Classification of complex simple Lie algebras via

arXiv:math/9902102v2[math.AG]19Oct2001ConstructionandclassificationofcomplexsimpleLiealgebrasviaprojectivegeometryJ.M.LandsbergandLaurentManivelAbstractWeconstructthecomplexsimpleLiealgebrasusingelementaryalgebraicgeometry.WeuseourconstructiontoobtainanewproofoftheclassificationofcomplexsimpleLiealgebrasthatdoesnotappealtotheclassificationofrootsystems.1OverviewWefirstpresentanalgorithmthatconstructstheminusculevarietiesusingelementaryalge-braicgeometry.Theminusculevarietiesareapreferredclassofhomogeneousvarieties.TheyareessentiallythehomogeneousprojectivevarietiesthatadmitanirreducibleHermitiansym-metricmetric;seebelowfortheprecisedefinition.ThealgorithmproceedsiterativelybybuildingalargerspaceX⊂PNfromasmallerspaceY⊂PnviaarationalmapPn99KPN,definedusingtheidealsofthesecantvarietiesofY,beginningwithY=CP1.Asabyprod-uct,weobtainelementaryconstructionsofallcomplexsimpleLiealgebras(exceptfore8whichhasnominusculehomogenousspace)andtheirminusculerepresentations,withoutanyreferencetoLiegroupsorLiealgebras.Nextwepresentanalgorithmthatconstructsthefundamentaladjointvarietiesusingtheidealsofthetangentialandsecantvarietiesofcertainminusculevarieties.Byanadjointvariety,wemeantheuniqueclosedorbitintheprojectivizationPgofasimplecomplexLiealgebrag.Wesaythatanadjointvarietyisfundamentaliftheadjointrepresentationisfundamental.Inparticular,weconstructallcomplexsimpleLiealgebraswithoutanyreferencetoLietheory.ComplexsimpleLiealgebraswerefirstclassifiedbyCartanandKilling100yearsago.Theirclassificationproofproceedsbyreducingthequestiontoacombinatorialproblem:theclassificationofirreduciblerootsystems,andthenclassifyingrootsystems.WepresentanewproofoftheclassificationofminusculevarietiesandcomplexsimpleLiealgebrasbyshowingouralgorithmsproduceallminuscule(resp.fundamentaladjoint)varietieswithoutusingtheclassificationofrootsystems,althoughwedousesomepropertiesofrootsystems.Wealsoprovideaproofthattheonlynon-fundamentaladjointvarietiesaretheadjointvarietiesofAmandCm,andthusweobtainanewproofoftheclassificationofcomplexsimpleLiealgebras.Ourproofcanbetranslatedintoacombinatorialargument:theconstructionconsistsoftwosetsofrulesforaddingnewnodestomarkedDynkindiagrams.Asacombinatorialalgorithm,itislessefficientthanthestandardproof,whichproceedsbyrulingoutallbutashortlistofDynkindiagramsimmediately,andthenstudyingthefewremainingdiagramstoseewhichareactuallyadmissible.1Ourconstructionshaveapplicationsthatgowellbeyondtheclassificationproofpresentedinthisarticle.Thisisthesecondpaperinaseries.In[8],[9]and[10]wepresentgeometricandrepresentation-theoreticapplicationsofouralgorithms.In[8]westudythegeometryoftheexceptionalhomogeneousspacesusingtheconstructionsofthispaper.In[9]and[10]weapplytheresultsofthispaper,especiallyourobservationsabouttheCasimirinsection5,toobtaindecompositionanddimensionformulasfortensorpowersofsomepreferredrepresentations.2StatementsofmainresultsLetVbeacomplexvectorspaceandletX⊆PVbeavarietyintheassociatedprojectivespace.Letvd(X)⊂P(SdV)denoteitsd-thVeronesere-embedding.IfP1,...,PNisabasisofSdV∗,thespaceofhomogeneouspolynomialsofdegreedonV,thenthemapPV→P(SdV)is[x]7→[P1(x),...,PN(x)].IfX⊆PVandY⊆PW,weletSeg(X×Y)⊂P(V⊗W)denotetheirSegreproduct,givenby([x],[y])7→[x⊗y].TheSegreproductgeneralizestoanarbitrarynumberoffactors.WewillusethenotationhXi⊂PVtodenotethelinearspanofX.Definition2.1.CallavarietyX⊂PVaminusculevarietyifX=G/PαwhereGisacomplexsimpleLiegroup,αisaminusculeroot,PαisanassociatedmaximalparabolicsubgroupandXistheprojectivizedorbitofahighestweightvectorinV=Vωwhereωisthefundamentalweightdualtothecorootofα(sotheembeddingistheminimalequivariantembedding).CallXageneralizedminusculevarietyifXisaSegreproductof(Veronesere-embeddingsof)minusculevarieties.InthissituationwewillcallVaminuscule(resp.generalizedminuscule)G-module.ThegeneralizedminusculevarietiesarethosevarietiesadmittingaHermitiansymmetricmetricinducedfromaFubini-Studymetricontheambientprojectivespace.Theminusculevarietiesarethoseforwhichthemetricisirreducibleandtheembeddingisminimal(i.e.,notaVeronesere-embedding).Definition2.2.ForasmoothvarietyX⊂PV,letT(X)⊂G(2,V)⊂P(Λ2V)denotethevarietyofembeddedtangentlinesofX.Letτ(X)⊂PVdenotethetangentialvarietyofX(theunionofthepointsonembeddedtangentlines),andletσp(X)⊂PVdenotethevarietyofsecantPp−1’stoX,thatis,forx1,...,xp⊂PV,letPx1,...,xpdenotetheprojectivespacetheyspan(generallyaPp−1),thenσp(X)=∪x1,...,xp∈XPx1,...,xp.Weletσ(X)=σ2(X).2.1MinusculecaseTheminusculealgorithm.LetY=Seg(vd1(X1)×···×vdr(Xr))⊂Pn−1=PTwheretheXj⊆PNj’sareoutputsofpreviousrunsthroughthealgorithmorP1⊆P1.WewillcallYadmissibleifT(Y)islinearlynondegenerate,thatis,ifhT(Y)i=P(Λ2T).IfYisadmissible,thendefinearationalmapasfollows:letdbethepositiveintegersuchthatσd−1(Y)6=σd(Y)=Pn−1.LinearlyembedPn−1⊂Pnasthehyperplane{x0=0},andconsidertherationalmapφ:Pn99KPN⊂P(SdCn+1∗)[x0,...,xn]7→[xd0,xd−10T∗,xd−20I2(Y),xd−30I3(σ2(Y)),...,Id(σd−1(Y))],andcallX=φ(Pn)⊂PNanoutput.HereT∗andIk(Z)=Ik(Z,PT)areshorthandnotationrespectivelyforabasisofT∗andasetofgeneratorsoftheidealofZindegreek.2Remark2.3.Weshowbelo