Pre-Lie代数的扩张与非阿贝尔上同调 59页

万方数据中文摘要中文摘要本文研究了pre-Lie代数的扩张，定义了pre-Lie代数的二阶非阿贝尔上同调，并证明pre-Lie代数的扩张为其所分类.考虑阿贝尔扩张的分类，又得到其与二阶阿贝尔上同调的关系.然后，本文通过在pre-Lie代数的Chevalley-Eilenberg复形上定义一种pre-Lie代数结构，得到一个分次微分李代数L.Pre-Lie代数扩张的范畴等价于L上的Deligne群胚，从而为其连通分支之集合所分类.同时，二阶阿贝尔上同调可以通过考察L的切线复形而得到.最后，本文指出虽然pre-Lie代数的二阶非阿贝尔上同调并不天然是PL¥-代数的内在上同调，但pre-Lie代数的中心扩张能被一种二项PL¥-代数的内在上同调所刻画.关键词：非阿贝尔上同调；扩张；Deligne群胚；Pre-Lie代数；代数形变理论；内在上同调I AbstractAbstractInthispaper,thesecondnon-abelianpre-Liealgebracohomologyisintroducedbyconsideringtheclassificationofextensionsofpre-Liealgebras.Thisnotionisrelatedtothesecondabeliancohomology,whichclassifiesabelianextensions,andthusgetsitsname.Thentheauthorintroducesapre-LiestructureontheChevalley-Eilenbergcomplexofpre-LiealgebrasandconstructsadifferentialgradedLiealgebraL.Theauthorthenprovesthatthecategoryofextensionsofpre-LiealgebrasisequivalenttotheDelignegroupoidDel(L).Thenthesecondnon-abelianpre-LiealgebracohomologycomesnaturallyasthesetofconnectedcomponentsoftheDelignegroupoidDel(L).Further,thesecondabeliancohomologyarisesfromthetangentcomplexofL.Finally,theauthorpointsoutthatalthoughthesecondnon-abelianpre-Liealgebracohomologyisnotnaturallyanintrinsiccohomology,thecentralextensionsofpre-Liealgebrascanbeclassifiedbyanintrinsiccohomologyof2-termPL¥-algebras.KeyWords:Non-abeliancohomology;Extensions;Delignegroupoid;Pre-Liealgebras;Algebraicdeformationtheory;IntrinsiccohomologyII万方数据 CONTENTSCONTENTS中文摘要



























































IAbstract



























































IIChapter1Introduction















































1Chapter2Preliminaries












































42.1ExtensionsofLiealgebrasandthenon-abeliancohomology








42.2ThenotionofDelignegroupoid






































62.3DescribeHLie2intermsofDelignegroupoids























92.4Thenotionofhomotopyalgebras
































112.5DescribeHLie2intermsof2-termL¥-algebras




















15Chapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-Liealgebracohomology





























173.1Thenotionofpre-Liealgebraextensions


























173.2Towardnon-abelian2-cocycles



































193.3Theclassificationtheorem









































23Chapter4DescribeH2intermsofDelignegroupoids








27preLie4.1TheDGLAstructureonC+1(A;A)





























274.2ConstructionoftheDGLAL






































334.3HpreLie2=p0Del(L)















































354.4Abeliancohomologyastangentcomplex


























37Chapter5Towardtheintrinsiccohomology























405.1Thenotionof2-termPL¥-algebras
































405.2Centralextensionsofpre-Liealgebras





























42AppendixAglanceofhighercategorytheory























45A.1Thenotionofhighercategorytheoryandintrinsiccohomology


45A.2Intrinsiccohomologyin(2;1)-categoricalcontext

















47References



























































54III万方数据 CONTENTSIndex

































































56致谢



























































58个人简历



























































59IV万方数据 万方数据Chapter1IntroductionChapter1IntroductionExtensionsofgroupsareclassifiedbynon-abeliangroupcohomology.Specially,abelianextensionsareclassifiedbyabeliangroupcohomology.ThisfacthasbeenfoundbyEilenbergandMaclaneinthe1940s([11]).ThentherearealotofanalogousresultsforLiealgebras([1,16,17]),Liesuperalgebras([2,12])andLiealgebroids([5]).Pre-Liealgebras,alsocalledleft-symmetricalgebras,quasi-associativealgebras,Koszul-Vinbergalgebrasetc.,arenonassociativealgebraswhosecommutatorsformLiealgebrasandwhoseleftmultiplicationsformrepresentationsofthecommutatorLieal-gebras.TheyhaveappearedinCayley’sworkonrootedtreealgebrasin1896([9]).Thentheywereforgottenforalongtimebeforearisingfromthestudyofseveraltopicsingeometryandalgebrain1960s,suchasconvexhomogenouscones([28]),affineman-ifoldsandaffinestructuresonLiegroups([20,24]),deformationofassociativealgebras([14])etc.Asurveyofthishistoryandthealgebraictheoryofpre-LiealgebrashavebeengivenbyBurdein[7].In1980s,Kimstudiedtheextensionsofpre-Liealgebrasanddefinedthenon-abeliancohomologyforpre-Liealgebrasin[18,19].However,thecollectionofallextensionsofanalgebrabyanotherisnaturallyagroupoid,ratherthanaset,sothesecondnon-abeliancohomologyshouldbeviewedasthesetofconnectedcomponentsofagroupoid.Moreover,ifthisgroupoidarisesasanactiongroupoid,itisnaturallyconnectedtothedeformationtheory.In2012,Frégierfoundthatthesecondnon-abeliancohomologyclassifyingLiealgebraextensionscanalsobedescribedintermsoftheDelignegroupoid,seeing[13].ThenotionofDelignegroupoidcomesfromtheideasofDeligneondeformationtheory,whichweretransmittedvia[15].Irefer[23]formoreinformationaboutthisapproachofdeformationtheory.TheDelignegroupoidusedinFrégier’sworkarisesfromadifferentialgradedLiealgebra(DGLAforshort)relatedtotheChevalley-EilenbergcomplexofLiealgebras.However,thisDGLAisobtainedbytakingthegradedcommutatorofagradedpre-Liealgebra,thusitisnaturaltoaskiftheanalogousresultsholdforpre-Liealgebras.1 Chapter1IntroductionThereisageneralperspectiveoncohomology,thatistheintrinsiccohomologyforahighercategory.Itisnothingbutthesetofconnectedcomponentsofthehom-spacewithpossibleextrastructuresinducedfromthecoefficents.Thisideawasessentiallyestablishedin[6]anddevelopedrecentlybyLurie([21]).Acomprehensiveaccountcanbefoundin[26].In[30],theauthorpointedoutthattheextensionsofLiealgebrascanbedescribedbyahom-spaceof2-termL¥-algebrasandthesecondnon-abelianLiealgebracohomol-ogythereforecanbeviewedasaspecialcaseoftheintrinsiccohomologyofL¥-algebrasinanaturalway.However,thisapproachreliesontheantisymmetryofLiebracket.Soitisdoubtfulwhetherthiscanbedoneforotherkindofalgebras.Inthispaper,Istudytheextensionsofpre-Liealgebrasandthesecondnon-abelianpre-Liealgebracohomology.ByconstructingasuitableDGLA,IgettheanalogousresultsofFrégierforpre-Liealgebras.Lackingofenoughsymmetry,theapproachencodingthesecondnon-abeliancohomologyintoanintrinsiccohomologyfailsforpre-Liealgebras.However,Ishowthatthecentralextensionsofpre-Liealgebrascanbeclassifiedbyanintrinsiccohomology.Thewholearticleisorganizedasfollows.First,inchapter2,Isummarizesomewell-knownresultsonLiealgebraextensionswithalittledifferentstyleofpresentation,andmakesomeconventions.Forself-contained,IalsoexplainthenotionofDelignegroupoidandhomotopyalgebras.Somenotionsfromhighercategorytheoryareused.So,Iexplaintheminastrict2-categoricalcontextandputthispartasanappendix.Then,inchapter3,Istudytheextensionsofpre-Liealgebrasanddefinethesecondnon-abelianpre-Liealgebracohomologytoclassifythem.Ialsodiscusstherelationshipbetweenthenon-abeliancohomologywiththeabelianone,whichclassifiesabelianex-tensions.AlthoughthoseresultshavebeengivenbyKim([18,19]),forself-contained,Iexplainthemindetails.Then,inchapter4,Iintroduceapre-LiestructureontheChevalley-Eilenbergcom-plexofpre-LiealgebrasandconstructtheDGLAL.Iprovethatthecategoryofexten-sionsofpre-LiealgebrasisequivalenttotheDelignegroupoidDel(L)oftheDGLAL.Then,thesecondnon-abelianpre-LiealgebracohomologyarisesnaturallyasthesetofconnectedcomponentsofDel(L)andthegroupsofnon-abelian1-cocyclesarises2万方数据 Chapter1IntroductionnaturallyastheautomorphismgroupsinDel(L).Further,theabeliancohomologyarisesfromthetangentcomplexofL.Finally,inchapter5,Irecallthenotionof2-termPL¥-algebras.ThenIexplainwhytheapproachforLiealgebrasfailsforpre-Liealgebras.Afterall,Ishowthatthecentralextensionscanbeclassifiedbyanintrinsiccohomologyof2-termPL¥-algebras.3万方数据 万方数据Chapter2PreliminariesChapter2PreliminariesThroughoutthispaper,allvectorspacesareoveragivenfieldk.Tosimplifyno-tations,Ialsomakethefollowingconventions:1.Ifthereisnoambiguity,thecompositesymbol◦willbeomitted.2.IfrisalinearmapfromsomevectorspaceAtothegeneralizedlinearLiealgebragl(V)forsomevectorspaceV,thenforallx2A,r(x)willbealsowrittenasrxwhenitisviewedasalinearmaponV.3.ThesubscriptgofaLiebracket[;]gemphasizeswhichalgebrathisbracketisworking,andwillbeomittedifthereisnoambiguity.4.Thecyclicsumnotationåisusedfrequently,wherethesummationistakenx;y;zoverallcyclicpermutationofx;yandz.Someterminologiesandbasicfactsfromcategorytheoryandhomologyalgebra,suchas5-lemma,complex,groupoidetc.,arefrequentlyused.Therearemanytext-booksonthosetopics,like[8,22,29].2.1ExtensionsofLiealgebrasandthenon-abeliancohomology2.1.1Letg;bg;hbeLiealgebras.bgissaidtobeanextensionofgbyhifthereexistashortexactsequenceip0!h!bg!g!0:Asplittingofbgisalinearmaps:g!bgsuchthatp◦s=id.Amorphismq:bg!bg′oftwoextensionsisaLiealgebramorphismqsuchthatthefollowingdiagramcommutes:i/p′0/hbg/g/0qi′p′0/h/bg′/g/0By5-lemma,thisq,ifexists,mustbeanisomorphism.Ifthisisthecase,thetwoextensionsbgandbg′aresaidtobeisomorphic.4 Chapter2PreliminariesNowallextensionsofgbyh,togetherwiththemorphismsbetweenthem,formacategory,denotedbyExtLie(g;h).Moreover,thiscategoryisagroupoid,thatisacategorywhoseeverymorphismisinvertible.OnecanthenconsiderthesetExtLie(g;h)ofisomorphismclassesofExtLie(g;h).RemarkExtLie(g;h)isnaturallyapointedgroupoidasitcontainsaspecialobject,whichistheLiealgebradirectsumgh.ThusExtLie(g;h)isnaturallyapointedset.2.1.2Anon-abelian2-cocycleongwithvaluesinhisacouple(w;r)ofanalternatingbilinearmapw:^2g!handalinearmapr:g!Der(h)satisfyingthefollowingequalitiesforallx;y;z2g:[r(x);r(y)]r([x;y])=ad(w(x;y));årxw(y;z)w([x;y];z)=0:x;y;zThesetofthose2-cocyclesisdenotedbyZLie2(g;h).Two2-cocycles(w;r)and(w′;r′)areequivalent,ifthereexistsalinearmapφ:g!hsuchthatr′r=ad◦φ,andforallx;y2g,w′(x;y)w(x;y)=rxφ(y)ryφ(x)+[φ(x);φ(y)]φ([x;y]):ThesetHLie2(g;h)ofequivalenceclassesofnon-abelian2-cocycleiscalledthesecondnon-abeliancohomologyoftheLiealgebragwithvaluesinh.2.1.3ExtensionsofgbyhareclassifiedbyHLie2(g;h).Moreprecisely,any2-cocycle(w;r)definesaLiebracketonghvia[x+u;y+v]w;r:=[x;y]g+w(x;y)+rx(v)ry(u)+[u;v]h;8x;y2g;u;v2h:ThisgivesaLiealgebrastructureongh,calledthesemidirectproductgnw;rh,andonecanseeitisanextensionofgbyh.Conversely,givenanextensionbgofgbyh,bychoosingasplittingsandidentifyhwithitsimageinbg,onecandefineanalternatingbilinearmapw:^2g!handalinearmapr:g!Der(h)asfollows:w(x;y):=[s(x);s(y)]bgs([x;y]g);8x;y2g;rx(u):=[s(x);u]bg;8x2g;y2h:5万方数据 万方数据Chapter2PreliminariesOnecancheckthiscouple(w;r)isa2-cocycleandthecohomologicalclassofitisindependentofthechoiceofs.Finally,theequivalenceof2-cocyclescorrespondingtotheisomorphismofexten-sions.Thedetailscanbefoundin[1,16,17].2.2ThenotionofDelignegroupoidInthissection,IrecallthedefinitionofDelignegroupoid.Forself-contained,Iexplaintheapproachindetails.⊕2.2.1AZ-gradedvectorspaceisadirectsumV=n2ZVnofcountablemanyvectorspaces.AnelementinatermVniscalledahomogeneouselementofdegreejxj=n.Agradedlinearmapφ:V!Wofdegreekisalinearmapsatisfyingφ(Vn)Wn+k.ThegradedtensorproductoftwogradedvectorspacesVandWisagradedvectorspaceVWwithgrading:⊕(VW)n:=ViWj:i+j=nAgradedbilinearmapispreciselyagradedlinearmapfromgradedtensorproduct.AgradedvectorspaceisaZ-gradedvectorspacewhoseeveryterminnegativedegreevanishes.AZ-gradedvectorspaceissaidtobelower-boundedifitcanbeidentifiedwithagradedvectorspacebyadegreeshifting.⊕AgradedLiealgebraisagradedvectorspaceg=n>0gnequippedwithagradedbilinearmap[;]:gg!gofdegree0satisfies1.thegradedantisymmetry,i.e.[x;y]=(1)jxjjyj[y;x],2.thegradedJacobinidentity,i.e.å(1)jxjjzj[x;[y;z]]=0.x;y;zHerex;y;zareallhomogeneouselementsing.Foranyx2gn,thegradedadjointtransformationadxisdefinedasadx:=[x;].Thisisagradedlinearmapofdegreen.AdifferentialgradedLiealgebra(DGLAforshort),isagradedLiealgebragequippedwithacohomologicalderivationd:g!gofdegree1,thatmeansagradedlinearmapofdegree1satisfiesd2=0andd[x;y]=[dx;y]+(1)jxj[x;dy];forallhomogeneouselementsx;yofg.6 Chapter2Preliminaries2.2.2LetgbeanilpotentLiealgebra.OnecanusetheBaker-Campbell-Hausdorffformula111xy:=x+y+[x;y]+([x;[x;y]][y;[x;y]])[y;[x;[x;y]]]+


