Root Systems

Abstract

In this chapter, we consider root systems apart from their origins in semisimple Lie algebras. We establish numerous “factoids” about root systems, which will be used extensively in subsequent chapters. Here is one example of how results about root systems will be used. In Chapter 9, we construct each finite-dimensional, irreducible representation of a semisimple Lie algebra as a quotient of an infinite-dimensional representation known as a Verma module. To prove that the quotient representation is finite-dimensional, we prove that the weights of the quotient are invariant under the action of the Weyl group, that is, the group generated by the reflections about the hyperplanes orthogonal to the roots. It is not possible, however, to prove directly that the weights are invariant under all reflections, but only under reflections coming from a special subset of the root system known as the base. To complete the argument, then, we need to know that the Weyl group is actually generated by the reflections associated to the roots in the base. This claim is the content of Proposition 8.24.