Cohomological properties of the quantum shuffle pr

arXiv:math/0112141v1[math.QA]13Dec2001Cohomologicalpropertiesofthequantumshuffleproductandapplicationtotheconstructionofquasi-Hopfalgebras.CyrilleOspelLaboratoiredeMath´ematiques,Universit´eBlaisePascalCampusdesC´ezeaux,63177Aubi`ereCedexFrancee-mail:[e-mail]:Foracommutativealgebratheshuffleproductisamorphismofcomplexes.Wegeneralizethisresulttothequantumshuffleproduct,associatedtoaclassofnon-commutativealgebras(forexamplealltheHopfalgebras).AsafirstapplicationweshowthattheHochschild-Serreidentityisthedualstatementofourresult.Inparticular,weextendthisidentitytoHopfalgebras.Secondly,weclarifytheconstructionofaclassofquasi-Hopfalgebras.MathematicsSubjectClassifications(1991):16W30,18G15,17B37,16S40IntroductionTheshufflesweredefinedbyS.EilenbergandS.MacLanetogiveanexplicitformulafortheequivalenceofcomplexesoftheEilenberg-Zilbertheorem.Theywerelaterusedtoshowthatthehomologyofanabeliangroup(oracommutativealgebra)isanalgebrawiththeshuffleproduct.Ontheotherhand,theshuffleswereused,moreimplicitly,byG.HochschildandJ.P.Serre[7]inthedefinitionoftheHochschild-Serreidentityofagroup.AfterwardsN.Habegger,V.Jones,O.PinoOrtiz,J.Ratcliffe[6]gaveaformulationoftheidentityintermsofshuffles.Thesetworesultswereprovedseparately,byalongandtechnicalverificationofthetwotermsoftheequality.Inthispaper,weprovethattheseresultshaveastronginteraction.Infact,weshowthattheHochschild-Serreidentityisaconsequenceofthehomologicalpropertyoftheshufflemap.Moregenerally,weconsideralgebraswithanautomorphismσofthesquaretensorprod-uctandsomerelationsbetweentheproductandσ;suchacoupleiscalledabraidedσ-commutativealgebra.Forsuchalgebras,weshowfirstthatthequantumshuffleprod-uct,associatedtoσanddefinedbyM.Rosso[16],canbefactorizedbytheshufflemap.ThisfactorizationallowstoprovethatthequantumshuffleproductisamorphismofcomplexesfromabraidedtensorproductofchaincomplexestotheHochschildchaincomplexofthealgebra.ThedualstatementisaHochschild-Serreidentityforbraidedσ-commutativealgebras.1AclassofexamplesofsuchalgebrasistheclassofHopfalgebraswithinvertiblean-tipode.ThebraidindisgivenbytheWoronowiczbraiding[20].InparticularthecaseoftheHopfalgebraofagroupgivestheclassicalHochschild-Serreidentityofthegroup.Secondly,wegiveamultiplicativestatementoftheHochschild-Serreidentityforaco-commutativeHopfalgebra.Then,weusethisresulttoclarifysomeconstructionsofquasi-Hopfalgebras,associatedtotheDrinfeldDouble,anddefinedbyR.Dijkgraaf,V.Pasquier,P.Roche[3]andD.Bulacu,F.Panaite[2].In[14]weshowthatthehomologicalpropertyofthequantumshuffleproductallowstoextendthefirstiterationoftheabeliangrouphomologyconstruction[5]tonon-commutativeHopfalgebras.Themultiplicativecohomologyassociatedtothischaincomplexhasapplicationstothetheoryofinvariantsforlinksand3-manifolds.Forexample,the3-cocyclesofthiscohomologyareweightsystemsforlinks.AcknowledgmentsIwouldliketoexpressmygratitudetoProfessorM.Rossoforexplainingmethesubject,byhelpfuldiscussionsandbibliographicreferences,andforsuggestingmetheleadingidea.Notations:Kisacommutativefield.AllalgebrasconsideredareassociativealgebrasoverKwithunit.Theproductisdenotedbyμ.WeuseSweedler’snotationforcoproductΔ(a)=Pa(1)⊗a(2).LetVbeavectorspaceoverK.ThetensorvectorspaceT(V)ofVisdefinedbyT(V)=Mk≥0V⊗k.Letw∈V⊗n.Wedenotethedegreenofwby|w|.Σnisthesetofallpermutationsof{1,...,n}.ForallνinΣn,wedenotethesignofνby(−1)|ν|.1Anewconstructionofthequantumshuffleproduct.Theoriginaldefinitionofthequantumshuffleproductwasgivenintheframeworkofrepresentationtheory.Thispointofviewisnotusefultostudythehomologicalpropertiesoftheproduct.Sowewillgiveafactorizationofthisproductbymorphismsofcomplexes,inparticularbytheshufflemap.1.1Theoriginaldefinition.ThequantumshuffleproductwasfirstdefinedbyM.Rosso[15].ItdescribestheproductofthefollowingcotensorialHopfalgebra.LetHbeaHopfalgebraandManH-Hopfbimodule,withthebicomodulestructuregivenbyδLandδR.Thecotensorial2HopfalgebraTcH(M)wasdefinedbyW.Nichols[13],by:TcH(M)=H⊕Mn≥1Mn,whereMMisthekernelofδR⊗IdM−IdM⊗δL.ThisHopfalgebraisanH-Hopfbimodule.AsalgebraTcH(M)isthecrossedproductofHbytheleft-coinvariantsubspaceofthecotensorialHopfbimodule.Thealgebrastructureinducedontheleft-coinvariantsubspaceisgivenbythequantumshuffleproduct.Moreexplicitly,letVbeavectorspaceoverKandσ∈End(V⊗V)whichsatisfiesthebraidequation:σ2σ1σ2=σ1σ2σ1,(1)whereforn,inonnegativeintegerssuchthatni,σi∈End(V⊗n)isdefinedbyσi=Id⊗i−1⊗σ⊗Id⊗n−i−1.Forn≥2,wedenotebyTσtherepresentationofthebraidgroupBnonV⊗ndefinedonthegenerators(ωi)1≤i≤n−1by:Tσ(ωi)=σi,1≤i≤n−1.Forallnonnegativeintegersp,qsuchthatp+q=n,Sp,nisthesetofall(p,q)-shuffles,i.e.thesetofallw∈Pnsuchthatw(1)···w(p)andw(p+1)···w(n).Let(νi)1≤i≤n−1bethetranspositions(i,i+1)ofPn.Forw∈Pnandw=νi1...νirareduceddecompositionofwwedefinetheextensionofTσtoΣnby:Tσ(w)=σi1...σir.Thequantumshuffleproductϕσ:T(V)⊗T(V)→T(V),associatedtoσ,isdefinedforallpositiveintegersp+q=nby:ϕσ:V⊗p⊗V⊗q−→V⊗nv⊗v′7−→Xw∈Sp,nTσ(w)(v⊗v′).Ifp=0orq=0,theproductisjustthemultiplicationbyelementsofK.Withthisproductandtheunit1K,T(V)isanalgebra.Inthispaper,wewillalwaysuseaquantumshuffleproductwithsign,i.e.associatedtot