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1、LeibnizalgebroidassociatedwithaNambu-PoissonstructureR.IBANEZ˜1,M.deLEON2,J.C.MARRERO3andE.PADRON31DepartamentodeMatem´aticas,FacultaddeCiencias,UniversidaddelPaisVasco,Apartado644,48080Bilbao,Spain,mtpibtor@lg.ehu.es2InstitutodeMatem´aticasyF´ısicaFundamental,ConsejoSuperiordeInve
2、stigacionesCient´ıficas,Serrano123,28006Madrid,SPAIN,E-mail:mdeleon@fresno.csic.es3DepartamentodeMatem´aticaFundamental,FacultaddeMatem´aticas,UniversidaddelaLaguna,LaLaguna,Tenerife,CanaryIslands,SPAIN,E-mail:jcmarrer@ull.es,mepadron@ull.esFebruary5,2008AbstractarXiv:math-ph/990602
3、7v130Jun1999ThenotionofLeibnizalgebroidisintroduced,anditisshownthateachNambu-PoissonmanifoldhasassociatedacanonicalLeibnizalgebroid.ThisfactpermitstodefinethemodularclassofaNambu-Poissonmanifoldasanappropiatecohomologyclass,extendingthewell-knownmodularclassofPoissonmanifolds.Mathe
4、maticsSubjectClassification(1991):53C15,58F05,81S10.PACSnumbers:02.40.Ma,03.20.+i,0.3.65.-wKeywordsandphrases:Nambu-Poissonbrackets,Nambu-Poissonmanifolds,Leibnizalgebras,Leibnizcohomology,Leibnizalgebroid,modularclass.11IntroductionALiealgebroidisanaturalgeneralizationofthenotionof
5、Liealgebra,andalsoofthetangentbundleofamanifold.Therearemanyotherinterestingexamples,forinstance,thecotangentbundleofanyPoissonmanifoldpossessesanaturalstructureofLiealgebroid.Roughlyspeaking,aLiealgebroidoveramanifoldMisavectorbundleEoverMsuchthatitsspaceofsectionsΓ(E)hasastructur
6、eofLiealgebraplusamapping(theanchormap)fromEontoTMwhichprovidesaLiealgebrahomomorphismfromΓ(E)intotheLiealgebraofvectorfieldsX(M).TheactionofΓ(E)onC∞(M,R)definestheLiealgebroidcohomologyofM.ForaPoissonmanifoldM,theassociatedLiealgebroidisjustthetriple(T∗M,[[,]],#),where[[,]]isthebrac
7、ketof1-formsand#isthemappingfrom1-formsintotangentvectorsdefinedbythePoissontensor.ForanorientedPoissonmanifoldManditsassociatedLiealgebroid,A.Weinstein[29,30]hasdefinedtheso-calledmodularclassofM,whichisanelementofthecorrespondingLiealgebroidcohomology(infact,anelementoftheLichnerow
8、icz-Poissoncohomologyspace
