Semisimple Lie Algebras

Abstract

In this chapter, we will consider a class of Lie algebras (the complex semisimple ones) that are sufficiently similar to sl(3; ℂ) that their representations can be described, similarly to sl(3; ℂ), by a “theorem of the highest weight.” We will not come to the representations themselves until the next chapter; in this chapter, we develop the structures needed to state the theorem of the highest weight. Although this chapter could be understood simply as a description of the structure of semisimple Lie algebras, without any mention of representation theory, I think it is helpful to have the representations in mind. The representation theory, especially in light of our experience with sl(3; ℂ), motivates the notions of Cartan subalgebras, roots, and the Weyl group.