Representations of sl(2, C)

Abstract

In this chapter, we study the finite-dimensional irreducible representations of sl(2, C). In doing this, we shall see, in a stripped-down form, many of the ideas needed to study representations of an arbitrary semisimple Lie algebra. Later we will see that representations of sl(2, C) control a large part of the structure of all semisimple Lie algebras.

We shall use the basis of sl(2, C) introduced in Exercise 1.12 throughout this chapter. Recall that we set

EquationSource$$ e = \left( {\begin{array}{*{20}c} 0 & 1\\ 0 & 0\\ \end{array} } \right), f = \left( {\begin{array}{*{20}c} 0 & 0\\ 1 & 0\\ \end{array} } \right), h = \left( {\begin{array}{*{20}c} 1 & 0\\ 0 & { - 1}\\ \end{array} } \right). $$