On the homotopy Lie algebra of an arrangement

arXiv:math/0502417v2[math.AT]22Jul2005ONTHEHOMOTOPYLIEALGEBRAOFANARRANGEMENTGRAHAMDENHAM1ANDALEXANDERI.SUCIU2Abstract.LetAbeagraded-commutative,connectedk-algebrageneratedindegree1.ThehomotopyLiealgebragAisdefinedtobetheLiealgebraofprimitivesoftheYonedaalgebra,ExtA(k,k).UndercertainhomologicalassumptionsonAanditsqua-draticclosure,weexpressgAasasemi-directproductofthewell-understoodholonomyLiealgebrahAwithacertainhA-module.ThisallowsustocomputethehomotopyLiealgebraassociatedtothecohomologyringofthecomplementofacomplexhy-perplanearrangement,providedsomecombinatorialassumptionsaresatisfied.Asanapplication,wegiveexamplesofhyperplanearrangementswhosecomplementshavethesamePoincar´epolynomial,thesamefundamentalgroup,andthesameholonomyLiealgebra,yetdifferenthomotopyLiealgebras.1.Definitionsandstatementsofresults1.1.HolonomyandhomotopyLiealgebras.Fixafieldkofcharacteristic0.LetAbeagraded,graded-commutativealgebraoverk,withgradedpieceAk,k≥0.WewillassumethroughoutthatAislocallyfinite,connected,andgeneratedindegree1.Inotherwords,A=T(V)/I,whereVisafinite-dimensionalk-vectorspace,T(V)=Lk≥0V⊗kisthetensoralgebraonV,andIisatwo-sidedideal,generatedindegrees2andhigher.TosuchanalgebraA,onenaturallyassociatestwogradedLiealgebrasoverk(seeforinstance[3],[14]).Definition1.1.TheholonomyLiealgebrahAisthequotientofthefreeLiealgebraonthedualofA1,modulotheidealgeneratedbytheimageofthetransposeofthemultiplicationmapμ:A1∧A1→A2:(1)hA=Lie(A∗1)ideal(im(μ∗:A∗2→A∗1∧A∗1)).NotethathAdependsonlyonthequadraticclosureofA:ifweputA=T(V)/(I2),thenhA=hA.Definition1.2.ThehomotopyLiealgebragAisthegradedLiealgebraofprimitiveelementsintheYonedaalgebraofA:(2)gA=Prim(ExtA(k,k)).2000MathematicsSubjectClassification.Primary16E05,52C35;Secondary16S37,55P62.Keywordsandphrases.HolonomyandhomotopyLiealgebras,hyperplanearrangement,supersolv-able,hypersolvable,Yonedaalgebra,Koszulalgebra,Hopfalgebra,spectralsequence,homotopygroups.1PartiallysupportedbyagrantfromNSERCofCanada.2PartiallysupportedbyNSFgrantDMS-0311142.12G.DENHAMANDA.I.SUCIUInotherwords,theuniversalenvelopingalgebraofthehomotopyLiealgebraistheYonedaalgebra:(3)U(gA)=ExtA(k,k).ThealgebraU=ExtA(k,k)isabigradedalgebra;letuswriteUpqtodenoteco-homologicaldegreepandpolynomialdegreeq.ThenUpq=0,unless−q≥p.ThesubalgebraR=Lp≥0Up,−piscalledthelinearstrandofU.Forconvenience,wewillletUpq=Up,−p−q.Thelowerindexqiscalledtheinternaldegree.ThenUisagradedR-algebra,withR=U0.NotethatU+=Lq0UqisanidealinU,withU/U+∼=R.TherelationshipbetweentheholonomyandhomotopyLiealgebrasofAisprovidedbythefollowingwell-knownresultofL¨ofwall.Lemma1.3(L¨ofwall[19]).TheuniversalenvelopingalgebraoftheholonomyLiealge-bra,U(hA),equalsthelinearstrand,R=Lp≥0Up0,oftheYonedaalgebraU=U(gA).ParticularlysimpleisthecasewhenAisaKoszulalgebra.Bydefinition,thismeansthehomotopyLiealgebragAcoincideswiththeholonomyLiealgebrahA,i.e.,U=R.Alternatively,Aisquadratic(i.e.,A=A),anditsquadraticdual,A!=T(V)/(I⊥2),co-incideswiththeYonedaalgebra:A!=U.ForanexpositoryaccountofKoszulalgebras,see[13].Asasimple(yetbasic)example,takeE=VV,theexterioralgebraonV.ThenEisKoszul,anditsquadraticdualisE!=Sym(V∗),thesymmetricalgebraonthedualvectorspace.Moreover,gA=hAistheabelianLiealgebraonV.1.2.Mainresult.ThecomputationofthehomotopyLiealgebraofagivenalgebraAis,ingeneral,averyhardproblem.OurgoalhereistodeterminegAundercertainhomologicalhypothesis.First,weneedonemoredefinition.LetB=AbethequadraticclosureofA.ViewJ=ker(B։A)asagradedleftmoduleoverB.Definition1.4.ThehomotopymoduleofagradedalgebraAis(4)MA=ExtB(J,k),viewedasabigradedleftmoduleovertheringR=U(hA)=ExtB(k,k)viatheYonedaproduct.Theorem1.5.LetAbeagradedalgebraoverafieldk,withquadraticclosureB=A,andhomotopymoduleM=MA.AssumeBisaKoszulalgebra,andthereexistsanintegerℓsuchthatMq=0unlessℓ≤q≤ℓ+1.Then,asgradedHopfalgebras,(5)U(gA)∼=T(MA[−2])b⊗kU(hA).HereM[q]isthegradedR-modulewithM[q]r=Mq+r,whileT(M[−2])b⊗kRisthe“twisted”tensorproductofalgebras,withunderlyingvectorspaceT(M[−2])⊗kRandmultiplication(m⊗r)·(n⊗s)=(−1)|r||n|((m⊗n)⊗rs+(m⊗nr)⊗s).TakingtheLiealgebrasofprimitiveelementsintherespectiveHopfalgebras,weobtainthefollowing.HOMOTOPYLIEALGEBRAOFANARRANGEMENT3Corollary1.6.Undertheabovehypothesis,thehomotopyLiealgebraofAsplitsasasemi-directproductoftheholonomyLiealgebrawiththefreeLiealgebraonthe(shifted)homotopymodule,(6)gA∼=Lie(MA[−2])⋊hA,wheretheactionofhonLie(M)isgivenby[m,h]=−hmforh∈handm∈M.AspointedouttousbyS.Iyengar,Theorem1.5implies(underourhypothesis)thattheprojectionmapU(gA)→U(hA)isaGolodhomomorphism.Therefore,thesemi-directproductstructureofgAalsofollowsfromresultsofAvramov[1],[2].1.3.Hyperplanearrangements.LetA={H1,...,Hn}beanarrangementofhyper-planesinCℓ,withintersectionlatticeL(A)andcomplementX(A).ThecohomologyringA=H•(X(A),k)admitsacombinatorialdescription(intermsofL(A)),duetoOrlikandSolomon:(7)A=E/I,whereEistheexterioralgebraoverk,ongeneratorse1,...,enindegree1,andIistheidealgeneratedbyallelementsoftheformPrq=1(−1)q−1ei1···ceiq···eirforwhichrk(Hi1∩···∩Hir)r;see[22].TheholonomyLiealgebraoftheOrlik-Solomonalgebraalsoadmitsanexplicitpre-sent