University Press of Florida

Lie Algebras

Buy This Book ( Related URL )
MISSING IMAGE

Material Information

Title:
Lie Algebras
Physical Description:
Book
Language:
en-US
Creator:
Sternberg, Shlomo
Publication Date:

Subjects

Subjects / Keywords:
Maurer-Cartan, tensor product, Poincar´e-Birkhoff-Witt Theorem, Casimir element, Weyl group, orthogonal algebras, symplectic algebras, maximal root, minimal root, Harish-Chandra isomorphism, OGT+ ISBN:9781616100520
Calculus, Mathematics
Mathematics / Calculus

Notes

Abstract:
This book addresses the following topics: The Campbell Baker Hausdor formula; sl(2) and its Representations; classical simple algebras; Engel-Lie-Cartan-Weyl; conjugacy of Cartan subalgebras; simple finite dimensional algebras; cyclic highest weight modules; Serre's theorem; Clifford algebras and spin representations; The Kostant Dirac operator; The center of U(g); and Chevalley's theorem.
General Note:
Expositive
General Note:
Higher Education
General Note:
http://www.ogtp-cart.com/product.aspx?ISBN=9781616100520
General Note:
Adobe Reader
General Note:
Diagram, Figure, Graph, Narrative text, Textbook
General Note:
MAS 396 - TOPICS IN ALGEBRA I, MAS 397 - TOPICS IN ALGEBRA II
General Note:
http://florida.theorangegrove.org/og/file/b11aec6b-b27e-f0b8-b135-774f3a6f22b0/1/lie_algebras.pdf

Record Information

Source Institution:
University of Florida
Rights Management:
This work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike 2.0 Generic License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/2.0/. You are free to copy, distribute and transmit this work and to adapt this work under the following conditions: 1) Attribution. You must attribute the work in …
Resource Identifier:
isbn - 9781616100520
System ID:
AA00011701:00001


This item is only available as the following downloads:


Full Text

PAGE 1

LiealgebrasShlomoSternbergApril23,2004

PAGE 2

2

PAGE 3

Contents1TheCampbellBakerHausdorFormula71.1Theproblem..............................71.2ThegeometricversionoftheCBHformula.............81.3TheMaurer-Cartanequations....................111.4ProofofCBHfromMaurer-Cartan..................141.5Thedierentialoftheexponentialanditsinverse.........151.6Theaveragingmethod.........................161.7TheEulerMacLaurinFormula....................181.8Theuniversalenvelopingalgebra...................191.8.1Tensorproductofvectorspaces...............201.8.2Thetensorproductoftwoalgebras.............211.8.3Thetensoralgebraofavectorspace.............211.8.4Constructionoftheuniversalenvelopingalgebra......221.8.5ExtensionofaLiealgebrahomomorphismtoitsuniversalenvelopingalgebra.......................221.8.6Universalenvelopingalgebraofadirectsum........221.8.7Bialgebrastructure......................231.9ThePoincare-Birkho-WittTheorem................241.10Primitives................................281.11FreeLiealgebras...........................291.11.1Magmasandfreemagmasonaset.............291.11.2TheFreeLieAlgebraLX...................301.11.3ThefreeassociativealgebraAssX.............311.12AlgebraicproofofCBHandexplicitformulas............321.12.1AbstractversionofCBHanditsalgebraicproof......321.12.2ExplicitformulaforCBH...................322slanditsRepresentations.352.1LowdimensionalLiealgebras.....................352.2slanditsirreduciblerepresentations...............362.3TheCasimirelement.........................392.4slissimple.............................402.5Completereducibility.........................412.6TheWeylgroup............................423

PAGE 4

4CONTENTS3Theclassicalsimplealgebras.453.1Gradedsimplicity...........................453.2sln+1................................473.3Theorthogonalalgebras........................483.4Thesymplecticalgebras........................503.5Therootstructures..........................523.5.1An=sln+1.........................523.5.2Cn=spn;n2......................533.5.3Dn=on;n3......................543.5.4Bn=on+1n2.....................553.5.5Diagrammaticpresentation..................563.6Lowdimensionalcoincidences.....................563.7Extendeddiagrams..........................584Engel-Lie-Cartan-Weyl614.1Engel'stheorem............................614.2SolvableLiealgebras.........................634.3Linearalgebra.............................644.4Cartan'scriterion...........................664.5Radical.................................674.6TheKillingform............................674.7Completereducibility.........................695ConjugacyofCartansubalgebras.735.1Derivations...............................745.2Cartansubalgebras..........................765.3Solvablecase..............................775.4ToralsubalgebrasandCartansubalgebras..............795.5Roots..................................815.6Bases..................................855.7Weylchambers.............................875.8Length.................................885.9ConjugacyofBorelsubalgebras...................896Thesimplenitedimensionalalgebras.936.1SimpleLiealgebrasandirreduciblerootsystems..........946.2Themaximalrootandtheminimalroot...............956.3Graphs.................................976.4Perron-Frobenius............................986.5Classicationoftheirreducible..................1046.6Classicationoftheirreduciblerootsystems............1056.7TheclassicationofthepossiblesimpleLiealgebras........109

PAGE 5

CONTENTS57Cyclichighestweightmodules.1137.1Vermamodules.............................1147.2WhenisdimIrr<1?......................1157.3ThevalueoftheCasimir.......................1177.4TheWeylcharacterformula.....................1217.5TheWeyldimensionformula.....................1257.6TheKostantmultiplicityformula...................1267.7Steinberg'sformula..........................1277.8TheFreudenthal-deVriesformula.................1287.9Fundamentalrepresentations.....................1317.10Equalranksubgroups.........................1338Serre'stheorem.1378.1TheSerrerelations...........................1378.2Therstverelations.........................1388.3ProofofSerre'stheorem........................1428.4Theexistenceoftheexceptionalrootsystems............1449Cliordalgebrasandspinrepresentations.1479.1Denitionandbasicproperties...................1479.1.1Denition............................1479.1.2Gradation...........................1489.1.3^pasaCpmodule.....................1489.1.4Chevalley'slinearidenticationofCpwith^p......1489.1.5Thecanonicalantiautomorphism...............1499.1.6Commutatorbyanelementofp...............1509.1.7Commutatorbyanelementof^2p.............1519.2OrthogonalactionofaLiealgebra..................1539.2.1Expressionforintermsofdualbases...........1539.2.2TheadjointactionofareductiveLiealgebra........1539.3Thespinrepresentations.......................1549.3.1Theevendimensionalcase..................1559.3.2Theodddimensionalcase...................1589.3.3SpinadandV.........................15910TheKostantDiracoperator16310.1Antisymmetrictrilinearforms....................16310.2JacobiandCliord..........................16410.3OrthogonalextensionofaLiealgebra................16510.4Thevalueof[v2+Casr]0.....................16710.5Kostant'sDiracOperator.......................16910.6EigenvaluesoftheDiracoperator..................17210.7Thegeometricindextheorem.....................17810.7.1TheindexofequivariantFredholmmaps..........17810.7.2InducedrepresentationsandBott'stheorem........17910.7.3Landweber'sindextheorem..................180

PAGE 6

6CONTENTS11ThecenterofUg.18311.1TheHarish-Chandraisomorphism..................18311.1.1Statement...........................18411.1.2Exampleofsl........................18411.1.3UsingVermamodulestoprovethatH:Zg!UhW.18511.1.4Outlineofproofofbijectivity.................18611.1.5RestrictionfromSggtoShW.............18711.1.6FromSggtoShW.....................18811.1.7Completionoftheproof....................18811.2Chevalley'stheorem..........................18911.2.1Transcendencedegrees....................18911.2.2Symmetricpolynomials....................19011.2.3Fixedelds...........................19211.2.4Invariantsofnitegroups...................19311.2.5TheHilbertbasistheorem..................19511.2.6ProofofChevalley'stheorem.................196

PAGE 7

Chapter1TheCampbellBakerHausdorFormula1.1Theproblem.Recallthepowerseries:expX=1+X+1 2X2+1 3!X3+;log1+X=X)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2X2+1 3X3+:Wewanttostudytheseseriesinaringwhereconvergencemakessense;forex-ampleintheringofnnmatrices.Theexponentialseriesconvergeseverywhere,andtheseriesforthelogarithmconvergesinasmallenoughneighborhoodoftheorigin.Ofcourse,logexpX=X;explog+X=1+Xwheretheseseriesconverge,orasformalpowerseries.Inparticular,ifAandBaretwoelementswhicharecloseenoughto0wecanstudytheconvergentserieslog[expAexpB]whichwillyieldanelementCsuchthatexpC=expAexpB.TheproblemisthatAandBneednotcommute.Forexample,ifweretainonlythelinearandconstanttermsintheserieswendlog[+A++B+]=log1+A+B+=A+B+:Ontheotherhand,ifwegoouttotermssecondorder,thenon-commutativitybeginstoenter:log[+A+1 2A2++B+1 2B2+]=7

PAGE 8

8CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULAA+B+1 2A2+AB+1 2B2)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2A+B+2=A+B+1 2[A;B]+where[A;B]:=AB)]TJ/F11 9.963 Tf 9.962 0 Td[(BA.1isthecommutatorofAandB,alsoknownastheLiebracketofAandB.Collectingthetermsofdegreethreeweget,aftersomecomputation,1 12)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(A2B+AB2+B2A+BA2)]TJ/F8 9.963 Tf 9.963 0 Td[(2ABA)]TJ/F8 9.963 Tf 9.963 0 Td[(2BAB]=1 12[A;[A;B]]+1 12[B;[B;A]]:Thissuggeststhattheseriesforlog[expAexpB]canbeexpressedentirelyintermsofsuccessiveLiebracketsofAandB.Thisisso,andisthecontentoftheCampbell-Baker-Hausdorformula.Oneoftheimportantconsequencesofthemereexistenceofthisformulaisthefollowing.SupposethatgistheLiealgebraofaLiegroupG.ThenthelocalstructureofGneartheidentity,i.e.therulefortheproductoftwoelementsofGsucientlyclosedtotheidentityisdeterminedbyitsLiealgebrag.Indeed,theexponentialmapislocallyadieomorphismfromaneighborhoodoftheoriginingontoaneighborhoodWoftheidentity,andifUWisapossiblysmallerneighborhoodoftheidentitysuchthatUUW,thetheproductofa=expandb=exp,witha2Uandb2UisthencompletelyexpressedintermsofsuccessiveLiebracketsofand.Wewillgivetwoproofsofthisimportanttheorem.Onewillbegeometric-theexplicitformulafortheseriesforlog[expAexpB]willinvolveintegration,andsomakessenseovertherealorcomplexnumbers.WewillderivetheformulafromtheMaurer-Cartanequations"whichwewillexplaininthecourseofourdiscussion.Oursecondversionwillbemorealgebraic.Itwillinvolvesuchideasastheuniversalenvelopingalgebra,comultiplicationandthePoincare-Birkho-Witttheorem.Inbothproofs,manyofthekeyideasareatleastasimportantasthetheoremitself.1.2ThegeometricversionoftheCBHformula.Tostatethisformulaweintroducesomenotation.LetadAdenotetheoperationofbracketingontheleftbyA,soadAB:=[A;B]:Denethefunctionbyz=zlogz z)]TJ/F8 9.963 Tf 9.962 0 Td[(1whichisdenedasaconvergentpowerseriesaroundthepointz=1so+u=+ulog+u u=1+u)]TJ/F11 9.963 Tf 10.949 6.74 Td[(u 2+u2 3+=1+u 2)]TJ/F11 9.963 Tf 10.949 6.74 Td[(u2 6+:

PAGE 9

1.2.THEGEOMETRICVERSIONOFTHECBHFORMULA.9Infact,wewillalsotakethisasadenitionoftheformalpowerseriesforintermsofu.TheCampbell-Baker-HausdorformulasaysthatlogexpAexpB=A+Z10expadAexptadBBdt:.2Remarks.1.Theformulasaysthatwearetosubstituteu=expadAexptadB)]TJ/F8 9.963 Tf 9.963 0 Td[(1intothedenitionof,applythisoperatortotheelementBandthenintegrate.IncarryingoutthiscomputationwecanignorealltermsintheexpansionofintermsofadAandadBwhereafactorofadBoccursontheright,sinceadBB=0.Forexample,toobtaintheexpansionthroughtermsofdegreethreeintheCampbell-Baker-Hausdorformula,weneedonlyretainquadraticandlowerordertermsinu,andsou=adA+1 2adA2+tadB+t2 2adB2+u2=adA2+tadBadA+Z101+u 2)]TJ/F11 9.963 Tf 11.158 6.74 Td[(u2 6dt=1+1 2adA+1 12adA2)]TJ/F8 9.963 Tf 13.649 6.74 Td[(1 12adBadA+;wherethedotsindicateeitherhigherordertermsortermswithadBoccurringontheright.Soupthroughdegreethree.2giveslogexpAexpB=A+B+1 2[A;B]+1 12[A;[A;B]])]TJ/F8 9.963 Tf 13.649 6.74 Td[(1 12[B;[A;B]]+agreeingwithourprecedingcomputation.2.ThemeaningoftheexponentialfunctiononthelefthandsideoftheCampbell-Baker-Hausdorformuladiersfromitsmeaningontheright.Ontherighthandside,exponentiationtakesplaceinthealgebraofendomorphismsoftheringinquestion.Infact,wewillwanttomakeafundamentalreinter-pretationoftheformula.WewanttothinkofA;B,etc.aselementsofaLiealgebra,g.Thentheexponentiationsontherighthandsideof.2arestilltakingplaceinEndg.Ontheotherhand,ifgistheLiealgebraofaLiegroupG,thenthereisanexponentialmap:exp:g!G,andthisiswhatismeantbytheexponentialsontheleftof1.2.Thisexponentialmapisadieomorphismonsomeneighborhoodoftheorigining,anditsinverse,log,isdenedinsomeneighborhoodoftheidentityinG.Thisisthemeaningwewillattachtothelogarithmoccurringontheleftin.2.3.ThemostcrucialconsequenceoftheCampbell-Baker-HausdorformulaisthatitshowsthatthelocalstructureoftheLiegroupGthemultiplicationlawforelementsneartheidentityiscompletelydeterminedbyitsLiealgebra.4.Forexample,weseefromtherighthandsideof.2thatgroupmulti-plicationandgroupinverseareanalyticifweuseexponentialcoordinates.

PAGE 10

10CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULA5.Considerthefunctiondenedbyw:=w 1)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F10 6.974 Tf 6.227 0 Td[(w:.3Thisisafamiliarfunctionfromanalysis,asitentersintotheEuler-Maclaurinformula,seebelow.Itistheexponentialgeneratingfunctionof)]TJ/F8 9.963 Tf 7.748 0 Td[(1kbkwherethebkaretheBernoullinumbers.Thenz=logz:6.Theformulaisnamedafterthreemathematicians,Campbell,Baker,andHausdor.Butthisisamisnomer.Substantiallyearlierthantheworksofanyofthesethree,thereappearedapaperbyFriedrichSchur,NeueBegruendungderTheoriederendlichenTransformationsgruppen,"MathematischeAnnalen35,161-197.Schurwritesdown,asconvergentpowerseries,thecom-positionlawforaLiegroupintermsof"canonicalcoordinates",i.e.,intermsoflinearcoordinatesontheLiealgebra.Hewritesdownrecursiverelationsforthecoecients,obtainingaversionoftheformulaswewillgivebelow.IamindebtedtoProf.Schmidforthisreference.Ourstrategyfortheproofof.2willbetoproveadierentialversionofit:d dtlogexpAexptB=expadAexptadBB:.4SincelogexpAexptB=Awhent=0,integrating.4from0to1willprove.2.Letusdene)-278(=\050t=\050t;A;Bby)-278(=logexpAexptB:.5Thenexp)-278(=expAexptBandsod dtexp\050t=expAd dtexptB=expAexptBB=exp\050tBsoexp)]TJ/F8 9.963 Tf 7.749 0 Td[(\050td dtexp\050t=B:Wewillprove.4byndingageneralexpressionforexp)]TJ/F11 9.963 Tf 7.748 0 Td[(Ctd dtexpCtwhereC=CtisacurveintheLiealgebra,g,see.11below.

PAGE 11

1.3.THEMAURER-CARTANEQUATIONS.11Inourderivationof.4from.11wewillmakeuseofanimportantpropertyoftheadjointrepresentationwhichwemightaswellstatenow:Foranyg2G,denethelineartransformationAdg:g!g:X7!gXg)]TJ/F7 6.974 Tf 6.226 0 Td[(1:Ingeometricalterms,thiscanbethoughtofasfollows:ThedierentialofLeftmultiplicationbygcarriesg=TIGintothetangentspace,TgGtoGatthepointg.Rightmultiplicationbyg)]TJ/F7 6.974 Tf 6.227 0 Td[(1carriesthistangentspacebacktogandsothecombinedoperationisalinearmapofgintoitselfwhichwecallAdg.NoticethatAdisarepresentationinthesensethatAdgh=AdgAdh8g;h2G:Inparticular,foranyA2g,wehavetheoneparameterfamilyoflineartrans-formationsAdexptAandd dtAdexptAX=exptAAXexp)]TJ/F11 9.963 Tf 7.749 0 Td[(tA+exptAX)]TJ/F11 9.963 Tf 7.749 0 Td[(Aexp)]TJ/F11 9.963 Tf 7.748 0 Td[(tA=exptA[A;X]exp)]TJ/F11 9.963 Tf 7.749 0 Td[(tAsod dtAdexptA=AdexptAadA:ButadAisalineartransformationactingongandthesolutiontothedier-entialequationd dtMt=MtadA;M=IinthespaceoflineartransformationsofgisexptadA.ThusAdexptA=exptadA.Settingt=1givestheimportantformulaAdexpA=expadA:.6Asanapplication,considerthe)-333(introducedabove.Wehaveexpad\051=Adexp\051=AdexpAexptB=AdexpAAdexptB=expadAexpadtBhencead)-278(=logexpadAexpadtB:.71.3TheMaurer-Cartanequations.IfGisaLiegroupand=tisacurveonGwith=A2G,thenA)]TJ/F7 6.974 Tf 6.227 0 Td[(1isacurvewhichpassesthroughtheidentityatt=0.HenceA)]TJ/F7 6.974 Tf 6.227 0 Td[(10isatangentvectorattheidentity,i.e.anelementofg,theLiealgebraofG.

