Introduction

Summary

The book can be divided roughly into five parts.

Basic theory of general algebraic groups (Chapters 1–8)

The first eight chapters cover the general theory of algebraic group schemes (not necessarily affine) over a field. After defining them and giving some examples, we show that most of the basic theory of abstract groups (subgroups, normal subgroups, normalizers, centralizers, Noether isomorphism theorems, subnormal series, etc.) carries over with little change to algebraic group schemes. We relate affine algebraic group schemes to Hopf algebras, and we prove that all algebraic group schemes in characteristic zero are smooth. We study the linear representations of algebraic group schemes and their actions on algebraic schemes. We show that every algebraic group scheme is an extension of an étale group scheme by a connected algebraic group scheme, and that every smooth connected group scheme over a perfect field is an extension of an abelian variety by an affine group scheme (Barsotti–Chevalley theorem).

Beginning with Chapter 9, all group schemes are affine.

Preliminaries on affine algebraic groups (Chapters 9–11)

The next three chapters are preliminary to the more detailed study of affine algebraic group schemes in the later chapters. They cover basic Tannakian theory, in which the category of representations of an algebraic group scheme plays the role of the topological dual of a locally compact abelian group, Jordan decompositions, the Lie algebra of an algebraic group, and the structure of finite group schemes. Throughout this work we emphasize the Tannakian point of view in which the group and its category of representations are placed on an equal footing.

Solvable affine algebraic groups (Chapters 12–16)

The next five chapters study solvable algebraic group schemes. Among these are the diagonalizable groups, the unipotent groups, and the trigonalizable groups.

An algebraic group G is diagonalizable if every linear representation of G is a direct sum of one-dimensional representations; in other words if, relative to some basis, the image of G lies in the algebraic subgroup of diagonal matrices in GL_n. An algebraic group that becomes diagonalizable over an extension of the base field is said to be of multiplicative type.