21224togiveagroupstructureong.Thisgroupwillbedenotedbyexp(g).Thefullformulacanbefoundinsometextbookslike[3,ch.II].Alternatively,onecanalsoviewgasthetangentLiealgebraofaLiegroupG,thentheexponentialmapexpiswelldefined.Anyhow,onecangetaconnectedLiegroupexp(g)suchthatitstangentLiealgebraisgandthattheexponentialmapisbijective.Notethat,whengisabelian,thegroupexp(g)willcoincidewiththeunderlyingabeliangroupofg.2.2.3Let(g;d)beanad0-nilpotentDGLA,thatmeansforallx2g0,thegradedadjointtransformationadxisnilpotent.ThenthesetofMaurer-Cartanelementsisdefinedas{}1MC(g):=a2g1da+[a;a]=0:2Anya2g1definesagradedderivationofdegree1bytheformulada=d+ada,calledg-connection.Specially,Maurer-Cartanelementsdefineflatconnectionsinthesenseofda2=0.Ifthisisthecase,onecansee(g;da)becomesanotherDGLA,calledthetangentcomplexgaofgata.Thesubspaceg0ofgitselfisthenanilpotentLiealgebra.ThecorrespondingLiegroupexp(g0)actsongasexp(g0)!GL(g)adnexp(x)7!eadx:=x:ån!n>0Thenthisgroupactsongl(g)byconjugation.SuchanactionisthesameastheadjointactionofGL(g)ongl(g),thusonehasexp(x)φexp(x)=eadx◦φ◦eadx=eadadx(φ);8φ2gl(g):Herethenotationadφdenotestheadjointtransformationofgl(g).7万方数据 Chapter2PreliminariesRemark1InaDGLA(g;d),foranyx2g0;a2g1,onehasadadx(d+ada)=adadx(a)dx:Indeed,foranyy2g,onehas[adx;d+ada](y)=[adx;d](y)+[adx;ada](y)=[x;dy]d[x;y]+ad[x;a](y)=(ad[x;a]dx)(y);whichshowstherequiredequality.Remark2LetgbeaLiealgebraandf(X)beapolynomial,thenalinearmapϕongisahomomorphismifandonlyifthefollowingequalityholds:f(adϕ(x))(ϕ(y))=ϕ(f(adx)(y));8x;y2g:2.2.4Bytheaboveremarks,onecanwritedowntheactionofexp(g0)onthesetofg-connectionsexplicitlyasexp(x)(d+ada)exp(x)=eadadx(d+ada)eadadxid=d+ada+(adadx(a)dx)adadx()eadxid=d+ada+ad(adx(a)dx)adx()eadxid=d+ada+(adx(a)dx):adxThus,theactionofexp(g0)transformsg-connectionstog-connections,anddefinesso-calledgaugetransformationsonthesetofg-connections.Notethat(exp(x)(d+ad2=exp(x)(d+ad)2exp(x):a)exp(x))aThusthegaugetransformationspreserveflatconnections.Thegaugetransformationsonthesetofg-connectionstheninducethegaugeac-tionofexp(g0)ong1byeadxidexp(x):a=a+(adx(a)dx);adxandthisactionpreservesMaurer-Cartanelements.8万方数据 万方数据Chapter2Preliminaries2.2.5GivenanactionofagroupGonthesetS,theactiongroupoidS==Gisthegroupoidconsistsofthefollowingdata:•objectsaretheelementsofS,•morphismsaretriples(s;s′;g),wheres2Sisthe“source”,s′2Sisthe“target”andgisanelementofGsuchthatg:s=s′,•thecompositeisinducedbythemultiplicationofG.Foranad0-nilpotentDGLA(g;d),itsDelignegroupoidDel(g)isdefinedtobetheactiongroupoidunderthegaugeaction:Del(g):=MC(g)==exp(g0):2.3DescribeH2intermsofDelignegroupoidsLieBeforegoingforward,Imakesomeconventionontensorandexteriornotations.First,the0-thtensorpower0Valwaysmeanthegroundfieldk.Ialsoidentifytheexteriorproduct^nVasasubspaceofthetensorproductnVbysettingx1^


^xn:=åsgn(s)s(x1


xn);8x1;


;xn2V;s2SnwhereSndenotesthegroupofn-permutataionsandeachs2Snactsonthetensorsbypermutingitscomponents,i.e.s(x1


xn):=xs1(1)


xs1(n):Forthisreason,Iwillalsodenotetheresultedtensorbyxs1Inthisway,anyn-linearmapfinducesanalternatingn-linearmapf^,calleditsantisymmetrization,asfollows:notethatann-linearmapfisactuallyalinearmapfromnV,thusonecandefinef^asf^(x1;


;xn):=f(x1^


^xn)=åsgn(s)f(xs1(1)


xs1(n))s2Sn=åsgn(s)f(xs1(1);


;xs1(n)):s2Sn9 Chapter2PreliminariesForinstance,^2Visidentifiedasasubspaceof2Vviax^y=xyyx,andanybilinearmapwonVinducesitsantisymmetrizationw^viaw^(x;y)=w(x;y)w(y;x):Finally,anylinearendomorphismfonVcanbeextendedtoamultilinearmapvianf(x1;


;xn)=åx1


f(xk)


xn:i=12.3.1LetgbeaLiealgebraandMag-module.TheChevalley-EilenbergcomplexisthegradedvectorspaceC(g;M)ofalternatingmultilinearmapsfromgtoM:Cn(g;M):=Hom(^ng;M);withthedifferentialdwhichmapsanyf2Cn1(g;M)ton;


;xk(df)(x1n):=å(1)xkf(x1;


;xbk;


;xn)k=1(1)i+jf([x;x];x;


;xb;


;xb;


;x);(2.1)åij1ijni(gh;h)isdefinedasthesubcomplexofC(gh;h)byC(gh;h)=C(gh;h)C(h;h):>>OnecanverifythatC+1(gh;h)isasub-DGLAofC+1(gh;gh)endowedwith>theNijenhuis-RichardsongradedLiebracket.DenotethisDGLAbyL.OnecanprovefurtherthatL0isabelianandthatLisad0-nilpotent,see[13]formoredetails.ThenonecandefinetheMaurer-CartanelementsofLandfurthertheDelignegroupoidDel(L)following2.2.3–2.2.5.2.3.3Extensionsofgbyhareclassifiedbyp0Del(L).Moreprecisely,onecanprovethatZLie2(g;h)=MC(L)andthattwo2-cyclesareequivalentifandonlyiftheircorrespondingMaurer-CartanelementsareconnectedbyamorphisminDel(L).ThusHLie2(g;h)isisomorphictothesetofconnectedcomponentsofDel(L),i.e.p0Del(L).See[13]formoredetailsontheproof.2.3.4LetgbeaLiealgebraandMbeag-module.Denotetheantisymmetrizationofitsg-modulestructurebya,thena2MC(L)andonehasthetangentcomplexLa.OnecanfurtherprovethattheChevalley-EilenbergcomplexC(g;M)isasub-DGLAofit.2.4ThenotionofhomotopyalgebrasFromnowon,Iwillusesomenotionsfromhighercategorytheory.AsIonlyusetheminanelementaryway,onecanrefertheappendixfornecessaryinformationinsteadofthemonograph[21].2.4.1AchaincomplexCisagradedvectorspaceequippedwithagradedlinearmapdofdegree1satisfyingd2=0calledthedifferential.Achaincomplexissaidtobe11 Chapter2PreliminariesconcentratedinfirstntermsifCk=0forallk>n.AnychaincomplexCadmitsachaincomplexconcentratedinfirstnterms


!0!Zn!


!C0;whereZnisthekernelofthedifferentialdinCn.Thischaincomplexiscalledn-truncationofCandisdenotedbyt6nC.AchainmapF:C!Dbetweentwochaincomplexesisagradedlinearmapofdegree0compatiblewiththedifferentials,thatmeansthefollowingdiagramcommutes.


/Cn/Cn1/


FF


/Dn/Dn1/


LetF;G:C!Dbetwoparallelchainmaps.AchainhomotopyF:F)GbetweenthemisagradedlinearmapF:C!Dofdegree1suchthatFG=F◦d+d◦F:Theverticalcompositeoftwochainhomotopiesisthenthesumofthemandthehor-izontalcompositeoftwochainhomotopiesnisthecompositeofthemasgradedlinearmaps.Onecanverifythatthechaincomplexestogetherwithchainmapsbetweenthemas1-morphismsandchainhomotopiesbetweenchainmapsas2-morphismsforma2-category,denotedbyCh.Thisisfurthera(2;1)-category,thatisa2-categorywhoseevery2-morphismisinvertible.Dually,onehascochaincomplexes,whicharegradedvectorspacesequippedwithagradedlinearmapdofdegree1satisfyingd2=0.Onealsohascochainmapsandcochainhomotopies.2.4.2ThetensorproductoftwocochaincomplexesCandDisthecochaincomplex(CD;d),whereCDisthetensorproductofgradedvectorspacesandd(x;y)=(d(x);y)+(1)jxj(x;d(y));8x2C;y2D:ThismakesChbecomeamonoidalcategoryandthenmanyalgebrastructuresonvectorspacescanbegeneralizedonchaincomplexes.Forinstance,aDGLAisaLieaglebraonacochaincomplex,butonecanalsodefineitonachaincomplex.12万方数据 Chapter2PreliminariesRemark1Notethatthegradedtranspositionoperations12:CC!CCmapsxyto(1)jxjjyjyxinsteadofyxandgeneralpermutationoperationsssaregen-eratedbythegardedtranspositionoperationsviatheformulasst=ss◦st.Remark2Recallthatthetensorproductoftwolinearmapsf:V!Wandg:V′!W′isalinearmapfg:VV′!WW′whichmapsuvtof(u)g(v).2.4.3TheintervalchaincomplexIisthechaincomplex(id;id)


