Actions and Invariants of Algebraic Groups, Second Edition presents a self-contained introduction to geometric invariant theory starting from the basic theory of affine algebraic groups and proceeding towards more sophisticated dimensions." Building on the first edition, this book provides an introduction to the theory by equipping the reader with the tools needed to read advanced research in the field. Beginning with commutative algebra, algebraic geometry and the theory of Lie algebras, the book develops the necessary background of affine algebraic groups over an algebraically closed field, and then moves toward the algebraic and geometric aspects of modern invariant theory and quotients.
"[The new edition of this book] retains all of the attractive features of the first edition and adds new chapters on observable actions (Chapter 15) and geometric invariant theory (Chapter 16)."
- Frank D. Grosshans, Mathematical Reviews, August 2017
Algebraic geometry. Lie algebras. Algebraic groups: basic definitions. Algebraic groups: Lie algebras and representations. Algebraic groups: Jordan decomposition and applications. Actions of algebraic groups. Homogeneous spaces. Algebraic groups and Lie algebras in characteristic zero. Observable subgroups of affine algebraic groups. Observable actions of affine algebraic groups. Observable subgroups of affine monoids. Affine homogeneous spaces. Hilbert's 14th problem. Quotients. Appendix. Basic definitions and results.