The Hopf modules category and the Hopf equation

arXiv:math/9807003v1[math.QA]2Jul1998TheHopfmodulescategoryandtheHopfequationG.MilitaruUniversityofBucharestFacultyofMathematicsStr.Academiei14RO-70109Bucharest1,Romaniae-mail:[e-mail](A,Δ)beaHopf-vonNeumannalgebraandRbetheunitaryfundamentaloperatoronAdefinedbyTakesakiin[28]:R(a⊗b)=Δ(b)(a⊗1).ThenR12R23=R23R13R12(seelemma4.9of[28]).ThisoperatorRplaysavitalroleinthetheoryofdualityforvonNeumannalgebras(see[28]or[2]).IfVisavectorspaceoveranarbitraryfieldk,weshallstudywhatwehavecalledtheHopfequation:R12R23=R23R13R12inEndk(V⊗V⊗V).TakingW:=τRτ,theHopfequationisequivalentwiththepentagonalequation:W12W13W23=W23W12fromthetheoryofoperatoralgebras(see[2]),whereWareviewedasmapinL(K⊗K),foraHilbertspaceK.ForabialgebraH,weshallprovethattheclassiccategoryofHopfmodulesHMHplaysadecisiveroleindescribingallsolutionsoftheHopfequation.Moreprecisely,ifHisabialgebraoverkand(M,·,ρ)∈HMHisanH-Hopfmodule,thenthenaturalmapR=R(M,·,ρ)isasolutionfortheHopfequation.Conversely,themainresultofthispaperisaFRTtypetheorem:ifMisafinitedimensionalvectorspaceandR∈Endk(M⊗M)isasolutionfortheHopfequation,thenthereexistsabialgebraB(R)suchthat(M,·,ρ)∈B(R)MB(R)andR=R(M,·,ρ).Byapplyingthisresult,weconstructnewexamplesofnoncommutativeandnoncocommutativebialgebraswhicharedifferentfromtheonesarisingfromquantumgrouptheory.Inparticular,overafieldofcharacteristictwo,anexampleoffivedimensionalnoncommutativeandnoncocommutativebialgebraisgiven.0IntroductionLetHbeabialgebraoverafieldk.TherearetwofundamentalcategoriesinthetheoryofHopfalgebrasandquantumgroups:HMH,thecategoryofH-HopfmodulesandHYDH,thecategoryofquantumYetter-Drinfel’dmodules.Theobjectsinthesecategoriesarek-vector1spacesMwhichareleftH-modules(M,·),rightH-comodules(M,ρ),suchthatthefollowingquitedistinctcompatibilityrelationshold:ρ(h·m)=Xh(1)·m0⊗h(2)m1(1)inthecaseHMH,andrespectivelyXh(1)·m0⊗h(2)m1=X(h(2)·m)0⊗(h(2)·m)1h(1)(2)fortheYetter-Drinfel’dcategories.Traditionally,thesetwocategorieshavebeenstudiedforcompletelydifferentreasons:theclassicalcategoryHMH(orimmediategeneralisationsofit:AMH,AM(H)C)isinvolvedinthetheoryofintegralsforaHopfalgebra(see[1],[27]orthemorerecent[19]),Cliffordtheoryofrepresentations([17],[23],[25],[26])andHopf-Galoistheory([19],[24],etc.).ThecateoryHYDH,introducedin[31],playsanimportantroleinthequantumYang-Baxterequation,quantumgroups,lowdimensionaltopologyandknottheory(see[13],[14],[20],[21],or[29]).However,therearetwoconnectionsbetweenthesecategories.ThefirstonewasgivenbyP.Schauenburgin[22]:itwasproventhatthecategoryHYDHisequivalenttothecategoryHHMHHoftwo-sided,two-cosidedHopfmodules.Thesecondwasgivenrecentelyin[4].ForAanH-comodulealgebraandCanH-modulecoalgebra,Doi(see[7])andindependentlyKoppinen(see[12])definedAM(H)C,thecategoryofDoi-KoppinenHopfmodules,whoseobjectsareleftA-modulesandrightC-comodulesandsatisfyacompatibilityrelationwhichgeneralises(1).In[4]itwasproventhatHYDHisisomorphictoHM(Hop⊗H)H,whereHcanbeviewedasanHop⊗H-module(comodule)coalgebra(algebra).TheisomorhismistheidentityfunctorM→M.WeherebyobtainastronglinkbetweenthecategoriesHMHandHYDH:bothareparticularcasesofthesamegeneralcategoryAM(H)C.Thisledusin[5],[6]tostudytheimplicationsofthecategoryHYDHintheclassic,non-quanticpartofHopfalgebratheory.In[5]westartwiththefollowingclassictheorem(see[19]):anyfinitedimensionalHopfalgebraisFrobenius.Inthelanguageofcategories,thisresultisinterpretedasfollows:theforgetfulfunctorHMH→HMisFrobenius(i.e.,cf.[5],bydefinitionhasthesameleftandrightadjoint)ifandonlyifHisfinitedimensional.Thenextstepiseasytotake:wemustgeneralizethisresultfortheforgetfulfunctorAM(H)C→AMandthenapplyitinthecaseofYetter-Drinfel’dmodulesfortheforgetfulfunctorHYDH→HM.WethusobtainthefactthattheforgetfulfunctorHYDH→HMisFrobeniusifandonlyifHisfinitedimensionalandunimodular(seetheorem4.2of[5]).Thesametreatmentwasappliedin[6]fortheclassicMaschketheorem.OneofthemajorobstacleswastocorrectlydefinethenotionofintegralfortheDoi-Hopfdatum(H,A,C),suchastobeconnectedtotheclassicintegralonaHopfalgebra(correspondingtothecaseC=A=H),aswellastothenotionoftotalintegral(correspondingtothecaseC=A)definedbyDoiin[8].Thistechniquecanbelookeduponasa”quantisation”ofthetheoremsfromtheclassictheoryofHopfalgebras.Therearetwostepstoit:first,weseektogeneralizearesultforthecategoryAM(H)C,thentoapplyittotheparticularHYDHcase.Thereisalsoanotherapproachtothis”quantisation”technique,recentlyevidencedin[9]forthesameFrobeniustypetheorem.It2wasfirstproven,bygeneralizingtheclassicresult,thatanyfinitedimensionalHopfalgebrasextensionsisaβ-Frobeniusextension(oraFrobeniusextensionofsecondkind).Then,thistheoremwas”quantised”tothecaseofHopfalgebrasextensionsinHYDH.TheresultincludesthecaseofenvelopingalgebrasofLiecoloralgebras.Beginningwiththispaper,weshalltacklethereverseproblem:weshalltrytoinvolvethecategoryHMHinfieldsdominateduntilnowbyHYDH,i.e.trya”dequantisation”.Forthebeginning,itisenoughtoremindthatthecategoryHYDHisdeeplyinvolvedinthequantumYang-Baxterequation:R12R13R23=R23R13R12(3)whereR∈Endk(M⊗M),Mbeingak-vectorspace.Thestartingpoint