线性代数群

Ⅰ.AlgebraicGeometry

0.SomeCommutativeAlgebra

1.AffineandProjectiveVarieties

1.1IdealsandAflineVarieties

1.2ZariskiTopologyonAffineSpace

1.3IrreducibleComponents

1.4ProductsofAffineVarieties

1.5AffineAlgebrasandMorphisms

1.6ProjectiveVarieties

1.7ProductsofProjectiveVarieties

1.8FlagVarieties

2.Varieties

2.1LocalRings

2.2Prevarieties

2.3Morphisms

2.4Products

2.5HausdorffAxiom

3.Dimension

3.1DimensionofaVariety

3.2DimensionofaSubvariety

3.3DimensionTheorem

3.4Consequences

4.Morphisms

4.1FibresofaMorphism

4.2FiniteMorphisms

4.3ImageofaMorphism

4.4ConstructibleSets

4.5OpenMorphisms

4.6BijectiveMorphisms

4.7BirationalMorphisms

5.TangentSpaces

5.1ZariskiTangentSpace

5.2ExistenceofSimplePoints

5.3LocalRingofaSimplePoint

5.4DifferentialofaMorphism.

5.5DifferentialCriterionforSeparability

6.CompleteVarieties

6.1BasicProperties

6.2CompletenessofProjectiveVarieties

6.3VarietiesIsomorphictoP

6.4AutomorphismsofP

Ⅱ.AflineAlgebraicGroups

7.BasicConceptsandExamples

7.1TheNotionofAlgebraicGroup

7.2SomeClassicalGroups

7.3IdentityComponent

7.4SubgroupsandHomomorphisms.

7.5GenerationbyIrreducibleSubsets

7.6HopfAIgebras

8.ActionsofAlgebraicGroupsonVarieties

8.1GroupActions

8.2ActionsofAlgebraicGroups

8.3ClosedOrbits

8.4SemidirectProducts

8.5TranslationofFunctions

8.6LinearizationofAffineGroups

Ⅲ.LieAlgebras

9.LieAlgebraofanAlgebraicGroup

9.1LieAlgebrasandTangentSpaces

9.2Convolution

9.3Examples.

9.4SubgroupsandLieSubalgebras

9.5DualNumbers

10.Differentiation.

10.1SomeElementaryFormulas

10.2DifferentialofRightTranslation

10.3TheAdjointRepresentation

10.4DifferentialofAd

10.5Commutators

10.6Centralizers

10.7AutomorphismsandDerivations

Ⅳ.HomogeneousSpaces

11.ConstructionofCertainRepresentations

11.1ActiononExteriorPowers

11.2ATheoremofChevalley

11.3PassagetoProjectiveSpace

11.4CharactersandSemi-lnvariants.

11.5NormalSubgroups

12.Quotients

Ⅴ.CharacteristicOTheory

Ⅵ.SemisimpleandUnipotentElements

Ⅶ.SolvableGroups

Ⅷ.BorelSubgroups

Ⅸ.CentralizersofTori

Ⅹ.StructureofReductiveGroups

Ⅺ.RepresentatinsandClassficationofSemisimpleGroups

Ⅻ.SurveyofRationalityProperties

AppendixRootSystems

Bibliography

IndexofTerminology

IndexofSymbols

　　Forthisprinting,Ihavecorrectedsomeerrorsandmadenumerousminorchangesintheinterestofclarity.ThemostsignificantcorrectionsoccurinSections4.2,4.3,5.5,30.3,32.1,and32.3.Ihavealsoupdatedthebibliographytosomeextent.Thanksareduetoanumberofreaderswhotookthetroubletopointouterrors,orobscurities;especiallyhelpfulwerethedetailedcommentsofJoseAntonioVargas.