Cohomology of graded Lie algebras of maximal class

arXiv:math/0412325v1[math.RT]16Dec2004COHOMOLOGYOFGRADEDLIEALGEBRASOFMAXIMALCLASSALICEFIALOWSKIANDDMITRIMILLIONSCHIKOVAbstract.ItwasshownbyA.Fialowskithatanarbitraryinfinite-dimensionalN-graded”filiformtype”Liealgebrag=L∞i=1giwithone-dimensionalhomo-geneouscomponentsgisuchthat[g1,gi]=gi+1,∀i≥2overafieldofzerocharacteristicisisomorphictoone(andonlyone)Liealgebrafromthreegivenones:m0,m2,L1,wheretheLiealgebrasm0andm2aredefinedbytheirstruc-turerelations:m0:[e1,ei]=ei+1,∀i≥2andm2:[e1,ei]=ei+1,∀i≥2,[e2,ej]=ej+2,∀j≥3andL1isthe”positive”partoftheWittalgebra.InthepresentarticlewecomputethecohomologyH∗(m0)andH∗(m2)withtrivialcoefficients,giveexplicitformulasfortheirrepresentativecocyclesanddescribethemultiplicativestructureinthecohomology.Alsowediscusstherelationswithcombinatoricsandrepresentationtheory.ThecohomologyH∗(L1)wascalculatedbyL.Goncharovain1973.IntroductionN-gradedLiealgebrasarecloselyrelatedtonilpotentLiealgebras,forinstance,afinite-dimensionalN-gradedLiealgebragmusttobenilpotent.Infinite-dimensionalonesarealsocalledresidualnilpotentLiealgebras.M.Vergnestudiedin[12]nilpotentLiealgebraswiththemaximalpossiblenilindexs(g)=dimg−1(bynilindexs(g)wemeanthelengthofthedescendingcentralseries{Cig}ofg).M.VergnecalledthemfiliformLiealgebras.InherstudytheN-gradedfiliformLiealgebram0(n)playedaspecialrole.ThisLiealgebraisdefinedbyitsbasise1,...,enandnon-trivialcommutatorrelations:[e1,ei]=ei+1,i=2,...,n−1.Evidently,m0(n)isgeneratedbye1ande2.AnotherexampleofN-gradedtwo-generatedfiliformLiealgebraism2(n):[e1,ei]=ei+1,i=2,...,n−1,[e2,ei]=ei+2,i=3,...,n−2.A.Fialowskiclassifiedin[4]allinfinite-dimensionalN-gradedtwo-generatedLiealgebrasg=⊕igiwithone-dimensionalhomogeneouscomponentsgi.Inparticular,thereareonlythreealgebrasinherlistsatisfyingthe”filiformproperty”:[g1,gi]=gi+1,∀i.Theyarem0,m2,L1,wherem0,m2denoteinfinite-dimensionalanaloguesofm0(n),m2(n),respectivelyandL1isthe”positive”partoftheWittorVirasoroalgebra.Theclassificationoffinite-dimensionalN-gradedfiliformLiealgebrasoverafieldofzerocharacteristicwasdonein[9].A.ShalevandE.Zelmanovdefinedin[11]thecoclass(whichmightbeinfinity)ofafinitelygeneratedandresiduallynilpotentLiealgebrag,inanalogywiththecase1991MathematicsSubjectClassification.17B56,17B70,17B10,17B65,05A17.Keywordsandphrases.gradedLiealgebras,filiformalgebras,Liealgebrasofmaximalclass,cohomology,Dixmier’sexactsequence.TheresearchoftheauthorswaspartiallysupportedbygrantsOTKAT043641,T043034,RFBR02-01-00659,”RussianScientificSchools”2185.2003.1.ThefinalversionofthepaperwascompletedduringthestayofthesecondauthorattheE¨otv¨osLor´andUniversityinBudapest.12ALICEFIALOWSKIANDDMITRIMILLIONSCHIKOVof(pro-)p-groups,ascc(g)=Pi≥1(dim(Cig/Ci+1g)−1).Obviouslythecoclassofafiliformalgebraisequaltooneandthesameistruefortheinfinite-dimensionalalgebrasm0,m2,L1.Algebrasofcoclass1arealsocalledalgebrasofmaximalclass.TheyarealsonarroworthinLiealgebras(A.Shalev,A.Caranti,M.Newman,etal.).PartofFialowski’sclassificationin[4]canbereformulatedinthefollowingway:UptoanisomorphismthereareonlythreeN-gradedLiealgebrasofmaximalclasswithone-dimensionalhomogeneouscomponents:m0,m2,L1.Thelaststatementwasrediscoveredin[11].Algebrasofmaximalclassareinthecenterofattentionthesedaysbothinzeroandpositivecharacteristic.Therearemanyopenquestionsrelatedtothem.Onenaturalquestionistheircohomologywhichisthesubjectofthepresentpaper.L.Goncharovacalculatedin1973[7]theBettinumbersbq(L1)=dimHq(L1).Herresultimplies,asacorollary,thecelebratedEuleridentityincombinatorics:∞Yj=1(1−tj)=∞Xk=0(−1)k(t3k2−k2+t3k2+k2)(see[5]).TheBettinumbersdimHq(m0(n))(finite-dimensionalcase)werecalcu-latedin[1](seealso[2]),however,therearenoexplicitformulasforbasiccocyclesandnodescriptionofthemultiplicativestructureofH∗(m0(n))wasobtained.WegiveacompletedescriptionofthecohomologyalgebrasH∗(m0)andH∗(m2).ThemethodweuseisbasedonDixmier’sexactsequenceinLiealgebracohomology[3].Inourconsiderationsweusecombinatorics:partitionsandgeneratingfunctions.Thepaperisorganizedasfollows.InSections1–2wereviewallnecessarydefi-nitionsandfactsconcerningLiealgebracohomologyandN-gradedLiealgebras,inparticularwerecallDixmier’sexactsequenceinthecohomology[3].WestartourcomputationsinSection3withthealgebraH∗(m0)(Theorem3.4).InSection4wediscusstherelationsofourresultswithrepresentationstheory.ItturnsoutthatthebasiccocyclesrepresentingHq(m0)areatthesametimethehighestweightvectors(primitivevectors)oftheq-thexteriorpowerΛq(V(λ))oftheirreducibleone-dimensionalsl(2,K)-moduleV(λ)forsomeλ(Theorem4.1).InSection5weapplyDixmier’sexactsequenceandtheresultsofSection3tocomputeH∗(m2)(Theorem5.5).Section6isdevotedtofinite-dimensionalanalogsofthealgebrasconsideredabove.RecallthatallofthemarefiliformLiealgebras.TheBettinum-bersdimHq(m0(n))areknown[1],[2].SomeofdimHq(L1/Ln+1)wasfoundin[8].Thequestionsontheexplicitformulasforrepresentingcocyclesandthemul-tiplicativestructurearestillopenforthesealgebras.Attheendofthepaperweconsiderthecharacteristicpanalogofthealgebrasm0andm2.WebrieflyremarkthatothercomputationaltoolssuchasspectralsequencesortheHodgeLaplacianofthedifferentialdle