A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory

218 Pages·2013·1.302 MB·English

by  Bangming Deng, Jie Du, Qiang Fu

LONDONMATHEMATICALSOCIETYLECTURENOTESERIES
ManagingEditor:ProfessorM.Reid,MathematicsInstitute,UniversityofWarwick,CoventryCV47AL,
UnitedKingdom
Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat
http://www.cambridge.org/mathematics
287 TopicsonRiemannsurfacesandFuchsiangroups,E.BUJALANCE,A.F.COSTA&E.MARTÍNEZ(eds)
288 Surveysincombinatorics,2001,J.W.P.HIRSCHFELD(ed)
289 AspectsofSobolev-typeinequalities,L.SALOFF-COSTE
290 QuantumgroupsandLietheory,A.PRESSLEY(ed)
291 Titsbuildingsandthemodeltheoryofgroups,K.TENT(ed)
292 Aquantumgroupsprimer,S.MAJID
293 SecondorderpartialdifferentialequationsinHilbertspaces,G.DAPRATO&J.ZABCZYK
294 Introductiontooperatorspacetheory,G.PISIER
295 Geometryandintegrability,L.MASON&Y.NUTKU(eds)
296 Lecturesoninvarianttheory,I.DOLGACHEV
297 Thehomotopycategoryofsimplyconnected4-manifolds,H.-J.BAUES
298 Higheroperads,highercategories,T.LEINSTER(ed)
299 Kleiniangroupsandhyperbolic3-manifolds,Y.KOMORI,V.MARKOVIC&C.SERIES(eds)
300 IntroductiontoMöbiusdifferentialgeometry,U.HERTRICH-JEROMIN
301 StablemodulesandtheD(2)-problem,F.E.A.JOHNSON
302 DiscreteandcontinuousnonlinearSchrödingersystems,M.J.ABLOWITZ,B.PRINARI&A.D.TRUBATCH
303 Numbertheoryandalgebraicgeometry,M.REID&A.SKOROBOGATOV(eds)
304 GroupsStAndrews2001inOxfordI,C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
305 GroupsStAndrews2001inOxfordII,C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds)
306 Geometricmechanicsandsymmetry,J.MONTALDI&T.RATIU(eds)
307 Surveysincombinatorics2003,C.D.WENSLEY(ed.)
308 Topology,geometryandquantumfieldtheory,U.L.TILLMANN(ed)
309 Coringsandcomodules,T.BRZEZINSKI&R.WISBAUER
310 Topicsindynamicsandergodictheory,S.BEZUGLYI&S.KOLYADA(eds)
311 Groups:topological,combinatorialandarithmeticaspects,T.W.MÜLLER(ed)
312 Foundationsofcomputationalmathematics,Minneapolis2002,F.CUCKERetal(eds)
313 Transcendentalaspectsofalgebraiccycles,S.MÜLLER-STACH&C.PETERS(eds)
314 Spectralgeneralizationsoflinegraphs,D.CVETKOVIC´,P.ROWLINSON&S.SIMIC´
315 Structuredringspectra,A.BAKER&B.RICHTER(eds)
316 Linearlogicincomputerscience,T.EHRHARD,P.RUET,J.-Y.GIRARD&P.SCOTT(eds)
317 Advancesinellipticcurvecryptography,I.F.BLAKE,G.SEROUSSI&N.P.SMART(eds)
318 Perturbationoftheboundaryinboundary-valueproblemsofpartialdifferentialequations,D.HENRY
319 DoubleaffineHeckealgebras,I.CHEREDNIK
320 L-functionsandGaloisrepresentations,D.BURNS,K.BUZZARD&J.NEKOVÁRˇ(eds)
321 Surveysinmodernmathematics,V.PRASOLOV&Y.ILYASHENKO(eds)
322 Recentperspectivesinrandommatrixtheoryandnumbertheory,F.MEZZADRI&N.C.SNAITH(eds)
323 Poissongeometry,deformationquantisationandgrouprepresentations,S.GUTTetal(eds)
324 Singularitiesandcomputeralgebra,C.LOSSEN&G.PFISTER(eds)
325 LecturesontheRicciflow,P.TOPPING
326 ModularrepresentationsoffinitegroupsofLietype,J.E.HUMPHREYS
327 Surveysincombinatorics2005,B.S.WEBB(ed)
328 Fundamentalsofhyperbolicmanifolds,R.CANARY,D.EPSTEIN&A.MARDEN(eds)
329 SpacesofKleiniangroups,Y.MINSKY,M.SAKUMA&C.SERIES(eds)