PAGE 12

12CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULAInthisway,wehavedenedalineardierentialformonGwithvaluesing.IncaseGisasubgroupofthegroupofallinvertiblennmatricessayovertherealnumbers,wecanwritethisformas=A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA:WecanthenthinkoftheAoccurringaboveasacollectionofn2realvaluedfunctionsonGthematrixentriesconsideredasfunctionsonthegroupanddAasthematrixofdierentialsofthesefunctions.Theaboveequationgivingisthenjustmatrixmultiplication.Forsimplicity,wewillworkinthiscase,althoughthemaintheorem,equation1.8below,worksforanyLiegroupandisquitestandard.ThedenitionsofthegroupsweareconsideringamounttoconstraintsonA,andthendierentiatingtheseconstraintsshowthatA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAtakesvaluesing,andgivesadescriptionofg.Itisbesttoexplainthisbyexamples:On:AAy=I;dAAy+AdAy=0orA)]TJ/F7 6.974 Tf 6.226 0 Td[(1dA+)]TJ/F11 9.963 Tf 4.566 -8.069 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAy=0:onconsistsofantisymmetricmatrices.Spn:LetJ:=0I)]TJ/F11 9.963 Tf 7.749 0 Td[(I0andletSpnconsistofallmatricessatisfyingAJAy=J:ThendAJay+AJdAy=0orA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAJ+JA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAy=0:TheequationBJ+JBy=0denestheLiealgebraspn.LetJbeasaboveanddeneGln,CtoconsistofallinvertiblematricessatisfyingAJ=JA:ThendAJ=JdA=0:andsoA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAJ=A)]TJ/F7 6.974 Tf 6.227 0 Td[(1JdA=JA)]TJ/F7 6.974 Tf 6.226 0 Td[(1dA:

PAGE 13

1.3.THEMAURER-CARTANEQUATIONS.13Wereturntogeneralconsiderations:Letustaketheexteriorderivativeofthedeningequation=A)]TJ/F7 6.974 Tf 6.226 0 Td[(1dA.ForthisweneedtocomputedA)]TJ/F7 6.974 Tf 6.227 0 Td[(1:SincedAA)]TJ/F7 6.974 Tf 6.227 0 Td[(1=0wehavedAA)]TJ/F7 6.974 Tf 6.227 0 Td[(1+AdA)]TJ/F7 6.974 Tf 6.226 0 Td[(1=0ordA)]TJ/F7 6.974 Tf 6.227 0 Td[(1=)]TJ/F11 9.963 Tf 7.749 0 Td[(A)]TJ/F7 6.974 Tf 6.227 0 Td[(1dAA)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Thisisthegeneralizationtomatricesoftheformulainelementarycalculusforthederivativeof1=x.Usingthisformulawegetd=dA)]TJ/F7 6.974 Tf 6.227 0 Td[(1dA=)]TJ/F8 9.963 Tf 7.748 0 Td[(A)]TJ/F7 6.974 Tf 6.226 0 Td[(1dAA)]TJ/F7 6.974 Tf 6.227 0 Td[(1^dA=)]TJ/F11 9.963 Tf 7.749 0 Td[(A)]TJ/F7 6.974 Tf 6.226 0 Td[(1dA^A)]TJ/F7 6.974 Tf 6.226 0 Td[(1dAortheMaurer-Cartanequationd+^=0:.8Ifweusecommutatorinsteadofmultiplicationwewouldwritethisasd+1 2[;]=0:.9TheMaurer-Cartanequationisofcentralimportanceingeometryandphysics,farmoreimportantthantheCampbell-Baker-Hausdorformulaitself.Supposewehaveamapg:R2!G,withs;tcoordinatesontheplane.Pullbacktotheplane,sog=g)]TJ/F7 6.974 Tf 6.227 0 Td[(1@g @sds+g)]TJ/F7 6.974 Tf 6.226 0 Td[(1@g @tdtDene=s;t:=g)]TJ/F7 6.974 Tf 6.227 0 Td[(1@g @sand:=s;t=g)]TJ/F7 6.974 Tf 6.227 0 Td[(1@g @tsothatg=ds+dt:Thencollectingthecoecientofds^dtintheMaurerCartanequationgives@ @s)]TJ/F11 9.963 Tf 11.158 6.74 Td[(@ @t+[;]=0:.10ThisistheversionoftheMaurerCartanequationweshalluseinourproofoftheCampbellBakerHausdorformula.Ofcoursethisversioniscompletelyequivalenttothegeneralversion,sinceatwoformisdeterminedbyitsrestrictiontoalltwodimensionalsurfaces.

PAGE 14

14CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULA1.4ProofofCBHfromMaurer-Cartan.LetCtbeacurveintheLiealgebragandletusapply.10togs;t:=exp[sCt]sothats;t=g)]TJ/F7 6.974 Tf 6.226 0 Td[(1@g @s=exp[)]TJ/F11 9.963 Tf 7.748 0 Td[(sCt]exp[sCt]Ct=Cts;t=g)]TJ/F7 6.974 Tf 6.226 0 Td[(1@g @t=exp[)]TJ/F11 9.963 Tf 7.749 0 Td[(sCt]@ @texp[sCt]soby1.10@ @s)]TJ/F11 9.963 Tf 9.962 0 Td[(C0t+[Ct;]=0:Forxedtconsiderthelastequationasthedierentialequationinsd ds=)]TJ/F8 9.963 Tf 7.749 0 Td[(adC+C0;=0whereC:=Ct;C0:=C0t.Ifweexpands;tasaformalpowerseriesinsforxedt:s;t=a1s+a2s2+a3s3+andcomparecoecientsinthedierentialequationweobtaina1=C0,andnan=)]TJ/F8 9.963 Tf 7.749 0 Td[(adCan)]TJ/F7 6.974 Tf 6.226 0 Td[(1ors;t=sC0t+1 2s)]TJ/F8 9.963 Tf 7.749 0 Td[(adCtC0t++1 n!sn)]TJ/F8 9.963 Tf 7.748 0 Td[(adCtn)]TJ/F7 6.974 Tf 6.227 0 Td[(1C0t+:Ifwedenez:=ez)]TJ/F8 9.963 Tf 9.963 0 Td[(1 z=1+1 2!z+1 3!z2+andsets=1intheexpressionwederivedabovefors;twegetexp)]TJ/F11 9.963 Tf 7.749 0 Td[(Ctd dtexpCt=)]TJ/F8 9.963 Tf 7.748 0 Td[(adCtC0t:.11NowtotheproofoftheCampbell-Baker-Hausdorformula.SupposethatAandBarechosensucientlyneartheoriginsothat)-278(=\050t=\050t;A;B:=logexpAexptB

PAGE 15

1.5.THEDIFFERENTIALOFTHEEXPONENTIALANDITSINVERSE.15isdenedforalljtj1.Then,asweremarked,exp)-278(=expAexptBsoexpad)-278(=expadAexptadBandhencead)-278(=logexpadAexptadB:Wehaved dtexp\050t=expAd dtexptB=expAexptBB=exp\050tBsoexp)]TJ/F8 9.963 Tf 7.749 0 Td[(\050td dtexp\050t=Bandtherefore)]TJ/F8 9.963 Tf 7.748 0 Td[(ad\050t)]TJ/F13 6.974 Tf 13.975 4.114 Td[(0t=Bby.11so)]TJ/F8 9.963 Tf 9.409 0 Td[(logexpadAexptadB)]TJ/F13 6.974 Tf 17.849 4.114 Td[(0t=B:Nowforjz)]TJ/F8 9.963 Tf 9.963 0 Td[(1j<1)]TJ/F8 9.963 Tf 9.409 0 Td[(logz=e)]TJ/F7 6.974 Tf 7.587 0 Td[(logz)]TJ/F8 9.963 Tf 9.963 0 Td[(1 )]TJ/F8 9.963 Tf 9.409 0 Td[(logz=z)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F8 9.963 Tf 9.962 0 Td[(1 )]TJ/F8 9.963 Tf 9.409 0 Td[(logz=z)]TJ/F8 9.963 Tf 9.963 0 Td[(1 zlogzsoz)]TJ/F8 9.963 Tf 9.41 0 Td[(logz1wherez:=zlogz z)]TJ/F8 9.963 Tf 9.962 0 Td[(1so)]TJ/F13 6.974 Tf 6.227 4.114 Td[(0t=expadAexptadBB:Thisproves.4andintegratingfrom0to1proves.2.1.5Thedierentialoftheexponentialanditsinverse.Onceagain,equation.11,whichwederivedfromtheMaurer-Cartanequa-tion,isofsignicantimportanceinitsownright,perhapsmorethantheusewemadeofit-toprovetheCampbell-Baker-Hausdortheorem.Wewillrewritethisequationintermsofmorefamiliargeometricoperations,butrstsomepreliminaries:TheexponentialmapexpsendstheLiealgebragintothecorrespondingLiegroup,andisadierentiablemap.If2gwecanconsiderthedierentialofexpatthepoint:dexp:g=Tg!TGexp

PAGE 16

16CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULAwherewehaveidentiedgwithitstangentspaceatwhichispossiblesincegisavectorspace.Inotherwords,dexpmapsthetangentspacetogatthepointintothetangentspacetoGatthepointexp.At=0wehavedexp0=idandhence,bytheimplicitfunctiontheorem,dexpisinvertibleforsu-cientlysmall.NowtheMaurer-Cartanform,evaluatedatthepointexpsendsTGexpbacktog:exp:TGexp!g:Henceexpdexp:g!gandisinvertibleforsucientlysmall.Weclaimthatad)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(expdexp=id.12whereisasdenedabovein.3.Indeed,weclaimthat.12isanimmediateconsequenceof.11.Recallthedenition.3ofthefunctionasz=1=)]TJ/F11 9.963 Tf 7.749 0 Td[(z.Multiplybothsidesof.11byadCttoobtainadCtexp)]TJ/F11 9.963 Tf 7.749 0 Td[(Ctd dtexpCt=C0t:.13ChoosethecurveCsothat=Cand=C0.Thenthechainrulesaysthatd dtexpCtjt=0=dexp:Thusexp)]TJ/F11 9.963 Tf 7.749 0 Td[(Ctd dtexpCtjt=0=expdexp;theresultofapplyingtheMaurer-Cartanformatthepointexptotheimageofunderthedierentialofexponentialmapat2g.Then.13att=0translatesinto.12.QED1.6Theaveragingmethod.Inthissectionwewillgiveanotherimportantapplicationof1.10:Forxed2g,thedierentialoftheexponentialmapisalinearmapfromg=TgtoTexpG.ThedierentialoflefttranslationbyexpcarriesTexpGbacktoTeG=g.Letusdenotethiscompositebyexp)]TJ/F7 6.974 Tf 6.227 0 Td[(1dexp.Soexpdexp=dexp)]TJ/F7 6.974 Tf 6.227 0 Td[(1dexp:g!gisalinearmap.Weclaimthatforany2gexp)]TJ/F7 6.974 Tf 6.227 0 Td[(1dexp=Z10Adexp)]TJ/F10 6.974 Tf 6.226 0 Td[(sds:.14

PAGE 17

1.6.THEAVERAGINGMETHOD.17Wewillprovethisbyapplying.10togs;t=expt+s:Indeed,s;t:=gs;t)]TJ/F7 6.974 Tf 6.227 0 Td[(1@g @t=+sso@ @sand;t:Thelefthandsideof.14is;1wheres;t:=gs;t)]TJ/F7 6.974 Tf 6.227 0 Td[(1@g @ssowemayuse.10togetanordinarydierentialequationfor;t.Deningt:=;t;.10becomesd dt=+[;]:.15Forany2g,d dtAdexp)]TJ/F10 6.974 Tf 6.227 0 Td[(t=Adexp)]TJ/F10 6.974 Tf 6.227 0 Td[(t[;]=[Adexp)]TJ/F10 6.974 Tf 6.227 0 Td[(t;]:Soforconstant2g,Adexp)]TJ/F10 6.974 Tf 6.227 0 Td[(tisasolutionofthehomogeneousequationcorrespondingto.15.So,byLagrange'smethodofvariationofconstants,welookforasolutionof1.15oftheformt=Adexp)]TJ/F10 6.974 Tf 6.227 0 Td[(ttand.15becomes0t=Adexptort=Adexp)]TJ/F10 6.974 Tf 6.227 0 Td[(tZt0Adexpsdsisthesolutionof.15with=0.Settings=1gives=Adexp)]TJ/F10 6.974 Tf 6.227 0 Td[(Z10Adexpsdsandreplacingsby1)]TJ/F11 9.963 Tf 9.962 0 Td[(sintheintegralgives.14.

PAGE 18

18CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULA1.7TheEulerMacLaurinFormula.Wepausetoremindthereaderofadierentrolethatthefunctionplaysinmathematics.Wehaveseenin.12thatentersintotheinverseoftheexponentialmap.Inasense,thisformulaistakingintoaccountthenon-commutativityofthegroupmultiplication,soishelpingtorelatethenon-commutativetothecommutative.Butmuchearlierinmathematicalhistory,wasintroducedtorelatethediscretetothecontinuous:LetDdenotethedierentiationoperatorinonevariable.ThenifwethinkofDastheonedimensionalvectoreld@=@hitgeneratestheoneparametergroupexphDwhichconsistsoftranslationbyh.Inparticular,takingh=1wehave)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(eDfx=fx+1:Thisequationisequallyvalidinapurelyalgebraicsense,takingftobeapolynomialandeD=1+D+1 2D2+1 3!D3+:Thisseriesisinnite.Butifpisapolynomialofdegreed,thenDkp=0fork>Dsowhenappliedtoanypolynomial,theabovesumisreallynite.SinceDkeah=akeahitfollowsthatifFisanyformalpowerseriesinonevariable,wehaveFDeah=Faeah.16intheringofpowerseriesintwovariables.Ofcourse,undersuitableconvergenceconditionsthisisanequalityoffunctionsofh.Forexample,thefunctionz=z=)]TJ/F11 9.963 Tf 10.307 0 Td[(e)]TJ/F10 6.974 Tf 6.226 0 Td[(zconvergesforjzj<2since2iaretheclosestzerosofthedenominatorotherthan0totheorigin.Henced dhezh z=ezh1 1)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F10 6.974 Tf 6.227 0 Td[(z.17holdsfor0
PAGE 19

1.8.THEUNIVERSALENVELOPINGALGEBRA.19thefor0
PAGE 20

20CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULA2.IfAisanyassociativealgebrawithunitand:L!AisanyLiealgebrahomomorphismthenthereexistsauniquehomomorphismofassociativealgebrassuchthat=:ItisclearthatifULexists,itisuniqueuptoauniqueisomorphism.SowemaythentalkoftheuniversalalgebraofL.Wewillcallittheuniversalenvelopingalgebraandsometimesputinparenthesis,i.e.writeUL.IncaseL=gistheLiealgebraofleftinvariantvectoreldsonagroupG,wemaythinkofLasconsistingofleftinvariantrstorderhomogeneousdierentialoperatorsonG.ThenwemaytakeULtoconsistofallleftinvariantdierentialoperatorsonG.InthiscasetheconstructionofULisintuitiveandobvious.TheringofdierentialoperatorsDonanymanifoldislteredbydegree:Dnconsistingofthosedierentialoperatorswithtotaldegreeatmostn.Thequotient,Dn=Dn)]TJ/F7 6.974 Tf 6.226 0 Td[(1consistsofthosehomogeneousdierentialoperatorsofdegreen,i.e.homogeneouspolynomialsinthevectoreldswithfunctioncoecients.Forthecaseofleftinvariantdierentialoperatorsonagroup,thesevectoreldsmaybetakentobeleftinvariant,andthefunctioncoecientstobeconstant.Inotherwords,ULn=ULn)]TJ/F7 6.974 Tf 6.226 0 Td[(1consistsofallsymmetricpolynomialexpressions,homogeneousofdegreeninL.ThisisthecontentofthePoincare-Birkho-Witttheorem.Inthealgebraiccasewehavetodosomeworktogetallofthis.WerstmustconstructUL.1.8.1Tensorproductofvectorspaces.LetE1;:::;Embevectorspacesandf;Famultilinearmapf:E1Em!F.Similarlyg;G.If`isalinearmap`:F!G,andg=`fthenwesaythat`isamorphismoff;Ftog;G.Inthiswaywemakethesetofallf;Fintoacategory.Wantauniversalobjectinthiscategory;thatis,anobjectwithauniquemorphismintoeveryotherobject.Sowantapairt;TwhereTisavectorspace,t:E1Em!Tisamultilinearmap,andforeveryf;Fthereisauniquelinearmap`f:T!Fwithf=`ft.Uniqueness.Bytheuniversalpropertyt=`0tt0;t0=`0ttsot=`0t`t0t,butalsot=tid.So`0t`t0=id.Similarlytheotherway.Thust;T,ifitexists,isuniqueuptoauniquemorphism.Thisisastandardargumentvalidinanycategoryprovingtheuniquenessofinitialelements".Existence.LetMbethefreevectorspaceonthesymbolsx1;:::;xm;xi2Ei.LetNbethesubspacegeneratedbyallthex1;:::;xi+x0i;:::;xm)]TJ/F8 9.963 Tf 9.962 0 Td[(x1;:::;xi;:::;xm)]TJ/F8 9.963 Tf 9.962 0 Td[(x1;:::;x0i;:::;xmandallthex1;:::;;axi;:::;xm)]TJ/F11 9.963 Tf 9.963 0 Td[(ax1;:::;xi;:::;xm

PAGE 21

1.8.THEUNIVERSALENVELOPINGALGEBRA.21foralli=1;:::;m;xi;x0i2Ei;a2k.LetT=M=Nandtx1;:::;xm=x1;:::;xm=N:Thisisuniversalbyitsveryconstruction.QEDWeintroducethenotationT=TE1Em=:E1Em:TheuniversalityimpliesanisomorphismE1EmEm+1Em+n=E1Em+n:1.8.2Thetensorproductoftwoalgebras.IfAandBarealgebras,theyaretheyarevectorspaces,sowecanformtheirtensorproductasvectorspaces.WedeneaproductstructureonABbydeninga1b1a2b2:=a1a2b1b2:ItiseasytocheckthatthisextendstogiveanalgebrastructureonAB.IncaseAandBareassociativealgebrassoisAB,andifinadditionbothAandBhaveunitelements,then1A1BisaunitelementforAB.Wewillfrequentlydropthesubscriptsontheunitelements,foritiseasytoseefromthepositionrelativetothetensorproductsignthealgebratowhichtheunitbelongs.Inotherwords,wewillwritetheunitforABas11.WehaveanisomorphismofAintoABgivenbya7!a1whenbothAandBareassociativealgebraswithunits.SimilarlyforB.Noticethata1b=ab=1ba1:Inparticular,anelementoftheforma1commuteswithanelementoftheform1b.1.8.3Thetensoralgebraofavectorspace.LetVbeavectorspace.ThetensoralgebraofavectorspaceisthesolutionoftheuniversalproblemformapsofVintoanassociativealgebra:itconsistsofanalgebraTVandamap:V!TVsuchthatislinear,andforanylinearmap:V!AwhereAisanassociativealgebrathereexistsauniquealgebrahomomorphism:TV!Asuchthat=.WesetTnV:=VVn)]TJ/F8 9.963 Tf 9.963 0 Td[(factors:WedenethemultiplicationtobetheisomorphismTnVTmV!Tn+mV

