We present basic facts about Lie groups and Lie algebras. We describe semi-simple Lie algebras and their representations which can be characterized in terms of roots and weights. We discuss infinite-dimensional Lie algebras, called affine Kac—Moody algebras, which are at the heart of the study of field theoretical integrable systems. In particular we construct the so-called level one representations using the techniques of Fock spaces and vertex operators introduced in Chapter 9.

Lie groups and Lie algebras

A Lie group is a group G which is at the same time a differentiable manifold, and such that the group operation (g, h) → gh-1 is differentiable. Due to a theorem of Montgomery and Zippin, the differentiable structure is automatically real analytic.

The maps h → gh and h → hg are called respectively left and right translations by g. Their differentials at the point h map the tangent space Th(G) respectively to Tgh(G) and Thg(G). We will denote by g · X and X · g the images of X ∊ Th(G) by these maps. This notation is coherent because, differentiating the associativity condition in G, one gets (g · X) · h = g · (X · h), and g · (h · X) = (gh) · X.