Chapter 12 - Representations of Semi-simple Lie Algebras

Abstract

This chapter elaborates the representations of semisimple Lie algebras. The key idea in the representation theory of semisimple complex Lie algebras is that of weights. The close connection among the representations of a semisimple complex Lie algebra, the representations of its various real forms, and the representations of the Lie groups associated with these real Lie algebras means that certain statements proved in one situation can be readily transferred to the others. It is found that every reducible representation of a semisimple real or complex Lie algebra is completely reducible. Each irreducible representation is uniquely and completely specified by its highest weight, all of its properties, such as its dimension and the other weights being easily deducible from it. The full sets of weights of the lower-dimensional irreducible representations are presented. The basic properties of the second-order Casimir operator are summarized. The irreducible representation of a complex semisimple Lie algebra, which is determined by its highest weight is also elaborated.