PAGE 22

22CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULAobtainedbydroppingtheparentheses,"i.e.theisomorphismgivenattheendofthelastsubsection.ThenTV:=MTnVwithT0Vthegroundeldisasolutiontothisuniversalproblem,andhencetheuniquesolution.1.8.4Constructionoftheuniversalenvelopingalgebra.IfwetakeV=LtobeaLiealgebra,andletIbethetwosidedidealinTLgeneratedtheelements[x;y])]TJ/F11 9.963 Tf 9.963 0 Td[(xy+yxthenUL:=TL=IisauniversalalgebraforL.Indeed,anyhomomorphismofLintoanassocia-tivealgebraAextendstoauniquealgebrahomomorphism:TL!AwhichmustvanishonIifitistobeaLiealgebrahomomorphism.1.8.5ExtensionofaLiealgebrahomomorphismtoitsuni-versalenvelopingalgebra.Ifh:L!MisaLiealgebrahomomorphism,thenthecompositionMh:L!UMinducesahomomorphismUL!UMandthisassignmentsendingLiealgebrahomomorphismsintoassociativealgebrahomomorphismsisfunctorial.1.8.6Universalenvelopingalgebraofadirectsum.Supposethat:L=L1L2,withi:Li!ULi,and:L!ULthecanonicalhomomorphisms.Denef:L!UL1UL2;fx1+x2=1x11+12x2:Thisisahomomorphismbecausex1andx2commute.Itthusextendstoahomomorphism:UL!UL1UL2:Also,x17!x1isaLiealgebrahomomorphismofL1!ULwhichthusextendstoauniquealgebrahomomorphism1:UL1!UL

PAGE 23

1.8.THEUNIVERSALENVELOPINGALGEBRA.23andsimilarly2:UL2!UL.Wehave1x12x2=2x21x1;x12L1;x22L2since[x1;x2]=0.AstheixigenerateULi,theaboveequationholdswithxireplacedbyarbitraryelementsui2ULi;i=1;2.Sowehaveahomomorphism:UL1UL2!UL;u1u2:=1u12u2:Wehavex1+x2=x11+x2=x1+x2so=id,onLandhenceonULandx11+1x2=x11+1x2so=idonL11+1L2andhenceonUL1UL2.ThusUL1L2=UL1UL2:1.8.7Bialgebrastructure.ConsiderthemapL!ULUL:x7!x1+1x:Thenx1+1xy1+1y=xy1+xy+yx++1xy;andmultiplyinginthereverseorderandsubtractinggives[x1+1x;y1+1y]=[x;y]1+1[x;y]:Thusthemapx7!x1+1xdeterminesanalgebrahomomorphism:UL!ULUL:Dene":UL!k;"=1;"x=0;x2Landextendasanalgebrahomomorphism.Then"idx1+1x=1x;x2L:WeidentifykLwithLandsocanwritetheaboveequationas"idx1+1x=x;x2L:

PAGE 24

24CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULAThealgebrahomomorphism"id:UL!UListheidentityon1andonLandhenceistheidentity.Similarlyid"=id:AvectorspaceCwithamap:C!CC,calledacomultiplicationandamap":D!kcalledaco-unitsatisfying"id=idandid"=idiscalledaco-algebra.IfCisanalgebraandbothand"arealgebrahomo-morphisms,wesaythatCisabi-algebrasometimesshortenedtobigebra".SowehaveprovedthatUL;;"isabialgebra.Also[id]x=x11+1x1+11x=[id]xforx2LandhenceforallelementsofUL.Hencethecomultiplicationisiscoassociative.Itisalsoco-commutative.1.9ThePoincare-Birkho-WittTheorem.SupposethatVisavectorspacemadeintoaLiealgebrabydeclaringthatallbracketsarezero.ThentheidealIinTVdeningUVisgeneratedbyxy)]TJ/F11 9.963 Tf 9.367 0 Td[(yx,andthequotientTV=Iisjustthesymmetricalgebra,SV.SotheuniversalenvelopingalgebraofthetrivialLiealgebraisthesymmetricalgebra.ForanyLiealgebraLdeneUnLtobethesubspaceofULgeneratedbyproductsofatmostnelementsofL,i.e.byallproductsx1xm;mn:Forexample,,U0L=k;thegroundeldandU1L=kL:WehaveU0LU1LUnLUn+1LandUmLUnLUm+nL:

PAGE 25

1.9.THEPOINCARE-BIRKHOFF-WITTTHEOREM.25WedenegrnUL:=UnL=Un)]TJ/F7 6.974 Tf 6.226 0 Td[(1LandgrUL:=MgrnULwiththemultiplicationgrmULgrnUL!grm+nULinducedbythemultiplicationonUL.Ifa2UnLwelet a2grnULdenoteitsimagebytheprojectionUnL!UnL=Un)]TJ/F7 6.974 Tf 6.227 0 Td[(1L=grnUL.WemaywriteaasasumofproductsofatmostnelementsofL:a=Xmncx;1x;m:Then acanbewrittenasthecorrespondinghomogeneoussum a=Xm=nc x;1 x;m:Inotherwords,asanalgebra,grULisgeneratedbytheelements x;x2L.Butallsuchelementscommute.Indeed,forx;y2L,xy)]TJ/F11 9.963 Tf 9.963 0 Td[(yx=[x;y]:bythedeningpropertyoftheuniversalenvelopingalgebra.TherighthandsideofthisequationbelongstoU1L.Hence x y)]TJET1 0 0 1 29.211 8.667 cmq[]0 d0 J0.398 w0 0.199 m17.035 0.199 lSQ1 0 0 1 0 -8.667 cmBT/F11 9.963 Tf 0 0 Td[(y x=0ingr2UL.ThisprovesthatgrULiscommutative.Hence,bytheuniversalpropertyofthesymmetricalgebra,thereexistsauniquealgebrahomomorphismw:SL!grULextendingthelinearmapL!grUL;x7! x:Sincethe xgenerategrULasanalgebra,weknowthatthismapissurjective.ThePoincare-Birkho-Witttheoremassertsthatw:SL!grULisanisomorphism:.19Supposethatwechooseabasisxi;i2IofLwhereIisatotallyorderedset.Since xi xj= xj xiwecanrearrangeanyproductof xisoastobeinincreasingorder.ThisshowsthattheelementsxM:=xi1xim;M:=i1;:::;imi1imspanULasavectorspace.Weclaimthat.19isequivalentto

PAGE 26

26CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULATheorem1Poincare-Birkho-Witt.TheelementsxMformabasisofUL.Proofthat.19isequivalenttothestatementofthetheorem.ForanyexpressionxMasabove,wedenoteitslengthby`M=m.Theelements xMaretheimagesunderwofthemonomialbasisinSmL.Asweknowthatwissurjective,equation.19isequivalenttotheassertionthatwisinjective.Thisamountstothenon-existenceofarelationoftheformX`M=ncMxM=X`Mjwesetxizj=xjzi+[xi;xj]z;=zj;i+Xckijzk

PAGE 27

1.9.THEPOINCARE-BIRKHOFF-WITTTHEOREM.27where[xi;xj]=XckijxkistheexpressionfortheLiebracketofxiwithxjintermsofourbasis.TheseckijareknownasthestructureconstantsoftheLiealgebra,Lintermsofthegivenbasis.Noticethattherstofthesetwocasesisconsistentwithandforcedonusby1.20whilethesecondisforcedonusby.21.WenowhavedenedxizMforalliandallMwith`M1,andwehavedonesoinsuchawaythat.20holds,and.21holdswhereitmakessensei.e.for`M=0.SosupposethatwehavedenedxjzNforalljif`N<`Mandforalljj:ThismakessensesincexizNisalreadydenedasalinearcombinationofzL'swith`L`N+1=`Mandbecause[xi;xj]canbewrittenasalinearcombinationofthexkasabove.FurthermoreholdswithjandNreplacedbyM.Furthermore,.20holdsbyconstruction.Wemustcheck.21.Bylinearity,thismeansthatwemustshowthatxixjzN)]TJ/F11 9.963 Tf 9.962 0 Td[(xjxizN=[xi;xj]zN:Ifi=jbothsidesarezero.Also,sincebothsidesareanti-symmetriciniandj,wemayassumethati>j.IfjNandi>jthenthisequationholdsbydenition.Soweneedonlydealwiththecasewherej6NwhichmeansthatN=kPwithkPandi>j>k.Sowehave,bydenition,xjzN=xjzkP=xjxkzP=xkxjzP+[xj;xk]zP:NowifjPthenxjzP=zjPandk
PAGE 28

28CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULAeachequationcancelandwegetxixj)]TJ/F11 9.963 Tf 9.963 0 Td[(xjxizN=xkxixj)]TJ/F11 9.963 Tf 9.962 0 Td[(xjxizP+[xi;[xj;xk]])]TJ/F8 9.963 Tf 9.963 0 Td[([xj;[xi;xk]zP=xk[xi;xj]zP+[xi;[xj;xk]])]TJ/F8 9.963 Tf 9.963 0 Td[([xj;[xi;xk]zP=[xi;xj]xkzP+[xk;[xi;xj]]+[xi;[xj;xk]])]TJ/F8 9.963 Tf 9.963 0 Td[([xj;[xi;xk]zP=[xi;xj]zN:Inpassingfromthesecondlinetothethirdweused1.21appliedtozPbyinductionandfromthethirdtothelastweusedtheantisymmetryofthebracketandJacobi'sequation.QEDProofofthePBWtheorem.WehavemadeVintoanLandhenceintoaULmodule.Byconstruction,wehave,inductively,xMz;=zM:ButifXcMxM=0then0=XcMzM=XcMxMz;contradictingthefactthethezMareindependent.QEDInparticular,themap:L!ULisaninjection,andsowemayidentifyLasasubspaceofUL.1.10Primitives.Anelementxofabialgebraiscalledprimitiveifx=x1+1x:SotheelementsofLareprimitivesinUL.Weclaimthatthesearetheonlyprimitives.FirstprovethisforthecaseLisabeliansoUL=SL.ThenwemaythinkofSLSLaspolynomialsintwicethenumberofvariablesasthoseofSLandfu;v=fu+v:Theconditionofbeingprimitivesaysthatfu+v=fu+fv:Takinghomogeneouscomponents,thesameequalityholdsforeachhomogeneouscomponent.Butiffishomogeneousofdegreen,takingu=vgives2nfu=2fu

PAGE 29

1.11.FREELIEALGEBRAS29sof=0unlessn=1.Takinggr,thisshowsthatforanyLiealgebratheprimitivesarecontainedinU1L.Butc+x=c1+x1+1xsotheconditiononprimitivityrequiresc=2corc=0.QED1.11FreeLiealgebras1.11.1MagmasandfreemagmasonasetAsetMwithamap:MM!M;x;y7!xyiscalledamagma.Thusamagmaisasetwithabinaryoperationwithnoaxiomsatallimposed.LetXbeanyset.DeneXninductivelybyX1:=XandXn=ap+q=nXpXqforn2.ThusX2consistsofallexpressionsabwhereaandbareelementsofX.Wewriteabinsteadofa;b.AnelementofX3iseitheranexpressionoftheformabcoranexpressionoftheformabc.AnelementofX4hasoneoutofveforms:abcd;abcd;abcd;abcdorabcd.SetMX:=1an=1Xn:Anelementw2MXiscalledanon-associativeword,anditslength`wistheuniquensuchthatw2Xn.Wehaveamultiplication"mapMXMXgivenbytheinclusionXpXq,!Xp+q:ThusthemultiplicationonMXisconcatenationofnon-associativewords.IfNisanymagma,andf:X!Nisanymap,wedeneF:MX!NbyF=fonX1,byF:X2!N;Fab=fafbandinductivelyF:XpXq!N;Fuv=FuFv:AnyelementofXnhasauniqueexpressionasuvwhereu2Xpandv2Xqforauniquep;qwithp+q=n,sothisinductivedenitionisvalid.ItisclearthatFisamagnahomomorphismandisuniquelydeterminedbytheoriginalmapf.ThusMXisthefreemagmaonX"ortheuniversal

PAGE 30

30CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULAmagmaonX"inthesensethatitisthesolutiontotheuniversalproblemassociatedtoamapfromXtoanymagma.LetAXbethevectorspaceofniteformallinearcombinationsofelementsofMX.SoanelementofAXisanitesumPcmmwithm2MXandcminthegroundeld.ThemultiplicationinMXextendsbybi-linearitytomakeAXintoanalgebra.IfwearegivenamapX!BwhereBisanyalgebra,wegetauniquemagnahomomorphismMX!BextendingthismapwherewethinkofBasamagmaandthenauniquealgebramapAX!Bextendingthismapbylinearity.NoticethatthealgebraAXisgradedsinceeveryelementofMXhasalengthandthemultiplicationonMXisgraded.HenceAXisthefreealgebraonXinthesensethatitsolvestheuniversalproblemassociatedwithmapsofXtoalgebras.1.11.2TheFreeLieAlgebraLX.InAXletIbethetwo-sidedidealgeneratedbyallelementsoftheformaa;a2AXandabc+bca+cab;a;b;c2AX.WesetLX:=AX=IandcallLXthefreeLiealgebraonX.AnymapfromXtoaLiealgebraLextendstoauniquealgebrahomomorphismfromLXtoL.WeclaimthattheidealIdeningLXisgraded.Thismeansthatifa=PanisadecompositionofanelementofIintoitshomogeneouscomponents,theneachoftheanalsobelongtoI.Toprovethis,letJIdenotethesetofalla=PanwiththepropertythatallthehomogeneouscomponentsanbelongtoI.ClearlyJisatwosidedideal.WemustshowthatIJ.Forthisitisenoughtoprovethecorrespondingfactforthegeneratingelements.Clearlyifa=Xap;b=Xbq;c=Xcrthenabc+bca+cab=Xp;q;rapbqcr+bqcrap+crapbq:Butalsoifx=Pxmthenx2=Xx2n+Xm
PAGE 31

1.11.FREELIEALGEBRAS311.11.3ThefreeassociativealgebraAssX.LetVXbethevectorspaceofallniteformallinearcombinationsofelementsofX.DeneAssX=TVX;thetensoralgebraofVX.AnymapofXintoanassociativealgebraAextendstoauniquelinearmapfromVXtoAandhencetoauniquealgebrahomomorphismfromAssXtoA.SoAssXisthefreeassociativealgebraonX.WehavethemapsX!LXand:LX!ULXandhencetheircom-positionmapsXtotheassociativealgebraULXandsoextendstoauniquehomomorphism:AssX!ULX:Ontheotherhand,thecommutatorbracketgivesaLiealgebrastructuretoAssXandthemapX!AssXthusgiverisetoaLiealgebrahomomorphismLX!AssXwhichdeterminesanassociativealgebrahomomorphism:ULX!AssX:bothcompositionsandaretheidentityonXandhence,byunique-ness,theidentityeverywhere.WeobtaintheimportantresultthatULXandAssXarecanonicallyisomorphic:ULX=AssX:.23NowthePoincare-Birkho-Witttheoremguaranteesthatthemap:LX!ULXisinjective.Soundertheaboveisomorphism,themapLX!AssXisinjective.Ontheotherhand,byconstruction,themapX!VXinducesasurjectiveLiealgebrahomomorphismfromLXintotheLiesubalgebraofAssXgeneratedbyX.Soweseethattheundertheisomorphism.23LXULXismappedisomorphicallyontotheLiesubalgebraofAssXgeneratedbyX.NowthemapX!AssXAssX;x7!x1+1xextendstoauniquealgebrahomomorphism:AssX!AssXAssX:Undertheidentication.23thisisnoneotherthanthemap:ULX!ULXULXandhenceweconcludethatLXisthesetofprimitiveelementsofAssX:LX=fw2AssXjw=w1+1w:g.24undertheidentication.23.

