Actions and Invariants of Algebraic Groups presents a self-contained introduction to geometric invariant theory that links the basic theory of affine algebraic groups to Mumford's more sophisticated theory. The authors systematically exploit the viewpoint of Hopf algebra theory and the theory of comodules to simplify and compactify many of the relevant formulas and proofs.
The first two chapters introduce the subject and review the prerequisites in commutative algebra, algebraic geometry, and the theory of semisimple Lie algebras over fields of characteristic zero. The authors' early presentation of the concepts of actions and quotients helps to clarify the subsequent material, particularly in the study of homogeneous spaces. This study includes a detailed treatment of the quasi-affine and affine cases and the corresponding concepts of observable and exact subgroups.
Among the many other topics discussed are Hilbert's 14th problem, complete with examples and counterexamples, and Mumford's results on quotients by reductive groups. End-of-chapter exercises, which range from the routine to the rather difficult, build expertise in working with the fundamental concepts. The Appendix further enhances this work's completeness and accessibility with an exhaustive glossary of basic definitions, notation, and results.
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The authors' stated goal in this book is to provide an 'introduction to geometric invariant theory' but in the process they provide a fine introduction to the developments just mentioned and the tools necessary to bring the reader to the point where the current literature can be understood. Each chapter ends with a set of exercises designed to expand on the text and deepen the readers' knowledge.
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