Simple Lie Algebras

Abstract

This chapter presents a survey of the theory of Lie algebras. This might appear somewhat remote from our main subject of interest: affine Lie algebras and their applications to conformal field theory. However, it turns out that in many respects the theory of affine Lie algebras is a natural extension of the theory of simple Lie algebras, and as such cannot be studied efficiently in isolation. This is an immediate motivation for devoting a complete chapter to Lie algebras. But as subsequent developments will show, conformal field theories with nonaffine additional symmetries, such as W algebras, parafermions, and son on, as well as related exactly solvable statistical models, also have a deep Lie-algebraic underlying structure, which can only be appreciated with a minimal background on simple Lie algebras.