Quantization of some Poisson-Lie dynamical r-matri

arXiv:math/0403283v2[math.QA]14Jun2004QUANTIZATIONOFSOMEPOISSON-LIEDYNAMICALr-MATRICESANDPOISSONHOMOGENEOUSSPACESBENJAMINENRIQUEZ,PAVELETINGOF,ANDIANMARSHALLTothememoryofJosephDoninAbstract.Poisson-Lie(PL)dynamicalr-matricesaregeneralizationsofdynamicalr-ma-trices,wherethebaseisaPoisson-Liegroup.Weproveanaloguesofbasicresultsfortheser-matrices,namelyconstructionsof(quasi)PoissongroupoidsandofPoissonhomogeneousspaces.WeintroduceaclassofPLdynamicalr-matrices,associatedtonondegenerateLiebialgebraswithasplitting;thisisageneralizationoftrigonometricr-matriceswithanabelianbase.WeproveacompositiontheoremforPLdynamicalr-matrices,andconstructquan-tizationsofthepolarizedPLdynamicalr-matrices.Thisway,weobtainquantizationsofPoissonhomogeneousstructuresonG/L(GasemisimpleLiegroup,LaLevisubgroup),therebygeneralizingearlierconstructions.IntroductionInthispaper,wecontinuethestudyofPoisson-Lie(PL)dynamicalr-matrices,whichwasstartedin[FM].Weconstructnewexamplesofsuchr-matrices,togetherwiththeirquantiza-tions.WeapplythistothequantizationofPoissonhomogeneousspaces,whichwereintroducedin[DGS].InSection1,wedefinePLdynamicalr-matricesandgivesomeexamples.Asin[Lu],PLdynamicalr-matricesgiveriseto(quasi)Poissongroupoids.InSection2,weintroducethenotionofanondegenerateLiebialgebrawithasplittingg=l⊕u.WeassociatetothisdatumaPLdynamicalr-matrixσgl.ThisconstructionisaPLanalogueof[EE2](seealso[FGP],Proposition1,and[Xu],Theorem2.3).ThemainexampleistheinclusionofaLevisubalgebralinasimpleLiealgebrag.WhenlcoincideswiththeCartansubalgebrah⊂g,σglisthestandardtrigonometricr-matrix(see[EV1]).Ingeneral,σglisaningredientinacompositiontheoremforPLdynamicalr-matrices,generalizing[EE2](seealso[EV1],Theorem3.14and[FGP],Proposition1)and[Mu](whotreatedthecasewhenl⊂gistheinclusionofaCartaninaLevisubalgebra).InSection3,weintroducethenotionofapolarizednondegenerateLiebialgebrawithasplitting.WeconstructaquantizationΨglofσglinthissituation.Thisconstructionisagen-eralizationof[EE2],andisbasedonanonabeliananalogueoftheinversionoftheShapovalovpairing;fromarepresentationtheoreticviewpoint,thismaybeformulatedintermsofinter-twiners,asin[EE2].Thisideaisalreadypresentin[DM];however,inordertocarryouttheanalogueoftheconstructionof[EE2],oneneedstoconstructaleftcoidealU~(u+)⊂U~(g)(seeSections3.1,3.2).WealsoproveaquantumcompositionformulaforthetwistsΨgl(Section3.8).Thesecondpartofthepaper(Sections4,5,6)isdevotedtoapplyingtheseconstructionstothequantizationofPoissonhomogeneousspacesoftheformG/L,wheregisasimpleLiealgebraandl⊂gisaLevisubalgebra.InSection4,werecalltheclassificationofthePoissonhomogeneousstructuresonG/L([DGS]).ThesetofallthesePoissonstructuresisanalgebraicvarietyP.Weintroduce12BENJAMINENRIQUEZ,PAVELETINGOF,ANDIANMARSHALLaZariskiopensubvarietyP0⊂P,andshowthatthePoissonstructurescorrespondingtoelementsofP0areexactlythosewhichmaybeobtainedeitherusingthedynamicalr-matrixrgl,orusingitsPoisson-Lieanalogueσgl.InSection5,weconstructquantizationsofallthePoissonhomogeneousstructurescorre-spondingtotheelementsofP0.Forthis,weproveanalgebraicityresultforΨgl.ThisresultisbasedonthecomputationoftheShapovalovpairingforUq(g)([DCK]),andonthequantumcompositionformulaforthetwistsΨgl.InSection6,wecomparethequantizationsofG/LobtainedinSection5andin[EE2].In[EE2],therewasconstructedthequantizationofafamilyofPoissonstructures,indexedbyananalyticopensubsetUofP0;thisworkisbasedontheKnizhnik-Zamolodchikovassociatorandisthereforenotpurelyalgebraic,contrarytotheconstructionofSection5.WeshowthatwhentheparameterofthePoissonstructurebelongstoU,bothquantizationsareequivalent.ThePoissonhomogeneousspaceG/L,equippedwithastructurefromP0,maybeviewedasadressingorbitofG∗.Similarly,intherationalcase,theg-invarianthomogeneousstructuresonG/LarethecoadjointorbitsG/L֒→g∗.InSection7,wequantizetheseG-(org-)spaceembeddings(intherationalcase,quantizationswereconstructedin[AL,EE2]).WealsoprovethatthequantumfunctionalgebraofG/LisasubalgebraofU~(g)/Ann(M~χ)(intherationalcase,ofU(g)/Ann(Mχ)[[~]]),whereM~χ(resp.,Mχ)isthegeneralizedVermamoduleoverU~(g)(resp.,overU(g)).SystemsofgeneratorsforAnn(Mχ),whoseclassicallimitsaresystemsofgeneratorsforthedefiningidealoftheorbitGχ⊂g∗,areconstructedin[O]whengisreductiveofclassicaltype.Acknowledgments.Ourmotivationinthisworkcame,toanimportantextent,fromthepapers[DGS]and[DM],aswellasfromveryusefulpersonalexplanationsbyJosephDonin.Wehadplannedtoshowhimthefinalresultsofourworkanddiscusswhattodonext.Sadly,thiswasnottobe.Wededicatethispapertohismemory.WewouldliketothankL.Feh´erandA.Mudrovfordiscussions.TheworkofP.E.waspartiallysupportedbytheNSFgrantDMS-9988796.P.E.andI.M.areindebtedtoIRMA(Strasbourg)forhospitality.1.Poisson-Liedynamicalr-matricesandquasi-Poissonstructures:generalfactsInthissection,werecallthenotionofdressingactionsofPoisson-Liegroups.WedefinePoisson-Lie(PL)dynamicalr-matricesandgivesomeexamples.Weshowthatsuchr-matricesgiveriseto(quasi)Poissongroupoids,anddiscussquantizationoftheseconstructions.ThematerialofthissectionisaPLgeneralizationofbasicconstructionsinvolvingdynamicalr-