Infinitesimal deformations of restricted simple Li

arXiv:math/0612861v3[math.RA]2Jul2008INFINITESIMALDEFORMATIONSOFRESTRICTEDSIMPLELIEALGEBRASIFILIPPOVIVIANIAbstract.Wecomputetheinfinitesimaldeformationsoftwofamiliesofre-strictedsimplemodularLiealgebrasofCartan-type:theWitt-JacobsonandtheSpecialLiealgebras.1.IntroductionSimpleLiealgebrasoveranalgebraicallyclosedfieldofcharacteristiczerowereclassifiedatthebeginningoftheXIXcenturybyKillingandCartan.Theyusedthenon-degeneracyoftheKillingformtodescribethesimpleLiealgebrasintermsofrootsystemswhicharethenclassifiedbyDynkindiagrams.ThismethodbreaksdowninpositivecharacteristicbecausetheKillingformmaydegenerate.Indeedtheclassificationproblemremainedopenforalongtimeuntilitwasrecentlysolved,ifthecharacteristicofthebasefieldisgreaterthan3,byWilson-Block(see[BW88]),Strade(see[STR89],[STR92],[STR91],[STR93],[STR94],[STR98])andPremet-Strade(see[PS97],[PS99],[PS01]).Theclassifica-tionremainsstillopenincharacteristic2and3(see[STR04,page209]).Accordingtothisclassification,simplemodular(thatisoverafieldofpositivecharacteristic)Liealgebrasaredividedintotwobigfamilies,calledclassical-typeandCartan-typealgebras.Thealgebrasofclassical-typeareobtainedbythesim-pleLiealgebrasincharacteristiczerobyfirsttakingamodelovertheintegers(viaChevalleybases)andthenreducingmodulop(see[SEL67]).ThealgebrasofCartan-typewereconstructedbyKostrikin-Shafarevichin1966(see[KS66])asfinite-dimensionalanaloguesoftheinfinite-dimensionalcomplexsimpleLiealge-bras,whichoccurredinCartan’sclassificationofLiepseudogroups,andaredividedintofourfamilies,calledWitt-Jacobson,Special,HamiltonianandContactalgebras.TheWitt-JacobsonLiealgebrasarederivationalgebrasoftruncateddividedpoweralgebrasandtheremainingthreefamiliesarethesubalgebrasofderivationsfixingavolumeform,aHamiltonianformandacontactform,respectively.Moreoverincharacteristic5thereisoneexceptionalsimplemodularLiealgebracalledtheMelikianalgebra(introducedin[MEL80]).WeareinterestedinaparticularclassofmodularLiealgebrascalledrestricted.ThesecanbecharacterizedasthosemodularLiealgebrassuchthatthep-powerofaninnerderivation(whichincharacteristicpisaderivation)isstillinner.Impor-tantexamplesofrestrictedLiealgebrasaretheonescomingfromgroupsschemes.Indeedthereisaone-to-onecorrespondencebetweenrestrictedLiealgebrasandfinitegroupschemeswhoseFrobeniusvanishes(see[DG70,Chap.2]).Date:2July2008.1991MathematicsSubjectClassification.Primary17B50;Secondary17B20,17B56.Keywordsandphrases.Deformations,restrictedLiealgebras,Cartan-typesimpleLiealgebras.Duringthepreparationofthispaper,theauthorwaspartiallysupportedbyagrantfromtheMittag-LefflerInstituteinStockholm.12FILIPPOVIVIANIBystandardfactsofdeformationtheory,theinfinitesimaldeformationsofaLiealgebraareparametrizedbythesecondcohomologyoftheLiealgebrawithvaluesintheadjointrepresentation(seeforexample[GER64]).Itisaclassicalresult(see[HS97])thatforasimpleLiealgebragoverafieldofcharacteristic0itholdsthatHi(g,g)=0foreveryi≥0,whichimpliesinparticularthatsuchLiealgebrasarerigid.Theproofofthisfactreliesonthenon-degeneracyoftheKillingformandthenon-vanishingofthetraceoftheCasimirelement,whichisequaltothedimensionoftheLiealgebra.ThereforethesameproofworksalsoforthesimplemodularLiealgebrasofclassicaltypeoverafieldofcharacteristicnotdividingthedeterminantoftheKillingformandthedimensionoftheLiealgebra.ActuallyRudakov(see[RUD71])showedthatsuchLiealgebrasarerigidifthecharacteristicofthebasefieldisgreaterthanorequalto5whileincharacteristic2and3therearenon-rigidclassicalLiealgebras(see[CHE05],[CK00],[CKK00]).ThepurposeofthisarticleistocomputetheinfinitesimaldeformationsofthefirsttwofamiliesofrestrictedsimpleLiealgebrasofCartantype:theWitt-JacobsonalgebrasW(n)andtheSpecialalgebrasS(n).Unliketheclassical-typesimplealgebras,itturnsoutthatthesetwofamiliesarenotrigid.MorepreciselywegetthefollowingtwoTheorems(werefertosubsections3.1and4.1forthestandardnotationsconcerningW(n)andS(n)andtosubsection2.3forthedefinitionofthesquaringoperatorsSq).Theorem1.1.AssumethatthecharacteristicpofthebasefieldFisdifferentfrom2.ThenwehaveH2(W(n),W(n))=nMi=1F·hSq(Di)iwiththeexceptionofthecasen=1andp=3whenitis0.Theorem1.2.AssumethatthecharacteristicofthebasefieldFisdifferentfrom2andmoreoveritisdifferentfrom3ifn=3.ThenwehaveH2(S(n),S(n))=nMi=1F·hSq(Di)iMF·hΘiwhereΘisdefinedbyΘ(Di,Dj)=Dij(xτ)andextendedby0outsideS(n)−1×S(n)−1.Inthetwoforthcomingpapers[VIV2,VIV3],wecomputetheinfinitesimalde-formationsoftheremainingrestrictedsimpleLiealgebrasofCartan-type,namelytheHamiltonian,theContactandtheexceptionalMelikianalgebras.Moreover,inanotherpaper[VIV4],weapplytheseresultstothestudyoftheinfinitesimaldefor-mationsofthesimplefinitegroupschemescorrespondingtotherestrictedsimpleLiealgebrasofCartantype.LetusmentionthattheinfinitesimaldeformationsofsimpleLiealgebrasofCartan-type(inthegeneralnon-restrictedcase)havebeenconsideredalreadybyDˇzumadildaevin[DZU80,DZU81,DZU89]andDˇzumadildaev-Kostrikinin[DK78]butacompletepictureaswellasdetailedproofsweremissing.Moreprecisely:in[DK78]theauthorscomputetheinfinitesimaldeformationsoftheJacobson-Wittalgebrasofrank1,in[DZU80,Theorem4]thea