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1、EUCLIDEANLIEALGEBRASROBERTV.MOODYIntroduction.OuraiminthispaperistostudyacertainclassofLiealgebraswhicharosenaturallyin(4).In(4),weshowedthatbeginningwithanindecomposablesymmetrizablegeneralizedCartanmatrix(Atj)andafield<ï>ofcharacteristiczero,wecouldconstructaLiea
2、lgebraE{(Ai3))over<ï>patternedonthefinite-dimensionalsplitsimpleLiealgebras.WewereabletoshowthatE{{Ai3))issimpleprovidingthat(Ai3)doesnotfallinthelistgivenin(4,Table).Wedidnotprovetheconverse,however.Thediagramsofthetableof(4)appearinTable2.Callthematricesthattheyr
3、epresentEuclideanmatricesandtheircorrespondingalgebrasEuclideanLiealgebras.OurfirstobjectiveistoshowthatEuclideanLiealgebrasarenotsimple.ThisinvolvesacloselookattherootsystemsofEuclideanLiealgebras(§1)andtheconstruction(§2)ofacertainmoduleendomorphismofE{Etreatedas
4、anE-moduleinthecustomaryway).AlongthewaywediscoverthatthesetofnullrootsZisagroupandthesubgroupZ*of(4,§6)isofindex1,2,or3.Wecall[Z:Z*]thetiernumber,f,ofourLiealgebra.OursecondobjectiveistodescribecertainsimpleepimorphicimagesofaEuclideanLiealgebra.Bytheresultsof(4,§
5、7),everyproperidealofEisoffinitecodimension.Foreachx£$—{0}thereisanidealofminimalco-dimensionandthequotient,£(/*),ofEbythisidealisafinite-dimensionalcentralsimpleLiealgebraover<ï>.Forthe1-tieredalgebraswehave:(i)£(/x)~E(v)forall/x,vG$-{0},(ii)E~$(x)(x)$E(l),where<
6、ï>(x)istheassociativealgebraoffiniteLaurentseriesinanindeterminatexover<$,and(iii)£(1)issplit.In§4weshowthat(i),andhence(ii),cannotholdingeneralfor2-tieredalgebras.IndeedtheidentityoftheE(n)swhenEis2-tieredor3-tieredisratherobscureandoureffortsareconcentratedinwork
7、ingoutthetypeofeachE(JJL).TheprocedureisessentiallytocalculatedimE(/x)(whichisindependentof/x)and,althoughthisisnotverysophisticated,itdoesinvolvesecuringsomefurtherresultsontherootsystemswhichareboundtobeimportantinanyfurtherinvestigations.Exceptfor^4,2,whichdoesn
8、otlenditselftothisprocedure,wecansaythatforany/x,vÇ<£>—{0},•£(/*)andE{y)areofthesametype,thistypebeinggiveninTable2.ReceivedAugust12,1968.Mostoft
