Algebra II: Textbook for Students of Mathematics

377 Pages·2017·3.78 MB·English

by  Alexey L. Gorodentsev

Alexey L. Gorodentsev
Algebra II
Textbook for Students of Mathematics
Algebra II
Alexey L. Gorodentsev
Algebra II
Textbook for Students of Mathematics
123
AlexeyL.Gorodentsev
FacultyofMathematics
NationalResearchUniversity
“HigherSchoolofEconomics”
Moscow,Russia
OriginallypublishedinRussianas“Algebra.Uchebnikdlyastudentov-matematikov.Chast’
2”,©MCCME2015
ISBN978-3-319-50852-8 ISBN978-3-319-50853-5 (eBook)
DOI10.1007/978-3-319-50853-5
LibraryofCongressControlNumber:2017930683
MathematicsSubjectClassification(2010):11.01,12.01,13.01,14.01,15.01,16.01,18.01,20.01
©SpringerInternationalPublishingAG2017
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisSpringerimprintispublishedbySpringerNature
TheregisteredcompanyisSpringerInternationalPublishingAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
Thisisthesecondpartofa2-yearcourseofabstractalgebraforstudentsbeginning
a professional study of higher mathematics.1 This textbook is based on courses
givenattheIndependentUniversityofMoscowandattheFacultyofMathematics
at the NationalResearch University Higher Schoolof Economics.In particular,it
containsalargenumberofexercisesthatwerediscussedinclass,someofwhichare
providedwithcommentaryandhints,aswellasproblemsforindependentsolution
thatwereassignedashomework.Workingouttheexercisesisofcrucialimportance
inunderstandingthesubjectmatterofthisbook.
Moscow,Russia AlexeyL.Gorodentsev
1Throughoutthisbook,thefirstvolumewillbereferredtoasAlgebraI.
v
Contents
1 TensorProducts............................................................. 1
1.1 MultilinearMaps..................................................... 1
1.1.1 MultilinearMapsBetweenFreeModules................. 1
1.1.2 UniversalMultilinearMap................................. 3
1.2 TensorProductofModules.......................................... 4
1.2.1 ExistenceofTensorProduct ............................... 5
1.2.2 LinearMapsasTensors .................................... 7
1.2.3 TensorProductsofAbelianGroups ....................... 9
1.3 Commutativity,Associativity,andDistributivityIsomorphisms... 10
1.4 TensorProductofLinearMaps...................................... 13
1.5 TensorProductofModulesPresentedbyGenerators
andRelations......................................................... 15
ProblemsforIndependentSolutiontoChapter1........................... 17
2 TensorAlgebras............................................................. 21
2.1 FreeAssociativeAlgebraofaVectorSpace........................ 21
2.2 Contractions.......................................................... 22
2.2.1 CompleteContraction...................................... 22
2.2.2 PartialContractions......................................... 23
2.2.3 LinearSupportandRankofaTensor...................... 25
2.3 QuotientAlgebrasofaTensorAlgebra ............................. 26
2.3.1 SymmetricAlgebraofaVectorSpace..................... 26
2.3.2 SymmetricMultilinearMaps .............................. 27
2.3.3 TheExteriorAlgebraofaVectorSpace................... 29
2.3.4 AlternatingMultilinearMaps.............................. 30
2.4 SymmetricandAlternatingTensors................................. 31
2.4.1 SymmetrizationandAlternation........................... 32
2.4.2 StandardBases.............................................. 33
2.5 PolarizationofPolynomials ......................................... 35
2.5.1 EvaluationofPolynomialsonVectors..................... 36
2.5.2 CombinatorialFormulaforCompletePolarization....... 37
vii
viii Contents
2.5.3 Duality ...................................................... 38
2.5.4 DerivativeofaPolynomialAlongaVector ............... 38
2.5.5 PolarsandTangentsofProjectiveHypersurfaces......... 40
2.5.6 LinearSupportofaHomogeneousPolynomial........... 43
2.6 PolarizationofGrassmannianPolynomials......................... 45
2.6.1 Duality ...................................................... 45
2.6.2 PartialDerivativesinanExteriorAlgebra................. 46
2.6.3 Linear Support of a Homogeneous
GrassmannianPolynomial ................................. 47
2.6.4 GrassmannianVarietiesandthePlückerEmbedding..... 49
2.6.5 TheGrassmannianasanOrbitSpace...................... 49
ProblemsforIndependentSolutiontoChapter2........................... 51
3 SymmetricFunctions....................................................... 57
3.1 SymmetricandSignAlternatingPolynomials...................... 57
3.2 ElementarySymmetricPolynomials................................ 60
3.3 CompleteSymmetricPolynomials.................................. 61
3.4 Newton’sSumsofPowers........................................... 62
3.4.1 GeneratingFunctionforthep ............................. 62
k
3.4.2 Transitionfrome andh top ............................. 63
k k k
3.5 Giambelli’sFormula ................................................. 65
3.6 Pieri’sFormula ....................................................... 67
3.7 TheRingofSymmetricFunctions .................................. 69
ProblemsforIndependentSolutiontoChapter3........................... 71