!0!k!kk;wherethetermkkisindegree0.ThenthetensorproductofIandCisthechaincomplexdd


!C2C2C1!C1C1C0!C0C00;wherethetermC0C00isindegree0andthedifferentialdisd(x1;x2;y)=(d(x1)+y;d(x2)y;d(y));8x1;x22Cn+1;y2Cn;8n2N:2.4.4LetF;G:C!Dbetwoparallelchainmaps.AchainhomotopyF:F)Gbetweenthemcanalsobeencodedintoachainmapasfollows:ForanyF;Gtwogradedlinearmapsofdegree0andFagradedlinearmapofdegree1,onehasthegradedlinearmap(F;G;F):CCC1!Dofdegree0.NotethatCI=CCC1asgradedvectorspaces.Then(F;G;F)isachainmapifandonlyifthefollowingdiagramcommutesforalln2N.(F;G;F)Cn+1Cn+1Cn/Dn+1dd(F;G;F)CnCnCn1/DnRestrictedonC00and0C0,thisconditionsaysthatF;Garechainmaps.Restrictedon00C,theconditionbecomesd◦F=FGF◦d;whichsaysFisachainhomotopybetweenFandG.13万方数据 Chapter2Preliminaries2.4.5Aschainhomotopiescanbeencodedintochainmaps,itmakessensetodefinechainhomotopiesbetweenthem.Inthisway,onehaschainmapsas1-morphisms,chainhomotopiesbetweenchainmapsas2-morphisms,chainhomotopiesbetweenthechainhomotopiesbetweenchainmapsas3-morphisms,etc.Therefore,Chisactuallyan(¥;1)-category.Agradedalgebra(forinstanceaLiealgebraonachaincomplex)canbeviewedasachaincomplexCequippedagradedbinaryoperation,whichisachainmapCC!C,satisfyingagradedequality,whichisacommutativediagramconsistingofchainmapsinducedbythegradedbinaryoperation.Thentheideaofhomotopyalgebrasarise:ahomotopyalgebraisachaincomplexCequippedafamilyofgradedlinearmapsln:nC!Cofdegreen2foralln>2.Herel2:CC!Cisthegradedbinaryoperation,l3:3C!Cisachainhomotopybetweenthetwochainmapscomingfromthecommutativediagramwheretheyshouldbeequal.Thenl2;l3formalargediagramwhichshouldbeautomaticallycommutativewhenl3istrivial,andnowbeencodedintoahomotopyl4.Continuethisprogress,onegetsallln.IftheunderlyingchaincomplexCisconcentratedinfirstnterms,thishomotopyalgebrawillbecalledan-termoneandinthiscaselkwillbetrivialwhenk>n+2.Notethatwhenn=1,allhomotopiesbecometrivial,thusthe1-termhomotopyalgebrasarepreciselytheoriginalalgebras.2.4.6Asanexample,a2-termL¥-algebraisachaincomplexCconcentratedinfirst2termsandequippedagradedantisymmetricbracketm:CC!Candachainhomotopyl:m◦(idm)◦(id+s+s2))0;3123123whichistotalgradedantisymmetricinthesensethatl3◦ss=(sgns)l3:andsatisfiesdl3=0;wheredisdefinedbyEq.(2.1)withg=C0,M=C1andtheactionisgivenbyadjoint.14万方数据 万方数据Chapter2PreliminariesAmorphismF:C!Dbetween2-termL¥-algebrasconsistsofachainmapF:C!DandachainhomotopyF2:F◦m)m◦(FF)whichisgradedantisym-metricandsatisfyingF(l3(x;y;z))l3(F(x);F(y);F(z))=F2(x;[y;z])+F2(y;[z;x])+F2(z;[x;y])+[F(x);F2(y;z)]+[F(y);F2(z;x)]+[F(z);F2(x;y)]forallx;y;z2C.AhomotopyF:F)F′betweentwomorphismsF;F′:C!DisachainhomotopyF:F)F′satisfyingF′(x;y)F(x;y)=[F(x);F(y)]+[F(x);F(y)]+[F(x);dF(y)]F([x;y]);8x;y2C:220Onecanalsodefinethecompositesofmorphismsandhomotopiesandverifythat2-termL¥-algebrasforma(2;1)-category.RemarkNotethattheDGLAsaredefinedoncochaincomplexes,whileL¥-algebrasaredefinedonchaincomplexes.OnecanalsodefineDGLAsonchaincomplexes,thenDGLAsarespecialkindofL¥-algebras.Butbytakingthen-truncationofaDGLAgandchangingthedegreeviak7!nk,onealsoobtainsan-termL¥-algebra.Likewise,morphismsbetweenconcentratedDGLAscanbeidentifiedwithmorphismsbetweenconcentratedL¥-algebras.Forinstance,the2-truncationoftheDGLA(C+1(g;g);[;];d)0d1C(g;g)!Z(g;g);whereZ1(g;g)=kerdC1(g;g),canbeviewedasaconcentratedchaincomplexbyputtingC0(g;g)ondegree1andZ1(g;g)ondegeree0.Thischaincomplex,togetherwithm=[;]andl3=0,forma2-termL¥-algebra.NotethatZ1(g;g)isthesetder(g)ofderivationsong,sothis2-termL¥-algebraisdenotedbyDer(g).2.5DescribeH2intermsof2-termL-algebrasLie¥2.5.1Recallthat,ina(2;1)-category,thehom-setHom(A;B)isnaturallyagroupoidsincethereareequivalencesbetweenmorphisms.DenotethisgroupoidbyHom(A;B)againandcallithom-spacetoemphasizethatitismorethenaset.15 Chapter2PreliminariesAsagroupoid,itisnaturaltoconsiderthesetp0Hom(A;B)ofconnectedcom-ponentsofHom(A;B).InthecasethatBisapointedobject,Hom(A;B)isnaturallyapointedgroupoidandthenp0Hom(A;B)isnaturallyapointedset.Thissetp0Hom(A;B)iscalledtheintrinsiccohomologyofAwithvaluesinBanddenotedbyH(A;B).2.5.2NotethatHLie2(g;h)canbedescribedintermsofDelignegroupoidsDel(L).Bythecanonicalisomorphism^(gh)=^g^hOnecanidentifyeachtermCn+1(gh;h)as>n+1n+1Hom(^ig^n+1ih;h)=Hom(^ig;Cni+1(h;h)):ÕÕi=1i=1Specially,onehasL10+1(h;h))Hom(^2g;h):=Hom(g;CViewgasacomplexconcentratedindegree0,thentherightsideispreciselythesetofpairsofachainmapandagradedlinearmapofdegree1fromgtoC+1(h;h).Moreover,theMaurer-Cartanelementsarepreciselythedataformmorphismsbetween2-termL¥-algebras0!gandDer(h)andthegaugeactiononMC(L)coincideswiththehomotopiesbetweenthosemorphisms.Consequently,onehasthefollowingequivalencesofgroupoidsExtLie(g;h)=Del(L)=Hom(g;Der(h));andisomorphismsofpointedsetsH2(g;h)=pDel(L)=H(g;Der(h)):Lie0RemarkOnecanseethatHLie2(g;h)isnaturalinginthesensethatg7!HLie2(g;h)inducesacontravariantfunctorfromLiealgebrastopointedsets.However,HLie2(g;h)isnotnaturalinhsinceDerisnotafunctor.16万方数据 万方数据Chapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-LiealgebracohomologyChapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-LiealgebracohomologyMostoftheresultsinthischapterhavebeengivenbyKimin[18,19].Forself-contained,Iexplainthemindetails.3.1Thenotionofpre-Liealgebraextensions3.1.1Apre-LiealgebraisavectorspaceAequippedwithabilinearproduct
suchthattheassociator(x;y;z):=(x
y)
zx
(y
z)issymmetricinx;y,i.e.(x;y;z)=(y;x;z);8x;y;z2A:(3.1)Thecommutator[x;y]:=x
yy
xdefinesaLiealgebrastructureonA,whichiscalledthesub-adjacentLiealgebraofAanddenotedbyg(A).TherearetwonaturallinearmapsL;R:A!gl(A)onAprovidedbythepre-Liealgebrastructure,thatistheleftandrightmultiplications:Lx(y):=x
y;Rx(y):=y
x;8x;y2A:OnecanseeEq.(3.1)equalstothefollows:[Lx;Ly]=L[x;y];8x;y2A:(3.2)ExampleEveryvectorspaceMadmitsatrivialpre-Liealgebrastructurebyu
v=0;8u;v2M:ExampleIfasubspaceBofapre-LiealgebraAisclosedunderthemultiplication
ofA,thenBhasamultiplicationgivenbyrestricting
onBandisapre-Liealgebra,calledasubalgebraofA.RemarkIfthereisnoambiguity,thesymbols
,LandRalwaysdenotethemultiplica-tion,leftandrightmultiplicationsofagivenpre-Liealgebra.17 Chapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-Liealgebracohomology3.1.2Arepresentationofapre-LiealgebraA,oranAmodule,isavectorspaceMequippedwithaLiealgebraactionlofg(A)onMandalinearmapr:A!gl(M)suchthatthefollowingequalityholds:[lx;ry]=rx
yryrx;8x;y2A:Inthiscase,(l;r)isalsocalledanactionofAonM.ExampleThepre-LiealgebraAtogetherwithLandRisarepresentationofAitself.3.1.3AnidealIofapre-LiealgebraAisasubspaceofAsuchthatforallx2Aandy2I,onehasx
y2Iandy
x2I.Thisconditionisequivalenttosay(I;L;R)isanAmodule.LetIbeanidealofA,thenthemultiplicationonAinducesamultiplicationonthequotientA=Iandmakesitbeingapre-Liealgebra.ExampleThesetofzero-divisorsofAisdefinedtobeAnn(A):=kerLkerR=fx2AjLx=Rx=0g:OnecanverifythatitisanidealofA.3.1.4LetA;Ab;Bbepre-Liealgebras.AbissaidtobeanextensionofAbyBifthereexistashortexactsequenceip0!B!Ab!A!0:AsplittingofAbisalinearmaps:A!Absuchthatp◦s=id.Amorphismq:Ab!Ab′oftwoextensionsisapre-Liealgebramorphismqsuchthatthefollowingdiagramcommutes:i/Abp/0/BA/0qi′p′0/B/Ab′/A/0By5-lemma,thisq,ifexists,mustbeanisomorphism.Ifthisisthecase,thetwoextensionsAbandAb′aresaidtobeisomorphic.18万方数据 万方数据Chapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-LiealgebracohomologyNowallextensionsofAbyB,togetherwiththemorphismsbetweenthem,formagroupoidExtpreLie(A;B).OnecanthenconsiderthesetExtpreLie(A;B)ofconnectedcomponents,i.e.isomorphismclasses,ofthisgroupoid.Considerthecommutators,onecanseethatifAbisapre-LiealgebraextensionofAbyB,theng(Ab)isaLiealgebraextensionofg(A)byg(B),andthatifthemorphismq:Ab!Ab′givesrisetoamorphismsbetweenpre-LiealgebraextensionsofAbyB,thenitalsogivesrisetoamorphismsbetweenLiealgebraextensionsofg(A)byg(B).Thusonehasafunctorg():ExtpreLie(A;B)!ExtLie(g(A);g(B)).Thisfunctorisingeneralnotsurjectiveoressentiallysurjective.Indeed,evenaLiealgebraextensionbgofg(A)byg(B)isisomorphictoasub-adjacentLiealgebraofsomepre-Liealgebraonthesamespace,thispre-Liealgebramaynotbeapre-LiealgebraextensionofAbyB.SoIonlyconsidertheimageofthisfunctoranddenoteitbyExtg(A;B).3.1.5Whenthepre-LiealgebrastructureonBistrivial,theextensionissaidtobeabelian.3.2Towardnon-abelian2-cocycles3.2.1Let(Ab;)beanextensionofAbyB,ands:A!AbasplittingofAb.AfteridentifyingBwithitsimageinAb,onecandefinethebilinearmapw:2A!Bandlinearmapsl;r:A!gl(B)asfollows:w(x;y):=s(x)s(y)s(x