330 Noncommutativelocalizationinalgebraandtopology,A.RANICKI(ed)
331 Foundationsofcomputationalmathematics,Santander2005,L.MPARDO,A.PINKUS,E.SÜLI&M.J.TODD
(eds)
332 Handbookoftiltingtheory,L.ANGELERIHÜGEL,D.HAPPEL&H.KRAUSE(eds)
333 Syntheticdifferentialgeometry(2ndEdition),A.KOCK
334 TheNavier–Stokesequations,N.RILEY&P.DRAZIN
335 Lecturesonthecombinatoricsoffreeprobability,A.NICA&R.SPEICHER
336 Integralclosureofideals,rings,andmodules,I.SWANSON&C.HUNEKE
337 MethodsinBanachspacetheory,J.M.F.CASTILLO&W.B.JOHNSON(eds)
338 Surveysingeometryandnumbertheory,N.YOUNG(ed)
339 GroupsStAndrews2005I,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
340 GroupsStAndrews2005II,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds)
341 Ranksofellipticcurvesandrandommatrixtheory,J.B.CONREY,D.W.FARMER,F.MEZZADRI&
N.C.SNAITH(eds)
342 Ellipticcohomology,H.R.MILLER&D.C.RAVENEL(eds)
343 AlgebraiccyclesandmotivesI,J.NAGEL&C.PETERS(eds)
344 AlgebraiccyclesandmotivesII,J.NAGEL&C.PETERS(eds)
345 Algebraicandanalyticgeometry,A.NEEMAN
346 Surveysincombinatorics2007,A.HILTON&J.TALBOT(eds)
347 Surveysincontemporarymathematics,N.YOUNG&Y.CHOI(eds)
348 Transcendentaldynamicsandcomplexanalysis,P.J.RIPPON&G.M.STALLARD(eds)
349 ModeltheorywithapplicationstoalgebraandanalysisI,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY&
A.WILKIE(eds)
350 ModeltheorywithapplicationstoalgebraandanalysisII,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY&
A.WILKIE(eds)
351 FinitevonNeumannalgebrasandmasas,A.M.SINCLAIR&R.R.SMITH
352 Numbertheoryandpolynomials,J.MCKEE&C.SMYTH(eds)
353 Trendsinstochasticanalysis,J.BLATH,P.MÖRTERS&M.SCHEUTZOW(eds)
354 Groupsandanalysis,K.TENT(ed)
355 Non-equilibriumstatisticalmechanicsandturbulence,J.CARDY,G.FALKOVICH&K.GAWEDZKI
356 EllipticcurvesandbigGaloisrepresentations,D.DELBOURGO
357 Algebraictheoryofdifferentialequations,M.A.H.MACCALLUM&A.V.MIKHAILOV(eds)
358 Geometricandcohomologicalmethodsingrouptheory,M.R.BRIDSON,P.H.KROPHOLLER&I.J.LEARY
(eds)
359 Modulispacesandvectorbundles,L.BRAMBILA-PAZ,S.B.BRADLOW,O.GARCÍA-PRADA&
S.RAMANAN(eds)
360 Zariskigeometries,B.ZILBER
361 Words:Notesonverbalwidthingroups,D.SEGAL
362 Differentialtensoralgebrasandtheirmodulecategories,R.BAUTISTA,L.SALMERÓN&R.ZUAZUA
363 Foundationsofcomputationalmathematics,HongKong2008,F.CUCKER,A.PINKUS&M.J.TODD(eds)
364 Partialdifferentialequationsandfluidmechanics,J.C.ROBINSON&J.L.RODRIGO(eds)
365 Surveysincombinatorics2009,S.HUCZYNSKA,J.D.MITCHELL&C.M.RONEY-DOUGAL(eds)
366 Highlyoscillatoryproblems,B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds)
367 Randommatrices:Highdimensionalphenomena,G.BLOWER
368 GeometryofRiemannsurfaces,F.P.GARDINER,G.GONZÁLEZ-DIEZ&C.KOUROUNIOTIS(eds)
369 Epidemicsandrumoursincomplexnetworks,M.DRAIEF&L.MASSOULIÉ
370 Theoryofp-adicdistributions,S.ALBEVERIO,A.YU.KHRENNIKOV&V.M.SHELKOVICH
371 Conformalfractals,F.PRZYTYCKI&M.URBAN´SKI
372 Moonshine:Thefirstquartercenturyandbeyond,J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds)
373 Smoothness,regularityandcompleteintersection,J.MAJADAS&A.G.RODICIO
