量子群

Preface
PartOneQuantumSL(2)
IPreliminaries
1AlgebrasandModules
2FreeAlgebras
3TheAffineLineandPlane
4MatrixMultiplication
5DeterminantsandInvertibleMatrices
6GradedandFilteredAlgebras
7OreExtensions
8NoetherianRings
9Exercises
10Notes
II
TensorProducts
1TensorProductsofVectorSpaces
2TensorProductsofLinearMaps
3DualityandTraces
4TensorProductsofAlgebras
5TensorandSymmetricAlgebras
6Exercises
7Notes
III
TheLanguageofHopfAlgebras
1Coalgebras
2Bialgebras
3HopfAlgebras
4RelationshipwithChapterI.TheHopfAlgebrasGL(2)
andSL(2)
5ModulesoveraHopfAlgebra
6Comodules
7Comodule-Algebras.CoactionofSL(2)ontheAffinePlane
8Exercises
9Notes
IV
TheQuantumPlaneandItsSymmetries
ITheQuantumPlane
2GaussPolynomialsandtheq-BinomialFormula
3TheAlgebraMq(2)
4Ring-TheoreticalPropertiesofMq(2)
5BialgebraStructureonMq(2)
6TheHopfAlgebrasGLq(2)andSLq(2)
7CoactionontheQuantumPlane
8Hopf*-Algebras
9Exercises
10Notes
V
TheLieAlgebraofSL(2)
1LieAlgebras
2EnvelopingAlgebras
3TheLieAlgebrasl(2)
4Representationsofsl(2)
5TheClebsch-GordanFormula
6Module-AlgebraoveraBialgebra.Actionofsl(2)onthe
AffinePlane
7DualitybetweentheHopfAlgebrasU(sl(2))andSL(2)
8Exercises
9Notes
VI
TheQuantumEnvelopingAlgebraof5[(2)
1TheAlgebraUq(sl(2))
2RelationshipwiththeEnvelopingAlgebraof5[(2)
3RepresentationsofUq
4TheHarish-ChandraHomomorphismandtheCentreofUq
5CasewhenqisaRootofUnity
6Exercises
7Notes
VII
AHopfAlgebraStructureonUq(Sl(2))
1Comultiplication
2Semisimplicity
3ActionofUq(sl(2))ontheQuantumPlane
4DualitybetweentheHopfAlgebrasUq(sl(2))andSLq(2)
5DualitybetweenUq(sl(2))-ModulesandSLq(2)-Comodules
6ScalarProductsonUq(sl(2))-Modules
7QuantumClebsch-Gordan
8Exercises
9Notes
PartTwoUniversalR-Matrices
VIII
TheYang-BaxterEquationand(Co)BraidedBialgebras
1TheYang-BaxterEquation
2BraidedBialgebras
3HowaBraidedBialgebraGeneratesR-Matrices
4TheSquareoftheAntipodeinaBraidedHopfAlgebra
5ADualConcept:CobraidedBialgebras
6TheFRTConstruction
7ApplicationtoGLq(2)andSLq(2)
8Exercises
9Notes
IX
Drinfeld'sQuantumDouble
1BicrossedProductsofGroups
2BicrossedProductsofBialgebras
3VariationsontheAdjointRepresentation
4Drinfeld'sQuantumDouble
5Representation-TheoreticInterpretationofthe
QuantumDouble
6ApplicationtoUq(sl(2))
7R-MatricesforUq
8Exercises
9Notes
PartThreeLow-DimensionalTopologyand
TensorCategories
X
Knots,Links,Tangles,andBraids
1KnotsandLinks
2ClassificationofLinksuptoIsotopy
3LinkDiagrams
4TheJones-ConwayPolynomial
5Tangles
6Braids
7Exercises
8Notes
9Appendix.TheFundamentalGroup
XI
TensorCategories
1TheLanguageofCategoriesandFunctors
2TensorCategories
3ExamplesofTensorCategories
4TensorFunctors
5TurningTensorCategoriesintoStrictOnes
6Exercises
7Notes
XII
TheTangleCategory
1PresentationofaStrictTensorCategory
2TheCategoryofTangles
3TheCategoryofTangleDiagrams
4RepresentationsoftheCategoryofTangles
5ExistenceProofforJones-ConwayPolynomial
6Exercises
7Notes
XIII
Braidings
1BraidedTensorCategories
2TheBraidCategory
3UniversalityoftheBraidCategory
4TheCentreConstruction
5ACategoricalInterpretationoftheQuantumDouble
6Exercises
7Notes
XIV
DualityinTensorCategories
1RepresentingMorphismsinaTensorCategory
2Duality
3RibbonCategories
4QuantumTraceandDimension
5ExamplesofRibbonCategories
6RibbonAlgebras
7Exercises
8Notes
XV
Quasi-Bialgebras
1Quasi-Bialgebras
2BraidedQuasi-Bialgebras
3GaugeTransformations
4BraidGroupRepresentations
5Quasi-HopfAlgebras
6Exercises
7Notes
PartFourQuantumGroupsandMonodromy
XVI
GeneralitiesonQuantumEnvelopingAlgebras
ITheRingofFormalSeriesandh-AdicTopology
2TopologicallyFreeModules
3TopologicalTensorProduct
4TopologicalAlgebras
5QuantumEnvelopingAlgebras
6SymmetrizingtheUniversalR-Matrix
7Exercises
8Notes
9Appendix.InverseLimits
XVII
DrinfeldandJimbo'sQuantumEnvelopingAlgebras
1SemisimpleLieAlgebras
2Drinfeld-JimboAlgebras
3QuantumGroupInvariantsofLinks
4TheCaseofsi(2)
5Exercises
6Notes
XVIII
CohomologyandRigidityTheorems
1CohomologyofLieAlgebras
2RigidityforLieAlgebras
3VanishingResultsforSemisimpleLieAlgebras
4ApplicationtoDrinfeld-JimboQuantumEnvelopingAlgebras
5CohomologyofCoalgebras
6ActionofaSemisimpleLieAlgebraontheCobarComplex
7ComputationsforSymmetricCoalgebras
8UniquenessTheoremforQuantumEnvelopingAlgebras
9Exercises
10Notes
11Appendix.ComplexesandResolutions
XIX
MonodromyoftheKnizhnik-ZamolodchikovEquations
1Connections
2BraidGroupRepresentationsfromMonodromy
3TheKnizhnik-ZamolodchikovEquations
4TheDrinfeld-KohnoTheorem
5EquivalenceofUh(g)andAg.t
6Drinfeld'sAssociator
7ConstructionoftheTopologicalBraidedQuasi-BialgebraAg.t
8VerificationoftheAxioms
9Exercises
10Notes
11Appendix.IteratedIntegrals
XX
Postlude.AUniversalKnotInvariant
1KnotInvariantsofFiniteType
2ChordDiagramsandKontsevich'sTheorem
3AlgebraStructuresonChordDiagrams
4InfinitesimalSymmetricCategories
5AUniversalCategoryforInfinitesimalBraidings
6FormalIntegrationofInfinitesimalSymmetricCategories
7ConstructionofKontsevich'sUniversalInvariant
8RecoveringQuantumGroupInvariants
9Exercises
10Notes
References
Index