y);8x;y2A;(3.3)lx(v):=s(x)v;8x2A;v2B;(3.4)ry(u):=us(y);8y2A;u2B:(3.5)Considertheactionsofthemoncommutators,theantisymmetrizationofwdefinesanalternatingbilinearmapw^:^2g(A)!g(B),andthedifferencelrdefinesalinearmapr:g(A)!gl(g(B)).Thiscouple(w^;r)ispreciselytheLiealgebra2-cocycledefinedbytheLiealgebraextensiong(Ab)ofg(A)byg(B)togetherwiththechosenofthesplittings.Givenasplitting,onehasAb=ABandthepre-LiealgebrastructureonAbcanbetransferredtoABbydefineLbtobeLbx+u(y+v)=Lx(y)+w(x;y)+lx(v)+ry(u)+Lu(v);8x;y2A;8u;v2B:(3.6)19 Chapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-LiealgebracohomologyItisarepresentationoftheLiealgebrastructureonABgivenbytheLiealgebra2-cocycle(w^;r):[x+u;y+v]w^;r:=[x;y]A+w^(x;y)+rx(v)ry(u)+[u;v]B;8x;y2A;u;v2B:(3.7)Conversely,onehasProposition3.2.2LetA;Bbetwopre-Liealgebras,w:2A!Bbeabilinearmapandl;r:A!gl(B)betwolinearmaps,thenEq.(3.6)definesapre-Liealgebrastruc-tureonABifandonlyifw;l;rsatisfythefollowingequalitiesforallx;y;z2A;u;v2B:rx([u;v])=u
rx(v)v
rx(u)(3.8)lx(u
v)=rx(u)
v+u
lx(v)(3.9)[lx;ly]l[x;y]=Lw^(x;y)(3.10)[lx;ry]rx
y+ryrx=Rw(x;y)(3.11)w([x;y];z)w(x;y
z)+w(y;x
z)=lxw(y;z)lyw(x;z)rz(w^(x;y)):(3.12)Ifthisisthecase,thecouple(w^;r),wherew^istheantisymmetrizationofwandr=lr,willbeaLiealgebra2-cocyleandEq.(3.7)definesaLiealgebrastructureonABmakingLbbeingitsrepresentation.Proof.NotethatthebracketdefinedbyEq.(3.7)isactuallythecommutatorofthemultiplicationdefinedbyEq.(3.6),thusthelaststatementsfollowimmediatelyfromthepreviousone.ByEq.(3.2),onecanseeEq.(3.6)definesapre-LiealgebrastructureonABifandonlyifthefollowingequalitieshold:[Lbx;Lby]=Lb[x;y]w^;r;8x;y2A;(3.13)[Lbx;Lbu]=Lb[x;u]w;8x2A;8u2B;(3.14)^;r[Lbu;Lbv]=Lb[u;v]w;8u;v2B:(3.15)^;r20万方数据 Chapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-LiealgebracohomologyApplytwosidesofEq.(3.13)onanyz2A,onehas[Lbx;Lby](z)=Lbx(Ly(z)+w(y;z))Lby(Lx(z)+w(x;z))=LxLy(z)+w(x;Ly(z))+lxw(y;z)LyLx(z)w(y;Lx(z))lyw(x;z);Lb[x;y]w^;r(z)=Lb[x;y]+w^(x;y)(z)=L[x;y](z)+w([x;y];z)+rz(w^(x;y)):Thusw([x;y];z)w(x;y
z)+w(y;x
z)=lxw(y;z)lyw(x;z)rz(w^(x;y));whichshowsEq.(3.12).ApplytwosidesofEq.(3.13)onanyu2B,onehas[Lbx;Lby](u)=lxly(u)lylx(u)=[lx;ly](u);Lb[x;y]w^;r(u)=Lb[x;y]+w^(x;y)(u)=l[x;y]u+w^(x;y)
u:Thus[lx;ly]l[x;y]=Lw^(x;y);whichshowsEq.(3.10).ApplytwosidesofEq.(3.14)onanyy2A,onehas[Lbx;Lbu](y)=Lbx(ry(u))Lbu(Lx(y)+w(x;y))=lxry(u)rx
y(u)u
w(x;y);Lb[x;u]w^;r(y)=Lbrx(u)(y)=ryrx(u):Thus[lx;ry]rx
y+ryrx=Rw(x;y);whichshowsEq.(3.11).ApplytwosidesofEq.(3.14)onanyv2B,onehas[Lbx;Lbu](v)=lx(u
v)u
lx(v);Lb[x;u]w^;r(v)=Lbrx(u)(v)=rx(u)
v:21万方数据 Chapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-LiealgebracohomologyThuslx(u
v)=rx(u)
v+u
lx(v);whichshowsEq.(3.9).ApplytwosidesofEq.(3.15)onanyx2A,onehas[Lbu;Lbv](x)=u
rx(v)v
rx(u);Lb[u;v]w(x)=rx([u;v]):^;rThusrx([u;v])=u
rx(v)v
rx(u);whichshowsEq.(3.8).Finally,sincetherestrictionofLbonBcoincideswithitsoriginalleftmultiplicationL,applytwosidesofEq.(3.15)onanyw2Bwillgiveatrivialequality.RemarkThepre-LiealgebradefinedbyEq.(3.6)iscalledthesemidirectproductandisdenotedbyAnw;l;rB.Onecanseeg(Anw;l;rB)=g(A)nw^;lrg(B).ExampleThereisanaturalpre-LiealgebrastructureonAB,itisgivenby(x+u)(y+v):=x
y+u
v;8x;y2A;u;v2B:Inotherwords,itisAn0;0;0B.Thispre-LiealgebraiscalledthedirectsumofAandBandisdenotedbyABagain.Directsumsgiverisetothesplitextensions.3.2.3Anon-abelian2-cocycleonAwithvaluesinBisatriple(w;l;r)ofabilinearmapw:2A!Bandtwolinearmapsl;r:A!gl(B),satisfiesEq.(3.8)–(3.12).ThesetofthemisdenotedbyZ2preLie(A;B).Two2-cocycle(w;l;r)and(w′;l′;r′)areequivalent,ifthereexistsalinearmapφ:A!Bsuchthatthefollowingequalitiesholdforallx;y2A:l′l=L◦φ;(3.16)r′r=R◦φ;(3.17)′(x;y)w(x;y)=lwxφ(y)+ryφ(x)+φ(x)
φ(y)φ(x
y):(3.18)22万方数据 万方数据Chapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-LiealgebracohomologyThesecondnon-abeliancohomologyHpreLie2(A;B)isthendefinedtobethequo-tientofZ2preLie(A;B)bytheaboveequivalencerelation.Anypre-Liealgebra2-cocyle(w;l;r)onAwithvaluesinBprovidesaLiealgebra2-cocyle(w^;lr)ong(A)withvaluesing(B),twopre-Liealgebra2-cocylesaresaidtobepre-equaliftheyprovidethesameLiealgebra2-cocyle,andpre-equivalentiftheyprovideequivalentLiealgebra2-cocyles.ThequotientofZ2preLie(A;B)bythepre-equalrelationisdenotedbyZg2(A;B),andthequotientofHpreLie2(A;B),andthusofZ2preLie(A;B),bythepre-equivalentrelationisdenotedbyHg2(A;B).Combineproposition3.2.2and2.1.3,onehasCorrollay3.2.4LetA;Bbetwopre-Liealgebras,thenZg2(A;B)canbeidentifiedwithasubsetofZLie2(g(A);g(B))via(w;l;r)7!(w^;lr),whoseelementscorrespondtotheobjectsofExtg(A;B).Consequently,Hg2(A;B)canbeidentifiedwithasubsetofHLie2(g(A);g(B)),whichclassifiesExtg(A;B).3.3TheclassificationtheoremLikeLiealgebracase,onehasTheorem3.3.1LetA;Bbetwopre-Liealgebras,theextensionsofAbyBareclassifiedbythesecondnon-abeliancohomologyHpreLie2(A;B).Proof.LetAbbeanextensionofAbyB.Bychoosingasplittings:A!Ab,oneobtaina2-cocycle(w;l;r)viaEq.(3.3)–(3.5).Firstofall,thecohomologicalclassofthis2-cocyleisindependentofthechoiceofsplittings.Indeed,lets;s′betwodifferentsplittingsand(w;l;r)and(w′;l′;r′)becorresponding2-cocycles.Setφ=s′s,thenitsimageliesinB,andforallx;y2A;u2B,onehasl′(u)l′xx(u)=s(x)us(x)u=φ(x)u;r′(u)r(u)=us′(x)us(x)=uφ(x);xxw′(x;y)w(x;y)=s′(x)s′(y)s′(x
y)s(x)s(y)+s(x
y)=lxφ(y)+ryφ(x)+φ(x)
φ(y)φ(x
y);whichshowsEq.(3.16)–(3.18).23 Chapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-LiealgebracohomologySecondly,morphismsofextensionsgiverisetoequivalenceof2-cocycles.Letq:Ab!Ab′beamorphismofpre-LiealgebraextensionsofAbyBands;s′betwosplittingssuchthatthefollowingdiagramcommutes.s0/B/Abvi_U/A/0q0/B/Ab′Ui_i/A/0s′Let(w;l;r)and(w′;l′;r′)bethecorresponding2-cocyclesof(Ab;s)and(Ab′;s′).Setφ=q1s′sandnotethatqjB=id.onehasl′(u)=s′(x)u=q1(s′(x)u)=q1s′(x)ux=s(x)u+φ(x)u=lx(u)+Lφ(x)(u):Therefore,l′l=L◦φ.Similarly,r′r=R◦φ.AsforEq.(3.18),considerthatforallx;y2A,onehasw′(x;y)=q1w′(x;y)=q1(s′(x)s′(y)s′(x
y))=q1s′(x)q1s′(y)q1s′(x
y)=(s+φ)(x)(s+φ)(y)(s+φ)(x
y)=w(x;y)+lxφ(y)+ryφ(x)+φ(x)
φ(y)φ(x
y);whichshowsEq.(3.18).Conversely,onehasseenthatnon-abelian2-cocyclesdefinespre-Liealgebrastruc-turesonABandthusprovideextensions.Let(w;l;r)and(w′;l′;r′)betwoequivalent2-cocycles,andφ:A!BbethemapsatisfyingEq.(3.16)–(3.18).Defineq:AB!ABasfollows:q(x+u)=xφ(x)+u;8x2A;u2B:OnecanverifythatthismapgivesrisetoamorphismoftheextensionsfromAnw;l;rBtoAnw′;l′;r′B.Thisfinishestheproof.24万方数据 Chapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-Liealgebracohomology3.3.2Fromtheaboveproof,onecanseethemorphismsfromAnw;l;rBtoAnw′;l′;r′Bareone-onecorrespondingtothelinearmapsφ:A!BsatisfyingEq.(3.16)–(3.18).Specially,automorphismsofAnw;l;rBareone-onecorrespondingtothelinearmapsφ:A!Ann(B)satisfyinglxφ(y)+ryφ(x)φ(x
y)=0:(3.19)Anon-abelian1-cocycleonAwithvaluesinBrespecttothe2-cocycle(l;r;w)isalinearmapφ:A!Ann(B)satisfyingEq.(3.19).ThesetofthemisdenotedbyZ1preLie(A;B;(w;l;r)).Notethatthenon-abelian1-cocyclesareindependentonw.3.3.3Nowonecanconsidertheabelianextensions.Inthiscase,Eq.(3.8)and(3.9)triviallyhold.ThenEq.(3.10)and(3.11)showthatthecouple(l;r)givesanA-modulestructureonB.Finally,Eq.(3.12)isequivalenttosaywisa2-cocycleofAwithvaluesintheA-module(B;l;r)inthesenseofusualabeliancohomologyofpre-Liealgebras.Irefer[7]and[4]forthedefinitionanddetails.Notethatthetwodefinitionsaredifferentalthoughtheyagreeatlowdegreeterms.If(B;l;r)isanA-module,onemaybeinterestingonlyontheextensionswhichinducethesameA-modulestructureonBwiththeoriginalone(l;r).ThissubcategoryisdenotedbyExtpreLie(A;(B;l;r))andisclassifiedbythesecondabeliancohomologyHpreLie2(A;(B;l;r)).Notethatwhenthepre-LiealgebrastructureonBistrivial,Eq.(3.16)and(3.17)forceequivalentnon-abelian2-cocylesgivethesameA-modulestructureonB,thusonehas⊔ExtpreLie(A;B)=ExtpreLie(A;(B;l;r));(l;r)⊔H2(A;B)=H2(A;(B;l;r));preLiepreLie(l;r)wheretheunionisdisjointandtakenoverallpossibleA-modulestructures(l;r)onB.3.3.4Nowonecanconsidertheautomorphismsofabelianextensions.Let(B;l;r)beanA-moduleandAnw;l;rBanextensionofAbyBbelongingtoExtpreLie(A;(B;l;r)).Thentheautomorphismsofthisextensionisone-onecorrespondingtothenon-abelian25万方数据 Chapter3Frompre-Liealgebraextensionstothesecondnon-abelianpre-Liealgebracohomology1-cocyclesonAwithvaluesinBrespecttothe2-cocycle(l;r;w),whicharepreciselythe1-cocyclesofAwithvaluesintheA-module(B;l;r)inthesenseofusualabeliancohomologyofpre-Liealgebras.26万方数据 万方数据Chapter4DescribeH2intermsofDelignegroupoidspreLieChapter4DescribeH2intermsofDelignegroupoidspreLie4.1TheDGLAstructureonC+1(A;A)ToconstructtheDGLAL,oneneedtheChevalley-Eilenbergcomplexforpre-Liealgebras.However,therearetwopossiblecomplexes,Iusetheonedefinedin[4].4.1.1LetAbeapre-Liealgebraandl;r:A!gl(M)arepresentationofA.Thenonecandefinetheaction(l;r)ofAonHom(nA;M)asfollows:(l(f))(x)=l(f(x))f(L(x));(r(f))(x)=r(f(x));8x2nA:(4.1)aaaaaNotethatwhenn=0,onehas0A=k,soHom(0A;M)=M,andthattheaction(l;r)coincideswith(l;r).Foranyf2Hom(nA;M),onecandefinedf2Hom(n+1A;M)bytheformulan(df)(x;


