) Conversely, we saw in §11.4.1 that a root system is determined up to isomorphism by its Dynkin diagram.

From the point of view of classifying complex semisimple Lie algebras, there is no need to distinguish between isomorphic root systems. Hence the problem of finding all root systems can be reduced to the problem of finding all Dynkin diagrams; this gives us a very convenient way to organise the classification.

We shall prove that apart from the four infinite families of root systems associated to the classical Lie algebras there are just five more root systems, the so-called exceptional root systems. We end this chapter by saying a little about how they may be constructed.