374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction,S.ALINHAC
375 Triangulatedcategories,T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds)
376 Permutationpatterns,S.LINTON,N.RUŠKUC&V.VATTER(eds)
377 AnintroductiontoGaloiscohomologyanditsapplications,G.BERHUY
378 Probabilityandmathematicalgenetics,N.H.BINGHAM&C.M.GOLDIE(eds)
379 Finiteandalgorithmicmodeltheory,J.ESPARZA,C.MICHAUX&C.STEINHORN(eds)
380 Realandcomplexsingularities,M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds)
381 Symmetriesandintegrabilityofdifferenceequations,D.LEVI,P.OLVER,Z.THOMOVA&P.WINTERNITZ
(eds)
382 Forcingwithrandomvariablesandproofcomplexity,J.KRAJÍCˇEK
383 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryI,R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII,R.CLUCKERS,
J.NICAISE&J.SEBAG(eds)
385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems,B.MARCUS,K.PETERSEN&
T.WEISSMAN(eds)
386 Independence-friendlylogic,A.L.MANN,G.SANDU&M.SEVENSTER
387 GroupsStAndrews2009inBathI,C.M.CAMPBELLetal(eds)
388 GroupsStAndrews2009inBathII,C.M.CAMPBELLetal(eds)
389 Randomfieldsonthesphere,D.MARINUCCI&G.PECCATI
390 Localizationinperiodicpotentials,D.E.PELINOVSKY
391 FusionsystemsinalgebraandtopologyM.ASCHBACHER,R.KESSAR&B.OLIVER
392 Surveysincombinatorics2011,R.CHAPMAN(ed)
393 Non-abelianfundamentalgroupsandIwasawatheory,J.COATESetal(eds)
394 VariationalProblemsinDifferentialGeometry,R.BIELAWSKI,K.HOUSTON&M.SPEIGHT(eds)
395 Howgroupsgrow,A.MANN
396 ArithmeticDifferentialOperatorsoverthep-adicIntegers,C.C.RALPH&S.R.SIMANCA
397 HyperbolicgeometryandapplicationsinquantumChaosandcosmology,J.BOLTE&F.STEINER(eds)
398 Mathematicalmodelsincontactmechanics,M.SOFONEA&A.MATEI
399 Circuitdoublecoverofgraphs,C.-Q.ZHANG
400 Densespherepackings:ablueprintforformalproofs,T.HALES
401 AdoubleHallalgebraapproachtoaffinequantumSchur–Weyltheory,B.DENG,J.DU&Q.FU
LondonMathematicalSocietyLectureNoteseries:401
A Double Hall Algebra Approach to
Affine Quantum Schur–Weyl Theory
BANGMING DENG
BeijingNormalUniversity
JIE DU
UniversityofNewSouthWales,Sydney
QIANG FU
TongjiUniversity,Shanghai
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,SãoPaulo,Delhi,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107608603
(cid:2)c B.Deng,J.DuandQ.Fu2012
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedandiboundiintheUnitedKingdombyitheMPGiBooksiGroup
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
ISBN978-1-107-60860-3Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
Wededicatethebooktoourteachers:
PeterGabriel
ShaoxueLiu
LeonardScott
JianpanWang
2010MathematicsSubjectClassification.Primary17B37,20G43,20C08;
Secondary16G20,20G42,16T20
Keywordsandphrases.affineHeckealgebra,affinequantumSchuralgebra,
cyclicquiver,Drinfelddouble,loopalgebra,quantumgroup,Schur–Weyl
duality,Ringel–Hallalgebra,simplerepresentation
Abstract
Overitsone-hundredyearhistory,thetheoryofSchur–Weylduality
and its quantum analogue have had and continue to have profound
influences in several areas of mathematics such as Lie theory, rep-
resentationtheory,invarianttheory,combinatorialtheory,andsoon.
