Lie algebras and the classification of semisimple algebraic groups

The Lie algebra of an algebraic group is the (first) linear approximation to the group. The study of Lie algebras is much more elementary than that of algebraic groups. For example, most of the results on Lie algebras that we shall need are proved already in the undergraduate text Erdmann and Wildon 2006. After many preliminaries, in 7 we describe the structure and classification of the semisimple Lie algebras and of their representations. The category of representations of a Lie algebra is a neutral tannakian category, and so there exists an affine group G.g/ such that RepG.g/D Rep.g/. We show that, when g is semisimple and the base field has characteristic zero, G.g/ is an algebraic group with Lie algebra g, and that every other connected algebraic group G with Lie algebra g is uniquely a quotient of G.g/ by a finite subgroup of its centre. In other words, G.g/ is the simply connected semisimple algebraic group with Lie algebra g. Once we have determined the centre of G.g/ in terms of g and its root system, we are able to read off the structure and classification of the semisimple algebraic groups and of their representations from the similar results for Lie algebras. Throughout this chapter k is a field.