Note aux C R Acad Sci Paris acceptee Algebra Algebre

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Note aux C. R. Acad. Sci. Paris (acceptee) Algebra / Algebre Koszul Duality for PROPs Bruno Vallette1 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 Dualite de Koszul des PROPs. La notion de PROP modelise les operations a plusieurs entrees et plusieurs sorties, agissant sur certaines structures algebriques comme les bigebres et les bigebres de Lie. Nous montrons une theorie de dualite de Koszul pour les PROPs qui generalise celle des algebres associatives et des operades. Version franc¸aise abregee. On travaille sur un corps de caracteristique nulle. Suivant J.-P. Serre dans [S], on regroupe sous le terme de gebre differentes structures algebriques comme les algebres, les cogebres et les bigebres. L'ensemble P(m, n) des operations a n entrees et m sorties agissant sur un certain type de gebres est un module a gauche sur le groupe symetrique Sm et a droite sur Sn. Ces deux actions sont compatibles. On appelle S-bimodule toute collection (P(m, n))m, n?N? de tels modules. Nous definissons un produit dans la categorie des S-bimodules qui represente les compositions d'operations a plusieurs entrees et plusieurs sorties.

  • dual coprop

  • uu uu

  • koszul duality

  • ww ww

  • pp pp

  • bb bb

  • koszul dual

  • hh hh

  • nn nn

  • gg gg


Publié le : lundi 18 juin 2012
Lecture(s) : 43
Source : math.unice.fr
Nombre de pages : 6
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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..
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