Recent new developments include, e.g., walled Brauer algebras and
rational Schur algebras, quantum Schur superalgebras, and the inte-
gral Schur–Weyl duality for types other than A. This book takes an
algebraicapproachtotheaffinequantumSchur–Weyltheory.
The book begins with a study of extended Ringel–Hall algebras
associated with the cyclic quiver of n vertices and the Green–Xiao
Hopf structure on their Drinfeld double—the double Ringel–Hall
algebra.ThisalgebraispresentedintermsofChevalleytypeandcen-
tral generators and is proved to be isomorphic to the quantum loop
algebraofthegenerallinearLiealgebra.Therestofthebookinves-
tigates the affine quantum Schur–Weyl duality on three levels. This
includes
• theaffinequantumSchur–Weylreciprocity;
• the bridging role played by the affine quantum Schur algebra
between the quantum loop algebra and the corresponding affine
Heckealgebra;
• Moritaequivalenceofcertainrepresentationcategories;
• thepresentationofaffinequantumSchuralgebras;and
• therealizationconjectureforthedoubleRingel–Hallalgebrawhich
isprovedtobetrueintheclassicalcase.
Connections with various existing works by Lusztig, Varagnolo–
Vasserot,Schiffmann,Hubery,Chari–Pressley,Frenkel–Mukhin,and
othersarealsodiscussedthroughoutthebook.
Contents
Introduction page1
1 Preliminaries 9
(cid:2)
1.1 Theloopalgebragl andsomenotation 9
n
1.2 RepresentationsofcyclicquiversandRingel–Hallalgebras 12
1.3 ThequantumloopalgebraU((cid:2)sl ) 17
n
1.4 Threetypesofgeneratorsandassociatedmonomialbases 20
1.5 HopfstructureonextendedRingel–Hallalgebras 25
2 DoubleRingel–Hallalgebrasofcyclicquivers 31
2.1 DrinfelddoublesandtheHopfalgebraD(cid:2)(n) 31
2.2 Schiffmann–Huberygenerators 37
2.3 PresentationofD(cid:2)(n) 41
2.4 Someintegralforms 45
2.5 ThequantumloopalgebraU(g(cid:2)l ) 49
n
2.6 Semisimplegeneratorsandcommutatorformulas 55
3 Affine quantum Schur algebras and the Schur–Weyl
reciprocity 62
3.1 Cyclicflags:thegeometricdefinition 63
3.2 AffineHeckealgebrasoftype A:thealgebraicdefinition 68
3.3 Thetensorspaceinterpretation 74
3.4 BLMbasesandmultiplicationformulas 77
3.5 TheD(cid:2)(n)-H(cid:2)(r)-bimodulestructureontensorspaces 79
3.6 AcomparisonwiththeVaragnolo–Vasserotaction 88
3.7 TriangulardecompositionsofaffinequantumSchuralgebras 95
3.8 AffinequantumSchur–Weylduality,I 102
3.9 Polynomialidentityarisingfromsemisimplegenerators 106
3.10 Appendix 115
vii
viii Contents
4 RepresentationsofaffinequantumSchuralgebras 121
4.1 AffinequantumSchur–Weylduality,II 122
4.2 Chari–Pressleycategoryequivalenceandclassification 126
4.3 Classification of simple S(cid:2)(n,r)C-modules:
theupwardapproach 132
4.4 IdentificationofsimpleS(cid:2)(n,r)C-modules:then >r case 136
4.5 Application:thesetS(n) 141
r
4.6 Classification of simple S(cid:2)(n,r)C-modules:
thedownwardapproach 143
4.7 ClassificationofsimpleU(cid:2)(n,r)C-modules 150
5 Thepresentationandrealizationproblems 153
5.1 McGerty’spresentationforU(cid:2)(n,r) 154
5.2 StructureofaffinequantumSchuralgebras 157
5.3 PresentationofS(cid:2)(r,r) 162
5.4 Therealizationconjecture 169
5.5 Lusztig’stransfermapsonsemisimplegenerators 172
6 Theclassical(v =1)case 179
6.1 TheuniversalenvelopingalgebraU(g(cid:2)l ) 179
n
6.2 MoremultiplicationformulasinaffineSchuralgebras 185
6.3 ProofofConjecture5.4.2atv =1 190
6.4 Appendix:ProofofProposition6.2.3 194
Bibliography 201
Index 205