4 CalculusofArrays,Tableaux,andDiagrams ........................... 75
4.1 Arrays................................................................. 75
4.1.1 NotationandTerminology ................................. 75
4.1.2 VerticalOperations......................................... 76
4.1.3 CommutationLemma ...................................... 77
4.2 Condensing........................................................... 79
4.2.1 CondensedArrays.......................................... 79
4.2.2 BidenseArraysandYoungDiagrams ..................... 80
4.2.3 YoungTableaux............................................. 81
4.2.4 YamanouchiWords......................................... 82
4.2.5 FiberProductTheorem..................................... 83
4.3 ActionoftheSymmetricGrouponDU-Sets ....................... 86
4.3.1 DU-SetsandDU-Orbits.................................... 86
4.3.2 ActionofS DAut.J/..................................... 86
m
4.4 CombinatorialSchurPolynomials................................... 88
4.5 TheLittlewood–RichardsonRule ................................... 91
4.5.1 TheJacobi–TrudiIdentity.................................. 93
4.5.2 Transitionfrome(cid:2)andh(cid:2)tos(cid:2) ............................ 93
4.6 TheInnerProductonƒ.............................................. 95
ProblemsforIndependentSolutiontoChapter4........................... 96
Contents ix
5 BasicNotionsofRepresentationTheory................................. 99
5.1 RepresentationsofaSetofOperators............................... 99
5.1.1 AssociativeEnvelope....................................... 99
5.1.2 Decomposabilityand(Semi)/Simplicity .................. 100
5.1.3 HomomorphismsofRepresentations...................... 103
5.2 RepresentationsofAssociativeAlgebras ........................... 104
5.2.1 DoubleCentralizerTheorem............................... 104
5.2.2 Digression:ModulesOverNoncommutativeRings ...... 106
5.3 IsotypicComponents................................................. 107
5.4 RepresentationsofGroups........................................... 109
5.4.1 DirectSumsandTensorConstructions.................... 109
5.4.2 RepresentationsofFiniteAbelianGroups................. 111
5.4.3 ReynoldsOperator.......................................... 113
5.5 GroupAlgebras....................................................... 114
5.5.1 CenterofaGroupAlgebra................................. 115
5.5.2 IsotypicDecompositionofaFiniteGroupAlgebra....... 115
5.6 SchurRepresentationsofGeneralLinearGroups .................. 121
5.6.1 ActionofGL.V/(cid:2)S onV˝n ............................. 122
n
5.6.2 TheSchur–WeylCorrespondence ......................... 124
ProblemsforIndependentSolutiontoChapter5........................... 124
6 RepresentationsofFiniteGroupsinGreaterDetail.................... 131
6.1 OrthogonalDecompositionofaGroupAlgebra.................... 131
6.1.1 InvariantScalarProductandPlancherel’sFormula....... 131
6.1.2 IrreducibleIdempotents.................................... 133
6.2 Characters............................................................. 134
6.2.1 Definition,Properties,andExamplesofComputation.... 134
6.2.2 TheFourierTransform ..................................... 137
6.2.3 RingofRepresentations.................................... 140
6.3 InducedandCoinducedRepresentations............................ 141
6.3.1 Restricted and InducedModules Over
AssociativeAlgebras....................................... 141
6.3.2 InducedRepresentationsofGroups........................ 142
6.3.3 TheStructureofInducedRepresentations................. 143
6.3.4 CoinducedRepresentations ................................ 146
ProblemsforIndependentSolutiontoChapter6........................... 148
7 RepresentationsofSymmetricGroups................................... 151
7.1 ActionofS onFilledYoungDiagrams............................. 151
n
7.1.1 RowandColumnSubgroupsAssociatedwith
aFilling ..................................................... 151
7.1.2 YoungSymmetrizerss Dr (cid:3)c ......................... 153
T T T
7.1.3 YoungSymmetrizerss0 Dc (cid:3)r ......................... 155
T T T
7.2 ModulesofTabloids ................................................. 157
x Contents
7.3 SpechtModules ...................................................... 159
7.3.1 DescriptionandIrreducibility.............................. 159
7.3.2 StandardBasisNumberedbyYoungTableaux............ 160
7.4 RepresentationRingofSymmetricGroups......................... 161
7.4.1 Littlewood–RichardsonProduct ........................... 162
7.4.2 ScalarProducton<........................................ 163
7.4.3 TheIsometricIsomorphism<⥲ƒ....................... 164
7.4.4 DimensionsofIrreducibleRepresentations............... 168
ProblemsforIndependentSolutiontoChapter7........................... 170
8 sl2-Modules ................................................................. 173
8.1 LieAlgebras.......................................................... 173
8.1.1 UniversalEnvelopingAlgebra............................. 173
8.1.2 RepresentationsofLieAlgebras........................... 174
8.2 Finite-DimensionalSimplesl2-Modules............................ 176
