Semisimple Algebraic Groups in Characteristic Zero

It is shown that the classification theorems for semisimple algebraic groups in characteristic zero can be derived quite simply and naturally from the corresponding theorems for Lie algebras by using a little of the theory of tensor categories. This article is extracted from Milne 2007. Introduction The classical approach to classifying the semisimple algebraic groups over C (see Borel 1975, §1) is to: classify the complex semisimple Lie algebras in terms of reduced root systems (Killing, E. Cartan, et al.); classify the complex semisimple Lie groups with a fixed Lie algebra in terms of certain lattices attached to the root system of the Lie algebra (Weyl, E. Cartan, et al.); show that a complex semisimple Lie group has a unique structure of an algebraic group compatible with its complex structure. Chevalley (1956-58, 1960-61) proved that the classification one obtains is valid in all characteristics, but his proof is long and complicated.1 Here I show that the classification theorems for semisimple algebraic groups in characteristic zero can be derived quite simply and naturally from the corresponding theorems for Lie algebras by using a little of the theory of tensor categories. In passing, one also obtains a classification of their finite-dimensional representations. Beyond its simplicity, the advantage of this approach is that it makes clear the relation between semisimple Lie algebras, semisimple algebraic groups, and tensor categories in characteristic zero. The idea of obtaining an algebraic proof of the classification theorems for semisimple algebraic groups in characteristic zero by exploiting their representations is not new — in a somewhat primitive form it can be found already in Cartier’s announcement (1956) — but I have not seen an exposition of it in the literature. Throughout, k is a field of characteristic zero and “representation” of a Lie algebra or affine group means “finite-dimensional linear representation”. I assume that the reader is familiar with the elementary parts of the theories of algebraic groups and tensor categories and with the classification of semisimple Lie algebras; see Milne 2007 for a more detailed account. ∗ c ©2007 J.S. Milne See Humphreys 1975, Chapter XI, and Springer 1998, Chapters 10 & 11. Despite its fundamental importance, many books on algebraic groups, e.g., Borel 1991, don’t prove the classification, and some, e.g., Tauvel and Yu 2005, don’t even state it.