PAGE 32

32CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULA1.12AlgebraicproofofCBHandexplicitfor-mulas.Werecallourconstructsofthepastfewsections:Xdenotesaset,LXthefreeLiealgebraonXandAssXthefreeassociativealgebraonXsothatAssXmaybeidentiedwiththeuniversalenvelopingalgebraofLX.SinceAssXmaybeidentiedwiththenon-commutativepolynomialsindexedbyX,wemayconsideritscompletion,FX,thealgebraofformalpowerseriesindexedbyX.SincethefreeLiealgebraLXisgradedwemayalsoconsideritscompletionwhichweshalldenotebyLX.FinallyletmdenotetheidealinFXgeneratedbyX.Themapsexp:m!1+m;log:1+m!marewelldenedbytheirformalpowerseriesandaremutualinverses.Thereisnoconvergenceissuesinceeverythingiswithintherealmofformalpowerseries.Furthermoreexpisabijectionofthesetof2msatisfying=1+1tothesetofall21+msatisfying=.1.12.1AbstractversionofCBHanditsalgebraicproof.Inparticular,sincethesetf21+mj=gformsagroup,weconcludethatforanyA;B2LXthereexistsaC2LXsuchthatexpC=expAexpB:ThisistheabstractversionoftheCampbell-Baker-Hausdorformula.Itde-pendsbasicallyontwoalgebraicfacts:ThattheuniversalenvelopingalgebraofthefreeLiealgebraisthefreeassociativealgebra,andthatthesetofprimitiveelementsintheuniversalenvelopingalgebrathosesatisfying=1+1ispreciselytheoriginalLiealgebra.1.12.2ExplicitformulaforCBH.Denethemap:mAssX!LX;x1:::xn:=[x1;[x2;:::;[xn)]TJ/F7 6.974 Tf 6.226 0 Td[(1;xn]]=adx1adxn)]TJ/F7 6.974 Tf 6.226 0 Td[(1xn;andlet:AssX!EndLXbethealgebrahomomorphismextendingtheLiealgebrahomomorphismad:LX!EndLX.Weclaimthatuv=uv;8u2AssX;v2mAssX:.25Proof.Itisenoughtoprovethisformulawhenuisamonomial,u=x1xn.Wedothisbyinductiononn.Forn=0theassertionisobviousandforn=1

PAGE 33

1.12.ALGEBRAICPROOFOFCBHANDEXPLICITFORMULAS.33itfollowsfromthedenitionof.Supposen>1.Thenx1xnv=x1x2xnv=x1x2:::xnv=x1xnv:QEDLetLnXdenotethen)]TJ/F8 9.963 Tf 7.748 0 Td[(thgradedcomponentofLX.SoL1XconsistsoflinearcombinationsofelementsofX,L2XisspannedbyallbracketsofpairsofelementsofX,andingeneralLnXisspannedbyelementsoftheform[u;v];u2LpX;v2LqX;p+q=n:Weclaimthatu=nu8u2LnX:.26Forn=1thisisimmediatefromthedenitionof.Sobyinductionitisenoughtoverifythisonelementsoftheform[u;v]asabove.Wehave[u;v]=uv)]TJ/F11 9.963 Tf 9.963 0 Td[(vu=uv)]TJ/F8 9.963 Tf 9.962 0 Td[(vu=quv)]TJ/F11 9.963 Tf 9.962 0 Td[(pvubyinduction=q[u;v])]TJ/F11 9.963 Tf 9.963 0 Td[(p[v;u]sincew=adwforw2LX=p+q[u;v]QED.Wecannowwritedownanexplicitformulaforthen)]TJ/F8 9.963 Tf 7.749 0 Td[(thtermintheCampbell-Baker-Hausdorexpansion.ConsiderthecasewhereXconsistsoftwoelementsX=fx;yg;x6=y.Letuswritez=logexpxexpyz2LX;z=1X1znx;y:Wewantanexplicitexpressionforznx;y.Weknowthatzn=1 nzn

PAGE 34

34CHAPTER1.THECAMPBELLBAKERHAUSDORFFFORMULAandznisasumofnon-commutativemonomialsofdegreeninxandy.Nowexpxexpy=1Xp=0xp p!!1Xq=0yq q!!=1+Xp+q1xpyq p!q!soz=logexpxexpy=1Xm=1)]TJ/F8 9.963 Tf 7.749 0 Td[(1m+1 m0@Xp+q1xpyq p!q!1Am=Xpi+qi1)]TJ/F8 9.963 Tf 7.749 0 Td[(1m+1 mxp1yq1xp2yq2xpmyqm p1!q1!pm!qm!:Wewanttoapply1 ntothetermsinthislastexpressionwhichareoftotaldegreensoastoobtainzn.Soletusexaminewhathappenswhenweapplytoanexpressionoccurringinthenumerator:Ifqm2weget0sincewewillhaveadyy=0.Similarlywewillget0ifqm=0;pm2.Hencetheonlytermswhichsurvivearethosewithqm=1orqm=0;pm=1.Accordinglywedecomposeznintothesetwotypes:zn=1 nXp+q=nz0p;q+z00p;q;.27wherez0p;q=X)]TJ/F8 9.963 Tf 7.749 0 Td[(1m+1 madxp1adyq1adxpmy p1!q1!pm!summedoverallp1++pm=p;q1++qm)]TJ/F7 6.974 Tf 6.226 0 Td[(1=q)]TJ/F8 9.963 Tf 9.963 0 Td[(1;qi+pi1;pm1andz00p;q=X)]TJ/F8 9.963 Tf 7.749 0 Td[(1m+1 madxp1adyq1adyqm)]TJ/F6 4.981 Tf 5.396 0 Td[(1x p1!q1!qm)]TJ/F7 6.974 Tf 6.227 0 Td[(1!summedoverp1++pm)]TJ/F7 6.974 Tf 6.227 0 Td[(1=p)]TJ/F8 9.963 Tf 9.963 0 Td[(1;q1++qm)]TJ/F7 6.974 Tf 6.227 0 Td[(1=q;pi+qi1i=1;:::;m)]TJ/F8 9.963 Tf 9.963 0 Td[(1qm)]TJ/F7 6.974 Tf 6.226 0 Td[(11:Therstfourtermsare:z1x;y=x+yz2x;y=1 2[x;y]z3x;y=1 12[x;[x;y]]+1 12[y;[y;x]]z4x;y=1 24[x;[y;[x;y]]]:

PAGE 35

Chapter2slanditsRepresentations.InthischapterandinmostofthesucceedingchaptersallLiealgebrasandvectorspacesareoverthecomplexnumbers.2.1LowdimensionalLiealgebras.AnyonedimensionalLiealgebramustbecommutative,since[X;X]=0inanyLiealgebra.IfgisatwodimensionalLiealgebra,saywithbasisX;Ythen[aX+bY;cX+dY]=ad)]TJ/F11 9.963 Tf 10.16 0 Td[(bc[X;Y],sothattherearetwopossibilities:[X;Y]=0inwhichcasegiscommutative,or[X;Y]6=0,callitB,andtheLiebracketofanytwoelementsofgisamultipleofB.SoifCisnotamultipleofB,wehave[C;B]=cBforsomec6=0,andsettingA=c)]TJ/F7 6.974 Tf 6.227 0 Td[(1CwegetabasisA;Bofgwiththebracketrelations[A;B]=B:ThisisaninterestingLiealgebra;itistheLiealgebraofthegroupofallanetransformationsoftheline,i.e.alltransformationsoftheformx7!ax+b;a6=0:Forthisreasonitissometimescalledtheax+bgroup".Sinceab01x1=ax+b1wecanrealizethegroupofanetransformationsofthelineasagroupoftwobytwomatrices.Writinga=exptA;b=tB35

PAGE 36

36CHAPTER2.SLANDITSREPRESENTATIONS.sothata001=exptA000;1b01=expt0B00weseethatouralgebragwithbasisA;Band[A;B]=BisindeedtheLiealgebraoftheax+bgroup.Inasimilarway,wecouldlistallpossiblethreedimensionalLiealgebras,byrstclassifyingthemaccordingtodim[g;g]andthenanalyzingthepossibilitiesforeachvalueofthisdimension.Ratherthangoingthroughallthedetails,welistthemostimportantexamplesofeachtype.Ifdim[g;g]=0thealgebraiscommutativesothereisonlyonepossibility.Averyimportantexampleariseswhendim[g;g]=1andthatistheHeisen-bergalgebra,withbasisP;Q;Zandbracketrelations[P;Q]=Z;[Z;P]=[Z;Q]=0:UptoconstantssuchasPlanck'sconstantandithesearethefamousHeisen-bergcommutationrelations.Indeed,wecanrealizethisalgebraasanalgebraofoperatorsonfunctionsofonevariablex:LetP=D=dierentiation,letQconsistofmultiplicationbyx.Since,foranyfunctionf=fxwehaveDxf=f+xf0weseethat[P;Q]=id,sosettingZ=id,weobtaintheHeisenbergalgebra.Asanexamplewithdim[g;g]=2wehavethecomplexicationoftheLiealgebraofthegroupofEuclideanmotionsintheplane.Herewecanndabasish;x;yofgwithbracketsgivenby[h;x]=y;[h;y]=)]TJ/F11 9.963 Tf 7.749 0 Td[(x;[x;y]=0:Moregenerallywecouldstartwithacommutativetwodimensionalalgebraandadjoinanelementhwithadhactingasanarbitrarylineartransformation,Aofourtwodimensionalspace.Theitemofstudyofthischapteristhealgebraslofalltwobytwomatricesoftracezero,where[g;g]=g.2.2slanditsirreduciblerepresentations.Indeedslisspannedbythematrices:h=100)]TJ/F8 9.963 Tf 7.748 0 Td[(1;e=0100;f=0010:Theysatisfy[h;e]=2e;[h;f]=)]TJ/F8 9.963 Tf 7.749 0 Td[(2f;[e;f]=h:Thuseveryelementofslcanbeexpressedasasumofbracketsofelementsofsl,inotherwords[sl;sl]=sl:

PAGE 37

2.2.SLANDITSIRREDUCIBLEREPRESENTATIONS.37Thebracketrelationsabovearealsosatisedbythematrices2h:=0@20000000)]TJ/F8 9.963 Tf 7.748 0 Td[(21A;2e:=0@0200010001A;2f:=0@0001000201A;thematrices3h:=0BB@3000010000)]TJ/F8 9.963 Tf 7.749 0 Td[(10000)]TJ/F8 9.963 Tf 7.749 0 Td[(31CCA;3e:=0BB@03000020000100001CCA;3f:=0BB@00001000020000301CCA;and,moregenerally,then+1n+1matricesgivenbynh:=0BBBBB@n000n)]TJ/F8 9.963 Tf 9.963 0 Td[(20...............00)]TJ/F11 9.963 Tf 30.995 0 Td[(n+2000)]TJ/F11 9.963 Tf 71.844 0 Td[(n1CCCCCA;ne=0BBBBB@0n000n)]TJ/F8 9.963 Tf 9.963 0 Td[(10...............0010001CCCCCA;nf:=0BBB@000100...............00n01CCCA:Theserepresentationsofslareallirreducible,asisseenbysuccessivelyapplyingnetoanynon-zerovectoruntilavectorwithnon-zeroelementintherstpositionandallotherentrieszeroisobtained.Thenkeepapplyingnftolluptheentirespace.Theseareallthenitedimensionalirreduciblerepresentationsofslascanbeseenasfollows:InUslwehave[h;fk]=)]TJ/F8 9.963 Tf 7.748 0 Td[(2kfk;[h;ek]=2kek.1[e;fk]=)]TJ/F11 9.963 Tf 7.748 0 Td[(kk)]TJ/F8 9.963 Tf 9.963 0 Td[(1fk)]TJ/F7 6.974 Tf 6.227 0 Td[(1+kfk)]TJ/F7 6.974 Tf 6.227 0 Td[(1h:.2Equation.1followsfromthefactthatbracketingbyanyelementisaderiva-tionandthefundamentalrelationsinsl.Equation.2isprovedbyinduc-tion:Fork=1itistruefromthedeningrelationsofsl.Assumingitfork,wehave[e;fk+1]=[e;f]fk+f[e;fk]=hfk)]TJ/F11 9.963 Tf 9.962 0 Td[(kk)]TJ/F8 9.963 Tf 9.963 0 Td[(1fk+kfkh=[h;fk]+fkh)]TJ/F11 9.963 Tf 9.962 0 Td[(kk)]TJ/F8 9.963 Tf 9.962 0 Td[(1fk+kfkh=)]TJ/F8 9.963 Tf 7.749 0 Td[(2kfk)]TJ/F11 9.963 Tf 9.962 0 Td[(kk)]TJ/F8 9.963 Tf 9.963 0 Td[(1fk+k+1fkh=)]TJ/F8 9.963 Tf 7.749 0 Td[(k+1kfk+k+1fkh:

PAGE 38

38CHAPTER2.SLANDITSREPRESENTATIONS.Wemayrewrite.2ase;1 k!fk=)]TJ/F11 9.963 Tf 7.749 0 Td[(k+11 k)]TJ/F8 9.963 Tf 9.963 0 Td[(1!fk)]TJ/F7 6.974 Tf 6.227 0 Td[(1+1 k)]TJ/F8 9.963 Tf 9.962 0 Td[(1!fk)]TJ/F7 6.974 Tf 6.227 0 Td[(1h:.3InanynitedimensionalmoduleV,theelementhhasatleastoneeigenvector.Thisfollowsfromthefundamentaltheoremofalgebrawhichassertthatanypolynomialhasatleastoneroot;inparticularthecharacteristicpolynomialofanylineartransformationonanitedimensionalspacehasaroot.Sothereisavectorwsuchthathw=wforsomecomplexnumber.Thenhew=[h;e]w+ehw=2ew+ew=+2ew:Thusewisagainaneigenvectorofh,thistimewitheigenvalue+2.Successivelyapplyingeyieldsavectorvsuchthathv=v;ev=0:.4ThenUslvisaninvariantsubspace,henceallofV.WesaythatvisacyclicvectorfortheactionofgonVifUgv=V,Wearethusledtostudyallmodulesforslwithacyclicvectorvsatis-fying.4.Inanysuchspacetheelements1 k!fkvspan,andareeigenspacesofhofweight)]TJ/F8 9.963 Tf 8.829 0 Td[(2k.Forany2Cwecanconstructsuchamoduleasfollows:Letb+denotethesubalgebraofslgeneratedbyhande.ThenUb+,theuniversalenvelopingalgebraofb+canberegardedasasubalgebraofUsl.WecanmakeCintoab+module,andhenceaUb+modulebyh1:=;e1:=0:ThenthespaceUslUb+CwitheactingonCas0andhactingviamultiplicationbyisacyclicmodulewithcyclicvectorv=11whichsatises.4.Itisauniversal"suchmoduleinthesensethatanyothercyclicmodulewithcyclicvectorsatisfying.4isahomomorphicimageoftheonewejustconstructed.ThisspaceUslUb+Cisinnitedimensional.Itisirreducibleunlessthereissome1 k!fkvwithe1 k!fkv=0wherekisaninteger1.Indeed,anynon-zerovectorwinthespaceisanitelinearcombinationofthebasiselements1 k!fkv;choosektobethelargestintegersothatthecoecientofthecorrrespondingelementdoesnotvanish.Thensuccessiveapplicationoftheelementek-timeswillyieldamultipleof

PAGE 39

2.3.THECASIMIRELEMENT.39v,andifthismultipleisnon-zero,thenUslw=Uslvisthewholespace.Bute1 k!fkv=e;1 k!fkv=)]TJ/F11 9.963 Tf 9.963 0 Td[(k+1 k)]TJ/F8 9.963 Tf 9.962 0 Td[(1!fk)]TJ/F7 6.974 Tf 6.226 0 Td[(1v:Thisvanishesonlyifisanintegerandk=+1,inwhichcasethereisauniquenitedimensionalquotientofdimensionk+1.QEDThenitedimensionalirreduciblerepresentationshavingzeroasaweightareallodddimensionalandhaveonlyevenweights.Wewillcallthemeven".Theyarecalledintegerspin"representationsbythephysicists.Theothersareodd"orhalfspin"representations.2.3TheCasimirelement.InUslconsidertheelementC:=1 2h2+ef+fe.5calledtheCasimirelementorsimplytheCasimir"ofsl.Sinceef=fe+[e;f]=fe+hinUslwealsocanwriteC=1 2h2+h+2fe:.6Thisimpliesthatifvisahighestweightvector"inaslmodulesatisfyingev=0;hv=vthenCv=1 2+2v:.7NowinUslwehave[h;C]=2[h;f]e+f[h;e]=2)]TJ/F8 9.963 Tf 7.748 0 Td[(2fe+2fe=0and[C;e]=1 22eh+he+2e)]TJ/F8 9.963 Tf 9.963 0 Td[(2he=eh)]TJ/F11 9.963 Tf 9.963 0 Td[(he+2e=)]TJ/F8 9.963 Tf 7.748 0 Td[([h;e]+2e=0:Similarly[C;f]=0:

PAGE 40

40CHAPTER2.SLANDITSREPRESENTATIONS.Inotherwords,Cliesinthecenteroftheuniversalenvelopingalgebraofsl,i.e.itcommuteswithallelements.IfVisamodulewhichpossessesahighestweightvector"vasabove,andifVhasthepropertythatvisacyclicvector,meaningthatV=ULvthenCtakesontheconstantvalueC=+2 2IdsinceCiscentralandviscyclic.2.4slissimple.AnidealIinaLiealgebragisasubspaceofgwhichisinvariantundertheadjointrepresentation.Inotherwords,Iisanidealif[g;I]I.IfaLiealgebraghasthepropertythatitsonlyidealsare0andgitself,andifgisnotcommutative,wesaythatgissimple.Letusprovethatslissimple.Sinceslisnotcommutative,wemustprovethattheonlyidealsare0andslitself.Wedothisbyintroducingsomenotationwhichwillallowustogeneralizetheproofinthenextchapter.Letg=slandsetg)]TJ/F7 6.974 Tf 6.227 0 Td[(1:=Cf;g0:=Ch;g1:=Cesothatg,asavectorspace,isthedirectsumofthethreeonedimensionalspacesg=g)]TJ/F7 6.974 Tf 6.226 0 Td[(1g0g1:Correspondingly,writeanyx2gasx=x)]TJ/F7 6.974 Tf 6.227 0 Td[(1+x0+x1:Ifweletd:=1 2hthenwehavex=x)]TJ/F7 6.974 Tf 6.227 0 Td[(1+x0+x1;[d;x]=)]TJ/F11 9.963 Tf 7.749 0 Td[(x)]TJ/F7 6.974 Tf 6.226 0 Td[(1+0+x1;and[d;[d;x]]=x)]TJ/F7 6.974 Tf 6.227 0 Td[(1+0+x1:Sincethematrix0@111)]TJ/F8 9.963 Tf 7.749 0 Td[(1011011Aisinvertible,weseethatwecansolveforthecomponents"x)]TJ/F7 6.974 Tf 6.227 0 Td[(1;x0andx1intermsofx;[d;x];[d;[d;x]].ThismeansthatifIisanideal,thenI=I1I0I1

PAGE 41

2.5.COMPLETEREDUCIBILITY.41whereI)]TJ/F7 6.974 Tf 6.227 0 Td[(1:=Ig)]TJ/F7 6.974 Tf 6.227 0 Td[(1;I0:=Ig0;I1:=Ig1:NowifI06=0thend=1 2h2I,andhencee=[d;e]andf=)]TJ/F8 9.963 Tf 7.749 0 Td[([d;f]alsobelongtoIsoI=sl.IfI)]TJ/F7 6.974 Tf 6.226 0 Td[(16=0sothatf2I,thenh=[e;f]2IsoI=sl.Similarly,ifI16=0sothate2Ithenh=[e;f]2IsoI=sl.ThusifI6=0thenI=slandwehaveprovedthatslissimple.2.5Completereducibility.WewillusetheCasimirelementCtoprovethateverynitedimensionalrep-resentationWofsliscompletelyreducible,whichmeansthatifW0isaninvariantsubspacethereexistsacomplementaryinvariantsubspaceW00sothatW=W0W00.Indeedwewillprove:Theorem21.Everynitedimensionalrepresentationofsliscompletelyreducible.2.Eachirreduciblesubspaceisacyclichighestweightmodulewithhighestweightnwherenisanon-negativeinteger.3.Whentherepresentationisdecomposedintoadirectsumofirreduciblecomponents,thenumberofcomponentswithevenhighestweightisthemultiplicityof0asananeigenvectorofhand4.thenumberofcomponentswithoddhighestweightisthemultiplicityof1asaneigenvalueofh.Proof.Weknowthateveryirreduciblenitedimensionalrepresentationisacyclicmodulewithintegerhighestweight,thatthosewithevenhighestweightcontain0asaneigenvalueofhwithmultiplicityoneanddonotcontain1asaneigenvalueofh,andthatthosewithoddhighestweightcontain1asaneigenvalueofhwithmultiplicityone,anddonotcontain0asaneigenvalue.So2,3and4followfrom1.Wemustprove1.WerstproveProposition1Let0!V!W!k!0beanexactsequenceofslmodulesandsuchthattheactionofslonkistrivialasitmustbe,sinceslhasnonon-trivialonedimensionalmodules.Thenthissequencesplits,i.e.thereisalineinWsupplementarytoVonwhichslactstrivially.Thispropositionis,ofcourse,aspecialcaseofthetheoremwewanttoprove.Butweshallseethatitissucienttoprovethetheorem.Proofofproposition.ItisenoughtoprovethepropositionforthecasethatVisanirreduciblemodule.Indeed,ifV1isasubmodule,thenbyinductionondimVwemayassumethetheoremisknownfor0!V=V1!W=V1!k!0sothatthereisaonedimensionalinvariantsubspaceMinW=V1supplementary

