Algebraic Groups, Lie Groups, and their Arithmetic Subgroups

This work is a modern exposition of the theory of algebraic group schemes, Lie groups, and their arithmetic subgroups. It supersedes Algebraic Groups and Arithmetic Groups. This file only contains the front matter. For the rest, see Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Preface For one who attempts to unravel the story, the problems are as perplexing as a mass of hemp with a thousand loose ends. Algebraic groups are groups defined by polynomials. Those that we shall be concerned with in this book can all be realized as groups of matrices. For example, the group of matrices of determinant 1 is an algebraic group, as is the orthogonal group of a symmetric bilinear form. The classification of algebraic groups and the elucidation of their structure were among the great achievements of twentieth century mathematics (Borel, Chevalley, Tits and others, building on the work of the pioneers on Lie groups). Algebraic groups are used in most branches of mathematics, and since the famous work of Hermann Weyl in the 1920s they have also played a vital role in quantum mechanics and other branches of physics (usually as Lie groups). Arithmetic groups are groups of matrices with integer entries. They are an important source of discrete groups acting on manifolds. The first goal of the present work is to provide a modern exposition of the theory of algebraic groups. It has been clear for over forty years, that in the definition of an algebraic group, the coordinate ring should be allowed to have nilpotent elements, 1 but the standard expositions 2 do not allow this. 3 In recent years, the tannakian duality 4 between algebraic groups and their categories of representations has come to play a role in the theory of algebraic groups similar to that of Pontryagin duality in the theory of locally compact abelian groups. Chapter I develops the basic theory of algebraic groups, including tannakian duality. As Cartier (1956) noted, the relation between Lie algebras and algebraic groups in characteristic zero is best understood through their categories of representations. In Chapter II we review the classification of semisimple Lie algebras and their representations, and we exploit tannakian duality to deduce the classification of semisimple algebraic groups and their representations in characteristic zero. The only additional complication presented by algebraic groups is that of determining the centre of …