Book Review: Basic theory of algebraic groups and Lie algebras

Abstract

The author's aim is clearly expressed in his preface. He says:``The theory of algebraic groups results from the interaction ofvarious basic techniques from field theory, multilinear algebra,commutative ring theory, algebraic geometry and general algebraicrepresentation theory of groups and Lie algebras. It is thus anideally suitable framework for exhibiting algebra in action. Todo that is the principal concern of this text. Accordingly, itsemphasis is on developing the major general mathematical toolsused for gaining control over algebraic groups, rather than onsecuring the final definitive results, such as the classificationof the simple groups and their irreducible representations. Inthe same spirit, this exposition has been made entirely selfcontained; no detailed knowledge beyond the usual standardmaterial of the first one or two years of graduate study inalgebra is presupposed.'' The chapter headings are well chosen toindicate the content and approach. They are: Representativefunctions and Hopf algebras; Affine algebraic sets and groups;Derivations and Lie algebras; Lie algebras and algebraicsubgroups; Semisimplicity and unipotency; Solvable groups;Elementary Lie algebra theory; Structure theory on characteristic0; Algebraic varieties; Local theory; Coset varieties; Borelsubgroups; Applications of Galois cohomology; Algebraicautomorphism groups; The universal enveloping algebra; SemisimpleLie algebras; From Lie algebras to groups. The reviewer enjoyedreading the book, learned from it, and welcomes this addition tothe mathematical literature. In his view this book achieves itsaims