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1、Can.J.Math.,Vol.XXVIII,No.1,1976,pp.174-180ONDERIVATIONSOFLIEALGEBRASSTEPHENBERMANIntroduction.AwellknownresultinthetheoryofLiealgebras,duetoH.Zassenhaus,statesthatif££isafinitedimensionalLiealgebraoverthefieldKsuchthatthekillingformofJ$fisnon-degenerate,thenthederivationso
2、f«ifareallinner,[3,p.74].Inparticular,thisappliestothefinitedimensionalsplitsimpleLiealgebrasoverfieldsofcharacteristiczero.InthispaperweextendthisresulttoaclassofLiealgebraswhichgeneralizethesplitsimpleLiealgebras,andwhicharedefinedbyCartanmatrices(foradefinitionsee§1).Bec
3、auseofthefactthatthealgebrasweconsiderareusuallyinfinitedimensional,themethodweemployinourinvestigationisquitedifferentfromthestandardoneusedinthefinitedimensionalcase,andmakesnoreferencetoanyassocia•tivebilinearformonthealgebras.If«ifisoneoftheLiealgebrasundercon•sideratio
4、n,welet2)(^£)denotethederivationalgebraof«ifandJ(£)theidealofinnerderivations.OurmainresultstatesthatthedimensionofQf{^£)/e/(«if)equalsthenullityoftheCartanmatrixwhichdefinesif.InSection1wegiveabriefdescriptionofthealgebrasweconsiderandinSection2weproveourmainresult.Inthefi
5、nalsectionwepresentanapplica•tionofourresulttotheproblemofdeterminingtheisomorphismclassesofthealgebrasunderconsideration.Oneoftheresultsinthissection,Theorem2,wasobtainedinjointworkwithR.Moody,andisofindependentinterest.1.Descriptionofthealgebras.ACartanmatrixisany/X/integ
6、ralmatrix{Atj)suchthatAu=2,Atj^0ifi^j,andAtj•=0ifandonlyifAji=0,for1^i,j^/.WewillalwaysassumeourCartanmatrix(Afj)isindecomposable,whichisthesamethingasrequiringthattheDynkindiagramassociatedtoitisconnected.LetKbeanyfieldofcharacteristiczeroandletJ^~7、erKgeneratedbythe3/elementseuhufit1^i^/.Let/denotetheidealofJTifgeneratedbythefollowingelements,[hi,hj],[euhj]—Ajfeu[fuhj]+Ajift,[eufj]-àtjhufor1^i,j^/,andleti^'vdenotethefactoralgebra,ifc/iscalledtheuniversalheffalumpalgebraoverKattachedto(^4^).Welet6denotetheidealof££vgen
8、eratedReceivedApril10,1975.ThisresearchwassupportedbyaNationalResearchCouncilgrant