;x):=(1)k(l1n+1åxkf)(x1;


;xbk;


;xn+1)k=1n+(1)k(råxn+1f)(x1;


;xbk;


;xn;xk);(4.2)k=1wherecindicatestheomissionoftheunderneathterm.Notethatthisdefinitiononlyworksforn>1,asforu2M=Hom(0A;M),set(du)(x)tobelx(u)+rx(u).Onecanseethat(d2u)(x;y)=(lxlylx
y)(u),thusonecandefinethesetofJacobinelementsas{}J(M):=u2M(lxlylx
y)(u)=0;8x;y2A:Now,letC0(A;M)=J(M)andCn(A;M)=Hom(nA;M)forn>1.Thenonecanverifythatd2vanishesonC(A;M)andthus(C(A;M);d)becomesanon-negativecochaincomplex.4.1.2DenotethedifferentialofC+1(A;A)byd.Following2.3.1,oneshoulddefineagradedLiebracket[;]onC+1(A;A)suchthatd=adm,wherem2C1+1(A;A)isthepre-LiemultiplicationofA.27 Chapter4DescribeH2intermsofDelignegroupoidspreLieRecallthatin2.3.1,theNijenhuis-RichardsongradedLiebracketisgivenasthegradedcommutatorofthegradedcompositeoperation.Onecanseethisgradedcom-positeoperationgivesagradedpre-LiealgebrastructureontheChevalley-Eilenbergcomplexinthesensethattheassociator(f;g;h)hasthegradedleftsymmetryproperty:(f;g;h)=(1)jfjjgj(g;f;h):Therefore,itsufficestodefineagradedpre-LiemultiplicationonC+1(A;A).4.1.3Todothis,IfirstidentifythecomplexC+1(A;A)withC(A;gl(A))viaCn+1(A;A)!Cn(A;gl(A))f7!fewherefeisdefinedviafe(x)(y)=f(x;y);8x2nA;y2A:Now,Idefineabinaryoperation⋄onC(A;gl(A))by(f⋄g)(x;y):=g(f(x)(y))+g(y)◦f(x);wheref;g2C(A;gl(A))andx2jfjA;y2jgjA.Foranyf;g;h2C(A;gl(A))andx2jfjA;y2jgjA;z2jhjA,onehas((f⋄g)⋄h)(x;y;z)=h((f⋄g)(x;y)(z))+h(z)◦((f⋄g)(x;y))=h(g(f(x)(y))(z))+h(g(y)(f(x)(z)))+h(z)◦g(f(x)(y))+h(z)◦g(y)◦f(x);(f⋄(g⋄h))(x;y;z)=(g⋄h)(f(x)(y;z))+((g⋄h)(y;z))◦f(x)=(g⋄h)(f(x)(y);z)+(g⋄h)(y;f(x)(z))+((g⋄h)(y;z))◦f(x)=h(g(f(x)(y))(z))+h(z)◦g(f(x)(y))+h(g(y)(f(x)(z)))+h(f(x)(z))◦g(y)+h(g(y)(z))◦f(x)+h(z)◦g(y)◦f(x)28万方数据 Chapter4DescribeH2intermsofDelignegroupoidspreLieThereforetheassociatoris(f;g;h)(x;y;z)=h(g(y)(z))◦f(x)h(f(x)(z))◦g(y):Thus(f;g;h)(x;y;z)=(g;f;h)(y;x;z);(4.3)whichshowsthat⋄isnotagradedpre-Liemultiplicationasdesired.However,afterantisymmetrizing,onehas(f;g;h)jfjjgj^(x;y;z)=(g;f;h)^(y;x;z)=(1)(g;f;h)^(x;y;z):Thissuggestsonetodefinethecorrectgradedcompositeoffandgastheantisym-metrization(f⋄g)^,whichis(f⋄g)(x)=sgn(s)(f⋄g)(xjfj+jgjA:^ås1);8x2s2Sjfj+jgjHowever,thisbringstoomanyterms.Notethattoforce⋄becomesgradedleftsymmetric,oneonlyneedthevariable(x;y;z)hasgradedleftsymmetrypropertyonxandyasawhole,soIdefinethebinaryoperation⋆by(f⋆g)(x)=åsgn(s)(f⋄g)(xs1;xs1);12s2Sh(jfj;jgj)wheresistakenoverthesetofall(jfj;jgj)-shufflesandthustheresultedtensorxs1canbewrittenintotwoparts(x1;x1),wherex12jfjA;x12jgjAandineachssss1212ofthem,theorderofcomponentsfollowstheoriginalone.ExampleLetf2C1(A;gl(A))andg2C3(A;gl(A)),takingx=x1


x424Athenonehas(f⋆g)(x)=(f⋄g)(x1;x2;x3;x4)(f⋄g)(x2;x1;x3;x4)+(f⋄g)(x3;x1;x2;x4)(f⋄g)(x4;x1;x2;x3)=g(f(x1)(x2;x3;x4))+g(x2;x3;x4)◦f(x1)g(f(x2)(x1;x3;x4))g(x1;x3;x4)◦f(x2)+g(f(x3)(x1;x2;x4))+g(x1;x2;x4)◦f(x3)g(f(x4)(x1;x2;x3))g(x1;x2;x3)◦f(x4):29万方数据 Chapter4DescribeH2intermsofDelignegroupoidspreLieBeforegoingforward,Ineedalemmaonshuffles:Lemma4.1.4Thereexistcanonicalisomorphisms:Sh(n;m;l)=Sh(n;m)Sh(n+m;l);Sh(n;m;l)=Sh(m;l)Sh(n;m+l):HereSh(m;n;l)denotethesetofall(m;n;l)-shuffles,i.e.permutationsssatisfyings1(1)<


(AB;B)isdefinedasthesubcomplexofC(AB;B)byC(AB;B)=C(AB;B)C(B;B):>Proposition4.2.2C+1(AB;B)isasub-DGLAofC+1(AB;AB)endowedwith>thebracket[;]definedin4.1.6.33 Chapter4DescribeH2intermsofDelignegroupoidspreLieProof.Anelementf2Cn(AB;AB)liesinC>n(AB;B)ifandonlyifitvanishesonnBanditsimageliesinB.Forsuchanelementf,itisobviousthatdfvanishesonn+1BanditsimageliesinB,i.e.df2Cn+1(AB;B).Similarly,>C+1(AB;B)isclosedunder⋆,afortiori[;].>DenotethisDGLAbyL,onefurtherhasProposition4.2.3Lisad0-nilpotent,further,L0isabelian.Proof.NotethatL0=Hom(A;B),thus⋆givesatrivialpre-LiealgebrastructureonL0.Asforthead0-nilpotentproperty,consider{}Cn;m:=f2Cn+m(AB;B)fvanishesoutsideAn;mB;whereAn;mBisthesubspaceofn+m(AB)obtainedby⊕An;mB:=X


X;s1(1)s1(n+m)s2Sh(n;m)whereX1=


=Xn=AandXn+1=


=Xn+m=B.Onecansee⊕L=Cn;m:n2N;m2NForallf2L0,g2Cn;m,x2n+m1(AB)andy2AB,onehasadf(g)(x;y)=åsgn(s)(g(f(xs1;y))f(g(xs1;y)))s2Sjgj=åsgn(s)g(f(xs1;y)):s2SjgjNotethatfmapsAtoBandvanishesonB,thusadfinduceslinearmaps:adfn;madfn+1;m1





!C!C!Further,onehasadf(Cn;0)=0.Thisshowsthatadfisnilpotent.RemarkFromtheaboveproof,onecanseeadn+1(Ln)=0forallf2L0.f34万方数据 万方数据Chapter4DescribeH2intermsofDelignegroupoidspreLie4.3H2=pDel(L)preLie04.3.1AstheDGLALisad0-nilpotent,onecandefinetheMaurer-CartanelementsofitandfurthertheDelignegroupoidDel(L)following2.2.3–2.2.5.NotethatL1=Hom(2A;B)Hom(AB;B)Hom(BA;B).Onecanthusviewanytriple(w;l;r)ofabilinearmapw:2A!Btogetherwithtwolinearmapsl;r:A!gl(B)asanelementofL1via:l(x;u):=lx(u);r(u;x):=rx(u);8x2A;u2B:Proposition4.3.2LetA;Bbetwopre-Liealgebras.Atriple(w;l;r)ofabilinearmapw:2A!Bandtwolinearmapsl;r:A!gl(B)isanon-abelian2-cocycle,thusEq.(3.6)definesapre-LiealgebrastructureonAB,ifandonlyifw+l+r2MC(L).Proof.Letc=w+l+r,thenc2MC(L)ifandonlyifQ=dc+1[c;c]vanishes2ifandonlyifthefollowingequalitiesholdforallx;y;z2Aandu;v;w2B:Q(x;y;z)=0;Q(x;y;u)=0;Q(x;u;y)=0;(4.7)Q(x;u;v)=0;Q(u;v;x)=0;Q(u;v;w)=0:(4.8)Tocalculatethem,oneshouldfirstwritedowntheexplicitformulaofQ.Foralle1;e2;e32AB,onehasQ(e1;e2;e3)=(dc+c⋆c)(e1;e2;e3)=e1
c(e2;e3)+c(e1
e2;e3)+c(e2;e1
e3)+e2
c(e1;e3)c(e2
e1;e3)c(e1;e2
e3)c(e2;e1)
e3+c(e1;e2)
e3+c(c(e1;e2);e3)+c(e2;c(e1;e3))c(c(e2;e1);e3)c(e1;c(e2;e3)):Bytheaboveformulas,onecanexpandtheleftsidesofEq.(4.7)–(4.8)asfollows:Q(x;y;z)=w([x;y];z)w(x;y
z)+w(y;x
z)lxw(y;z)+lyw(x;z)+rzw^(x;y)Q(x;y;u)=l[x;y](u)+w^(x;y)
u+lylx(u)lxly(u)Q(x;u;y)=rx
y(u)+u
w(x;y)+rylx(u)ryrx(u)lxry(u);Q(x;u;v)=u
lx(v)lx(u
v)rx(u)
v+lx(u)
v;Q(u;v;x)=u
rx(v)+rx([u;v])+v
rx(u);Q(u;v;w)=0:35 Chapter4DescribeH2intermsofDelignegroupoidspreLieOnecanthenimmediatelyseethatc2MC(L)ifandonlyifEq.(3.8)–(3.12)holds.Thisfinishestheproof.RemarkByproposition4.3.2,onecanidentifyZ2preLie(A;B)withMC(L).However,thereisanequivalencerelationonZ2preLie(A;B)andthereisthegaugeactiononMC(L).Thusitisnaturaltoconsidertherelationshipbetweenthem.Ontheotherhand,combineproposition4.3.2,proposition3.2.2and3.2.1,onegetsthefollowingsurjectivemap:FMC(L)!ExtpreLie(A;B)w+l+r7!Anw;l;rB:SinceMC(L)isthesetofobjectsofDel(L),onemayaskifthismapcanbefurtherafunctor.4.3.3NotethatL0isabelian,thusexp(L0)=(L0;+)=Hom(A;B).Forallφ2L0,sincead2(L1)=0,thegaugeactiononL1thusread:φ11φ:a=a+adφ(a)dφadφ(dφ);8a2L:2Foranya=w+l+r2MC(L)withw2Hom(2A;B),l2Hom(AB;B)andr2Hom(BA;B),thegaugeactionofφtranslateittoφ:a.Towritedownit,takinganyx;y2Aandu2B,onehas1(φ:a)(x;y)=w(x;y)+l(x;φ(y))+r(φ(x);y)dφ(x;y)(φ⋆dφ)(x;y)2=w(x;y)+lxφ(y)+ryφ(x)φ(x
y)+φ(x)
φ(y);(4.9)1(φ:a)(x;u)=l(x;u)dφ(x;u)(φ⋆dφ)(x;u)=lx(u)+φ(x)
u;(4.10)21(φ:a)(u;x)=r(u;x)dφ(u;x)(φ⋆dφ)(u;x)=rx(u)+u
φ(x):(4.11)2ComparethemwithEq.(3.16)–(3.15),onecanseethatφ:aliesinthesamecohomologyclassofa.Further,bytheproofoftheorem3.3.1,theaboveactionofφinducesamorphismqofextensionsofAbyBviaq(x+u)=xφ(x)+u;8x2A;u2B:(4.12)Thus,bysettingF(a!φ:a)=q,themapFbecomesafunctorF:Del(L)!ExtpreLie(A;B):36万方数据 万方数据Chapter4DescribeH2intermsofDelignegroupoidspreLieTheorem4.3.4F:Del(L)!ExtpreLie(A;B)isanequivalenceofcategories.Proof.onehasseenFissurjective,afortioriessentiallysurjective,thusitsuf-ficestoshowFisfullyfaithful.LetF(φ)=q,thenEq.(4.12)showsthatφ(x)=xq(x)forallx2A.Thefaithfulnessthusfollows.Bytheorem3.3.1,anymorphismofextensionsofAbyBgivestoequivalenceofcorresponding2-cocycles.Thus,toshowFisfull,itsufficestoshowequivalent2-cocyclesareconnectedbysomegaugetransformation.Indeed,let(w;l;r)and(w′;l′;r′)betwoequivalent2-cocycles.Thenthereexistssomeφ2Hom(A;B)=L0suchthatEq.(3.16)–(3.15)hold.ThenEq.(4.9)–(4.11)showthatφ:(w+l+r)=w′+l′+r′asdesired.Takingthesetp0ofconnectedcomponentsofthetwocategories,onehasCorrollay4.3.5ExtensionsofAbyBareclassifiedbyp0Del(L).RemarkForanya2MC(L),considerthestabilizerSa:=fφ2L0jφ:a=agofa,whichisasubgroupofL0andonehasSa=AutDel(L)(a)bydefinition.ButthelateroneisisomorphictoAutExtpreLie(A;B)(AnaB)bytheorem4.3.4.Ifa′isanotherelementinMC(L)andthereexistsφ2L0suchthatφ:a′=a,then1=AutonehasSa′=φSaφandsothatAutExtpreLie(A;B)(Ana′B)ExtpreLie(A;B)(AnaB).Writtenaasw+l+rsothat(w;l;r)isa2-cocycle.ThenEq.(4.9)–(4.11)showthatSa=Z1preLie(A;B;(w;l;r)).4.4Abeliancohomologyastangentcomplex4.4.1NowIconsidertheabelianextensions.LetAbeapre-Liealgebraand(B;l;r)anA-module.AsthesemiproductAn0;l;rBisalreadyanextensionofAbyB,onehasl+r2MC(L).Denotel+rbya,thenonegetsthetangentcomplexLaata.LetddenotethedifferentialofL,thusthedifferentialofLaisda=d+ada.Proposition4.4.2(C+1(A;B);d)isasub-DGLAof(La;da).Proof.Onecanseethebracketdefinedin4.1.6vanishesonC+1(A;B),thusitisanabeliansubalgebraofLa.Itremainstoshowthatthetwodifferentialsdanddacoincide.37 Chapter4DescribeH2intermsofDelignegroupoidspreLieForanyf2Cn1+1(A;B)andx2n+1A,notethatfvanishesoutsidenA,onehasndf(x)=(1)k1f(L(x;


