Lie algebras have played a crucial role in the formulation of the electromagnetic, nuclear weak and strong forces as well as many other areas of theoretical physics. As string theory aims to provide a single theory of physics it is not unlikely that Lie algebras will play a central role. With this in mind we first give, in this chapter, a review of finite-dimensional semi-simple Lie algebras, that is, those in the list of Cartan. A proper understanding of these Lie algebras then allows us to define Kac-Moody algebras and discuss their properties, in particular a subclass of these algebras called Lorentzian algebras. We illustrate the general theory in the context of a Kac-Moody algebra called E11. Finally, we show how string theory and its vertex operators lead very naturally to Lie algebras.

Finite dimensional and affine Lie algebras

A review of finite-dimensional Lie algebras and lattices

It is beyond the scope of this book to give a complete account of Lie algebras; however, in this and next section we will give an account that contains many of the main results. Although someone who is unfamiliar with Lie algebras could read this chapter it might be desirable to gain some familiarity with this subject first; some useful accounts are [5.5, 16.1]. Some textbooks covering the same material on finite dimensional Lie algebras are [5.4, 16.2, 16.3]. We will begin with an exposition which is familiar to physicists and then develop a point of view which is more often encountered in the mathematics literature, namely the Serre presentation of semi-simple Lie algebras. This starts from a very concise formulation of Lie algebras and deduces their structure and all their properties in a very elegant and efficient way. This viewpoint is essential for understanding the latter sections on Kac–Moody algebras. The main aim of these first two sections is to enable physicists to bridge the gap between the physics account of Lie algebras and that found in the mathematics literature.