Semisimple Lie Algebras

Abstract

In this chapter we introduce a class of Lie algebras, the semisimple algebras, for which we can classify the irreducible representations using a strategy similar to the one we used for $\mathsf{sl}(3; \mathbb{C})$. In this chapter, we develop the relevant structures of semisimple Lie algebras. In Chapter 8, we look into the properties of the set of roots. Then in Chapter 9, we construct and classify the irreducible, finite-dimensional representations of semisimple Lie algebras. Finally, in Chapter 10, we consider several additional properties of the representations constructed in Chapter 9. Meanwhile, in Chapters 11 and 12, we consider representation theory from the closely related viewpoint of compact Lie groups.