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內容簡介

For this printing， I have corrected some errors and made numerous minor changes in the interest of clarity. The most significant corrections occur in Sections 4.2， 4.3， 5.5， 30.3， 32.1， and 32.3. I have also updated the biblio-graphy to some extent. Thanks are due to a number of readers who took the trouble to point out errors， or obscurities; especially helpful were the detailed comments of Jose Antonio Vargas.

內頁插圖

目錄

I.AlgebraicGeometry
0.SomeCommutativeAlgebra
1.AffineandProjectiveVarieties
1.1 IdealsandAflineVarieties
1.2 ZariskiTopologyonAffineSpace
1.3 IrreducibleComponents
1.4 ProductsofAffineVarieties
1.5 AffineAlgebrasandMorphisms
1.6 ProjectiveVarieties
1.7 ProductsofProjectiveVarieties
1.8 FlagVarieties

2.Varieties
2.1 LocalRings
2.2 Prevarieties
2.3 Morphisms
2.4 Products
2.5 HausdorffAxiom

3.Dimension
3.1 DimensionofaVariety
3.2 DimensionofaSubvariety
3.3 DimensionTheorem
3.4 Consequences

4.Morphisms
4.1 FibresofaMorphism
4.2 FiniteMorphisms
4.3 ImageofaMorphism
4.4 ConstructibleSets
4.5 OpenMorphisms
4.6 BijectiveMorphisms
4.7 BirationalMorphisms

5.TangentSpaces
5.1 ZariskiTangentSpace
5.2 ExistenceofSimplePoints
5.3 LocalRingofaSimplePoint
5.4 DifferentialofaMorphism
5.5 DifferentialCriterionforSeparability

6.CompleteVarieties
6.1 BasicProperties
6.2 CompletenessofProjectiveVarieties
6.3 VarietiesIsomorphictoP
6.4 AutomorphismsofP
II.AflineAlgebraicGroups

7.BasicConceptsandExamples
7.1 TheNotionofAlgebraicGroup
7.2 SomeClassicalGroups
7.3 IdentityComponent
7.4 SubgroupsandHomomorphisms
7.5 GenerationbyIrreducibleSubsets
7.6 HopfAIgebras

8.ActionsofAlgebraicGroupsonVarieties
8.1 GroupActions
8.2 ActionsofAlgebraicGroups
8.3 ClosedOrbits
8.4 SemidirectProducts
8.5 TranslationofFunctions
8.6 LinearizationofAffineGroups
III.LieAlgebras

9.LieAlgebraofanAlgebraicGroup
9.1 LieAlgebrasandTangentSpaces
9.2 Convolution
9.3 Examples
9.4 SubgroupsandLieSubalgebras
9.5 DualNumbers

10.Differentiation
10.1 SomeElementaryFormulas
10.2 DifferentialofRightTranslation
10.3 TheAdjointRepresentation
10.4 DifferentialofAd
10.5 Commutators
10.6 Centralizers
10.7 AutomorphismsandDerivations
IV.HomogeneousSpaces

11.ConstructionofCertainRepresentations
11.1 ActiononExteriorPowers
11.2 ATheoremofChevalley
11.3 PassagetoProjectiveSpace
11.4 CharactersandSemi-lnvariants
11.5 NormalSubgroups

12.Quotients
12.1 UniversalMappingProperty
12.2 TopologyofY
12.3 FunctionsonY
12.4 Complements
12.5 Characteristic0
V.Characteristic0Theory

13.CorrespondenceBetweenGroupsandLieAlgebras
13.1 TheLatticeCorrespondence
13.2 InvariantsandInvariantSubspaces
13.3 NormalSubgroupsandIdeals
13.4 CentersandCentralizers
13.5 SemisimpleGroupsandLieAlgebras

14.SemisimpleGroups
14.1 TheAdjointRepresentation
14.2 SubgroupsoraSemisimpleGroup
14.3 CompleteReducibilityofRepresentations
VI.SemisimpleandUnipotentElements

15.Jordan-ChevalleyDecomposition
15.1 DecompositionofaSingleEndomorphism
15.2 GL（n，K）andgl（n，K）
15.3 JordanDecompositioninAlgebraicGroups
15.4 CommutingSetsofEndomorphisms
15.5 StructureofCommutativeAlgebraicGroups

16.DiagonalizableGroups
16.1 Charactersandd-Groups
16.2 Tori
16.3 RigidityofDiagonalizableGroups
16.4 WeightsandRoots
VII.SolvableGroups

17.NilpotentandSolvableGroups
17.1 AGroup-TheoreticLemma
17.2 CommutatorGroups
17.3 SolvableGroups
17.4 NilpotentGroups
17.5 UnipotentGroups
17.6 Lie-KolchinTheorem

18.SemisimpleElements
18.1 GlobalandInfinitesimalCentralizers
18.2 ClosedConjugacyClasses
18.3 ActionofaSemisimpleElementonaUnipotentGroup
18.4 ActionofaDiagonalizableGroup

19.ConnectedSolvableGroups
19.1 AnExactSequence
19.2 TheNilpotentCase
19.3 TheGeneralCase
19.4 NormalizerandCentralizer
19.5 SolvableandUnipotentRadicals