PAGE 42

42CHAPTER2.SLANDITSREPRESENTATIONS.toV=V1onwhichtheactionistrivial.LetNbetheinverseimageofMinW.Byanotherapplicationoftheproposition,thistimetothesequence0!V1!N!M!0wendaninvariantline,P,inNcomplementarytoV1.SoN=V1P.SinceW=V1=V=V1MwemusthavePV=f0g.ButsincedimW=dimV+1,wemusthaveW=VP.InotherwordsPisaonedimensionalsubspaceofWwhichiscomplementarytoV.NextwearereducedtoprovingthepropositionforthecasethatslactsfaithfullyonV.Indeed,letI=thekerneloftheactiononV.Sinceslissimple,eitherI=slorI=0.SupposethatI=sl.Forallx2slwehave,byhypothesis,xWV,andforx2I=slwehavexV=0.Hence[sl;sl]=slactstriviallyonallofWandthepropositionisobvious.SowearereducedtothecasethatVisirreducibleandtheaction,,ofslonVisinjective.WehaveourCasimirelementCwhoseimageinEndWmustmapW!Vsinceeveryelementofsldoes.Ontheotherhand,C=1 2nn+2Id6=0sinceweareassumingthattheactionofslontheirreduciblemoduleVisnottrivial.Inparticular,therestrictionofCtoVisanisomorphism.HencekerC:W!VisaninvariantlinesupplementarytoV.Wehaveprovedtheproposition.Proofoftheoremfromproposition.Let0!E0!Ebeanexactsequenceofslmodules,andwemayassumethatE06=0.WewanttondaninvariantcomplementtoE0inE.DeneWtobethesubspaceofHomkE;E0whoserestrictiontoE0isascalartimestheidentity,andletVWbethesubspaceconsistingofthoselineartransformationswhoserestrictionstoE0iszero.EachoftheseisasubmoduleofEndE.Wegetasequence0!V!W!k!0andhenceacomplementarylineofinvariantelementsinW.Inparticular,wecanndanelement,Twhichisinvariant,mapsE!E0,andwhoserestrictiontoE0isnon-zero.ThenkerTisaninvariantcomplementarysubspace.QED2.6TheWeylgroup.Wehaveexpe=1101andexp)]TJ/F11 9.963 Tf 7.749 0 Td[(f=10)]TJ/F8 9.963 Tf 7.748 0 Td[(11soexpeexp)]TJ/F11 9.963 Tf 7.748 0 Td[(fexpe=110110)]TJ/F8 9.963 Tf 7.749 0 Td[(111101=01)]TJ/F8 9.963 Tf 7.749 0 Td[(10:Sinceexpadx=Adexpx

PAGE 43

2.6.THEWEYLGROUP.43weseethat:=expadeexpad)]TJ/F11 9.963 Tf 7.749 0 Td[(fexpadeconsistsofconjugationbythematrix01)]TJ/F8 9.963 Tf 7.749 0 Td[(10:Thush=01)]TJ/F8 9.963 Tf 7.749 0 Td[(10100)]TJ/F8 9.963 Tf 7.749 0 Td[(10)]TJ/F8 9.963 Tf 7.749 0 Td[(110=)]TJ/F8 9.963 Tf 7.748 0 Td[(1001=)]TJ/F11 9.963 Tf 7.749 0 Td[(h;e=01)]TJ/F8 9.963 Tf 7.749 0 Td[(1001000)]TJ/F8 9.963 Tf 7.749 0 Td[(110=00)]TJ/F8 9.963 Tf 7.748 0 Td[(10=)]TJ/F11 9.963 Tf 7.749 0 Td[(fandsimilarlyf=)]TJ/F11 9.963 Tf 7.749 0 Td[(e.Inshort:e7!)]TJ/F11 9.963 Tf 20.479 0 Td[(f;f7!)]TJ/F11 9.963 Tf 20.479 0 Td[(e;h7!)]TJ/F11 9.963 Tf 20.479 0 Td[(h:Inparticular,inducesthereection"h7!)]TJ/F11 9.963 Tf 20.479 0 Td[(honChandhencethereection7!)]TJ/F11 9.963 Tf 20.479 0 Td[(whichweshallalsodenotebysontheonedimensionaldualspace.InanynitedimensionalmoduleVofsltheactionoftheelement=expeexp)]TJ/F11 9.963 Tf 7.749 0 Td[(fexpeisdened,and)]TJ/F7 6.974 Tf 6.227 0 Td[(1h=Ad)]TJ/F11 9.963 Tf 4.567 -8.07 Td[()]TJ/F7 6.974 Tf 6.227 0 Td[(1h=s)]TJ/F7 6.974 Tf 6.227 0 Td[(1h=shsoifhu=uthenhu=)]TJ/F7 6.974 Tf 6.227 0 Td[(1hu=shu=)]TJ/F11 9.963 Tf 7.748 0 Td[(u=su:SoifV:fu2Vjhu=ugthenV=Vs:.8ThetwoelementgroupconsistingoftheidentityandtheelementsactingasareectionasaboveiscalledtheWeylgroupofsl.ItsgeneralizationtoanarbitrarysimpleLiealgebra,togetherwiththegeneralizationofformula.8willplayakeyroleinwhatfollows.

PAGE 44

44CHAPTER2.SLANDITSREPRESENTATIONS.

PAGE 45

Chapter3Theclassicalsimplealgebras.Inthischapterweintroducetheclassical"nitedimensionalsimpleLieal-gebras,whichcomeinfourfamilies:thealgebrassln+1consistingofalltracelessn+1n+1matrices,theorthogonalalgebras,onevenandodddimensionalspacesthestructurefortheevenandoddcasesaredierentandthesymplecticalgebraswhosedenitionwewillgivebelow.Wewillprovethattheyareindeedsimplebyauniformmethod-themethodthatweusedintheprecedingchaptertoprovethatslissimple.Soweaxiomatizethismethod.3.1Gradedsimplicity.WeintroducethefollowingconditionsontheLiealgebrag:g=1Mi=)]TJ/F7 6.974 Tf 6.226 0 Td[(1gi.1[gi;gj]gi+j.2[g1;g)]TJ/F7 6.974 Tf 6.227 0 Td[(1]=g0.3[g)]TJ/F7 6.974 Tf 6.227 0 Td[(1;x]=0x=0;8x2gi;8i0.4Thereexistsad2g0satisfying[d;x]=kx;x2gk;8k;.5andg)]TJ/F7 6.974 Tf 6.226 0 Td[(1isirreducibleundertheadjointactionofg0:.6Condition.4meansthatifx2gi;i0issuchthat[y;x]=0forally2g)]TJ/F7 6.974 Tf 6.227 0 Td[(1thenx=0.Wewishtoshowthatanynon-zerogsatisfyingthesesixconditionsissimple.Weknowthatg)]TJ/F7 6.974 Tf 6.227 0 Td[(1;g0andg1areallnon-zero,since06=d2g0by.5and45

PAGE 46

46CHAPTER3.THECLASSICALSIMPLEALGEBRAS.[g)]TJ/F7 6.974 Tf 6.227 0 Td[(1;g1]=g0by.3.Sogcannotbetheonedimensionalcommutativealgebra,andhencewhatwemustshowisthatanynon-zeroidealIofgmustbeallofg.WerstshowthatanyidealImustbeagradedideal,i.e.thatI=I)]TJ/F7 6.974 Tf 6.227 0 Td[(1I0I1;whereIj:=Igj:Indeed,writeanyx2gasx=x)]TJ/F7 6.974 Tf 6.227 0 Td[(1+x0+x1++xkandsuccessivelybracketbydtoobtainx=x)]TJ/F7 6.974 Tf 6.227 0 Td[(1+x0+x1++xk[d;x]=)]TJ/F11 9.963 Tf 7.749 0 Td[(x)]TJ/F7 6.974 Tf 6.226 0 Td[(1+0+x1++kxk[d;[d;x]]=x)]TJ/F7 6.974 Tf 6.227 0 Td[(1+0+x1++k2xk.........addkx=)]TJ/F8 9.963 Tf 7.749 0 Td[(1kx)]TJ/F7 6.974 Tf 6.227 0 Td[(1+0+x1++kkxkaddk+1x=)]TJ/F8 9.963 Tf 7.749 0 Td[(1k+1x)]TJ/F7 6.974 Tf 6.226 0 Td[(1+0+x1++kk+1xk:Thematrix0BBBBB@1111)]TJ/F8 9.963 Tf 7.749 0 Td[(101k............)]TJ/F8 9.963 Tf 7.749 0 Td[(1k01kk)]TJ/F8 9.963 Tf 7.749 0 Td[(1k+101kk+11CCCCCAisnonsingular.Indeed,itisavanderMondematrix,thatisamatrixoftheform0BBBBB@1111t1t21tk+2............tk1tk21tkk+2tk+11tk+121tk+1k+21CCCCCAwhosedeterminantisYi
PAGE 47

3.2.SLN+147that[g)]TJ/F7 6.974 Tf 6.226 0 Td[(1;y]=0andhencethaty=0by.4.ThusI0=0.SupposethatweknowthatIj)]TJ/F7 6.974 Tf 6.227 0 Td[(1=0.Thenthesameargumentshowsthatanyy2Ijsatises[g)]TJ/F7 6.974 Tf 6.227 0 Td[(1;y]=0andhencey=0.SoIj=0forallj,andsinceIisthesumofalltheIjweconcludethatI=0.NowsupposethatI)]TJ/F7 6.974 Tf 6.227 0 Td[(1=g)]TJ/F7 6.974 Tf 6.226 0 Td[(1.Theng0=[g)]TJ/F7 6.974 Tf 6.227 0 Td[(1;g1]=[I)]TJ/F7 6.974 Tf 6.227 0 Td[(1;g1]I.Fur-thermore,sinced2g0IweconcludethatgkIforallk6=0sinceeveryelementyofsuchagkcanbewrittenasy=1 k[d;y]2I.HenceI=g.QEDForexample,theLiealgebraofallpolynomialvectorelds,wheregk=fXXi@ @xiXihomogenouspolynomialsofdegreek+1gisasimpleLiealgebra.HeredistheEulervectoreldd=x1@ @x1++xn@ @xn:Thisalgebraisinnitedimensional.WeareprimarilyinterestedinthenitedimensionalLiealgebras.3.2sln+1Writethemostgeneralmatrixinsln+1as)]TJ/F8 9.963 Tf 9.409 0 Td[(trAwvAwhereAisanarbitrarynnmatrix,visacolumnvectorandw=w1;:::;wnisarowvector.Letg)]TJ/F7 6.974 Tf 6.227 0 Td[(1consistofmatriceswithjustthetoprow,i.e.withv=A=0.Letg1consistofmatriceswithjusttheleftcolumn,i.e.withA=w=0.Letg0consistofmatriceswithjustthecentralblock,i.e.withv=w=0.Letd=1 n+1)]TJ/F11 9.963 Tf 7.748 0 Td[(n00IwhereIisthennidentitymatrix.Thusg0actsong)]TJ/F7 6.974 Tf 6.227 0 Td[(1asthealgebraofallendomorphisms,andsog)]TJ/F7 6.974 Tf 6.227 0 Td[(1isirreducible.Wehave[00v0;0w00]=hw;vi00vw;wherehw;videnotesthevalueofthelinearfunctionwonthevectorv,andthisispreciselythetraceoftherankonelineartransformationvw.Thusallouraxiomsaresatised.Thealgebrasln+1issimple.

PAGE 48

48CHAPTER3.THECLASSICALSIMPLEALGEBRAS.3.3Theorthogonalalgebras.Thealgebraoisonedimensionalandhencecommutative.InourrealEuclideanthreedimensionalspace,thealgebraohasabasisX;Y;Zin-nitesimalrotationsabouteachoftheaxeswithbracketrelations[X;Y]=Z;[Y;Z]=X;[Z;X]=Y;theusualformulaeforvectorproduct"inthreedimensions".Butweareoverthecomplexnumbers,socanconsiderthebasisX+iY;)]TJ/F11 9.963 Tf 7.749 0 Td[(X+iY;iZandndthat[iZ;X+iY]=X+iY;[iZ;)]TJ/F11 9.963 Tf 7.749 0 Td[(X+iY]=)]TJ/F8 9.963 Tf 7.748 0 Td[()]TJ/F11 9.963 Tf 7.749 0 Td[(X+iY;[X+iY;)]TJ/F11 9.963 Tf 7.749 0 Td[(X+iY]=2iZ:Thesearethebracketrelationsforslwithe=X+iY;f=)]TJ/F11 9.963 Tf 7.749 0 Td[(X+iY;h=iZ.Inotherwords,thecomplexicationofourthreedimensionalworldistheirreduciblethreedimensionalrepresentationofslsoo=slwhichissimple.Tostudythehigherdimensionalorthogonalalgebrasitisusefultomaketworemarks:IfVisavectorspacewithanon-degeneratesymmetricbilinearform;,wegetanisomorphismofVwithitsdualspaceVsendingeveryu2Vtothelinearfunction`uwhere`uv=v;u.ThisgivesanidenticationofEndV=VVwithVV:Underthisidentication,theelementsofoVbecomeidentiedwiththeanti-symmetrictwotensors,thatiswithelementsof^2V.Intermsofanor-thonormalbasis,amatrixAbelongstooVifandonlyifitisanti-symmetric.Explicitly,anelementu^vbecomesidentiedwiththelineartransformationAu^vwhereAu^vx=x;vu)]TJ/F8 9.963 Tf 9.962 0 Td[(u;xv:Thishasthefollowingconsequence.Supposethatz2Vwithz;z6=0,andletwbeanyelementofV.ThenAw^zz=z;zw)]TJ/F8 9.963 Tf 9.963 0 Td[(z;wzandsoUoVz=V.Ontheotherhand,supposethatu2Vwithu;u=0.Wecanndv2Vwithv;v=0andv;u=1.NowsupposeinadditionthatdimV3.Wecanthenndaz2Vorthogonaltotheplanespannedbyuandvandwithz;z=1.ThenAz^vu=z;soz2UoVeandhenceUoVu=V.Wehaveproved:1IfdimV3,theneverynon-zerovectorinViscyclic,i.etherepresenta-tionofoVonVisirreducible.

PAGE 49

3.3.THEORTHOGONALALGEBRAS.49Intwodimensionsthisisfalse-thelinespannedbyavectorewithe;e=0isaonedimensionalinvariantsubspace.Wenowshowthat2oVissimplefordimV5.Forthis,beginbywritingdownthebracketrelationsforelementsofoVintermsoftheirparametrizationbyelementsof^2V.Directcomputationshowsthat[Au^v;Ax^y]=v;xAu^y)]TJ/F8 9.963 Tf 9.963 0 Td[(u;xAv^y)]TJ/F8 9.963 Tf 9.963 0 Td[(v;yAu^x+u;yAv^x:.7Nowletn=dimV)]TJ/F8 9.963 Tf 9.962 0 Td[(2andchooseabasisu;v;x1;:::;xnofVwhereu;u=u;xi=v;v=v;xi=08i;u;v=1;xi;xj=ij:Letg:=oVandwriteWforthesubspacespannedbythexi.Setd:=Au^vandg)]TJ/F7 6.974 Tf 6.227 0 Td[(1:=fAv^x;x2Wg;g0:=oWCd;g1:=fAu^x;x2Wg:Itthenfollowsfrom.7thatdsatises.5.Thespacesg)]TJ/F7 6.974 Tf 6.227 0 Td[(1andg1looklikecopiesofWwiththeoWpartofg0actingasoW,henceirreduciblysincedimW3.Allourremainingaxiomsareeasilyveried.HenceoVissimplefordimV5.Wehaveseenthato=slissimple.Howeveroisnotsimple,beingisomorphictoslsl:Indeed,ifZ1andZ2arevectorspacesequippedwithnon-degenerateanti-symmetricbilinearformsh;i1andh;i2thenZ1Z2hasanon-degeneratesymmetricbilinearform;determinedbyu1u2;v1v2=hu1;v1i1hu2;v2i2:Thealgebraslactingonitsbasictwodimensionalrepresentationinnitesi-mallypreservestheantisymmetricformgivenbyx1x2;y1y2=x1y2)]TJ/F11 9.963 Tf 9.963 0 Td[(x2y1:Hence,ifwetakeZ=Z1=Z2tobethistwodimensionalspace,weseethatslslactsasinnitesimalorthogonaltransformationsonZZwhichisfourdimensional.Butoissixdimensionalsotheembeddingofslslinoisinfactanisomorphismsince3+3=6.

PAGE 50

50CHAPTER3.THECLASSICALSIMPLEALGEBRAS.3.4Thesymplecticalgebras.Weconsideranevendimensionalspacewithcoordinatesq1;q2;:::;p1;p2;:::.ThepolynomialshaveaPoissonbracketff;gg:=X@f @pi@g @qi)]TJ/F11 9.963 Tf 12.065 6.74 Td[(@f @qi@g @pi:.8Thisisclearlyanti-symmetric,anddirectcomputationwillshowthattheJa-cobiidentityissatised.HereisamoreinterestingproofofJacobi'sidentity:Noticethatiffisaconstant,thenff;gg=0forallg.Soindoingbracketcomputationswecanignoreconstants.Ontheotherhand,ifwetakegtobesuccessivelyq1;:::;qn;p1;:::;pnin.8weseethatthepartialderivativesoffarecompletelydeterminedbyhowitbracketswithallg,infactwithalllinearg.Ifwexf,themaph7!ff;hgisaderivation,i.e.itislinearandsatisesff;h1h2g=ff;h1gh2+h1ff;h2g:Thisfollowsimmediatelyfromfromthedenition.8.NowJacobi'sidentityamountstotheassertionthatfff;gg;hg=ff;fg;hgg)-222(fg;ff;hgg;i.e.thatthederivationh7!fff;gg;hgisthecommutatoroftheofthederivationsh7!ff;hgandh7!fg;hg:Itisenoughtocheckthisonlinearpolynomialsh,andhenceonthepolynomialsqjandpk.Ifwetakeh=qjthenff;qjg=@f @pj;fg;qjg=@g @pjsoff;fg;qjgg=X@f @pi@2g @qi@pj)]TJ/F11 9.963 Tf 12.065 6.739 Td[(@f @qi@2g @pi@pjff;ff;qjgg=X@g @pi@2f @qi@pj)]TJ/F11 9.963 Tf 12.486 6.74 Td[(@g @qi@2f @pi@pjsoff;fg;qjgg)-222(fg;ff;qjgg=@ @pjff;gg=fff;gg;qjgasdesired,withasimilarcomputationforpk.

