Actions and invariants of algebraic groups / Walter Ricardo Ferrer Santos, Alvaro Rittatore
- Author:
- Ferrer Santos, Walter Ricardo
- Published:
- Boca Raton, FL : CRC Press, Taylor & Francis Group, [2017]
- Edition:
- Second edition.
- Physical Description:
- xx, 459 pages ; 26 cm.
- Additional Creators:
- Rittatore, Alvaro
- Series:
- Contents:
- Machine generated contents note: 1.Algebraic geometry: basic definitions and results -- 1.Introduction -- 2.Commutative algebra -- 2.1.Ring and field extensions -- 2.2.Hilbert's Nullstellensatz -- 2.3.Separability -- 2.4.Faithfully flat ring extensions -- 2.5.Regular local rings -- 3.Algebraic subsets of the affine space -- 3.1.Basic definitions -- 3.2.The Zariski topology -- 3.3.Polynomial maps. Morphisms -- 4.Algebraic varieties -- 4.1.Sheaves on topological spaces -- 4.2.The maximal spectrum -- 4.3.Affine algebraic varieties -- 4.4.Algebraic varieties -- 5.Exercises -- 2.Algebraic varieties -- 1.Introduction -- 2.Morphisms of algebraic varieties -- 3.Complete varieties -- 4.Singular points and normal varieties -- 5.The Proj variety associated to a graded algebra -- 6.Deeper results on morphisms -- 7.Algebraic varieties and k-schemes -- 8.Exercises -- 3.Lie algebras -- 1.Introduction -- 2.Definitions and basic concepts -- 3.The theorems of F. Engel and S. Lie -- 4.Semisimple Lie algebras -- 5.Cohomology of Lie algebras -- 6.The theorems of H. Weyl and F. Levi -- 7.p-Lie algebras -- 8.Exercises -- 4.Algebraic groups: basic definitions -- 1.Introduction -- 2.Definitions and basic concepts -- 3.Subgroups and homomorphisms -- 4.Actions of affine groups on algebraic varieties -- 5.Subgroups and semidirect products -- 6.Exercises -- 5.Algebraic groups: Lie algebras and representations -- 1.Introduction -- 2.Hopf algebras and algebraic groups -- 3.Rational G-modules -- 4.The category of rational G-modules -- 5.Representations of SL2 -- 6.Characters and semi-invariants -- 7.The Lie algebra associated to an affine algebraic group -- 8.Explicit computations -- 9.Exercises -- 6.Algebraic groups: Jordan decomposition and applications -- 1.Introduction -- 2.The Jordan decomposition of a single operator -- 3.The Jordan decomposition of an algebra homomorphism and of a derivation -- 4.The Jordan decomposition for coalgebras -- 5.The Jordan decomposition for an affine algebraic group -- 6.Unipotency and semisimplicity -- 7.The solvable and the unipotent radical -- 8.Structure of solvable groups -- 9.The classical groups -- 9.1.The general linear group GLn -- 9.2.The special linear group SLn (case A) -- 9.3.The projective general linear group PGLn (k) (case A) -- 9.4.The special orthogonal group SOn (cases B, D) -- 9.5.The symplectic group Spn, n = 2m (case C) -- 10.Exercises -- 7.Actions of algebraic groups -- 1.Introduction -- 2.Actions: examples and first properties -- 3.Basic facts about the geometry of the orbits -- 4.Categorical and geometric quotients -- 5.Affinized quotients -- 6.The subalgebra of invariants -- 7.Induction and restriction of representations -- 8.Exercises -- 8.Homogeneous spaces -- 1.Introduction -- 2.Embedding H-modules inside G-modules -- 3.Definition of subgroups in terms of semi-invariants -- 4.The coset space G/H as a geometric quotient -- 5.Quotients by normal subgroups -- 6.Applications and examples -- 7.Exercises -- 9.Algebraic groups and Lie algebras in characteristic zero -- 1.Introduction -- 2.Correspondence between subgroups and subalgebras -- 3.Algebraic Lie algebras -- 4.Exercises -- 10.Reductivity -- 1.Introduction -- 2.Linear and geometric reductivity -- 3.Examples of linearly and geometrically reductive groups -- 4.Reductivity and the structure of the group -- 5.Reductive groups are linearly reductive in characteristic zero -- 6.Exercises -- 11.Observable subgroups of affine algebraic groups -- 1.Introduction -- 2.Basic definitions -- 3.Induction and observability -- 4.Split and strong observability -- 5.The geometric characterization of observability -- 6.Exercises -- 12.Affine homogeneous spaces -- 1.Introduction -- 2.Geometric reductivity and observability -- 3.Exact subgroups -- 4.From quasi-affine to affine homogeneous spaces -- 5.Exactness, Reynolds operators, total integrals -- 6.Affine homogeneous spaces and exactness -- 7.Affine homogeneous spaces and reductivity -- 8.Exactness and integrals for unipotent groups -- 9.Exercises -- 13.Hilbert's 14th problem -- 1.Introduction -- 2.A counterexample to Hilbert's 14th problem -- 3.Reductive groups and finite generation of invariants -- 4.V. Popov's converse to Nagata's theorem -- 5.Partial positive answers to Hilbert's 14th problem -- 6.Geometric characterization of Grosshans pairs -- 7.Exercises -- 14.Quotient varieties: basic results -- 1.Introduction -- 2.Actions by reductive groups: the semigeometric quotient -- 3.Actions by reductive groups: the geometric quotient -- 4.Canonical forms of matrices: a geometric perspective -- 5.Rosenlicht's theorem -- 6.Induced actions and homogeneous fiber bundles -- 7.Revisiting affinized quotients -- 8.Further results on invariants of finite groups -- 8.1.Invariants of graded algebras -- 8.2.Polynomial subalgebras of polynomial algebras -- 8.3.The case of a group generated by reflections -- 8.4.The degree of the fundamental invariants for a finite group -- 9.Exercises -- 15.Observable actions of affine algebraic groups -- 1.Introduction -- 2.Basic definitions -- 3.Observable actions and unipotency -- 4.The geometry of observable actions -- 5.The algebraic viewpoint on observable actions -- 6.Observable actions of reductive groups -- 7.Exercises -- 16.Quotient varieties: an introduction to geometric invariant theory -- 1.Introduction -- 2.One parameter subgroups and actions of Gm -- 2.1.One parameter subgroups -- 2.2.Actions of Gm on affine varieties -- 3.Reductive groups acting on affine algebraic varieties -- 3.1.Stable points --- affine case -- 3.2.Semistable points --- affine case -- 3.3.Hilbert-Mumford criterion in the affine case -- 4.Actions of reductive groups on projective varieties -- 4.1.Linear actions on the projective space -- 4.2.Actions on projective varieties -- 5.Exercises -- Appendix: basic definitions and results -- 1.Introduction -- 2.Notations -- 2.1.Category theory -- 2.2.General topology -- 2.3.Linear algebra -- 2.4.Group theory -- 3.Rings and modules -- 4.Representations.
- Subject(s):
- ISBN:
- 9781482239157 hardcover ; alkaline paper
1482239159 hardcover ; alkaline paper - Bibliography Note:
- Includes bibliographical references and indexes.
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