;xb;


;x));(a⋆f)(x)=0;åxk1kn+1k=1andn()(f⋆a)(x)=(1)nkråxn+1f(x1;


;xbk;


;xn;xk)+lxkf(x1;


;xbk;


;xn+1):k=1Thusdn1af=df+[a;f]=df(1)f⋆a=df:asdesired.4.4.3Bytheorem4.3.4,anelementw2C1+1(A;B)givesrisetoanabelianextensionwhichinducesthesameA-modulestructureonBwiththeoriginaloneifandonlyifw+a2MC(L).LetMC(L)adenotetheset{}w2C1+1(A;B)w+a2MC(L):ByEq.(4.9)–(4.11),thegaugeactionofφ2Hom(A;B)mapsw+atoφ:(w+a)=wdφ[a;φ]+a=wda(φ)+a:(4.13)ThereforeMC(L)aispreservedunderthegaugeactionandthusinducesasubgroupoidDel(L)aofDel(L).Bytheorem4.3.4,thefunctorFinducesanequivalenceofcategories:FDel(L)a!ExtpreLie(A;(B;l;r)):Ontheotherhand,notethatw⋆w=0,onehas11d(w+a)+[w+a;w+a]=dw+[a;w]+da+[a;a]=dw+[a;w]:22Thusw+a2MC(L)ifandonlyifw2kerda=kerd,inotherwords,wisanabelian2-cocycleofAwithvaluesintheA-module(B;l;r).Moreover,Eq.(4.13)showsthatthegaugeactiononMC(L)agivesrisetotheabelian2-coboundaries,i.e.elementsofd(C1(A;B)).Therefore,2d3pDel(L)=ker(C(A;B)!C(A;B))=:H2(A;(B;l;r)):0apreLieim(C1(A;B)!dC2(A;B))38万方数据 Chapter4DescribeH2intermsofDelignegroupoidspreLieThus,ExtpreLie(A;(B;l;r))isclassifiedbytheabeliancohomologyHpreLie2(A;(B;l;r))asmentionedbefore.4.4.4ConsiderAutDel(L)a(w)forallw2MC(L)a.ByEq.(4.13),φ2Hom(A;B)belongstoAutDel(L)a(w)ifandonlyifda(φ)=0,inotherwords,φisanabelian1-cocycleofAwithvaluesintheA-module(B;l;r).ThereforeAut(w)=Z1(A;(B;l;r)):Del(L)apreLieThus,automorphismsofanabelianextensionofapre-LiealgebraAbyanA-module(B;l;r)isone-onecorrespondingtotheabelian1-cocyclesasmentionedbefore.39万方数据 万方数据Chapter5TowardtheintrinsiccohomologyChapter5Towardtheintrinsiccohomology5.1Thenotionof2-termPL¥-algebrasTheterminologyofaPL¥-algebrahasbeenintroducedin[10].Itisaright-symmetricalgebrauptohomotopy.Byaslightmodification,onecouldobtainaleft-symmetriconeuptohomotopy,whichisthehomotopicversionofthenotionofpre-Liealgebraswhichusedinthispaper.Sincetofollow2.5.2,oneonlyneeds2-termPL¥-algebras,soonecanrestrictinthesubcategoryof2-termPL¥-algebras,whichareobtainedbytruncationofPL¥-algebras.Irefer[27]formoreinformationonthe2-termPL¥-algebrasandtheirprop-erties.5.1.1A2-termPL¥-algebraconsistsofd•theunderlyingcomplexA:


!0!A1!A0,whichisachaincomplexcon-centratedinfirst2terms,•themultiplicationm:AA!A,whichischainmap,•theleft-symmetrylawg:a)a12,whichisachainhomotopybetweenthetwoparallelchainmapsa;a12:AAA!A,whereadenotestheassociatorofmanda12isdefinedbya12(x;y;z)=a(y;x;z);8x;y;z2A;satisfyingthefollowingequalities:g=g12;d(g)=0;wheredisdefinedbyEq.(4.2)withA=A0,M=A1andl;rtheleftandrightmulti-plicationinducedbym.NotethatalthoughingeneralA0isnotapre-Liealgebraandl;r:A0!gl(A1)isnotarepresentationofA0,theformulaEq.(4.2)stillmakessense.A2-termPL¥-algebra(A;d;m;l3)issaidtobestrictifl3=0.40 Chapter5TowardtheintrinsiccohomologyFromnowon,ifthereisnoambiguity,a2-termPL¥-algebra(A;d;m;l3)willbesimplydenotedasAandthemultiplicationm(x;y)willbewrittenasx
y.5.1.2AmorphismF:(A;d;m;l3)!(A′;d′;m′;l3′)between2-termPL¥-algebrascon-sistsofachainmorphismFbetweenthe2-termchaincomplexes(A;d)and(A′;d′)andachainhomotopyF2:m′◦(FF))F◦msatisfyingF◦l(x;y;z)l′(F(x);F(y);F(z))33=F2(m(x;y)m(y;x);z)F2(x;m(y;z))+F2(y;m(x;z))m′(F(x);F(y;z))+m′(F(y);F(x;z))+m′(F(x;y)F(y;x);F(z)):2222forallx;y;z2A0.LetF:A!A′andG:A′!A′′betwomorphismsbetween2-termPL¥-algebras,thecompositeofthemisthemorphismG◦F:A!A′′consistingofthechainmapG◦Fandthechainhomotopy(G◦F)2definedby(G◦F)2:=G2◦(FF)+G◦F25.1.3AhomotopyF:F)F′betweentwomorphismsF;F′:A!A′isachainhomotopyF:F)F′satisfyinganextraconditionthatitinducesahomotopyFbe-tweenthechainhomotopiesFandF′.Towritedownthisextracondition,notethatthechainhomotopyF2,regardedasachainmap,is(2A)2/0d(2A)2(2A)2(2A)1/0d222F2/A′(A)1(A)1A01dd′(m◦(FF);F◦m)2A02A0/A′0where(2A)1=A1A0A0A1and(2A)2=A1A1.NotethatthisFisagradedlinearmapfrom(2A)ItoA′ofdegree1andsatisfyingd◦F=(m′◦(FF);F◦m)(m′◦(F′F′);F′◦m);F◦d=FF′:2241万方数据 万方数据Chapter5TowardtheintrinsiccohomologyThefirstequalitysuggestsonetodefinethisFas()F=m′◦(F′F+FF′+FdF);F◦m;andthenthesecondoneprovidestheexplicitformulafortheextraconditionm′(F′(x);F(y))+m′(F(x);F′(y))+m′(F(x);dF(y))F(m(x;y))=F(x;y)F′(x;y);8x;y2A:(5.1)220Theverticalandhorizontalcompositeofhomotopiesarethesameaschainhomotopies.5.1.4The2-termPL¥-algebras,togetherwithmorphismsbetweenthemandhomo-topiesbetweenmorphisms,forma2-category,denotedby2PreL¥.Moreover,thisisa(2;1)categorysinceeveryhomotopyFhasaninverseF.2PreL¥isthenasubcat-egoryofthe(¥;1)categoryPreL¥ofallPL¥-algebras.Thenasa(2;1)-categories,theintrinsiccohomologyof2-termPL¥-algebrasisthesetofconnectedcomponentoftheirhom-spaceH(A;A′):=pHom(A;A′):05.2Centralextensionsofpre-LiealgebrasAlthoughitisnaturaltoquestionifonecanfollow2.5.2toencodethesecondnon-abelianpre-Liealgebracohomologyintoanintrinsiccohomology.Butthiscannotbedownsinceforpre-Liealgebras,L1isnotcanonicallyisomorphictoHom(2A;B)Hom(A;C1(B;B)).OnewaytodropofftheextratermHom(BA;B)istoconsideronlythecentralextensions.5.2.1RecallthecenterZ(A)ofapre-LiealgebraAisdefinedtobethesetofelementscommutewitheveryelementsofA,inotherwords,Z(A)=ker(LR).LetAbeapre-Liealgebra,ancentralextensionisanpre-LiealgebraextensionAbofAbyBsuchthattheimageofBliesinthecenterofAb.ThefullsubcategoryofcentralextensionsofAbyBisdenotedbyExtcen(A;B).Notethat,theinjectivityofB!AbforcesBtobeacommutativealgebra.Ifthisisnotthecase,setExtcen(A;B)=∅.Following3.2.1,bychoosingasplittingandidentifyBwithitsimage,onecanidentifyAbwithsomesemiproductAnw;l;rB.Moreover,sinceBliesinthecenterofAb,onehasl=r.Thenproposition3.2.2becomes42 Chapter5TowardtheintrinsiccohomologyProposition5.2.2LetAbeapre-LiealgebrasandBacommutativealgebra,apair(w;l)ofabilinearmapw:2A!Bandalinearmapl:A!gl(B)producesacentralextensionAnw;l;lBviaEq.(3.6)ifandonlyifitsatisfiesthefollowingequalitiesforallx;y;z2A;u;v2B:lx(u
v)=u
lx(v)(5.2)lxlylx
y=Lw(x;y)(5.3)w([x;y];z)w(x;y
z)+w(y;x
z)=lxw(y;z)lyw(x;z)lz(w^(x;y)):(5.4)Suchkindoftriples(w;l;l)arecalledcentral2-cocycles,obviouslytheyare2-cocyclesandonethusgetasubsetZcen2(A;B)ofZ2preLie(A;B).Moreover,itisclosedundertheequivalenceof2-cocyclesandthusprovidesaquotientHcen2(A;B),whichisasubsetofHpreLie2(A;B)andclassifyingExtcen(A;B).Aside,by3.3.2,theautomorphismsofacentralextensioncorrespondingtothecentral2-cocycle(w;l;l)aredescribedbyZ1preLie(A;B;(w;l;l)).5.2.3TheEq.(5.2)meanslbelongstothemultiplierM(B)ofthecommutativealgebraB,thatisthesubspaceofgl(B)definedasM(B):=fl2gl(B)jl(u
v)=l(u)
v;8u;v2Bg:OnecanseeitcontainstheleftmultiplicationLofBandisclosedunderthecompositeoperationoflinearmaps.Therefore,M(B)isasub-associativealgebraofgl(B).Moreover,M(B)canberegardedasa2-termPL¥-algebraM(B)asfollows:L•theunderlyingcomplexis


