Quasi-Hopf Algebras: A Categorical Approach (Encyclopedia of Mathematics and its Applications)

546 Pages·2019·4.352 MB·English

by  Daniel Bulacu, Stefaan Caenepeel, Florin Panaite, Freddy Van Oystaeyen

QUASI-HOPF ALGEBRAS
ThisisthefirstbooktobededicatedentirelytoDrinfeld’squasi-Hopfalgebras.
Idealforgraduatestudentsandresearchersinmathematicsandmathematical
physics,thistreatmentislargelyself-contained,takingthereaderfromthebasics,
withcompleteproofs,tomuchmoreadvancedtopics,withalmostcompleteproofs.
Manyoftheproofsarebasedongeneralcategoricalresults;thesameapproachcan
thenbeusedinthestudyofotherHopf-typealgebras,forexampleTuraevorZunino
Hopfalgebras,Hom-Hopfalgebras,Hopfishalgebras,andingeneralanyalgebrafor
whichthecategoryofrepresentationsismonoidal.
Newcomerstothesubjectwillappreciatethedetailedintroductionto(braided)
monoidalcategories,(co)algebrasandtheothertoolstheywillneedinthisarea.
Moreadvancedreaderswillbenefitfromhavingrecentresearchgatheredinone
place,withopenquestionstoinspiretheirownresearch.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationin
mathematicsormathematicalscienceandforwhichadetaileddevelopmentofthe
abstracttheoryislessimportantthanathoroughandconcreteexplorationofthe
implicationsandapplications.
BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir
subjectscomprehensively.Lessimportantresultsmaybesummarizedasexercisesat
theendsofchapters.Fortechnicalities,readerscanbereferredtothebibliography,
whichisexpectedtobecomprehensive.Asaresult,volumesareencyclopedic
referencesormanageableguidestomajorsubjects.
EncyclopediaofMathematicsanditsApplications
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122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabischetal.AggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.TuckerTopicsinTopological
GraphTheory
129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
131 H.MorimotoStochasticControlandMathematicalModeling
132 G.SchmidtRelationalMathematics
133 P.KornerupandD.W.MatulaFinitePrecisionNumberSystemsandArithmetic
134 Y.CramaandP.L.Hammer(eds.)BooleanModelsandMethodsinMathematics,ComputerScience,
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135 V.Berthe´andM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
136 A.Krista´ly,V.D.Ra˘dulescuandC.VargaVariationalPrinciplesinMathematicalPhysics,Geometry,
andEconomics
137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications
138 B.CourcelleandJ.EngelfrietGraphStructureandMonadicSecond-OrderLogic
139 M.FiedlerMatricesandGraphsinGeometry
140 N.VakilRealAnalysisthroughModernInfinitesimals
141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation
142 Y.CramaandP.L.HammerBooleanFunctions
143 A.Arapostathis,V.S.BorkarandM.K.GhoshErgodicControlofDiffusionProcesses
144 N.Caspard,B.LeclercandB.MonjardetFiniteOrderedSets
145 D.Z.ArovandH.DymBitangentialDirectandInverseProblemsforSystemsofIntegraland
DifferentialEquations
146 G.DassiosEllipsoidalHarmonics
147 L.W.BeinekeandR.J.Wilson(eds.)withO.R.OellermannTopicsinStructuralGraphTheory
148 L.Berlyand,A.G.KolpakovandA.NovikovIntroductiontotheNetworkApproximationMethodfor
MaterialsModeling
149 M.BaakeandU.GrimmAperiodicOrderI:AMathematicalInvitation
150 J.Borweinetal.LatticeSumsThenandNow
151 R.SchneiderConvexBodies:TheBrunn–MinkowskiTheory(SecondEdition)
152 G.DaPratoandJ.ZabczykStochasticEquationsinInfiniteDimensions(SecondEdition)
153 D.Hofmann,G.J.SealandW.Tholen(eds.)MonoidalTopology
154 M.CabreraGarc´ıaandA´.Rodr´ıguezPalaciosNon-AssociativeNormedAlgebrasI:TheVidav–Palmerand
Gelfand–NaimarkTheorems
155 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables(SecondEdition)
156 L.W.BeinekeandR.J.Wilson(eds.)withB.ToftTopicsinChromaticGraphTheory
157 T.MoraSolvingPolynomialEquationSystemsIII:AlgebraicSolving
158 T.MoraSolvingPolynomialEquationSystemsIV:BuchbergerTheoryandBeyond
159 V.Berthe´andM.Rigo(eds.)Combinatorics,WordsandSymbolicDynamics
160. B.RubinIntroductiontoRadonTransforms:WithElementsofFractionalCalculusandHarmonicAnalysis
161 M.GherguandS.D.TaliaferroIsolatedSingularitiesinPartialDifferentialInequalities
162 G.MolicaBisci,V.D.RadulescuandR.ServadeiVariationalMethodsforNonlocalFractionalProblems
163 S.WagonTheBanach–TarskiParadox(SecondEdition)
164 K.BroughanEquivalentsoftheRiemannHypothesisI:ArithmeticEquivalents
165 K.BroughanEquivalentsoftheRiemannHypothesisII:AnalyticEquivalents
166 M.BaakeandU.Grimm(eds.)AperiodicOrderII:CrystallographyandAlmostPeriodicity
167 M.CabreraGarc´ıaandA´.Rodr´ıguezPalaciosNon-AssociativeNormedAlgebrasII:Representation
TheoryandtheZel’manovApproach
168 A.Yu.Khrennikov,S.V.KozyrevandW.A.Zu´n˜iga-GalindoUltrametricPseudodifferentialEquations
andApplications
169 S.R.FinchMathematicalConstantsII
170 J.Kraj´ıcˇekProofComplexity
171 D.Bulacu,S.Caenepeel,F.PanaiteandF.VanOystaeyenQuasi-HopfAlgebras

