Coalgebraic Approach to the Loday Infinity Categor

arXiv:0809.4328v1[math.RA]25Sep2008CoalgebraicApproachtotheLodayInfinityCategory,StemDifferentialfor2n-aryGradedandHomotopyAlgebrasMouradAmmar,NorbertPoncin∗September25,2008AbstractWedefineagradedtwisted-coassociativecoproductonthetensoralgebraTWofanyZn-gradedvectorspaceW.IfWisthedesuspensionspace↓VofagradedvectorspaceV,thecoderivations(resp.quadratic“degree1”codifferentials,arbitraryoddcodifferentials)ofthiscoalgebraare1-to-1withsequencesπs,s≥1,ofs-linearmapsonV(resp.Zn-gradedLodaystructuresonV,sequencesthatwecallLodayinfinitystructuresonV).WeproveaminimalmodeltheoremforLodayinfinityalgebras,investigateLodayinfinitymorphisms,andobservethattheLod∞categorycontainstheL∞categoryasasubcategory.Moreover,thegradedLiebracketofcoderivationsgivesrisetoagradedLie“stem”bracketonthecochainspacesofgradedLoday,Lodayinfinity,and2n-arygradedLodayalgebras(thelatterextendthecorrespondingLiealgebrasinthesenseofMichorandVinogradov).Thesealgebraicstructureshavesquarezerowithrespecttothestembracket,sothatweobtainnaturalcohomologicaltheoriesthathavegoodpropertieswithrespecttoformaldeformations.ThestembracketrestrictstothegradedNijenhuis-Richardsonand—uptoisomorphism—totheGrabowski-Marmobrackets(thelastbracketextendstheSchouten-Nijenhuisbrackettothespaceofgradedantisymmetricfirstorderpolydifferentialoperators),anditencodes,beyondthealreadymentionedcohomologies,thoseofgradedLie,gradedPoisson,gradedJacobi,Lieinfinity,aswellasthatof2n-arygradedLiealgebras.MathematicsSubjectClassification(2000):16W30,16E45,17B56,17B70.Keywords:GradeddualLeibnizcoalgebra,gradedLoday/Lie/Poisson/Jacobistructure,stronglyhomotopyalgebra,square-zeroelementmethod,gradedcohomology,Schouten-Nijenhuis/Nijenhuis-Richardson/Grabowski-Marmobracket,deformationtheory.1IntroductionTheprevailingapproachtocohomologyconsistsinassociatinginacanonicalwayadifferentialmoduletotheinvestigatedalgebraicstructure.Forinstance,in[LLMP99]and[ILLMP01](resp.in[Oud97])theauthor(s)define(s)Lichnerowicz-JacobiandNambu-Poissoncohomologies(resp.thecohomologyofgradedLeibnizalgebraswithtrivialcoefficients)essentiallyasthecohomologyofaLiealgebroid(resp.asthehomologyofadifferentialgradeddualLeibnizalgebra)thatisassoci-atedwithanyJacobiorNambu-Poissonmanifold[itsufficestosolvetheLiealgebrahomomorphismequationforthenaturalanchormap](resp.thatisinducedbytheconsideredLeibnizalgebra[viadualization]).∗UniversityofLuxembourg,CampusLimpertsberg,MathematicsResearchUnit,162A,avenuedelaFaïencerie,L-1511LuxembourgCity,Grand-DuchyofLuxembourg,E-mail:[e-mail],[e-mail]2However,themostnaturalwaytoobtainadifferentialspacefromanalgebraicstructureisthesquare-zeroorcanonicalelementmethodthatwasinitiatedbyDeWildeandLecomtein[DWL88].TheylookforagradedLiealgebra,suchthatthereisa1-to-1correspondencebetweenthestudiedalgebraicstructuresandthedegree1elementsoftheLiealgebraspacethatsquaretozerowithrespecttotheLiebracket.TheadjointactionbythegivenstructurethenprovidesadifferentialgradedLiealgebra,thecohomologyofwhichhastheusualpropertieswithrespecttoformaldeformationsofthealgebraicstructure.TwoefficienttoolsallowfindingthementionedgradedLiealgebra:-Thecoalgebraictechniqueconsistsintheidentificationofthecochainspaceoftheinvestigatedalgebrawithcertaincoderivations,insuchawaythatthealgebraicstructurecanbeviewedasanoddcodifferential.TheseidentificationsenableconstructingagradedLiebracketonthespaceofcochainsbytransferofthecommutatorbracketofcoderivations;theconsideredalgebraicstructuresarethencanonicalelementsofthistransferredbracket.TheprocedurewasappliedbyStasheff[Sta93]inthecaseofassociativeandLiealgebrasandbyPenkava[Pen01]forstronglyhomotopyassociativeandLiealgebras.-Theoperadictheory,asdevelopedin[MSS02],showsthatStasheff’sapproachisapplicableforanyquadraticoperadP.TheauthorsassociatetoPacofreecoalgebraoverthedualoperad,whosequadraticcodifferentialscorrespondtotheP−algebrastructuresonagradedvectorspaceV.ThisconstructionyieldsthehomologyandcohomologytheoriesofP−algebrasonVandallowsdefiningstronglyhomotopyP−algebrastructuresonVasarbitrarycodifferentials.InthepresentworkweprovideatensorcoalgebrathatinducestheproperconceptsofLodayinfinityalgebrasandmorphisms,anddefinethecohomologiesofZn-gradedLoday,Lodayinfinity,and2p-arygradedLodayalgebras.ThisleadstoagradedLie“stem”bracket,inwhichareencrypted,inadditiontotheprecedingcohomologies,thecoboundaryoperatorsofgradedLie,gradedPoisson,gradedJacobi,Lieinfinity,and2p-arygradedLiealgebras[MV97].Ourapproachisself-contained,resultsareexplicit,andtheysuggestthattheoperadictheoryisnotconfinedtothequadraticcase.Thepaperisorganizedasfollows.InSection2,westudythepropertiesofcohomologyalgebras,whichareimplementedbysquare-zeroelementsinagradedLiealgebra,withrespecttoformaldeformationsoftheseelements.OurupshotsextendsimilarpropertiesfortheadjointHochschild(resp.Chevalley-Eilenberg,Leibniz)cohomologyanddeformationsofassociative(resp.Lie,Loday)structures,whichwereprovedin[Ger64](resp.[NR67],[Bal96]),andrecoveredin[Bal97].Section