The Lie algebra structure on the first Hochschild

arXiv:math/0111060v2[math.RT]11Dec2001TheLiealgebrastructureonthefirstHochschildcohomologygroupofamonomialalgebraClaudiaStrametzD´epartementdeMath´ematiques,Universit´edeMontpellier2,F-34095Montpelliercedex5,FranceE-mail:[e-mail]´esum´eNous´etudionslastructured’alg`ebredeLiedupremiergroupedelacoho-mologiedeHochschildd’unealg`ebremonomialededimensionfinieΛ,enter-mescombinatoiresdesoncarquois,enquelconquecaract´eristique.Celanouspermetaussid’examinerlacomposantedel’identit´edugroupealg´ebriquedesautomorphismesext´erieursdeΛencaract´eristiquez´ero.Nousdonnonsdescrit`erespourla(semi-)simplicit´etlar´esolubilit´e.AbstractWestudytheLiealgebrastructureofthefirstHochschildcohomologygroupofafinitedimensionalmonomialalgebraΛ,intermsofthecombi-natoricsofitsquiver,inanycharacteristic.ThisallowsusalsotoexaminetheidentitycomponentofthealgebraicgroupofouterautomorphismsofΛincharacteristiczero.Criteriaforthe(semi-)simplicity,thesolvability,thereductivity,thecommutativityandthenilpotencyaregiven.2000MathematicsSubjectClassification:16E4016W20Keywords:cohomology,Hochschild,Lie,monomial1IntroductionTheHochschildcohomologyH∗(Λ,Λ)ofanyassociativealgebraΛoverafieldkhasthestructureofaGerstenhaberalgebra(see[5]).Inparticular,thefirstHochschildcohomologygroupH1(Λ,Λ)isaLiealgebra,afactwhichcanbeverifieddirectly.Notethatinthefinitedimensionalcaseincharacteristic0thisLiealgebracanberegardedastheLiealgebraofthealgebraicgroupofouterautomorphismsOut(Λ)=Aut(Λ)/Inn(Λ)ofthealgebraΛ.IthasbeentreatedbyGuil-AsensioandSaor´ınin[8].Huisgen-ZimmermannandSaor´ınprovedin[11]thattheidentitycomponentOut(Λ)◦oftheouterautomorphismgroupofΛisinvariantunderde-rivedequivalenceandKeller[13]showedthatHochschildcohomologyispreservedunderderivedequivalenceasagraded(super)Liealgebra.ConsequentlytheLiealgebraH1(Λ,Λ)isinvariantunderderivedequivalence.ThepurposeofthispaperistostudytheLiealgebrastructureofH1(Λ,Λ)inthecaseoffinitedimensionalmonomialalgebraswithoutanyrestrictiononthecharacteristicofthefieldkusingonlyalgebraicmethods.TherelationshipbetweenH1(Λ,Λ)andOut(Λ)allowsustotransfertheresultsobtainedinthiswaytotheidentitycomponentOut(Λ)◦ofthealgebraicgroupOut(Λ)incharacteristic10.ThuswegiveadifferentproofofGuil-AsensioandSaor´ın’scriterionforthesolvabilityofOut(Λ)◦andwegeneralizesomeresultstheyobtainedusingalgebraicgrouptheoryandmethodsinalgebraicgeometry.Thispaperisorganizedinthefollowingway:inthefirstsectionwewillusetheminimalprojectiveresolutionofamonomialalgebraΛ(asaΛ-bimodule)givenbyBardzellin[1]togetahandydescriptionoftheLiealgebraH1(Λ,Λ)intermsofparallelpaths.ThepurposeofthesecondsectionistolinkthisdescriptiontoGuil-AsensioandSaor´ın’sworkonthealgebraicgroupofouterautomorphismsofmonomialalgebras(see[9]).InsectionthreewecarryoutthestudyoftheLiealgebraH1(Λ,Λ).Inparticularwegivecriteriaforthe(semi-)simplicity,thesolv-ability,thereductivity,thecommutativityandthenilpotencyofthisLiealgebraandconsequentlyoftheconnectedalgebraicgroupOut(Λ)◦,intermsofthecom-binatoricsofthequiverofΛ.FinallyaMoritaequivalencegivenbyGabrielin[3]incasechark=p0betweengroupalgebraskGwherethefinitegroupGadmitsanormalcyclicSylowp-subgroupandcertainmonomialalgebrasallowsustoap-plysomeofourresultstotheLiealgebraH1(kG,kG).NotethattheHochschildcohomologyisMoritainvariantasaGerstenhaberalgebra(see[6]).Wegivesomenotationandterminologywhichwekeepthroughoutthepaper.LetQdenoteafinitequiver(thatisafiniteorientedgraph)andkanalgebraicallyclosedfield.Foralln∈NletQnbethesetoforientedpathsoflengthnofQ.NotethatQ0isthesetofverticesandthatQ1isthesetofarrowsofQ.Wedenotebys(γ)thesourcevertexofan(oriented)pathγofQandbyt(γ)itsterminusvertex.ThepathalgebrakQisthek-linearspanofthesetofpathsofQwherethemultiplicationofβ∈Qiandα∈Qjisprovidedbytheconcatenationβα∈Qi+jift(α)=s(β)and0otherwise.WedenotebyΛafinitedimensionalmonomialk-algebra,thatisafinitedimensionalk-algebrawhichisisomorphictoaquotientofapathalgebrakQ/Iwherethetwo-sidedidealIofkQisgeneratedbyasetZofpathsoflength≥2.WeshallassumethatZisminimal,i.e.nopropersubpathofapathinZisagaininZ.LetBbethesetofpathsofQwhichdonotcontainanyelementofZasasubpath.Itisclearthatthe(classesmoduloI=hZiof)elementsofBformabasisofΛ.WeshalldenotebyBnthesubsetQn∩BofBformedbythepathsoflengthn.LetE≃kQ0betheseparablesubalgebraofΛgeneratedbythe(classesmoduloIofthe)verticesofQ.WehaveaWedderburn-MalcevdecompositionΛ=E⊕rwhererdenotestheJacobsonradicalofΛ.ThisworkwillformpartofaPh.D.thesisunderthesupervisionofClaudeCibils.IwouldliketothankhimforhiscommentsandhisencouragementwhichIappre-ciated.2ProjectiveresolutionsandtheLiebracketTheHochschildcohomologyH∗(Λ,Λ)=Ext∗Λe(Λ,Λ)ofak-algebraΛcanbecomputedusingdifferentprojectiveresolutionsofΛoveritsenvelopingalgebraΛe=Λ⊗kΛop.ThestandardresolutionPHochis···→Λ⊗nkδ→Λ⊗n−1k→···→Λ⊗kΛε→Λ→0whereε(x1⊗kx2)=x1x2andδ(x1⊗k···⊗kxn)=n−1Xi=1(−1)i+1x1⊗k···⊗kxixi+1⊗k···⊗kxnforx1,...,xn∈Λ.ApplyingthefunctorHomΛe(,Λ)toPHochandidentifyingHomΛe(Λ⊗kΛ⊗nk⊗kΛ,Λ)withHomk(Λ⊗nk,Λ)foralln∈N,yieldsthecochain2complexCHochdefinedbyHochschild:0→Λd0−→Homk(Λ,Λ)→···→Homk(Λ⊗nk,Λ)dn−→H