Appendix B: Weyl’s Theorem

Abstract

We want to show that any finite-dimensional representation of a complex semisimple Lie algebra is a direct sum of irreducible representations. This is known as Weyl’s Theorem; it is a fundamental result in the representation theory of Lie algebras. We used it several times in Chapter 9 to decompose a representation of sl(2, C) into a direct sum of irreducible representations.

As usual, all Lie algebras and representations in this chapter are finitedimensional.