Chapter 11 - The Structure of Semi-simple Lie Algebras

Abstract

This chapter focuses on the structure of semisimple Lie algebras. The theory of semisimple Lie algebras has considerable physical applications, particularly in elementary particle theory. The process of complexification, which is of going from a real Lie algebra to a complex Lie algebra, is investigated. A Lie algebra is said to be semisimple if it does not possess an Abelian invariant subalgebra. It is suggested that as all one-dimensional Lie algebras are Abelian, simple and semisimple Lie algebras must have dimension greater than one. It is found that the Killing form provides not only a very convenient criterion for distinguishing semisimple Lie algebras but also plays an important part in the analysis of the structure of such algebras. The key to the whole theory of semisimple Lie algebras is provided by Cartan's criterion for semisimplicity. It is suggested that the deduction of the real forms of a complex Lie algebra is not a trivial matter, even if the complex Lie algebra is simple. The Cartan subalgebras and roots of semisimple complex Lie algebras are also elaborated.