20.OneDimensionalGroups
20.1 CommutativityofG
20.2 VectorGroupsande-Groups
20.3 Propertiesofp-Polynomials
20.4 AutomorphismsofVectorGroups
20.5 TheMainTheorem
VIII.BorelSubgroups

21.FixedPointandConjugacyTheorems
21.1 ReviewofCompleteVarieties
21.2 FixedPointTheorem
21.3 ConjugacyofBorelSubgroupsandMaximalTori
21.4 FurtherConsequences

22.DensityandConnectednessTheorems
22.1 TheMainLemma
22.2 DensityTheorem
22.3 ConnectednessTheorem
22.4 BorelSubgroupsofCG（S）
22.5 CartanSubgroups：Summary

23.NormalizerTheorem
23.1 StatementoftheTheorem
23.2 ProofoftheTheorem
23.3 TheVarietyG/B
23.4 Summary
IX.CentralizersofTori

24.RegularandSingularTori
24.1 WeylGroups
24.2 RegularTori
24.3 SingularToriandRoots
24.4 Regular1-ParameterSubgroups

25.ActionofaMaximalTorusonG/B
25.1 Actionofa1-ParameterSubgroup
25.2 ExistenceofEnoughFixedPoints
25.3 GroupsofSemisimpleRank1
25.4 WeylChambers

26.TheUnipotentRadical
26.1 CharacterizationofRu（G）
26.2 SomeConsequences
26.3 TheGroupsUa
X.StructureofReductiveGroups

27.TheRootSystem
27.1 AbstractRootSystems
27.2 TheIntegralityAxiom
27.3 SimpleRoots
27.4 TheAutomorphismGroupofaSemisimpleGroup
27.5 SimpleComponents

28.BruhatDecomposition
28.1 T-StableSubgroupsofBu
28.2 GroupsofSemisimpleRank1
28.3 TheBruhatDecomposition
28.4 NormalForminG
28.5 Complements

29.TitsSystems
29.1 Axioms
29.2 BruhatDecomposition
29.3 ParabolicSubgroups
29.4 GeneratorsandRelationsforW
29.5 NormalSubgroupsofG

30.ParabolicSubgroups
30.1 StandardParabolicSubgroups
30.2 LeviDecompositions
30.3 ParabolicSubgroupsAssociatedtoCertainUnipotentGroups
30.4 MaximalSubgroupsandMaximalUnipotentSubgroups
XI.RepresentationsandClassificationofSemisimpleGroups

31.Representations
31.1 Weights
31.2 MaximalVectors
31.3 IrreducibleRepresentations
31.4 ConstructionofIrreducibleRepresentations
31.5 MultiplicitiesandMinimalHighestWeights
31.6 ContragredientsandInvariantBilinearForms

32.IsomorphismTheorem
32.1 TheClassificationProblem
32.2 ExtensionofψTtoN（T）
32.3 ExtensionofψTtoZa
32.4 ExtensionofψTtoTUa
32.5 ExtensionofψTtoB
32.6 Multiplicativityofψ

33.RootSystemsofRank2
33.1 Reformulationof（A），（B），（C）
33.2 SomePreliminaries
33.3 TypeA2
33.4 TypeB2
33.5 TypeG2
33.6 TheExistenceProblem
XII.SurveyofRationalityProperties

34.FieldsofDefinition
34.1 Foundations
34.2 ReviewofEarlierChapters
34.3 Tori
34.4 SomeBasicTheorems
34.5 Borei-TitsStructureTheory
34.6 AnExample：OrthogonalGroups

35.SpecialCases
35.1 SplitandQuasisplitGroups
35.2 FiniteFields
35.3 TheRealField
35.4 LocalFields
35.5 Classification
Appendix.RootSystems
Bibliography
IndexofTerminology
IndexofSymbols

精彩書摘

Over the last two decades the Borel-Chevalley theory of Iinear algebraic groups（as further developed by Borel，Steinberg，Tits，and others）has made possible significant progress In a aurabef of areas：scmisimple Lie groups and arithmetic subgroups，p-adic groups，classical linear groups，finite simple groups，invariant theory。etc.Unfortunately，the subject has not been as accessible as it ought to be.in part due to the fairly substantial background in algebraic geometry assumed by Chevalley ，Borei ， Borel，Tits .The difliculty of the theory also stems in Dart from the fact that the main results culminate a Iong series of arguments which are hard to“see through”from beginning to end.In writing this introductory text. aimed at the second year graduate level.I have tried to take these factors into account.
First.the requisite algebraic geometry has been treated in fullin Chapter I.modulo some more.or-less standard results from commutative algebra （quoted in§o），e.g.，the theorem that a regular local ring is an integrally closed domain.The treatment is intentionally somewhat crude and is not at all scheme-oriented.In fact.everything is done over an algebraically closed field K（of arbitrary characteristic）.even though most of the eventual applications involve a feld of definition k.I believe this c.an be iustified as follows.In order to work over k from the outset，it would be necessary to spend a good deal of time perfecting the foundations.and then the only rationality statements proved along the way would be Of a minor sort rcf （34.2）） 綫性代數群 [Linear Algebraic Groups] 下載 mobi epub pdf txt 電子書

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