Encyclopedia of Mathematics and its Applications
Quasi-Hopf Algebras
A Categorical Approach
DANIEL BULACU
UniversitateadinBucures¸ti,Romania
STEFAAN CAENEPEEL
VrijeUniversiteitBrussel,Belgium
FLORIN PANAITE
InstituteofMathematicsoftheRomanianAcademy
FREDDY VAN OYSTAEYEN
UniversiteitAntwerpen,Belgium
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DOI:10.1017/9781108582780
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Names:Bulacu,Daniel,1973–author.|Caenepeel,Stefaan,1956–author.|
Panaite,Florin,1970–author.|Oystaeyen,F.Van,1947–author.
Title:Quasi-Hopfalgebras:acategoricalapproach/DanielBulacu
(UniversitateadinBucureti,Romania),StefaanCaenepeel(Vrije
Universiteit,Amsterdam),FlorinPanaite(InstituteofMathematicsofthe
RomanianAcademy),FreddyvanOystaeyen(UniversiteitAntwerpen,Belgium).
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,[2019]|
Series:Encyclopediaofmathematicsanditsapplications;171|Includes
bibliographicalreferencesandindex.
Identifiers:LCCN2018034517|ISBN9781108427012(hardback)
Subjects:LCSH:Hopfalgebras.|Tensorproducts.|Tensoralgebra.
Classification:LCCQA613.8.B852019|DDC512/.55–dc23
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Contents
Preface pagexi
1 MonoidalandBraidedCategories 1
1.1 MonoidalCategories 1
1.2 ExamplesofMonoidalCategories 7
1.2.1 TheCategoryofSets 7
1.2.2 TheCategoryofVectorSpaces 7
1.2.3 TheCategoryofBimodules 7
1.2.4 TheCategoryofG-gradedVectorSpaces 8
1.2.5 TheCategoryofEndo-functors 13
1.2.6 AStrictCategoryAssociatedtoaMonoidalCategory 15
1.3 MonoidalFunctors 16
1.4 MacLane’sStrictificationTheoremforMonoidalCategories 25
1.5 (Pre-)BraidedMonoidalCategories 28
1.6 RigidMonoidalCategories 38
1.7 TheLeftandRightDualFunctors 43
1.8 BraidedRigidMonoidalCategories 48
1.9 Notes 54
2 AlgebrasandCoalgebrasinMonoidalCategories 55
2.1 AlgebrasinMonoidalCategories 55
2.2 CoalgebrasinMonoidalCategories 65
2.3 TheDualCoalgebra/AlgebraofanAlgebra/Coalgebra 70
2.4 CategoriesofRepresentations 78
2.5 CategoriesofCorepresentations 82
2.6 BraidedBialgebras 87
2.7 BraidedHopfAlgebras 95
2.8 Notes 101
3 Quasi-bialgebrasandQuasi-HopfAlgebras 103
3.1 Quasi-bialgebras 103
3.2 Quasi-HopfAlgebras 110
3.3 ExamplesofQuasi-bialgebrasandQuasi-HopfAlgebras 119
viii Contents
3.4 TheRigidMonoidalStructureof MfdandMfd 125
H H
3.5 TheReconstructionTheoremforQuasi-HopfAlgebras 128
3.6 SovereignQuasi-HopfAlgebras 131
3.7 DualQuasi-HopfAlgebras 135
3.8 FurtherExamplesof(Dual)Quasi-HopfAlgebras 141
3.9 Notes 146
4 Module(Co)Algebrasand(Bi)ComoduleAlgebras 147
4.1 ModuleAlgebrasoverQuasi-bialgebras 147
4.2 ModuleCoalgebrasoverQuasi-bialgebras 154
4.3 ComoduleAlgebrasoverQuasi-bialgebras 162
4.4 BicomoduleAlgebrasandTwo-sidedCoactions 168
4.5 Notes 176
5 CrossedProducts 177
5.1 SmashProducts 177
5.2 Quasi-smashProductsandGeneralizedSmashProducts 185
5.3 EndomorphismH-moduleAlgebras 188
5.4 Two-sidedSmashandCrossedProducts 191
∗
5.5 H -HopfBimodules 196
5.6 DiagonalCrossedProducts 201
5.7 L–R-smashProducts 214
5.8 ADualityTheoremforQuasi-HopfAlgebras 220
5.9 Notes 223
6 Quasi-HopfBimoduleCategories 225
6.1 Quasi-HopfBimodules 225
6.2 TheDualofaQuasi-HopfBimodule 230
6.3 StructureTheoremsforQuasi-HopfBimodules 235
6.4 TheCategories MH and M 239
H H H
6.5 AStructureTheoremforComoduleAlgebras 246
6.6 Coalgebrasin MH 249
H H
6.7 Notes 251
7 Finite-DimensionalQuasi-HopfAlgebras 253
7.1 FrobeniusAlgebras 253
7.2 IntegralTheory 261
7.3 SemisimpleQuasi-HopfAlgebras 268
7.4 SymmetricQuasi-HopfAlgebras 273
7.5 CointegralTheory 279
7.6 Integrals,CointegralsandtheFourthPoweroftheAntipode 288
7.7 AFreenessTheoremforQuasi-HopfAlgebras 299
7.8 Notes 303
8 Yetter–DrinfeldModuleCategories 305
8.1 TheLeftandRightCenterConstructions 305