!0!B!M(B)•themultiplicationisgivenbyf
g:=f◦g;f
u=u
f:=f(u);8f;g2M(B);u2B:•theleftsymmetrylawvanishes.NowEq.(5.2)–(5.4)isobviouslyequivalenttosay(l;w)isamorphismbetweenthe2-termPL¥-algebras0!AandM(B).Giventwomorphisms(l′;w′)and(l;w),ahomotopybetweenthemisachainhomotopyF:l′)lsatisfyingEq.(5.1).AchainhomotopyFbetweenl′andlisa43万方数据 Chapter5Towardtheintrinsiccohomologygradedlinearmapfrom0!AtoM(B)ofdegree1,whichisequivalenttoalinearmapφ:A!Bsatifyingl′=l=L◦φ;whichisEq.(3.16),andalsoEq.(3.17)bythecommutativityofB.ThentheconditionEq.(5.1)isequivalenttow′(x;y)w(x;y)=lxφ(y)+ryφ(x)+φ(x)
φ(y)φ(x
y);8x;y2A:Therefore,ahomotopybetweentwomorphisms(l′;w′)and(l;w)ispreciselyanequiv-alenceof2-cocycles.Consequently,onehasTheorem5.2.4Therefollowinggroupoidsareequivalent:Extcen(A;B)=Hom(A;M(B))Consequently,centralextensionsofAbyBareclassifiedbytheintrinsiccohomologyH(A;M(B)).5.2.5NowoneconsiderthesplitcentralextensionB!AB!A.By3.3.2,theau-tomorphismsofitarecorrespondingtothe1-cocyclesrespecttothe2-cocycle(0;0;0),whicharelinearmapsφ:A!Ann(B)satisfyingφ(x
y)=0forallx;y2A.OnecanseethosemapsarepreciselythehomomorphismsbetweenAandAnn(B).RemarkOnecanseeH(A;M(B))isnaturalinA,butnotinBsinceMisnotafunctorfromcommutativealgebrastoassociativealgebras.However,H(A;M(B))isnaturalinM(B).Then,bytheknowledgefromintrinsiccohomology,wheneverthereisafibrationsequenceof2-termPL¥-algebrasM(B1)!M(B2)!M(B3);Thenthereisalongexactsequenceofpointedsets0!H(A;ΩM(B1))!H(A;ΩM(B2))!H(A;ΩM(B3))!H(A;M(B1))!H(A;M(B2))!H(A;M(B3)):OnecanverifythatforanycommutativealgebraB,theloopingΩM(B)ofthe2-termPL¥-algebraM(B)ispreciselythesubalgebraAnn(B)ofBandH(A;Ann(B))isaabeliangroupundertheadditionoperation.44万方数据 万方数据AppendixAglanceofhighercategorytheoryAppendixAglanceofhighercategorytheoryA.1ThenotionofhighercategorytheoryandintrinsiccohomologyA.1.1Theoriginalnotionofcategorycanbeextendedtoinvolvehighermorphisms,suchas2-morphismsbetweentheoriginalmorphismsand3-morphismsbetween2-morphisms.An-categoryisthenanextendedcategoryinvolvingk-morphismsforall16k6n,andan¥-categoryisanextendedcategoryinvolvingk-morphismsforallk>1.Thosehighermorphismsshouldsatisfysuitableassociativeandunitalcomposi-tionlaws.Inthestrictcase,thecompositionlawsforhighermorphismsarethesameforthe1-morphismswiththeextrarequirementthatdifferentwaysofcompositesarecompatible.Forinstance,a(strict)2-categoryCconsistsof•0-morphisms,i.e.objects,•1-morphisms,i.e.theoriginalmorphisms,betweenobjects,andtheircompositesfgg◦f
!
!
/o/o/o/
!
•2-morphismsbetween1-morphisms,andtheverticalandhorizontalcompositesa_NR
-
/o/o/o/
b◦a
b__RRR
b
a
/o/o/o/
ba
___andsatisfiesthefollowingaxioms:1.thecompositessatisfytheassociativelaw,2.everyobjectAadmitsan1-morphismidAcalledtheidentityofA,whichistheidentityunderthecompositeoperationof1-morphisms,45 AppendixAglanceofhighercategorytheory3.every1-morphismfadmitsa2-morphismidfcalledtheidentityoff,whichistheidentityundertheverticalcompositeoperationof2-morphisms,4.foreveryobjectA,the2-morphismididAistheidentityunderthehorizontalcompositeoperationof2-morphisms,5.theverticalandhorizontalcompositessatisfytheinterchangelaw:forallquadruples(a;a′;b;b′)of2-morphismsoftheformaa′_N_N
-
-
bb′__thefollowingequalityholds.(b◦a)(b′◦a′)=(bb′)◦(aa′):However,thecompositesaswellascompositionlawscanbereplacedbysomeweakerones,thenonegetsvariousofweakhighercategories.Thetechnicaldefinitionscanbefoundin[21].Iomititsinceitisnotnecessaryinthispaper.A.1.2Inoriginalcategorytheory,therearefunctorsbetweencategoriesandnaturaltransformationsbetweenfunctors.Undersuitablesizehypothesis,allsmallcategoriestogetherwithfunctorsbetweenthemandnaturaltransformationsbetweenthosefunctorsformalarge2-categoryCat.Inhighercategorytheory,functorsshouldalsoworkonallkindofhighermor-phisms.Thereforetherearenotonlyfunctorsbetweencategoriesandnaturaltransfor-mationsbetweenfunctorsbutalso2-transformationsbetweennaturaltransformations,3-transformationsbetween2-transformationsetc.Undersuitablesizehypothesis,thosedataforsmalln-categoriesformalarge(n+1)-categorynCat.Herencanbetakenas¥bysetting¥+1=¥.A.1.3Inoriginalcategorytheory,anisomorphismfisaninvertiblemorphism,thatmeansithasaninverseg,whichisamorphismsatisfyingf◦g=idandg◦f=id.Inhighercategorytheory,thesamenotioncanbedefinedforallhighermorphisms.Soitmakessensetosaytwomorphismsareisomorphic.46万方数据 万方数据AppendixAglanceofhighercategorytheoryButthenaweakversionofinvertibilityarisessincetheequalitiesf◦g=idandg◦f=idcanbeweakenedtobeisomorphisms.Thenonecandefinevariousofweakversionsofinvertibilitybysimilarconsideration.Morphismssatisfyinganykindofinvertibilityarecalledequivalences.A(n;r)-categoryisan-categorywhoseeveryk-morphismisinvertibleforallk>r.Allsmall(n;r)-categoriesformasub(n+1)-categoryofnCat,denotedby(n;r)Cat.Specially,(n;0)-categoriescanbeviewedasthegeneralizationofgroupoidsandarecalledn-groupoids.The(n+1)-categoryofallsmalln-groupoidsisdenotedbynGrpd.RemarkThenotionof(n;r)-categoriescanbeweakenedbyreplacinginvertibilitywithweakversion,whichisnotnecessarythroughoutthispaper.Underthisconvention,theonlyweakversionofinvertibilityina(n;1)-categoryistheonementionedabove.A.1.4Foranypairofobjects(A;B)inahighercategoryC,thereexistsanaturalhighercategorystructureonthesetHomC(A;B)of1-morphismsbetweenthem:theobjectsare1-morphismsofCbetweenAandB,the1-morphismsare2-morphismsofCbetweenthose1-morphismsetc.ThishighercategoryHomC(A;B)iscalledthehom-spaceofAandB.Ina(n;1)-categoryC,ahom-spaceHom(A;B)isan-groupoid.Likegroupoids,onecantakingthesetofconnectedcomponentsofHom(A;B).ThissetiscalledtheintrinsiccohomologyofAwithvaluesinB,anddenotedbyH(A;B).Inotherwords,theintrinsiccohomologyisthecompositeofthefollowingfunctors:Hom(;)p0CC!nGrpd!Set:IfCsatisfiesgoodproperties,theintrinsiccohomologywillbehavewell.A.2Intrinsiccohomologyin(2;1)-categoricalcontextInthissection,Irecallsomefactsaboutintrinsiccohomologyandprovethemin(2;1)-categoricalcontext.Thosefactsholdingeneralhighercategories,butitisnotnecessaryforthispaper.47 AppendixAglanceofhighercategorytheoryA.2.1Recallthatinoriginalcategorytheory,asquarediagramf′
/
g′gf/

issaidtocommuteiff◦g′=g◦f′.Inhighercategorytheory,thisdiagramissaidtocommuteuptohomotopyiff◦g′andg◦f′aredifferentbyanequivalence.Notethat,ina(2;1)-category,thismeansf◦g′andg◦f′areisomorphic.A.2.2Recallthatinanoriginalcategory,acartesiandiagramisacommutativesquarediagramf′ACB/Bg′gfA/Csuchthatforanycommutativesquarediagramf′′D/Bg′′gfA/Cthereexistsauniquemorphismusuchthatf′′=f′◦uandg′′=g′◦u.Ifthisisthecase,f′(resp.g′)issaidtobethepullbackoff(resp.g)alongg(resp.f)andACBiscalledthefibredproductoffandg.Acategoryissaidtobehavingpullbacksifforeverypairofmorphisms
!

,thepullbacksexist.Notethatacommutativesquarediagramoftheformf′′D/Bg′′gfA/CisnothingbutanelementofthefibredproductHom(D;A)Hom(D;C)Hom(D;B)ofhom-sets.ThentheaboveuniversalpropertycanbeencodedintothefollowingnaturalisomorphisminSetHom(D;ACB)=Hom(D;A)Hom(D;C)Hom(D;B):Thosenotionscanbegeneralizedintohighercategoriesandcanbeweakenedbyreplacingcommutativediagramswithdiagramscommutinguptohomotopy.48万方数据 AppendixAglanceofhighercategorytheoryA.2.3Recallthat,intheoriginalcategorytheory,thecommacategory(F#G)oftwofunctorsF:A!CandG:B!Cisthecategoryinwhich•objectsaretriples(A;f;B)ofanobjectAinA,anobjectBinBandamorphismf:F(A)!G(B)inC,•morphismsfrom(A;f;B)to(A′;f′;B′)arepairs(g;h)ofamorphismg:A!A′inAandamorphismh:B!B′inBmakingthefollowingdiagramcommutefF(A)/G(B)F(g)G(h)f′F(A′)/G(B′)•compositeof(g;h)and(g′;h′)is(g◦g′;h◦h′),•identityof(A;f;B)is(idA;idB).fgConsiderapairofmorphismsA!CBina(2;1)-category,letfandgfgdenotetheinducedfunctorsHom(D;A)!Hom(D;C)Hom(D;B).Onecanseethat(f#g)ispreciselythecategoryofhomotoycommutativesquarediagramsoftheformD/B5Ig••••fA/CThusthehomotopyfibredproductshouldbedefinedastheobjectAhBsuchCthatitinducesanaturalequivalenceofgroupoidsHom(D;AhB)=(f#g):CAsaspecialcase,thecommafibredproductisdefinedtobetheobjectAhBinducingCanaturalisomorphisminsteadofmerelyequivalence.Tosimplifythediscussion,Ionlyconsiderthecommafibredproductsandcallthemhomotopyfibredproducts.A.2.4Now,onecanwritedowntheexplicitdefinitionofhomotopyfibredproducts.Ahomotopycartesiandiagramisasquarediagramwhichcommutesuptohomotopyf′AhB/BC′ayyyy2FggfA/Ctogetherwiththeequivalencea:f◦g′)g◦f′satisfying49万方数据 AppendixAglanceofhighercategorytheory•1-universalproperty:foranysquarediagramwhichcommutesuptohomotopyf′′D/B′′a′5Igg••••fA/Ctogetherwiththeequivalencea′:f◦g′′)g◦f′′,thereexistsauniquemorphismusuchthatf′′=f′◦u,g′′=g′◦uanda′=au.Hereaudenotesthehorizontalcompositeofaandidu.•2-universalproperty:foranytwomorphismsu;v:D!AhBand2-morphismsCϕ:f′◦u)f′◦vandy:g′◦u)g′◦vsatisfying(gϕ)◦(au)=(av)◦(fy);thenthereexistsaunique2-morphismq:u)vsuchthatϕ=f′q;y=g′q.Ifthisisthecase,f′(resp.g′)issaidtobethehomotopypullbackoff(resp.g)alongg(resp.f)andAhBiscalledthehomotopyfibredproductoffandg.AhighercategoryCissaidtobehavinghomotopypullbacksifforeverypairofmorphisms
!

