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NoteauxC.R.Acad.Sci.Paris(acceptee) Algebra /rbeelgA
Koszul Duality for PROPs
1 Bruno Vallette
Abstract The notion of PROP models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. We prove a Koszul duality theory for PROPs generalizing the one for associative algebras and for operads. Resume DualitedeKoszuldesPROPs.rseupasisularenoitdlePOomseposileLanodePRtion entreesetplusieurssorties,agissantsurcertainesstructuresalgebriquescommelesbigebresetles bigebresdeLie.NousmontronsunetheoriededualitedeKoszulpourlesPROPsquigeneralise celledesalgebresassociativesetdesoperades.
Versionfrancaiseabregee.
Ontravaillesuruncorpsdecaracteristiquenulle.
Suivant J.-P. Serre dans [S], on regroupe sous le terme deerbeg iductrretuenersstebrelgsasueiq commelesalgebres,lescogebresetlesbigebres.
L’ensembleP(m, ntaresnoiade)psoneesetertnmsorties agissant sur un certain type de gebresestunmoduleagauchesurlegroupesymetriqueSmet a droite surSn. Cesdeux actions sont compatibles. On appelleS-bimoduletoute collection (P(m, n))m, nNesdis nesulou.Ntelemsdosnosd  un produitiedeegorcatnslaadsStaoisnauqrilusemidob-omscleteenesreprepodsnoitisop plusieursentreesetplusieurssorties.Ceproduitestbasesurlesgraphesdiriges(cf.Figure 1).
 On de nitunPROPcomme unS-bimodule muni d’une compositionPPP →associative. On donne les exemples du PROPBiLie(eiLesdreebgbiesdcf.[D]), du PROPBiLie0des bigebresdeLiecombinatoires(cf.[C]) et du PROPInfBisbde nitesimalesgierbseedoHfpni (cf.appelle[A]). OnPebre-g, tout module sur le PROPPruevselgeterbosed ensietsidonO.rn classiques. Parexemple, uneBiLieg-nemetcaxetseerbeebredeLitunebig.e
Nousetendonslesde nitionsdebaretcobarconstructionsdesalgebresetdesoperadesaux PROPs,etnousgeneralisonsleslemmesdecomparaisondeB.Fresse[F]auxPROPs.Remarquons quelesdemonstrationsoperadiquesnesontpasreconductiblesici,carcesdernieresreposentsur lesproprietescombinatoiresdesarbres.
1 InstitutdeRechercheMathematiqueAvancee,UniversiteLouisPasteuretCNRS,7,RueReneDescartes,67084 Strasbourg Cedex, France. E-mail :vallette@math.u-strasbg.fr URL :http://www-irma.u-strasbg.fr/vallette 1
2 ApartirdunPROPgradueparunpoids,parexemplequadratiquecest-a-direde nipardes ¡ generateursetdesrelationsquadratiques,onconstruitunecoPROPdualPet uncomplexe de ¡ Koszul(PP, dK).
Leprincipaltheoremedecettetheorieestlesuivant: Theoreme.SoitPROPdunPrenti eguemeialrgdatneepuunaridpopas(exerelpmPnu,POR quadratique),lespropositionssuivantessontequivalentes ¡ (1)le complexe de Koszul(PP, dK)est acyclique, ¡ (2)la cobar construction sur le coPROP dualPuonrtinuftionduPROePresoluP:  c¡ B(P)P →. Danscecas,laresolutionobtenueestlenimelamiomleddePet elle permet de de nirla notion dePserpeipotmohoareebg-.
Nous montrons que le PROPBiLiedesbigebreLeds(eicf.[D]), le PROPBiLie0ersgbedbies de Lie combinatoires (cf.[C]) et le PROPInfBimasis(le nineitbreigsbpfHodeesedcf.[A]) sontdesPROPsdeKoszul.Cequipermetdedonnerlesde nitionsdebigebresdeLie,bigebres deLiecombinatoiresetbigebresdeHopfin nitesimalesahomotopiepres.
