Representation Theory of Lie Algebras

Abstract

Even though representation theory is not in the focus of this book, we provide in the present chapter the basic theory for Lie algebras as it repeatedly plays an important role in structural questions. In this chapter, we first introduce the universal enveloping algebra ${\mathcal{U}}({\mathfrak{g}})$ of a Lie algebra ${\mathfrak{g}}$. This is a unital associative algebra containing ${\mathfrak{g}}$ as a Lie subalgebra and is generated by ${\mathfrak{g}}$. It has the universal property that each representation of ${\mathfrak{g}}$ extends uniquely to ${\mathcal{U}}({\mathfrak{g}})$, so that any ${\mathfrak{g}}$-module becomes a ${\mathcal{U}}({\mathfrak{g}})$-module. We may thus translate freely between Lie algebra modules and algebra modules, which is convenient for several representation theoretic constructions. The Poincaré–Birkhoff–Witt (PBW) Theorem 7.1.9 provides crucial information on the structure of ${\mathcal{U}}({\mathfrak{g}})$, including the injectivity of the natural map ${\mathfrak{g}}\to {\mathcal{U}}({\mathfrak{g}})$.