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1、JOURNALOFALGEBRA10,211-230(19681ANewClassofLieAlgebrasROBERTV.MOODYDepartmentofMathematics,NewMexicoStateUniversity,LasCruces,NewMexico88001CommunicatedbyNathanJacobsonReceivedOctober10,1967;revisedDecember10,1967Thisisthefirstoftwopapersdevotedtothestudyofcertain
2、typesofLiealgebras(generallyinfinite-dimensional)whichareconstructedfrommatrices(calledgeneralizedCartanmatrices)closelyresemblingCartanmatrices.InthefirstthreesectionsweconstructtheLiealgebras,derivetheirbasicproperties,andconstructasymmetricinvariantformonthoseL
3、iealgebrasderivedfromtheso-calledsymmetrizablegeneralizedCartanmatrices.InSection5weshowthatidealsinourLiealgebrasmustbeprincipaland,infact,mostofthetimethealgebrasaresimple.Sections6,7,and8describetheideallatticesofthenon-simpleLiealgebras(calledEuclideanLiealgeb
4、ras)andterminatewithaclassificationofsuchalgebras.FinallyinSection9weseethattheonIyfinite-dimensionalLiealgebrasobtainedfromgeneralizedCartanmatricesarethesplitsemisimpleLiealgebras.ThepapertofollowthiswillbedevotedexclusivelytoEuclideanLiealgebras,primarilydirect
5、edtowardsdescribingfactorsofsuchalgebrasbymaximalideals(see[3]).1.CONSTRUCTIONOFTHEALGEBRASThematerialofthissectionisverysimilartothefirstpartofChapterVIIof[2].Consequentlywewillbefairlybrief.Let@beafieldofcharacteristiczeroandlettherationals,Q,beidentifiedwithits
6、primesubfield.ZandNwilldenotetheintegersandnaturalnumbers,respectively.Let(A,)beanIxIintegralmatrixwiththeproperties(i)&=2i=l,...,Z,(ii)A,<0ifi#j,(iii)Aii=0oAji=0foralli,j=l,...,1.SuchamatrixwillbecalledageneralizedCurtanmatrix.LetL={l,...,I}.211212MOODYLetFbethef
7、reeLiealgebraover@generatedbythe31freegeneratorshi,ei,fi,iEL.DefineH=@hi+.*.+@hrandletAbeanZ-dimensionalvectorspaceover@withabasis01~,...,01~.Wedefinealinearmapping,-,ofAintoH*(thedualspaceofH)by&(hj)=Aijforalli,jEL.Inthesequelwewilldroptheovertildesandwrite/3(h)f
8、or&h).Itshouldbenoted,however,thatthelinearmappingisnotingenerall-landitisimportantforourpurposestodistinguishbetweenelementsofAevenwhentheyinducethesam