Introduction The Koszul duality theory for algebras, proved by S. Priddy in [P] has been generalized to the operads by V. Ginzburg and M. M. Kapranov in [GK]. An operad models the operations acting on a certain type of algebras (associative, commutative and Lie algebras for instance).Since these operations have multiple inputs but only one output, their compositions can be represented by trees.This theory has a lot of applications.It gives the minimal model of an operadP, the notion ofP-algebras up to homotopy and a natural homology theory for theP-algebras. Tostudyalgebraicstructuresde nedbyoperationswithmultipleinputsandmultipleoutputs, like bialgebras or Lie bialgebras for instance, one needs to generalize the notion of operad and introduce the notion of PROP. ItisnaturaltotrytogeneralizetheKoszuldualityforPROPs.A rstresultinthedirection is due to W. L. Gan in [G], see also M. Markl and A. A. Voronov in [MV].
Weworkovera eldkThe symmetric group onof characteristic 0.nelements is denoted bySn. 1.PROPs andP-gebras Over a vector space, various algebraic structures can be considered like algebras, coalgebras, bialgebras. FollowingJ.-P. Serre in [S], we callgebraany one of these structures.The setP(m, n) of the operations ofninputs andmoutputs acting on a gebraAis a module overSmon the left and overSnon the right. We have the following morphisms ofSm-modules : nm P(m, n)SnA →A . op fSS.  Definition(S-bimodule).AnS-bimodule(P(m, n))m, nNis a collection om n-modules 1.1.The composition product.We introduce a product onS-bimodules which describes the composition of operations.
Agrowhaithwapyanbobgloalwehtfegdeerasevigrientheoonsotatiasrgieherpawh (from the top to the bottom, for instance).LetGahtiwshpargetinf toseheetbeow.W suppose that the inputs and the outputs of each vertex are labeled by integers.When the vertices of a graphgcan be dispatched on two levels, we denoteNi(i= 1,2) the set of vertices belonging th 2 to theidenote bylevel. WeGthe set of such graphs (cf.Figure 1).
1 2 34 5 @ @ @ ~@ ~ @ ~@ ~ @ ~@ ~ @ ~@ ~ @ ~@ ~ @3~ @~ ~ ~ '&!1%$"#11P2 1 21 2 _ __ _ _ _ _ _ _ _ __ _ _ P2 2n B1 P n | Pn B | Pn B P n | Pn B P | Pn B n | Pn B | Pn B P | PB n | nP B n P |BnB B |B | B |B | B |B | B |B | B |B | B |B | B |B | | | '&!2$%#"34 1 21 2 _ __ _ _ _ _ _ _ _ __ _ _ 3@1 ~ @ 1~2@ ~ @ ~ @ ~ @ ~ @ ~ @ ~4 12 3
Figure 1.Example of a 2-levels graph.
3
Definition(Product).Given twoS-bimodulesPandQal edew,priehtenbyctduromuorefth   , M OO   PQ=P(|Out()|,|I n()|) Q(|Out()|,|I n()|), 2 g∈G∈N2∈N1 where the relationis generated by > G > Gu  Gu > G u > G u > G u >2G(2)u 1G u>  > 3(1)u(3) 1  =  H  =w H 1 3  =(1)w H(3) 2= wH  =w(2)H w H  =w H  =w H  =w H {{w$$ . Here|Out()|and|I n()|are the numbers of the outgoing and the incoming edges of the vertex . This product has an algebraic writing using the symmetric groups (cf.[V]). 1.2.PROPs.The notion of PROP models the operations acting on a certain type of gebras and their compositions. e We denote byIthe identitySby the formula-bimodule de ned ( e I(n, n) =k[Sn], e I(m, nelsewhere) = 0. Definition(PROP).A structure ofPROPover anS-bimodulePis given by the following data  an associativecompositionPP →P,  e aunitI→ P. Remark.(eMabyancLngioenivedehtin nelatottPisequivonofaPROds einitTihcf.[McL]) nmn Examples.For any vector spaceV, the sets (H om(VV ,))m, nNof morphisms fromV  m toVwith the composition of morphisms asform a PROP, denotedEnd(V). The associative algebras and the operads are examples of PROPs. Dually,wede nethenotionofcoPROP, which is a PROP in the opposite category of the category ofS-bimodules equipped with the product..