PAGE 51

3.4.THESYMPLECTICALGEBRAS.51Thesymplecticalgebraspnisdenedtobethesubalgebraconsistingofallhomogeneousquadraticpolynomials.Wedividethesepolynomialsintothreegroupsasfollows:Letg1consistofhomogeneouspolynomialsintheq'salone,sog1isspannedbytheqiqj.Letg)]TJ/F7 6.974 Tf 6.227 0 Td[(1bethequadraticpolynomialsinthep'salone,andletg0bethemixedterms,sospannedbytheqipj.Itiseasytoseethatg0glnandthat[g)]TJ/F7 6.974 Tf 6.227 0 Td[(1;g1]=g0.Tocheckthatg)]TJ/F7 6.974 Tf 6.227 0 Td[(1isirreducibleunderg0,observethat[p1qj;pkp`]=0ifj6=kor`,and[p1qj;pjp`]isamultipleofp1p`.Sowecanbyoneortwobracketscarryanynon-zeroelementofg)]TJ/F7 6.974 Tf 6.227 0 Td[(1intoanon-zeromultiplyofp21,andthengetanymonomialfromp21bybracketingwithpiq1appropriately.Theelementdisgivenby1 2p1q1++pnqn.Wehaveshownthatthesymplecticalgebraissimple,butwehaven'treallyexplainedwhatitis.ConsiderthespaceofVofhomogenouslinearpolynomials,i.eallpolynomialsoftheform`=a1q1++anqn+b1pq++bnpn:Deneananti-symmetricbilinearform!onVbysetting!`;`0:=f`;`0:Fromtheformula.8itfollowsthatthePoissonbracketoftwolinearfunctionsisaconstant,so!doesindeeddeneanantisymmetricbilinearformonV,andweknowthatthisbilinearformisnon-degenerate.Furthermore,iffisahomogenousquadraticpolynomial,and`islinear,thenff;`gisagainlinear,andifwedenotethemap`7!ff;`gbyA=Af,thenJacobi'sidentitytranslatesinto!A`;`0+!`A`0=0.9sincef`;`0gisaconstant.Condition.9canbeinterpretedassayingthatAbelongstotheLiealgebraofthegroupofalllineartransformationsRonVwhichpreserve!,i.e.whichsatisfy!R`;R`0=!`;`0:Thisgroupisknownasthesymplecticgroup.Theform!inducesanisomor-phismofVwithVandhenceofHomV;V=VVwithVV,andthistimetheimageofthesetofAsatisfying.9consistsofallsymmetricten-sorsofdegreetwo,i.e.ofS2V.Justasintheorthogonalcasewegottheanti-symmetrictensors.ButthespaceS2Visthesameasthespaceofho-mogenouspolynomialsofdegreetwo.Inotherwords,thesymplecticalgebraasdenedaboveisthesameastheLiealgebraofthesymplecticgroup.Itisaneasytheoreminlinearalgebra,thatifVisavectorspacewhichcarriesanon-degenerateanti-symmetricbilinearform,thenVmustbeevendimensional,andifdimV=2nthenitisisomorphictothespaceconstructedabove.Wewillnotpausetoprovethistheorem.

PAGE 52

52CHAPTER3.THECLASSICALSIMPLEALGEBRAS.3.5Therootstructures.Wearegoingtochooseabasisforeachoftheclassicalsimplealgebraswhichgeneralizesthebasise;f;hthatwechoseforsl.Indeed,foreachclassicalsimplealgebragwewillrstchooseamaximalcommutativesubalgebrahallofwhoseelementsaresemi-simple=diagonizableintheadjointrepresentation.Sincetheadjointactionofalltheelementsofhcommute,thismeansthattheycanbesimultaneouslydiagonalized.Thuswecandecomposegintoadirectsumofsimultaneouseigenspacesg=hMg.10where06=2handg:=fx2gj[h;x]=hx8h2hg:Thelinearfunctionsarecalledrootsoriginallybecausetheharerootsofthecharacteristicpolynomialofadh.Thesimultaneouseigenspacegiscalledtherootspacecorrespondingto.Thecollectionofallrootswillusuallybedenotedby.Letusseehowthisworksforeachoftheclassicalsimplealgebras.3.5.1An=sln+1.Wechoosehtoconsistofthediagonalmatricesinthealgebrasln+1ofalln+1n+1matriceswithtracezero.Asabasisofhwetakeh1:=0BBB@10000)]TJ/F8 9.963 Tf 7.749 0 Td[(100...............00001CCCAh2:=0BBBBB@0000010000)]TJ/F8 9.963 Tf 7.749 0 Td[(10...............00001CCCCCA...:=...hn:=0BBB@0000...............0010000)]TJ/F8 9.963 Tf 7.749 0 Td[(11CCCA:LetLidenotethelinearfunctionwhichassignstoeachdiagonalmatrixitsi-thdiagonalentry,

PAGE 53

3.5.THEROOTSTRUCTURES.53LetEijdenotethematrixwithoneinthei;jpositionandzero'selsewhere.Then[h;Eij]=Lih)]TJ/F11 9.963 Tf 9.962 0 Td[(LjhEij8h2hsothelinearfunctionsoftheformLi)]TJ/F11 9.963 Tf 9.962 0 Td[(Lj;i6=jaretheroots.Wemaysubdividethesetofrootsintotwoclasses:thepositiveroots+:=fLi)]TJ/F11 9.963 Tf 9.963 0 Td[(Lj;i
PAGE 54

54CHAPTER3.THECLASSICALSIMPLEALGEBRAS.sotherootsareLi+Ljalli;jandLi)]TJ/F11 9.963 Tf 9.962 0 Td[(Lji6=j:Wecandividetherootsintopositiveandnegativerootsbysetting+=fLi+Ljgallij[fLi)]TJ/F11 9.963 Tf 9.962 0 Td[(Ljgi
PAGE 55

3.5.THEROOTSTRUCTURES.55WecanchooseaspositiverootstheLk+L`;Lk)]TJ/F11 9.963 Tf 9.963 0 Td[(L`;k<`andseti:=Li)]TJ/F11 9.963 Tf 9.962 0 Td[(Li+1;i=1;:::;n)]TJ/F8 9.963 Tf 9.963 0 Td[(1;n:=Ln)]TJ/F7 6.974 Tf 6.227 0 Td[(1+Ln:Everypositiverootisasumofthesesimpleroots.Ifwesethi:=Auivi)]TJ/F11 9.963 Tf 9.962 0 Td[(Aui+1vi+1;i=1;:::n)]TJ/F8 9.963 Tf 9.962 0 Td[(1;andhn=Aun)]TJ/F6 4.981 Tf 5.397 0 Td[(1vn)]TJ/F6 4.981 Tf 5.396 0 Td[(1+Aunvnthenihi=2andfori6=jihj=0j6=i1;i=1;:::n)]TJ/F8 9.963 Tf 9.962 0 Td[(2ihi1=)]TJ/F8 9.963 Tf 7.748 0 Td[(1i=1;:::;n)]TJ/F8 9.963 Tf 9.963 0 Td[(2n)]TJ/F7 6.974 Tf 6.226 0 Td[(1hn)]TJ/F7 6.974 Tf 6.227 0 Td[(2=)]TJ/F8 9.963 Tf 7.748 0 Td[(1.13nhn)]TJ/F7 6.974 Tf 6.227 0 Td[(2=)]TJ/F8 9.963 Tf 7.749 0 Td[(1nhn)]TJ/F7 6.974 Tf 6.227 0 Td[(1=0:Fori=1;:::;n)]TJ/F8 9.963 Tf 9.88 0 Td[(1theelementshi;Auivi+1;Aui+1viformasubalgebraisomor-phictoslasdohn;Aun)]TJ/F6 4.981 Tf 5.396 0 Td[(1un;Avn)]TJ/F6 4.981 Tf 5.397 0 Td[(1vn.3.5.4Bn=on+1n2.Wechooseabasisu1;:::;un;v1;:::;vn;xofourorthogonalvectorspaceVsuchthatui;uj=vi;vj=0;8i;j;ui;vj=ij;andx;ui=x;vi=08i;x;x=1:AsintheevendimensionalcasewelethbethesubalgebraofoVspannedbytheAuivi,i=1;:::;nandtakeAu1v1;:::;AunvnasabasisofhandletL1;:::;Lnbethedualbasis.ThenLiLji6=j;Liareroots.WetakeLiLj;1i
PAGE 56

56CHAPTER3.THECLASSICALSIMPLEALGEBRAS.tobethepositiveroots,andi:=Li)]TJ/F11 9.963 Tf 9.962 0 Td[(Li+1;i=1;:::;n)]TJ/F8 9.963 Tf 9.963 0 Td[(1;n:=Lntobethesimpleroots.Welethi:=Auivi)]TJ/F11 9.963 Tf 9.963 0 Td[(Aui+1vi+1;i=1;:::n)]TJ/F8 9.963 Tf 9.963 0 Td[(1;asintheevencase,butsethn:=2Aunvn:Theneverypositiverootcanbewrittenasasumofthesimpleroots,ihi=2;i=1;:::n;andfori6=jihj=0j6=i1;i=1;:::nihi1=)]TJ/F8 9.963 Tf 7.749 0 Td[(1i=1;:::;n)]TJ/F8 9.963 Tf 9.963 0 Td[(2;n.14n)]TJ/F7 6.974 Tf 6.227 0 Td[(1hn=)]TJ/F8 9.963 Tf 7.748 0 Td[(2Noticethatinthiscasen)]TJ/F7 6.974 Tf 6.227 0 Td[(1+2n=Ln)]TJ/F7 6.974 Tf 6.227 0 Td[(1+Lnisaroot.Finallywecanconstructsubalgebrasisomorphictosl,withtherstn)]TJ/F8 9.963 Tf 10.431 0 Td[(1asintheevenorthogonalcaseandthelastslspannedbyhn;Aunx;)]TJ/F11 9.963 Tf 7.748 0 Td[(Avnx.3.5.5Diagrammaticpresentation.Theinformationofthelastfoursubsectionscanbesummarizedineachofthefollowingfourdiagrams:Thewaytoreadthisdiagramisasfollows:eachnodeinthediagramstandsforasimpleroot,readingfromlefttoright,startingwith1attheleft.InthediagramD`thetworightmostnodesare`)]TJ/F7 6.974 Tf 6.227 0 Td[(1and`,saythetop`)]TJ/F7 6.974 Tf 6.227 0 Td[(1andthebottom`.Twonodesiandjareconnectedbyoneormoreedgesifandonlyifihj6=0.Inallcases,thedierence,i)]TJ/F11 9.963 Tf 9.196 0 Td[(jisneveraroot,and,fori6=j;ihj0andisaninteger.If,fori6=j,ihj<0theni+jisaroot.IntwoofthecasesB`andC`ithappensthatihj=)]TJ/F8 9.963 Tf 7.749 0 Td[(2.Theni+jandi+2jareroots,andwedrawadoublebondwithanarrowpointingtowardsj.Inthiscase2isisthemaximumintegersuchthati+kjisaroot.Inallothercases,thismaximumintegerkisoneifthenodesareconnectedandzeroittheyarenot.3.6Lowdimensionalcoincidences.Wehavealreadyseenthatoslsl.Wealsohaveosl:

PAGE 57

3.6.LOWDIMENSIONALCOINCIDENCES.57 ...... HHD``4 ...... < C``2 ...... >B``3 ............ A`Figure3.1:Dynkindiagramsoftheclassicalsimplealgebras.Bothalgebrasarefteendimensionalandbotharesimple.Sotorealizethisisomorphismweneedonlyndanorthogonalrepresentationofslonasixdimensionalspace.IfweletV=C4withthestandardrepresentationofsl,wegetarepresentationofslon^2Vwhichissixdimensional.Sowemustdescribeanon-degeneratebilinearformon^2Vwhichisinvariantundertheactionofsl.Wehaveamap,wedgeproduct,of^2V^2V!^4V:Furthermorethismapissymmetric,andinvariantundertheactionofgl.Howeverslpreservesabasisanon-zeroelementof^4Vandsowemayidentify^4VwithC.Itiseasytocheckthatthebilinearformsoobtainedisnon-degenerateWealsohavetheidenticationspobothalgebrasbeingtendimensional.ToseethisletV=C4withanantisym-metricform!preservedbySp.Then!!inducesasymmetricbilinearformonVVaswehaveseen.SittinginsideVVasaninvariantsubspaceis^2Vaswehaveseen,whichissixdimensional.But^2Visnotirreducibleasarepresentationofsp.Indeed,!2^2Visinvariant,andhenceitskernelisavedimensionalsubspaceof^2Vwhichisinvariantundersp.Wethusgetanon-zerohomomorphismsp!owhichmustbeanisomorphismsincespissimple.

PAGE 58

58CHAPTER3.THECLASSICALSIMPLEALGEBRAS.Thesecoincidencescanbeseeninthediagrams.Ifweweretoallow`=2inthediagramforB`itwouldbeindistinguishablefromC2.Ifweweretoallow`=3inthediagramforD`itwouldbeindistinguishablefromA3.3.7Extendeddiagrams.ItfollowsfromJacobi'sidentitythatinthedecomposition.10,wehave[g;g0]g+0.15withtheunderstandingthattherighthandsideiszeroif+0isnotaroot.Ineachofthecasesexaminedabove,everypositiverootisalinearcombinationofthesimplerootswithnon-negativeintegercoecients.Sincethealgebraisnite,theremustbeamaximalpositiverootinthesensethat+iisnotarootforanysimpleroot.Forexample,inthecaseofAn=sln+1,theroot:=L1)]TJ/F11 9.963 Tf 8.575 0 Td[(Ln+1ismaximal.Thecorrespondinggconsistsofalln+1n+1matriceswithzeroseverywhereexceptintheupperrighthandcorner.Wecanalsoconsidertheminimalrootwhichisthenegativeofthemaximalroot,so0:=)]TJ/F11 9.963 Tf 7.749 0 Td[(=Ln+1)]TJ/F11 9.963 Tf 9.963 0 Td[(L1inthecaseofAn.Continuingtostudythiscase,leth0:=hn+1)]TJ/F11 9.963 Tf 9.963 0 Td[(h1:Thenwehaveihi=2;i=0;:::nand0h1=0hn=)]TJ/F8 9.963 Tf 7.749 0 Td[(1;0hi=0;i6=0;1;n:Thismeansthatifwewriteoutthen+1n+1matrixwhoseentriesareihj;i;j=0;:::nweobtainamatrixoftheform2I)]TJ/F11 9.963 Tf 9.963 0 Td[(MwhereMij=1ifandonlyifj=1withtheunderstandingthatn+1=0and)]TJ/F8 9.963 Tf 7.749 0 Td[(1=n,i.ewedothesubscriptarithmeticmodn.Inotherwords,Mistheadjacencymatrixofthecyclicgraphwithn+1verticeslabeled0;:::n.Also,wehaveh0+h1++hn=0:Ifweapplyitothisequationfori=0;:::nweobtainI)]TJ/F11 9.963 Tf 9.962 0 Td[(M1=0;where1isthecolumnvectorallofwhoseentriesare1.WecanwritethisequationasM1=21:

PAGE 59

3.7.EXTENDEDDIAGRAMS.59Inotherwords,1isaneigenvectorofMwitheigenvalue2.InthechaptersthatfollowweshallseethatanynitedimensionalsimpleLiealgebrahasroots,simpleroots,maximalrootsetc.givingrisetoamatrixMwithintegerentrieswhichisirreducibleinthesenseofnon-negativematrices-denitionlateronandwhichhasaneigenvectorwithpositiveintegerentrieswitheigenvalue2.ThiswillallowustoclassifythesimplenitedimensionalLiealgebras.

PAGE 60

60CHAPTER3.THECLASSICALSIMPLEALGEBRAS.

