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1、Izv.Akad.NaukSSSRMath.USSR-IzvestijaSer.Mat.Tom32(1968),No.6Vol.2(1968),No.6SIMPLEIRREDUCIBLEGRADEDLIEALGEBRASOFFINITEGROWTHV.G.KACUDC519.4WeclassifythesimplegradedLiealgebrasG=ΘGi,forwhichthedimensionofthei=—oospaceG;growsassomepowerof
2、£
3、,undertheadditionalassumptionthattheadjointr
4、epre-sentationofGQonG_,isirreducible.Fromtheseresultsweobtainaclassificationoftheprimitiveinfinite-dimensionalCartanpseudogroupsoftransformationsandaclassificationofsymmetricspaces.IntroductionInthepresentarticle,undercertainrestrictions,weclassifythesimplegradedLiealgebrasoveranalg
5、ebraicallyclosedfieldofcharacteristic0.Theresultscanbeappliedtoclassifytheprimitiveinfinite-dimensionalpseudogroupsoftransformationsand,entirelyunexpectedly,toclassifysymmetricspaces.+oo.WeconsideronlythosegradedLiealgebrasG=©Gi,whicharegeneratedbythesub-i——oospaceG_xφGQφG1.AgradedL
6、iealgebraiscalledsimpleifitcontainsnonontrivialhomoge-nousideals.+ooForasimplegradedLiealgebraG=ΦG{in"generalposition"thedimensionofthei=—cospaceGLgrowsexponentiallywithi.However,itturnsoutthatinexceptionalcases,thedimensionofG£growsasapowerim·TheseLiealgebrasarecalledLiealgebraso
7、ffinitegrowth;thenum-berr(G)=m+1iscalledthegrowth*oftheLiealgebraG.ItiseasytoshowthatthegrowthofGisequaltothedimensionofGinthesenseofGel'fandandKirillov[10],andconsequentlyisinde-pendentofthegradation.+00AgradedLiealgebraG=ΦGiiscalledirreducibleiftherepresentationoftheLiealge-i=—oob
8、raCoonthespaceG_tisirreducible.'Editor'snote:"Height"inthetranslationof[5],12711272V.G-KACInthepresentarticleweclassifythesimpleirreduciblegradedLiealgebrasoffinitegrowth.Theconsiderationslyingatthebasisofourargumentsareasfollows:a)ifthegrowthofasubalgebraorofafactor-algebraofaLieal
9、gebraGisinfinite,thenthegrowthofGisinfinite;b)thegrowthofacertain"standard"LiealgebraK2(seeChapterII,§6)isinfinite.WheneveritisnecessarytoprovethatthegrowthofaLiealgebraisinfinite,weturnupasub-algebrawhosefactor-algebraisisomorphictoK2.TheearliestexamplesofsimplegradedLiealgebrasoff
10、initegrowtharethewe
