Duality for Lie Rinehart algebras and Batalin Vilkovisky algebras

Johannes Huebschmann

Description

Duality for Lie-Rinehart algebras and Batalin-Vilkovisky algebras J. Huebschmann USTL, UFR de Mathematiques CNRS-UMR 8524 59655 Villeneuve d'Ascq Cedex, France Clermont-Ferrand, June 19, 2009 1

• lie-rinehart algebras


• ?al


• between left


• batalin- vilkovisky algebra


• pre-lie-rinehart algebra


• structures ∂


• module structure

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 28
• Langue: English

Extrait

DualityforLie-Rinehartalgebras Batalin-Vilkoviskyalgebras J.Huebschmann USTL,UFRdeMathe´matiques CNRS-UMR8524 59655Villeneuved’AscqCe´dex,France [e-mail] Clermont-Ferrand,June19,2009 1 nad

Abstract WewillunveilthestructureofaspecialkindofBatalin-Vilkovisky algebraintermsofLie-Rinehartalgebras:ForanyLie-Rinehart algebra(A,L),Batalin-Vilkoviskyalgebrastructures∂onthe exteriorA-algebraΛ A Lcorrespondbijectivelytoright(A,L)- modulestructuresonA;likewise,generatorsfortheGerstenhaber algebraΛ A Lcorrespondbijectivelytoright(A,L)-connections onA.WhenLisprojectiveasanA-module,givenaBatalin- Vilkoviskyalgebrastructure∂onΛ A L,theBatalin-Vilkovisky algebra(Λ A L,∂)coincideswiththestandardcomplexcomput- ingthehomologyofLwithcoefficientsinAwithreferenceto theright(A,L)-modulestructuredeterminedby∂.WhenL isalsooffiniterankn,viaduality,therearebijectivecorre- spondencesbetween(A,L)-connectionsonΛ n A Landright(A,L)- connectionsonAandbetweenleft(A,L)-modulestructureson Λ n A Landright(A,L)-modulestructuresonA.Hencetherearebi- jectivecorrespondencesbetween(A,L)-connectionsonΛ n A Land generatorsfortheGerstenhaberbracketonΛ A Landbetween (A,L)-modulestructuresonΛ n A LandBatalin-Vilkoviskyalgebra structuresonΛ A L.ThehomologyofsuchaBatalin-Vilkovisky algebra(Λ A L,∂)coincideswiththecohomologyofLwithcoeffi- cientsinΛ n A L,withreferencetotheleft(A,L)-modulestructure determinedby∂.Theseobservationshavevariousapplications toPoissonstructuresandtodifferentialgeometry. Thegeneralizationofthemutualstructureofinteractiontothe differentialgradedcontextleadstotwilledLie-Rinehartalgebras. Thesegeneralize,intheLie-Rinehartcontext,complexstructures onsmoothmanifolds.Analmostcomplexmanifolddeterminesan “almosttwilledpre-Lie-Rinehartalgebra”,whichisatruetwilled Lie-Rinehartalgebraifandonlyifthealmostcomplexstructure isintegrable.TheGerstenhaberalgebraarisingfromanalmost complexstructureisadifferentialGerstenhaberalgebraifand onlyifthealmostcomplexstructureisintegrable.Thiskindof Gerstenhaberalgebra,endowedwithageneratorturningitinto aBatalin-Vilkoviskyalgebra,arisesfromaCalabi-Yaumanifold andexplainsinparticulartherelationshipbetweenholomorphic 2