PAGE 61

Chapter4Engel-Lie-Cartan-WeylWereturntothegeneraltheoryofLiealgebras.Manyoftheresultsinthischapterarevalidoverarbitraryelds,indeedifweusetheaxiomstodeneaLiealgebraoveraringmanyoftheresultsarevalidinthisgenerality.Butsomeoftheresultsdependheavilyontheringbeinganalgebraicallyclosedeldofcharacteristiczero.Asacompromise,throughoutthischapterwedealwithelds,andwillassumethatallvectorspacesandallLiealgebraswhichappeararenitedimensional.Wewillindicatethenecessaryadditionalassumptionsonthegroundeldastheyoccur.ThetreatmentherefollowsSerreprettyclosely.4.1Engel'stheoremDeneaLiealgebragtobenilpotentif:9nj[x1;[x2;:::xn+1]:::]=08x1;:::;xn+12g:Example:n+:=n+gld:=allstrictlyuppertriangularmatrices.Noticethattheproductofanyd+1suchmatricesiszero.TheclaimisthatallnilpotentLiealgebrasareessentiallyliken+.Wecanreformulatethedenitionofnilpotentassayingthattheproductofanynoperatorsadxivanishes.OneversionofEngel'stheoremisTheorem3gisnilpotentifandonlyifadxisanilpotentoperatorforeachx2g.ThisfollowstakingV=gandtheadjointrepresentationfromTheorem4EngelLet:g!EndVbearepresentationsuchthatxisnilpotentforeachx2g.Thenthereexistsabasisintermsofwhichgn+gld,i.e.becomesstrictlyuppertriangular.Hered=dimV.Givenasinglenilpotentoperator,wecanalwaysndanon-zerovector,vwhichitsendsintozero.ThenonV=fvganon-zerovectorwhichtheinduced61

PAGE 62

62CHAPTER4.ENGEL-LIE-CARTAN-WEYLmapsendsintozeroetc.Sointermsofsuchaag,thecorrespondingmatrixisstrictlyuppertriangular.Thetheoremassertsthatwearecanndasingleagwhichworksforallx.Inviewoftheaboveproofforasingleoperator,Engel'stheoremfollowsfromthefollowingsimplerlookingstatement:Theorem5UnderthehypothesesofEngel'stheorem,ifV6=0,thereexistsanon-zerovectorv2Vsuchthatxv=08x2g.ProofofTheorem5inseveneasysteps.Replacegbyitsimage,i.e.assumethatgEndV.Thenadxy=Lxy)]TJ/F11 9.963 Tf 9.18 0 Td[(RxywhereLxisthelinearmapofEndVintoitselfgivenbyleftmultiplicationbyx,andRxisgivenbyrightmultiplicationbyx.BothLxandRxarenilpotentasoperatorssincexisnilpotent.Alsotheycommute.Hencebythebinomialformulaadxn=Lx)]TJ/F11 9.963 Tf 9.076 0 Td[(Rxnvanishesforsucientlylargen.WemayassumebyinductionthatforanyLiealgebra,m,ofsmallerdimensionthanthatofgandanyrepresentationthereexistsav2Vsuchthatxv=08x2m.Letkgbeasubalgebra,k6=g,andletN=Nk:=fx2gjadxkkgbeitsnormalizer.Theclaimisthat3Nkisstrictlylargerthank.Toseethis,observethateachx2kactsonkandong=kbynilpotentmaps,andhencethereisan06=^y2g=kkilledbyallx2k.Buttheny62k,and[y;x]=)]TJ/F8 9.963 Tf 7.749 0 Td[([x;y]2kforallx2k.Soy2Nk;y62k.Ifg6=0,thereisanidealigsuchthatdimg=i=1.Indeed,letibeamaximalpropersubalgebraofg.Itsnormalizerisstrictlylarger,henceallofg,soiisanideal.Theinverseimageingofalineing=iisasubalgebra,andisstrictlylargerthani.Henceitmustbeallofg.Choosesuchanideal,i.ThesubspaceWV;W=fvjxv=0;8x2igisinvariantunderg.Indeed,ify2g;w2Wthenxyw=yxw+[x;y]w=0.W6=0byinduction.Takey2g;y62i.ItpreservesWandisnilpotent.Hencethereisanon-zerov2Wwithyv=0.Sinceyandispang,wehavexv=08x2g:QEDNoassumptionsaboutthegroundeldwentintothis.

PAGE 63

4.2.SOLVABLELIEALGEBRAS.634.2SolvableLiealgebras.LetgbeaLiealgebra.DngisdenedinductivelybyD0g:=g;D1g:=[g;g];:::;Dn+1g:=[Dng;Dng]:Ifwetakebtoconsistofalluppertriangularnnmatrices,thenD1b=n+consistsofallstrictlytriangularmatricesandthensuccessivebracketseventuallyleadtozero.WeclaimthatthefollowingconditionsareequivalentandanyLiealgebrasatisfyingthemiscalledsolvable.1.9njDng=0.2.9nsuchthatforeveryfamilyof2nelementsofgthesuccessivebracketsofbracketsvanish;e.gforn=4thissays[[[[x1;x2];[x3;x4]];[[x5;x6];[x7;x8]]];[[[x9;x10];[x11;x12]];[[x13:x14];[x15;x16]]]]=0:3.Thereexistsasequenceofsubspacesg:=i1i2in=0sucheachisanidealintheprecedingandsuchthatthequotientij=ij+1isabelian,i.e.[ij;ij]ij+1.Proofoftheequivalenceoftheseconditions.[g;g]isalwaysanidealingsotheDjgformasequenceofidealsdemandedby3,andhence13.Wealsohavetheobviousimplications32and21.Soallthesedenitionsareequivalent.Theorem6[Lie.]LetgbeasolvableLiealgebraoveranalgebraicallyclosedeldkofcharacteristiczero,and;Vanitedimensionalrepresentationofg.ThenwecanndabasisofVsothatgconsistsofuppertriangularmatrices.ByinductionondimVthisreducestoTheorem7[Lie.]Underthesamehypotheses,thereexistsanon-zerocom-moneigenvectorvforallthey,i.e.thereisavectorv2Vandafunction:g!ksuchthatyv=yv8y2g:.1Lemma2Supposethatiisanidealofgand.1holdsforally2i.Then[x;h]=0;8x2gh2i:Proofoflemma.Forx2gletVibethesubspacespannedbyv;xv;:::;xi)]TJ/F7 6.974 Tf 6.227 0 Td[(1vandletn>0beminimalsuchthatVn=Vn+1.SoVnisnitedimensionalandxVnVn.AlsoVn=Vn+k8k.

PAGE 64

64CHAPTER4.ENGEL-LIE-CARTAN-WEYLAlso,forh2i,droppingthewehave:hv=hvhxv=xhv)]TJ/F8 9.963 Tf 9.962 0 Td[([x;h]vhxvmodV1hx2v=xhxv+[h;x]xvhx2v+uxv;modV1u2Ihx2v+uxvmodV1=hx2vmodV2......hxivhxivmodVi:ThusVnisinvariantunderiandforeachh2i,trjVnh=nh.InparticularbothxandhleaveVninvariantandtrjVn[x;h]=0sincethetraceofanycommutatoriszero.Thisprovesthelemma.Proofoftheorembyinductionondimg,whichwemayassumetobepositive.Letmbeanysubspaceofgwithgm[g;g].Then[g;m][g;g]msomisanidealing.Inparticular,wemaychoosemtobeasubspaceofcodimension1containing[g;g].Byinductionwecanndav2Vanda:m!ksuchthat4.1holdsforallelementsofm.LetW:=fw2Vjhw=hw8h2mg:Ifx2g,thenhxw=xhw)]TJ/F8 9.963 Tf 9.962 0 Td[([x;h]w=hxw)]TJ/F11 9.963 Tf 9.963 0 Td[([x;h]w=hxwsince[x;h]=0bythelemma.ThusWisstableunderallofg.Pickx2g;x62m,andletv2Wbeaneigenvectorofxwitheigenvalue,say.Thenvisasimultaneouseigenvectorforallofgwithextendedash+rx=h+r:QEDWehadtodividebynintheaboveargument.Infact,thetheoremisnottrueoveraeldofcharacteristic2,withslasacounterexample.Appliedtotheadjointrepresentation,Lie'stheoremsaysthatthereisaagofidealswithcommutativequotients,andhence[g;g]isnilpotent.4.3LinearalgebraLetVbeanitedimensionalvectorspaceoveranalgebraicallyclosedeldofcharacteristiczero,andletdetTI)]TJ/F11 9.963 Tf 9.963 0 Td[(u=YT)]TJ/F11 9.963 Tf 9.962 0 Td[(imi

PAGE 65

4.3.LINEARALGEBRA65bethefactorizationofitscharacteristicpolynomialwheretheiaredistinct.LetSTbeanypolynomialsatisfyingSTimodT)]TJ/F11 9.963 Tf 9.963 0 Td[(imi;ST0modT;whichispossiblebytheChineseremaindertheorem.ForeachiletVi:=thekernelofu)]TJ/F11 9.963 Tf 10.132 0 Td[(imi.ThenV=LViandonVi,theoperatorSuisjustthescalaroperatoriI.Inparticulars=Suissemi-simpleitseigenvectorsspanVand,sincesisapolynomialinuitcommuteswithu.Sou=s+nwheren=Nu;NT=T)]TJ/F11 9.963 Tf 9.963 0 Td[(STisnilpotent.Alsons=sn:Weclaimthatthesetwoelementsareuniquelydeterminedbyu=s+n;sn=ns;withssemisimpleandnnilpotent.Indeed,sincesn=ns;su=ussosu)]TJ/F11 9.963 Tf -335.962 -11.956 Td[(ik=u)]TJ/F11 9.963 Tf 8.404 0 Td[(ikssosViVi.Sinces)]TJ/F11 9.963 Tf 8.403 0 Td[(uisnilpotent,shasthesameeigenvaluesonViasudoes,i.e.i.Sosandhencenisuniquelydetermined.IfPTisanypolynomialwithvanishingconstantterm,thenifABaresubspaceswithuBAthenPuBA.So,inparticular,sBAandnBA.DeneVp;q:=VVVVVwithpcopiesofVandqcopiesofV.Letu2EndVactonVby)]TJ/F11 9.963 Tf 7.749 0 Td[(uandonVpqbyderivation,so,forexample,u12=u11)]TJ/F8 9.963 Tf 9.962 0 Td[(1u1)]TJ/F8 9.963 Tf 9.963 0 Td[(11u:Similarly,u11actsonV1;1=VVbyu11x`=ux`)]TJ/F11 9.963 Tf 9.963 0 Td[(xu`:UndertheidenticationofVVwithEndV,theelementx`actsony2Vbysendingitinto`yx:Sotheelementu11x`sendsyto`yux)]TJ/F8 9.963 Tf 9.963 0 Td[(u`yx=`yux)]TJ/F11 9.963 Tf 9.963 0 Td[(`uyx:Thisisthesameasthecommutatoroftheoperatoruwiththeoperatorcor-respondingtox`actingony.Inotherwords,undertheidenticationofVVwithEndV,thelineartransformationu11getsidentiedwithadu.

PAGE 66

66CHAPTER4.ENGEL-LIE-CARTAN-WEYLProposition2Ifu=s+nisthedecompositionofuthenupq=spq+npqisthedecompositionofupq.Proof.[spq;npq]=0andthetensorproductsofaneigenbasisforsisaneigenbasisforspq.Alsonpqisasumofcommutingnilpotentshencenilpotent.Themapu7!upqislinearhenceupq=spq+npq:QEDIf:k!kisamap,wedenesbysjVi=i.IfwechooseapolynomialsuchthatP=0;Pi=ithenPu=s.Proposition3Supposethatisadditive.Thenspq=spq:Proof.DecomposeVpqintoasumoftensorproductsoftheViorVj.Oneachsuchspacewehavesp;q=i1+)-222(=i1+::::=sp;qwherethemiddleequationisjusttheadditivity.QEDAsanimmediateconsequenceweobtainProposition4Notationasabove.IfABVp;qwithupqBAthenforanyadditivemap,spqBAProposition5overCLetu=s+nasabove.Iftrus=0fors= sthenuisnilpotent.Proof.trus=Pmii i=Pmijij2.Sotheconditionimpliesthatallthei=0.QED4.4Cartan'scriterion.LetgEndVbeaLiesubalgebrawhereVisnitedimensionalvectorspaceoverC.Thengissolvable,trxy=08x2g;y2[g;g]:Proof.Supposegissolvable.Chooseabasisforwhichgisuppertriangular.Theneveryy2[g;g]haszerosonthediagonal,Hencetrxy=0.Forthereverseimplication,itisenoughtoshowthat[g;g]isnilpotent,and,byEngel,thateachu2[g;g]isnilpotent.Soitisenoughtoshowthattru s=0,wheresisthesemisimplepartofu,byProposition5above.Ifitweretruethat s2gwewouldbedone,butthisneednotbeso.Writeu=X[xi;yi]:

PAGE 67

4.5.RADICAL.67Nowfora;b;c2EndVtr[a;b]c=trabc)]TJ/F11 9.963 Tf 9.963 0 Td[(bac=trbca)]TJ/F11 9.963 Tf 9.963 0 Td[(bac=trb[c;a]sotru s=Xtr[xi;yi] s=Xtryi[ s;xi]:Soitisenoughtoshowthatad s:g![g;g].Weknowthatadu:g![g;g],andwecan,byLagrangeinterpolation,ndapolynomialPsuchthatPu= s.TheresultnowfollowsfromProp.4:SinceEndVV1;1,takeA=[g;g]andB=g.Thenadu=u1;1sou1;1g[g;g[andhence s1;1g[g;g]or[ s;x]2[g;g]8x2g:QED4.5Radical.Ifiisanidealofgandg=iissolvable,thenDng=i=0impliesthatDngi.IfiitselfissolvablewithDmi=0,thenDm+ng=0.Sowehaveproved:Proposition6Ifigisanideal,andbothiandg=iaresolvable,soisg.Ifiandjaresolvableideals,theni+j=ji=ijissolvable,beingthehomomorphicimageofasolvablealgebra.So,bythepreviousproposition:Proposition7Ifiandjaresolvableidealsingsoisi+j.Inparticular,everyLiealgebraghasalargestsolvableidealwhichcontainsallothersolvableideals.Itisdenotedbyradgorsimplybyrwhengisxed.Analgebragiscalledsemi-simpleifradg=0.SinceDiisanidealwheneveriisbyJacobi'sidentity,ifr6=0thenthelastnon-zeroDnrisanabelianideal.Soanequivalentdenitionis:gissemi-simpleifithasnonon-zeroabelianideals.WeshallcallaLiealgebrasimpleifitisnotabelianandifithasnoproperideals.Weshallshowinthenextsectionthateverysemi-simpleLiealgebraisthedirectsumofsimpleLiealgebrasinauniqueway.4.6TheKillingform.Abilinearform;:gg!kiscalledinvariantif[x;y];z+y;[x;z]=08x;y;z2g:.2Noticethatif;isaninvariantform,andiisanideal,theni?isagainanideal.

PAGE 68

68CHAPTER4.ENGEL-LIE-CARTAN-WEYLOnewayofproducinginvariantformsisfromrepresentations:if;Visarepresentationofg,thenx;y:=trxyisinvariant.Indeed,[x;y];z+y;[x;z]=trfxy)]TJ/F11 9.963 Tf 9.963 0 Td[(yxz+yxz)]TJ/F11 9.963 Tf 9.963 0 Td[(zxg=trfxyz)]TJ/F11 9.963 Tf 9.962 0 Td[(yzxg=0:Inparticular,ifwetake=ad;V=gthecorrespondingbilinearformiscalledtheKillingformandwillbedenotedby;.Wewillalsosometimeswritex;yinsteadofx;y.Theorem8gissemi-simpleifandonlyifitsKillingformisnon-degenerate.Proof.Supposegisnotsemi-simpleandsohasanon-zeroabelianideal,a.Wewillshowthatx;y=08x2a;y2g.Indeed,let=adxady.Thenmapsg!aanda!0.Henceintermsofabasisstartingwithelementsofaandextending,itisuppertriangularandhas0alongthediagonal.Hencetr=0.HenceifgisnotsemisimplethenitsKillingformisdegenerate.Conversely,supposethatgissemi-simple.WewishtoshowthattheKillingformisnon-degenerate.Soletu:=g?=fxjtradxady=08y2gg:Ifx2u;z2gthentrfad[x;z]adyg=trfadxadzady)]TJ/F8 9.963 Tf 9.963 0 Td[(adzadxadyg=trfadxadzady)]TJ/F8 9.963 Tf 9.963 0 Td[(adyadzg=tradxad[z;y]=0;souisanideal.Inparticular,truadxuadyu=trgadgxadgyforx;y2u,ascanbeseenfromablockdecompositionstartingwithabasisofuandextendingtog.Ifwetakey2Du,weseethattraduDadu=0,soaduissolvablebyCartan'scriterion.Butthekernelofthemapu!aduisthecenterofu.Soifaduissolvable,soisu.QEDProposition8Letgbeasemisimplealgebra,ianyidealofg,andi?itsorthocomplementwithrespecttoitsKillingform.Thenii?=0.Indeed,ii?isanidealonwhichtradxady0henceissolvablebyCartan'scriterion.Sincegissemi-simple,therearenonon-trivialsolvableideals.QEDThereforeProposition9Everysemi-simpleLiealgebraisthedirectsumofsimpleLiealgebras.

PAGE 69

4.7.COMPLETEREDUCIBILITY.69Proposition10Dg=gforasemi-simpleLiealgebra.Sincethisistrueforeachsimplecomponent.Proposition11Let:g!sbeasurjectivehomomorphismofasemi-simpleLiealgebraontoasimpleLiealgebra.Thenifg=Lgiisadecompositionofgintosimpleideals,therestriction,ioftoeachsummandiszero,exceptforonesummandwhereitisanisomorphism.Proof.Sincesissimple,theimageofeveryiis0orallofs.Ifiissurjectiveforsomeithenitisanisomorphismsincegiissimple.Thereisatleastoneiforwhichitissurjectivesinceissurjective.Ontheotherhand,itcannotbesurjectiveforfortwoideals,gi;gji6=jforthen[gi;gj]=06=[s;s]=s:QED4.7Completereducibility.ThebasictheoremisTheorem9[Weyl.]Everynitedimensionalrepresentationofasemi-simpleLiealgebraiscompletelyreducible.Proof.1.If:g!EndVisinjective,thentheform;isnon-degenerate.Indeed,theidealconsistingofallxsuchthatx;y=08y2gissolvablebyCartan'scriterion,hence0.2.TheCasimiroperator.Leteiandfibebasesofgwhicharedualwithrespecttosomenon-degenerateinvariantbilinearform,;.Soei;fj=ij.Astheformisnon-degenerateandinvariant,itdenesamapofgg7!Endg;xyw=y;wx:Thismapisanisomorphismandisagmorphism.Underthismap,Xeifiw=Xw;fiei=wbythedenitionofdualbases.HenceundertheinversemapEndg7!ggtheidentityelement,id,correspondstoPeifiandsothisexpressionisindependentofthechoiceofdualbases.SinceidisannihilatedbycommutatorbyanyelementofEndg,weconcludethatPieifiisannihilatedbytheactionofalladx2=adx1+1adx;x2g.Indeed,forx;e;f;y2gwehaveadx2efy=adxef+eadxfy=f;y[x;e]+[x;f];ye=f;y[x;e])]TJ/F8 9.963 Tf 9.963 0 Td[(f;[x;y]eby.2=adxef)]TJ/F8 9.963 Tf 9.962 0 Td[(efadxy:

PAGE 70

70CHAPTER4.ENGEL-LIE-CARTAN-WEYLSetC:=Xieifi2UL:.3ThusCistheimageoftheelementPieifiunderthemultiplicationmapgg7!Ug,andisindependentofthechoiceofdualbases.Furthermore,CisannihilatedbyadxactingonUg.Inotherwords,itcommuteswithallelementsofg,andhencewithallofUg;itisinthecenterofUg.TheCcorrespondingtotheKillingformiscalledtheCasimirelement,itsimageinanyrepresentationiscalledtheCasimiroperator.3.Supposethat:g!EndVisinjective.Theimageofthecentralelementcorrespondingto;denesanelementofEndVdenotedbyCandtrC=trXeifi=trXeifi=Xiei;fi=dimgWiththesepreliminaries,wecanstatethemainproposition:Proposition12Let0!V!W!k!0beanexactsequenceofgmodules,wheregissemi-simple,andtheactionofgonkistrivialasitmustbe.Thenthissequencesplits,i.e.thereisalineinWsupplementarytoVonwhichgactstrivially.Theproofofthepropositionandofthetheoremisalmostidenticaltotheproofwegaveaboveforthespecialcaseofsl.Wewillneedonlyoneortwoadditionalarguments.Asinthecaseofsl,thepropositionisaspecialcaseofthetheoremwewanttoprove.Butweshallseethatitissucienttoprovethetheorem.Proofofproposition.ItisenoughtoprovethepropositionforthecasethatVisanirreduciblemodule.Indeed,ifV1isasubmodule,thenbyinductionondimVwemayassumethetheoremisknownfor0!V=V1!W=V1!k!0sothatthereisaonedimensionalinvariantsubspaceMinW=V1supplementarytoV=V1onwhichtheactionistrivial.LetNbetheinverseimageofMinW.Byanotherapplicationoftheproposition,thistimetothesequence0!V1!N!M!0wendaninvariantline,P,inNcomplementarytoV1.SoN=V1P.SinceW=V1=V=V1MwemusthavePV=f0g.ButsincedimW=dimV+1,wemusthaveW=VP.InotherwordsPisaonedimensionalsubspaceofWwhichiscomplementarytoV.