,thehomotopypullbacksexist.LemmaA.2.5(Pastinglemma)Inahighercategoryhavinghomotopypullbacks,con-siderthefollowingdiagramofmorphisms.g′f′
/
/
h′′h′hgf
/
/
Ifh′isthehomotopypullbackofhalongfandh′′isthehomotopypullbackofh′alongg,thenh′′isthehomotopypullbackofhalongf◦g.Proof.Ifthereexistequivalencesa:f◦h′)h◦f′andb:g◦h′′)h′◦g′makingtheabovetwosquaresbecomehomotopycartesiandiagrams,thenonecanverifythat(ag′)◦(fb):f◦g◦h′′)h◦f′◦g′isanequivalencemakingtheouterrectanglebecomeahomotopycartesiandiagram.A.2.6Recallthatinanoriginalcategory,theinitialobject(resp.terminalobject)istheobjectAsuchthatHom(A;X)(resp.Hom(X;A))isasingletonforallobjectsX.If50万方数据 AppendixAglanceofhighercategorytheoryaninitialobjectisalsoaterminalobject,thenitiscalledthezeroobjectandisusuallydenotedby0.Acategorywithzeroobjectiscalledapointedcategory.Inapointedcategoryhavingpullbacks,thekernelkerf!Aofamorphismf:A!Bisthepullbackof0!Balongf.Thekernelofakernelkerf!Aisthenapullbackof0!Balong0!B,whichis0!0.Nowconsiderapointedhighercategoryhavinghomotopypullbacks.Thehomo-topykernelkerf!Aofamorphismf:A!Bisdefinedtobethehomotopypullbackof0!Balongf.Thehomotopykernelofahomotopykernelkerf!Aisthenapullbackof0!Balong0!B.However,itisingeneralnot0!0.Remark1Forconvenience,onecanusethetermkerneltoinfereithertheobjectkerforthemorphismkerf!Aifthereisnoambiguity.Remark2AnycategoryChavingterminalobject0admitsapointedcategoryC0=,thecategoryofpointsofobjectsofC,whoseobjectsaremorphisms0!XinCandamorphismfrom0!Ato0!Bisamorphismf:A!BinCsuchthatthefollowingdiagramcommutes.0••????••fA/BfgAnobjectinthiscategoryiscalledapointedobjectinC.Let
!

bemorphismsbetweenpointedobjects,thentherealreadyexistsacommutativediagram.0/
gf/

ThereforeifXisthehomotoyfibredproductoffandginC,thentherewillbeauniquemorphism0!XmakingitbecomethehomotoyfibredproductinC0=.ExampleThecategorySethasnonontrivial2-morphisms,thushomotopypullbacksinSetarepreciselytheusualpullbacks.AstheterminalobjectinSetisthesingleton,apointedsetisjustasetequippedwithaspecificelement.Thekernelofamapfbetweenpointedsets(A;a)and(B;b)istheinverseimagef1(b)ofthatspecificelementb.Whenkerf=A,thismapissaidtobeazeromap.51万方数据 AppendixAglanceofhighercategorytheoryExampleThe2-categoryGrpdofgroupoidshasaterminalobject0,whichisthecate-goryhavingonlyoneobjectandonemorphism.ApointedobjectinGrpdisagroupoidGtogetherwithafunctorx:0!G,whichcanbeviewedasaspecificobjectinG.Thenthehomotopyfibredproductofxandxisthecommacategory(x#x),whichcanbeidentifiedwiththegroupofautomorphismsofxinG.A.2.7Inahighercategoryhavingpullbacksandterminalobject,theloopingΩXofapointedobject0!Xisthehomotopyfibredproductof0!Xanditself.Thepreviousexampleshowsthattheloopingofagroupoidisagroup,whichisnotanaccident.Indeed,let0!Xbeapointedobjectina(2;1)-categoryC,thenforanyobjectYinC,theuniversalpropertyofhomotopyfibredproductprovidesanaturalbijectionfromthesetHom(Y;ΩX)of1-morphismstothesetof2-automorphismsofY!0!X,whichisnaturallyagroup.ThusthebijectiongivesagroupstructureonHom(Y;ΩX).Moreover,thisgroupstructureisnaturalinthesensethatanymorphismY!Y′inC(resp.X!X′in0=C)inducesagrouphomomorphismHom(Y′;ΩX)!Hom(Y;ΩX)(resp.Hom(Y;ΩX)!Hom(Y;ΩX′))ratherthanamap.RecallthatanobjectGinacategoryCiscalledagroupobjectifforanyobjectsXinC,thereexistsagroupstructureonHom(X;G)andforanymorphismf:X!Y,themapHom(Y;G)!Hom(X;G)isagrouphomomorphism.Inotherwords,Hom(;G)inducesafunctorfromCtothecategoryGrpofgroups.Amorphismbetweengroupobjectsisamorphismf:G!G′inCsuchthatitinducesanaturaltransformationHom(;G))Hom(;G′)offunctorsfromCtoGrp.ThenoneobtainsafunctorΩ:C0=!Grp(C);whereGrp(C)denotesthefullsubcategoryofCconsistingofallgroupobjects.AnygroupobjectadmitsaspecialpointcorrespondingtotheunityofthegroupHom(0;G),therefore,onecancompositeΩandgetsfunctorsΩ2;Ω3;


.Inhighercategorytheory,thesimilarresultcanbegeneralizedbyusingthenotionof¥groupobjects.fgA.2.8AsequenceA!B!Cissaidtobeafibrationsequenceiffisthehomotopykernelofg.Ifthisisthecase,bylemmaA.2.5,thekerneloffwillbetheloopingofC.Thusthereexistsalongfibrationsequence2ΩfΩgfg


!ΩC!ΩA!ΩB!ΩC!A!B!C:52万方数据 AppendixAglanceofhighercategorytheorySincethehomotopypullbacksarepreservedbythecovarianthom-functors,oneobtainsthelongfibrationsequenceofpointedgroupoidsforeveryobjectX:


!Hom(X;ΩB)!Hom(X;ΩC)!Hom(X;A)!Hom(X;B)!Hom(X;C):ThenonecanusethefollowinglemmaA.2.9toobtainalongexactsequence


!H(X;ΩB)!H(X;ΩC)!H(X;A)!H(X;B)!H(X;C):Notethatherethisisalongexactsequenceofpointedsetsinthesensethatforanyuvadjacenttwomaps
!
!
,onehasimu=kerv.Aside,intheabovesequence,onlythefirstthreetermsH(X;A),H(X;B)andH(X;C)arenotgroups,theremainingsequenceisalongexactsequenceofgroups.Forthisreason,onecandefinethegradedintrinsiccohomologybyHn(X;A):=H(X;ΩnA):LemmaA.2.9Thefunctorp0mapsfibrationsequencesofpointedgroupoidstoexactsequenceofpointedsets.FGProof.LetA!B!Cbeafibrationsequencesofpointedgroupoids.ThenonecanidentifyAasthecommacategory(G#0).ThenF(A)ispreciselythesub-categoryofBconsistingofallthoseobjectswhoseimagesinCareisomorphictothespecificobjectofC.Thusim(p0F)=ker(p0G).53万方数据 ReferencesReferences[1]AlekseevskyD.,MichorP.W.andRuppertW.ExtensionsofLiealgebras.ArXive-prints,May2000.arXiv:math/0005042[math.DG].[2]AlekseevskyD.,MichorP.W.andRuppertW.A.F.ExtensionsofsuperLiealgebras.J.LieTheory,2005,15(1):125–134.[3]BourbakiN.LiegroupsandLiealgebras.Chapters1–3.Springer-Verlag,Berlin,1998:155–164.[4]BoyomM.N.ThecohomologyofKoszul-Vinbergalgebras.PacificJ.Math.2006,225(1):119–153.[5]BrahicO.ExtensionsofLiebrackets.J.Geom.Phys.2010,60(2):352–374.[6]BrownK.S.Abstracthomotopytheoryandgeneralizedsheafcohomology.Trans.Amer.Math.Soc.1973,186:419–458.[7]BurdeD.Left-symmetricalgebras,orpre-Liealgebrasingeometryandphysics.Cent.Eur.J.Math.2006,4(3):323–357.[8]CartanH.,EilenbergS.Homologicalalgebra.PrincetonUniversityPress,Prince-ton,NJ,1999:xvi+390.[9]CayleyA.OntheTheoryoftheAnalyticalFormscalledTrees.In:ThecollectedmathematicalpapersofArthurCayley.Volume3.CambridgeUniversityPress,Cambridge,2009:242–246.[10]ChapotonF.,LivernetM.Pre-Liealgebrasandtherootedtreesoperad.Internat.Math.Res.Notices,2001,(8):395–408.[11]EilenbergS.,MacLaneS.Cohomologytheoryinabstractgroups.II.Groupex-tensionswithanon-Abeliankernel.Ann.ofMath.(2),1947,48:326–341.[12]FialowskiA.,PenkavaM.Extensionsof(super)Liealgebras.Commun.Con-temp.Math.2009,11(5):709–737.[13]FrégierY.Non-abeliancohomologyofextensionsofLiealgebrasasDelignegroupoid.J.Algebra,2014,398:243–257.[14]GerstenhaberM.Thecohomologystructureofanassociativering.Ann.ofMath.(2),1963,78:267–288.[15]GoldmanW.M.,MillsonJ.J.Thedeformationtheoryofrepresentationsoffun-damentalgroupsofcompactKählermanifolds.Inst.HautesÉtudesSci.Publ.Math.1988,(67):43–96.[16]HochschildG.CohomologyclassesoffinitetypeandfinitedimensionalkernelsforLiealgebras.Amer.J.Math.1954,76:763–778.[17]InassaridzeN.,KhmaladzeE.andLadraM.Non-abeliancohomologyandexten-sionsofLiealgebras.J.LieTheory,2008,18(2):413–432.[18]KimH.Completeleft-invariantaffinestructuresonnilpotentLiegroups.J.Dif-ferentialGeom.1986,24(3):373–394.54万方数据 REFERENCES[19]KimH.Extensionsofleft-symmetricalgebras.AlgebrasGroupsGeom.1987,4(1):73–117.[20]KoszulJ.-L.Domainesbornéshomogènesetorbitesdegroupesdetransforma-tionsaffines.Bull.Soc.Math.France,1961,89:515–533.[21]LurieJ.Highertopostheory.PrincetonUniversityPress,Princeton,NJ,2009:xviii+925.[22]MacLaneS.Categoriesfortheworkingmathematician.Springer-Verlag,NewYork,1998:xii+314.[23]ManettiM.Lecturesondeformationsofcomplexmanifolds(deformationsfromdifferentialgradedviewpoint).Rend.Mat.Appl.(7),2004,24(1):1–183.[24]MatsushimaY.Affinestructuresoncomplexmanifolds.OsakaJ.Math.1968,5:215–222.[25]NijenhuisA.,RichardsonJr.R.W.Cohomologyanddeformationsofalgebraicstructures.Bull.Amer.Math.Soc.1964,70:406–411.[26]NikolausT.,SchreiberU.andStevensonD.Principalinfinity-bundles-Generaltheory.ArXive-prints,July2012.arXiv:1207.0248[math.AT].[27]ShengY.CategorificationofPre-LieAlgebrasandSolutionsof2-gradedClas-sicalYang-BaxterEquations.ArXive-prints,Mar.2014.arXiv:1403.2144[math-ph].[28]VinbergE.Thetheoryofconvexhomogeneouscones.Trans.Mosc.Math.Soc.1963,12:340–403.[29]WeibelC.A.Anintroductiontohomologicalalgebra.CambridgeUniversityPress,Cambridge,1994:xiv+450.[30]nLab.nonabelianLiealgebracohomology.http://ncatlab.org/nlab/show/nonabelian+Lie+algebra+cohomology,retrievedon02/19/2015.55万方数据