Algebraic groups play much the same role for algebraists as Lie groups
play for analysts. This book is the first comprehensive introduction to
the theory of algebraic group schemes over fields that includes the
structure theory of semisimple algebraic groups, and is written in the
language of modern algebraic geometry. The first eight chapters study
general algebraic group schemes over a field and culminate in a proof of
the Barsotti-Chevalley theorem, realizing every algebraic group as an
extension of an abelian variety by an affine group. After a review of
the Tannakian philosophy, the author provides short accounts of Lie
algebras and finite group schemes. The later chapters treat reductive
algebraic groups over arbitrary fields, including the Borel-Chevalley
structure theory. Solvable algebraic groups are studied in detail.
Prerequisites have also been kept to a minimum so that the book is
accessible to non-specialists in algebraic geometry.