On the Lie envelopping algebra of a pre-Lie algebr

arXiv:math/0404457v1[math.QA]26Apr2004OntheLieenveloppingalgebraofapre-LiealgebraJ.-M.OudomandD.Guinoudommath.univ-montp2.fr,dguinmath.univ-montp2.frI3M,UniversitØMontpellierII,ase051PlaeEugŁneBataillon34095MontpellierFebruary1,2008Forthelastveyears,manyombinatorisHopfalgebraswereintroduedindierentset-tings.OneanquotetheHopfalgebrasofC.BrouderandA.Frabetti[2℄,A.ConnesandD.Kreimer[6℄,L.Foissy[7℄,R.GrossmanandR.-G.Larson[10℄,J.-L.LodayandM.Rono[15℄,I.MoerdijkandP.vanderLaan[17℄[22℄.Inthispaper,ouraimistoshowthatalotoftheseHopfalgebrasarerelatedtogeneralalge-braionstrutions.Intheommutative(oroommutative)ase,thekeyalgebraistrutureisthepre-Liealgebraone,asnotiedbyChapotonandLivernet[3℄.Inthenonommutativeandnonoommutativease,itseemstobeplayedbybraealgebras.Intherstpart,wereallthedenitionofapre-Liealgebraandgivesomeexamples.Theseondpartisdevotedtotheonstrutionofanexpliit∗produtonthesymmetri(o)algebraS(L)ofanypre-LiealgebraL.Then,weprovethat(S(L),∗,Δ)isisomorphitotheenvelopingalgebraofLLie.Inthethirdpart,westudytheasewhereListhepre-Liealgebraofrootedtrees,andweshowthatouronstrutionorrespondstothedualofConnesandKreimer’sone.ThisgivesanotherproofofthedualitybetweentheConnes-KreimerandGrossman-LarsonHopfalgebras[6℄[19℄[12℄.Inthefollowingpart,weusetheproduts∗and◦oftherstpart,inordertoshowthatsymmetribraealgebrasintroduedin[14℄arenothingelsebutpre-Liealgebras.Finally,inthefthpart,werealltheonstrutionofaHopfalgebrastrutureonthetensor(o)algebraT(V)ofanybraealgebraVandweshowthatFoissy’sHopfalgebraofplanarrootedtreesbelongstothisgeneralsetting.Fromherewexaommutativeringkoverwhihmodules,algebras,tensorprodutsandlinearmapsaretaken.1Pre-Liealgebras.Denition1.1.Apre-LiealgebraisamoduleLequippedwithabilinearprodut◦whoseassoiatorissymmetriinthetwolastvariables:X◦(Y◦Z)−(X◦Y)◦Z=X◦(Z◦Y)−(X◦Z)◦Y.Thesealgebraswerealsoalledrightsymmetrialgebras,Koszul-VinbergorVinbergalgebras.Here,weusetheterminologyofF.ChapotonandM.Livernetin[3℄.There’snodiultytogiveagradedversionofpre-Liealgebrasbyreplaingtheaboveidentityby:X◦(Y◦Z)−(X◦Y)◦Z=(−1)|Y||Z|(X◦(Z◦Y)−(X◦Z)◦Y).1wherethethreevariablesarehomogeneousand|−|denotesthedegree.Letusnowreallawellknownfat,whihistherootoftheinterestthatpeoplehaveshowninpre-Liealgebras:Proposition1.2.Let(L,◦)beapre-Liealgebra.Thefollowingbraket[X,Y]=X◦Y−Y◦XmakesLintoaLiealgebra,whihwewilldenoteLLie.Inthefollowing,wereallsomelassialandmorereentexamples.1.1Firstexamples.Asitsassoiatorissymmetriinallthreevariables,anassoiativealgebrais,ofourse,apre-Liealgebra.Butthepre-Liestrutureisinfatrelatedtoaweakerkindofassoiativity:theassoiativityoftheompositionofmulti-variablesfuntions.M.GerstenhaberalreadynotedthisfatintheHohshildohomologysetting[8℄:Example1.1.1.Deformationomplexesofalgebras.LetVbeamoduleandletusdenoteCn(V,V)thespaeofalln-multilinearmapsfromVtoV.Forf∈Cp(V,V),g∈Cq(V,V)andi∈J1,pK,oneandene:f◦ig(x1,······,xp+q−1):=f(x1,···,xi−1,g(xi,···,xi+q−1),xi+q,···,xp+q−1)f◦g=pPi=1f◦igThentheprodut◦makesC•(V,V)apre-Liealgebra.In[8℄,M.GerstenhabergaveagradedversionofthisprodutontheHohshildohomologyomplexC•(A,A)ofanassoiativealgebraA:f◦ig(x1,···,xp+q−1):=(−1)(p−1)(i−1)f(x1,··,xi−1,g(xi,··,xi+q−1),xi+q,··,xp+q−1).Then,heshowedthat,inharateristidierentfrom2,thegradedLiealgebrainduedbythisgradedpre-LiestrutureontrolsthedeformationsofA.Moregenerally,D.Balavoinegavein[1℄asimilargradedLiealgebraonstrutionontheo-homologyomplexofanalgebraoveranyquadratioperad,whihontrolsthegivenalgebra’sdeformations.Example1.1.2.Operads.Anoperadisasequeneofmodules(P(n)n≥2,whereeahP(n)isak[Sn]-module.Foreveryibetween1andn,wehavea◦ioperation:◦i:P(n)⊗P(m)−→P(n+m−1)satisfyingthefollowingonditionsofassoiativity:f◦i(g◦jh)=(f◦ig)◦i+j−1h(f◦ig)◦|g|+i+j−1h=(f◦jh)◦|h|+i+j−1gThis◦i-operationsshouldmoreoverbesomewayompatiblewiththek[Sn]-ations.Thereisnodiultytohekthattheprodut:f◦g=|f|Xi=1f◦igdenesapre-LieprodutonLn≥2P(n).21.2Ageometriexample:anemanifolds.Ananemanifoldisamanifoldwithatorsionfreeandatonnetion∇.Equivalently,itisamanifoldequippedwithanatlaswithanetransitionfuntions.Thevetoreldsofsuhamanifoldisapre-Liealgebraforthefollowingirleprodut:X◦Y=∇X(Y),whoseassoiatedLiebraketistheusualone.1.3Theinspiringexample:rootedtrees.Denition1.3.1.Arootedtreeisatreewithadistinguishedvertex:itsroot.Aombina-torialdenitionouldbethefollowing:itisaniteposetwithaminimum(theroot)andnoritialpair:xyztyzzy=⇒orForanyrootedtreeT,wedenote|T|theunderlyingsetandweallitsvertiesofTitselements.LetXbeaset.AX-oloredrootedtreeisarootedtreeTequippedwithaolormap|T|→X,whihassoiatesitsolorinXtoeveryvertexofT.Wewillrepresenttreesbyusingplanargraphsandtheorderinduedbythegravity:==Herearetwodierentoloredtreeswiththesameunderlyingtree:6=FortwogivenrootedtreesT1andT2,andahosenvertexvofT1,weanglueT1onv.TherootedtreeT1◦vT2obtainedinsuhawayistheposetwhoseunderlyingsetisthedisjointunion|T1|∐|T2|.TheorderisinduedbytheordersofT1andT2andwvforallwinT2:vv:=◦vNotiethat,whenT1andT2areoloredbyasetX,thenT1◦vT2isaX-oloredrootedtree,3whoseolormapisthedisjointunionofthetwoolormapsofT1andT2.Propositi