8.3 SemisimplicityofFinite-Dimensionalsl2-Modules................ 179
ProblemsforIndependentSolutiontoChapter8........................... 183
9 CategoriesandFunctors................................................... 187
9.1 Categories............................................................. 187
9.1.1 ObjectsandMorphisms.................................... 187
9.1.2 Mono-,Epi-,andIsomorphisms........................... 189
9.1.3 ReversingofArrows........................................ 190
9.2 Functors............................................................... 191
9.2.1 CovariantFunctors ......................................... 191
9.2.2 Presheaves .................................................. 192
9.2.3 TheFunctorsHom.......................................... 195
9.3 NaturalTransformations............................................. 197
9.3.1 EquivalenceofCategories.................................. 198
9.4 RepresentableFunctors .............................................. 200
9.4.1 DefinitionsviaUniversalProperties....................... 203
9.5 AdjointFunctors ..................................................... 205
9.5.1 TensorProductsVersusHomFunctors.................... 206
9.6 LimitsofDiagrams................................................... 213
9.6.1 (Co)completeness.......................................... 217
9.6.2 FilteredDiagrams........................................... 218
9.6.3 FunctorialPropertiesof(Co)limits........................ 219
ProblemsforIndependentSolutiontoChapter9........................... 222
10 ExtensionsofCommutativeRings........................................ 227
10.1 IntegralElements..................................................... 227
10.1.1 DefinitionandPropertiesofIntegralElements............ 227
10.1.2 AlgebraicIntegers.......................................... 230
10.1.3 NormalRings............................................... 231
10.2 ApplicationstoRepresentationTheory ............................. 232
10.3 AlgebraicElementsinAlgebras..................................... 234
Contents xi
10.4 TranscendenceGenerators........................................... 236
ProblemsforIndependentSolutiontoChapter10 ......................... 239
11 AffineAlgebraicGeometry................................................ 241
11.1 SystemsofPolynomialEquations................................... 241
11.2 AffineAlgebraic–GeometricDictionary............................ 243
11.2.1 CoordinateAlgebra......................................... 243
11.2.2 MaximalSpectrum ......................................... 244
11.2.3 PullbackHomomorphisms................................. 246
11.3 ZariskiTopology ..................................................... 250
11.3.1 IrreducibleComponents.................................... 251
11.4 RationalFunctions ................................................... 253
11.4.1 TheStructureSheaf......................................... 254
11.4.2 PrincipalOpenSetsasAffineAlgebraicVarieties........ 255
11.5 GeometricPropertiesofAlgebraHomomorphisms................ 256
11.5.1 ClosedImmersions......................................... 257
11.5.2 DominantMorphisms ...................................... 257
11.5.3 FiniteMorphisms........................................... 258
11.5.4 NormalVarieties............................................ 259
ProblemsforIndependentSolutiontoChapter11 ......................... 261
12 AlgebraicManifolds........................................................ 265
12.1 DefinitionsandExamples............................................ 265
12.1.1 StructureSheafandRegularMorphisms.................. 268
12.1.2 ClosedSubmanifolds....................................... 268
12.1.3 FamiliesofManifolds...................................... 269
12.1.4 SeparatedManifolds........................................ 269
12.1.5 RationalMaps .............................................. 271
12.2 ProjectiveVarieties................................................... 272
12.3 ResultantSystems.................................................... 274
12.3.1 ResultantofTwoBinaryForms............................ 276
12.4 ClosenessofProjectiveMorphisms................................. 278
12.4.1 FiniteProjections........................................... 279
12.5 DimensionofanAlgebraicManifold ............................... 281
12.5.1 DimensionsofSubvarieties................................ 283
12.5.2 DimensionsofFibersofRegularMaps.................... 285
12.6 DimensionsofProjectiveVarieties.................................. 286
ProblemsforIndependentSolutiontoChapter12 ......................... 290
13 AlgebraicFieldExtensions ................................................ 295
13.1 FiniteExtensions..................................................... 295
13.1.1 PrimitiveExtensions........................................ 296
13.1.2 Separability ................................................. 297
13.2 ExtensionsofHomomorphisms..................................... 300
13.3 SplittingFieldsandAlgebraicClosures............................. 302
13.4 NormalExtensions................................................... 304