http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
Actions and invariants of algebraic groups
1 online resource (xx, 459 pages : illustrations
"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."--Provided by publisher
Chapter 1 Introduction -- chapter 2 Commutative algebra -- chapter 3 Algebraic subsets of the affine space -- chapter 4 Algebraic varieties -- chapter 5 Exercises -- chapter 1 Introduction -- chapter 2 Morphisms of algebraic varieties -- chapter 3 Complete varieties -- chapter 4 Singular points and normal varieties -- chapter 5 The Proj variety associated to a graded algebra -- chapter 6 Deeper results on morphisms -- chapter 7 Algebraic varieties and k-schemes -- chapter 8 Exercises -- chapter 1 Introduction -- chapter 2 Definitions and basic concepts -- chapter 3 The theorems of F. Engel and S. Lie -- chapter 4 Semisimple Lie algebras -- chapter 5 Cohomology of Lie algebras -- chapter 6 The theorems of H. Weyl and F. Levi -- chapter 7 p-Lie algebras -- chapter 8 Exercises -- chapter 1 Introduction -- chapter 2 Definitions and basic concepts -- chapter 3 Subgroups and homomorphisms -- chapter 4 Actions of affine groups on algebraic varieties -- chapter 5 Subgroups and semidirect products -- chapter 6 Exercises -- chapter 1 Introduction -- chapter 2 Hopf algebras and algebraic groups -- chapter 3 Rational G-modules -- chapter 4 The category of rational G-modules -- chapter 5 Representations of SL2 -- chapter 6 Characters and semi-invariants -- chapter 7 The Lie algebra associated to an affine algebraic group -- chapter 8 Explicit computations -- chapter 9 Exercises -- chapter 1 Introduction -- chapter 2 The Jordan decomposition of a single operator -- chapter 3 The Jordan decomposition of an algebra homomorphism and of a derivation -- chapter 4 The Jordan decomposition for coalgebras -- chapter 5 The Jordan decomposition for an affine algebraic group -- chapter 6 Unipotency and semisimplicity -- chapter 7 The solvable and the unipotent radical -- chapter 8 Structure of solvable groups -- chapter 9 The classical groups -- chapter 10 Exercises -- chapter 1 Introduction -- chapter 2 Actions: examples and first properties -- chapter 3 Basic facts about the geometry of the orbits -- chapter 4 Categorical and geometric quotients -- chapter 5 Affinized quotients -- chapter 6 The subalgebra of invariants -- chapter 7 Induction and restriction of representations -- chapter 8 Exercises -- chapter 1 Introduction -- chapter 2 Embedding H-modules inside G-modules -- chapter 3 Definition of subgroups in terms of semi-invariants -- chapter 4 The coset space G/H as a geometric quotient -- chapter 5 Quotients by normal subgroups -- chapter 6 Applications and examples -- chapter 7 Exercises -- chapter 1 Introduction -- chapter 2 Correspondence between subgroups and subalgebras -- chapter 3 Algebraic Lie algebras -- chapter 4 Exercises -- chapter 1 Introduction -- chapter 2 Linear and geometric reductivity -- chapter 3 Examples of linearly and geometrically reductive groups -- chapter 4 Reductivity and the structure of the group -- chapter 5 Reductive groups are linearly reductive in characteristic zero -- chapter 6 Exercises -- chapter 1 Introduction -- chapter 2 Basic definitions -- chapter 3 Induction and observability -- chapter 4 Split and strong observability -- chapter 5 The geometric characterization of observability -- chapter 6 Exercises -- chapter 1 Introduction -- chapter 2 Geometric reductivity and observability -- chapter 3 Exact subgroups -- chapter 4 From quasi-affine to affine homogeneous spaces -- chapter 5 Exactness, Reynolds operators, total integrals -- chapter 6 Affine homogeneous spaces and exactness -- chapter 7 Affine homogeneous spaces and reductivity -- chapter 8 Exactness and integrals for unipotent groups -- chapter 9 Exercises -- chapter 1 Introduction -- chapter 2 A counterexample to Hilbert's 14th problem -- chapter 3 Reductive groups and finite generation of invariants -- chapter 4 V. Popov's converse to Nagata's theorem -- chapter 5 Partial positive answers to Hilbert's 14th problem -- chapter 6 Geometric characterization of Grosshans pairs -- chapter 7 Exercises -- chapter 1 Introduction -- chapter 2 Actions by reductive groups: the semigeometric quotient -- chapter 3 Actions by reductive groups: the geometric quotient -- chapter 4 Canonical forms of matrices: a geometric perspective -- chapter 5 Rosenlicht's theorem -- chapter 6 Induced actions and homogeneous fiber bundles -- chapter 7 Revisiting affinized quotients -- chapter 8 Further results on invariants of finite groups -- chapter 9 Exercises -- chapter 1 Introduction -- chapter 2 Basic definitions -- chapter 3 Observable actions and unipotency -- chapter 5 The algebraic viewpoint on observable actions -- chapter 6 Observable actions of reductive groups -- chapter 7 Exercises -- chapter 1 Introduction -- chapter 2 One parameter subgroups and actions of Gm -- chapter 3 Reductive groups acting on affine algebraic varieties -- chapter 4 Actions of reductive groups on projective varieties -- chapter 5 Exercises -- chapter 1 Introduction -- chapter 3 Rings and modules -- chapter 4 Representations.
Walter Ricardo Ferrer Santos, Alvaro Rittatore.
Includes bibliographical references and indexes.
Boca Raton, FL : CRC Press, Taylor & Francis Group, [2017]
Monographs and research notes in mathematics