Lie Algebras, Algebraic Groups, and Lie Groups

These notes are an introduction to Lie algebras, algebraic groups, and Lie groups in characteristic zero, emphasizing the relationships between these objects visible in their categories of representations. Eventually these notes will consist of three chapters, each about 100 pages long, and a short appendix. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Bibliography 181 Index 185 4 Preface [Lie] did not follow the accepted paths.. . I would compare him rather to a pathfinder in a primal forest who always knows how to find the way, whereas others thrash around in the thicket.. . moreover, his pathway always leads past the best vistas, over unknown mountains and valleys. Lie algebras are an essential tool in studying both algebraic groups and Lie groups. Chapter I develops the basic theory of Lie algebras, including the fundamental theorems of Engel, Lie, Cartan, Weyl, Ado, and Poincaré-Birkhoff-Witt. The classification of semisim-ple Lie algebras in terms of the Dynkin diagrams is explained, and the structure of semisim-ple Lie algebras and their representations described. In Chapter II we apply the theory of Lie algebras to the study of algebraic groups in characteristic zero. As Cartier (1956) noted, the relation between Lie algebras and algebraic groups in characteristic zero is best understood through their categories of representations. For example, when g is a semisimple Lie algebra, the representations of g form a tan-nakian category Rep.g/ whose associated affine group G is the simply connected semisim-ple algebraic group G with Lie algebra g. In other words, Rep.G/ D Rep.g/ (1) with G a simply connected semisimple algebraic group having Lie algebra g. It is possible to compute the centre of G from Rep.g/, and to identify the subcategory of Rep.g/ corresponding to each quotient of G by a finite subgroup. This makes it possible to read off the entire theory of semisimple algebraic groups and their representations from the (apparently simpler) theory of semisimple Lie algebras. For a general Lie algebra g, we consider the category Rep nil .g/ of representations of g such that the elements in the largest nilpotent ideal of g act as nilpotent endomorphisms. Ado's theorem assures us that g has a faithful such representation, and from this we are able to deduce a correspondence between algebraic Lie algebras and algebraic groups with unipotent centre. Let G be a reductive algebraic group with a …