PAGE 71

4.7.COMPLETEREDUCIBILITY.71NextwecanreducetoprovingthepropositionforthecasethatgactsfaithfullyonV.Indeed,leti=thekerneloftheactiononV.Forallx2gwehave,byhypothesis,xWV,andforx2iwehavexV=0.HenceDiactstriviallyonW.Buti=Disinceiissemi-simple.HenceiactstriviallyonWandwemaypasstog=i.Thisquotientisagainsemi-simple,sinceiisasumofsomeofthesimpleidealsofg.SowearereducedtothecasethatVisirreducibleandtheaction,,ofgonVisinjective.ThenwehaveaninvariantelementCwhoseimageinEndWmustmapW!Vsinceeveryelementofgdoes.Wemayassumethatg6=0.Ontheotherhand,C6=0,indeeditstraceisdimg.TherestrictionofCtoVcannothaveanon-trivialkernel,sincethiswouldbeaninvariantsubspace.HencetherestrictionofCtoVisanisomorphism.HencekerC:W!VisaninvariantlinesupplementarytoV.Wehaveprovedtheproposition.Proofoftheoremfromproposition.Let0!E0!Ebeanexactsequenceofgmodules,andwemayassumethatE06=0.WewanttondaninvariantcomplementtoE0inE.DeneWtobethesubspaceofHomkE;E0whoserestrictiontoE0isascalartimestheidentity,andletVWbethesubspaceconsistingofthoselineartransformationswhoserestrictionstoE0iszero.EachoftheseisasubmoduleofEndE.Wegetasequence0!V!W!k!0andhenceacomplementarylineofinvariantelementsinW.Inparticular,wecanndanelement,Twhichisinvariant,mapsE!E0,andwhoserestrictiontoE0isnon-zero.ThenkerTisaninvariantcomplementarysubspace.QEDAsanillustrationofconstructionoftheCasimiroperatorconsiderg=slwithh=100)]TJ/F8 9.963 Tf 7.749 0 Td[(1;e=0100;f=0010:Thentradh2=8tradeadf=4sothedualbasistothebasish;e;fish=8;f=4;e=4,or,ifwedividethemetricby4,thedualbasisish=2;f;eandsotheCasimiroperatorCis1 2h2+ef+fe=1 2h2+h+2fe:ThiscoincideswiththeCthatweusedinChapterII.

PAGE 72

72CHAPTER4.ENGEL-LIE-CARTAN-WEYL

PAGE 73

Chapter5ConjugacyofCartansubalgebras.Itisastandardtheoreminlinearalgebrathatanyunitarymatrixcanbedi-agonalizedbyconjugationbyunitarymatrices.Ontheotherhand,itiseasytocheckthatthesubgroupTUnconsistingofallunitarymatricesisamaximalcommutativesubgroup:anymatrixwhichcommuteswithalldiagonalunitarymatricesmustitselfbediagonal;indeedifAisadiagonalmatrixwithdistinctentriesalongthediagonal,anymatrixwhichcommuteswithAmustbediagonal.NoticethatTisaproductofcircles,i.e.atorus.ThistheoremhasanimmediategeneralizationtocompactLiegroups:LetGbeacompactLiegroup,andletTandT0betwomaximaltori.SoTandT0areconnectedcommutativesubgroupshencenecessarilytoriandeachisnotstrictlycontainedinalargerconnectedcommutativesubgroup.Thenthereexistsanelementa2GsuchthataT0a)]TJ/F7 6.974 Tf 6.227 0 Td[(1=T.Toprovethis,chooseoneparametersubgroupsofTandT0whicharedenseineach.Thatis,choosexandx0intheLiealgebragofGsuchthatthecurvet7!exptxisdenseinTandthecurvet7!exptx0isdenseinT0.Ifwecouldnda2Gsuchthattheaexptx0a)]TJ/F7 6.974 Tf 6.227 0 Td[(1=exptAdax0commutewithalltheexpsx,thenaexptx0a)]TJ/F7 6.974 Tf 6.227 0 Td[(1wouldcommutewithallele-mentsofT,hencebelongtoT,andbycontinuity,aT0a)]TJ/F7 6.974 Tf 6.227 0 Td[(1Tandhence=T.Sowewouldliketondanda2Gsuchthat[Adax0;x]=0:Putapositivedenitescalarproduct;ong,theLiealgebraofGwhichisinvariantundertheadjointactionofG.ThisisalwayspossiblebychoosinganypositivedenitescalarproductandthenaveragingitoverG.Choosea2GsuchthatAdax0;xisamaximum.Lety:=Adax0:73

PAGE 74

74CHAPTER5.CONJUGACYOFCARTANSUBALGEBRAS.Wewishtoshowthat[y;x]=0:Foranyz2gwehave[z;y];x=d dtAdexptzy;xjt=0=0bythemaximality.But[z;y];x=z;[y;x]bytheinvarianceof;,hence[y;x]isorthogonaltoallghence0.QEDWewanttogiveanalgebraicproofoftheanalogueofthistheoremforLiealgebrasoverthecomplexnumbers.Incontrasttotheelementaryproofgivenaboveforcompactgroups,theproofinthegeneralLiealgebracasewillbequiteinvolved,andtheavoroftheproofwillbyquitedierentforthesolvableandsemi-simplecases.Nevertheless,someoftheingredientsoftheaboveproofchoosinggenericelements"analogoustothechoiceofxandx0forexamplewillmaketheirappearance.TheproofsinthischapterfollowHumphreys.5.1Derivations.LetbeaderivationoftheLiealgebrag.thismeansthat[y;z]=[y;z]+[y;z]8y;z2g:Then,fora;b2C)]TJ/F11 9.963 Tf 9.963 0 Td[(a)]TJ/F11 9.963 Tf 9.962 0 Td[(b[y;z]=[)]TJ/F11 9.963 Tf 9.962 0 Td[(ay;z]+[y;)]TJ/F11 9.963 Tf 9.963 0 Td[(bz])]TJ/F11 9.963 Tf 9.962 0 Td[(a)]TJ/F11 9.963 Tf 9.963 0 Td[(b2[y;z]=[)]TJ/F11 9.963 Tf 9.962 0 Td[(a2y;z]+2[)]TJ/F11 9.963 Tf 9.963 0 Td[(ay;)]TJ/F11 9.963 Tf 9.963 0 Td[(bz]+[y;)]TJ/F11 9.963 Tf 9.963 0 Td[(b2z])]TJ/F11 9.963 Tf 9.962 0 Td[(a)]TJ/F11 9.963 Tf 9.963 0 Td[(b3[y;z]=[)]TJ/F11 9.963 Tf 9.962 0 Td[(a3y;z]+3[)]TJ/F11 9.963 Tf 9.963 0 Td[(a2y;)]TJ/F11 9.963 Tf 9.963 0 Td[(bz]+3[)]TJ/F11 9.963 Tf 9.962 0 Td[(ay;)]TJ/F11 9.963 Tf 9.962 0 Td[(b2z]+[y;)]TJ/F11 9.963 Tf 9.963 0 Td[(b3z]......)]TJ/F11 9.963 Tf 9.963 0 Td[(a)]TJ/F11 9.963 Tf 9.962 0 Td[(bn[y;z]=Xnk[)]TJ/F11 9.963 Tf 9.963 0 Td[(aky;)]TJ/F11 9.963 Tf 9.962 0 Td[(bn)]TJ/F10 6.974 Tf 6.227 0 Td[(kz]:Consequences:Letga=gadenotethegeneralizedeigenspacecorrespondingtotheeigenvaluea,so)]TJ/F11 9.963 Tf 9.962 0 Td[(ak=0ongaforlargeenoughk.Then[ga;gb]g[a+b]:.1Lets=sdenotethediagonizablesemi-simplepartof,sothats=aonga.Then,fory2ga;z2gbs[y;z]=a+b[y;z]=[sy;z]+[y;sz]sosandhencealson=n,thenilpotentpartofarebothderivations.

PAGE 75

5.1.DERIVATIONS.75[;adx]=adx].Indeed,[;adx]u=[x;u])]TJ/F8 9.963 Tf 10.423 0 Td[([x;u]=[x;u].Inparticular,thespaceofinnerderivations,InngisanidealinDerg.IfgissemisimplethenInng=Derg.Indeed,splitoaninvariantcomple-menttoInnginDergpossiblebyWeyl'stheoremoncompletereducibil-ity.Foranyinthisinvariantcomplement,wemusthave[;adx]=0since[;adx]=adx.Thissaysthatxisinthecenterofg.Hencex=08xhence=0.Henceanyx2gcanbeuniquelywrittenasx=s+n;s2g;n2gwhereadsissemisimpleandadnisnilpotent.Thisisknownasthedecompositionintosemi-simpleandnilpotentpartsforasemi-simpleLiealgebra.Backtogeneralg.Letkbeasubalgebracontainingg0adxforsomex2g.Thenxbelongsg0adxhencetok,henceadxpreservesNgkbyJacobi'sidentity.Wehavex2g0adxkNgkgallofthesesubspacesbeinginvariantunderadx.Therefore,thecharacter-isticpolynomialofadxrestrictedtoNgkisafactorofthecharactristicpolynomialofadxactingong.Butallthezerosofthischaracteristicpolynomialareaccountedforbythegeneralizedzeroeigenspaceg0adxwhichisasubspaceofk.ThismeansthatadxactsonNgk=kwithoutzeroeigenvalue.Ontheotherhand,adxactstriviallyonthisquotientspacesincex2kandhence[Ngk;x]kbythedenitionofthenormalizer.HenceNgk=k:.2Wenowcometothekeylemma.Lemma3Letkgbeasubalgebra.Letz2kbesuchthatg0adzdoesnotstrictlycontainanyg0adx;x2k.Supposethatkg0adz:Theng0adzg0ady8y2k:Proof.Choosezasinthelemma,andletxbeanarbitraryelementofk.Byhypothesis,x2g0adzandweknowthat[g0adz;g0adz]g0adz.Therefore[x;g0adz]g0adzandhenceadz+cxg0adzg0adzforallconstantsc.Thusadz+cxactsonthequotientspaceg=g0adz.Wecanfactorthecharacteristicpolynomialofadz+cxactingongasPadz+cxT=fT;cgT;c

PAGE 76

76CHAPTER5.CONJUGACYOFCARTANSUBALGEBRAS.wherefisthecharacteristicpolynomialofadz+cxong0adzandgisthecharacteristicpolynomialofadz+cxong=g0adz.WritefT;c=Tr+f1cTr)]TJ/F7 6.974 Tf 6.226 0 Td[(1+frcr=dimg0adzgT;c=Tn)]TJ/F10 6.974 Tf 6.227 0 Td[(r+g1cTn)]TJ/F10 6.974 Tf 6.226 0 Td[(r)]TJ/F7 6.974 Tf 6.227 0 Td[(1++gn)]TJ/F10 6.974 Tf 6.226 0 Td[(rcn=dimg:Thefiandthegiarepolynomialsofdegreeatmostiinc.Since0isnotaneigenvalueofadzong=g0adz,weseethatgn)]TJ/F10 6.974 Tf 6.226 0 Td[(r6=0.Sowecanndr+1valuesofcforwhichgn)]TJ/F10 6.974 Tf 6.227 0 Td[(rc6=0,andhenceforthesevalues,g0adz+cxg0adz:Bytheminimality,thisforcesg0adz+cx=g0adzforthesevaluesofc.ThismeansthatfT;c=Trforthesevaluesofc,soeachofthepolynomialsf1;:::;frhasr+1distinctroots,andhenceisidenticallyzero.Henceg0adz+cxg0adzforallc.Takec=1;x=y)]TJ/F11 9.963 Tf 9.963 0 Td[(ztoconcludethetruthofthelemma.5.2Cartansubalgebras.ACartansubalgebraCSAisdenedtobeanilpotentsubalgebrawhichisitsownnormalizer.ABorelsubalgebraBSAisdenedtobeamaximalsolvablesubalgebra.ThegoalistoproveTheorem10AnytwoCSA'sareconjugate.AnytwoBSA'sareconjugate.Herethewordconjugatemeansthefollowing:DeneNg=fxj9y2g;a6=0;withx2gaadyg:NoticethateveryelementofNgisadnilpotentandthatNgisstableunderAutg.Asanyx2Ngisnilpotent,expadxiswelldenedasanautomorphismofg,andweletEgdenotethegroupgeneratedbytheseelements.Itisanormalsubgroupofthegroupofautomorphisms.Conjugacymeansthatthereisa2Egwithh1=h2whereh1andh2areCSA's.SimilarlyforBSA's.AsarststepwegiveanalternativecharacterizationofaCSA.Proposition13hisaCSAifandonlyifh=g0adzwhereg0adzcon-tainsnopropersubalgebraoftheformg0adx.

PAGE 77

5.3.SOLVABLECASE.77Proof.Supposeh=g0adzwhichisminimalinthesenseoftheproposition.Thenweknowby.2thathisitsownnormalizer.Also,bythelemma,hg0adx8x2h.Henceadxactsnilpotentlyonhforallx2h.Hence,byEngel'stheorem,hisnilpotentandhenceisaCSA.SupposethathisaCSA.Sincehisnilpotent,wehavehg0adx;8x2h.Chooseaminimalz.Bythelemma,g0adzg0adx8x2h:Thushactsnilpotentlyong0adz=h.Ifthisspacewerenotzero,wecouldndanon-zerocommoneigenvectorwitheigenvaluezerobyEngel'stheorem.Thismeansthatthereisay62hwith[y;h]hcontradictingthefacthisitsownnormalizer.QEDLemma4If:g!g0isasurjectivehomomorphismandhisaCSAofgthenhisaCSAofg0.Clearlyhisnilpotent.Letk=Kerandidentifyg0=g=ksoh=h+k.Ifx+knormalizesh+kthenxnormalizesh+k.Buth=g0adzforsomeminimalsuchz,andasanalgebracontainingag0adz,h+kisself-normalizing.Sox2h+k.QEDLemma5:g!g0besurjective,asabove,andh0aCSAofg0.AnyCSAhofm:=)]TJ/F7 6.974 Tf 6.226 0 Td[(1h0isaCSAofg.hisnilpotentbyassumption.Wemustshowitisitsownnormalizering.Bytheprecedinglemma,hisaCartansubalgebraofh0.Buthisnilpotentandhencewouldhaveacommoneigenvectorwitheigenvaluezeroinh0=h,contradictingtheselfnormalizingpropertyofhunlessh=h0.Soh=h0.Ifx2gnormalizesh,thenxnormalizesh0.Hencex2h0sox2msox2h.QED5.3Solvablecase.InthiscaseaBorelsubalgebraisallofgsowemustproveconjugacyforCSA's.Incasegisnilpotent,weknowthatanyCSAisallofg,sinceg=g0adzforanyz2g.Sowemayproceedbyinductionondimg.Leth1andh2beCartansubalgebrasofg.Wewanttoshowthattheyareconjugate.Chooseanabelianidealaofsmallestpossiblepositivedimensionandletg0=g=a.ByLemma4theimagesh01andh02ofh1andh2ing0areCSA'sofg0andhencethereisa02Eg0with0h01=h02.Weclaimthatwecanliftthistoa2Eg.Thatis,weclaimLemma6Let:g!g0beasurjectivehomomorphism.If02Eg0then

PAGE 78

78CHAPTER5.CONJUGACYOFCARTANSUBALGEBRAS.thereexistsa2Egsuchthatthediagramg)334()222()223()333(!g0??y??y0g)334()222()223()333(!g0commutes.Proofoflemma.Itisenoughtoprovethisongenerators.Supposethatx02gay0andchoosey2g;y=y0sogay=gay0,andhencewecanndanx2Ngmappingontox0.Thenexpadxisthedesiredintheabovediagramif0=expadx0.QEDBacktotheproofoftheconjugacytheoreminthesolvablecase.Letm1:=)]TJ/F7 6.974 Tf 6.226 0 Td[(1h01;m2:=)]TJ/F7 6.974 Tf 6.227 0 Td[(1h02.Wehaveawithm1=m2soh1andh2arebothCSA'sofm2.Ifm26=gwearedonebyinduction.Sotheonenewcaseiswhereg=a+h1=a+h2:Writeh2=g0adxforsomex2g.Sinceaisanideal,itisstableunderadxandwecansplititintoits0andnon-zerogeneralizedeigenspaces:a=a0adxaadx:Sinceaisabelian,adofeveryelementofaactstriviallyoneachsummand,andsinceh2=g0adxandaisanideal,thisdecompositionisstableunderh2,henceunderallofg.Byourchoiceofaasaminimalabelianideal,oneortheotherofthesesummandsmustvanish.Ifa=a0adxwewouldhaveah2sog=h2andgisnilpotent.Thereisnothingtoprove.Sotheonlycasetoconsiderisa=aadx.Sinceh2=g0adxwehavea=gadx:Sinceg=h1+a,writex=y+z;y2h1;z2gadx:Sinceadxisinvertibleongadx,writez=[x;z0];z02aadx.Sinceaisanabelianideal,adz02=0,soexpadz0=1+adz0.Soexpadz0x=x)]TJ/F11 9.963 Tf 9.963 0 Td[(z=y:Soh:=g0adyisaCSAofg,andsincey2h1wehaveh1g0ady=handhenceh1=h.Soexpadz0conjugatesh2intoh1.Writingz0assumofitsgeneralizedeigencomponents,andusingthefactthatalltheelementsofacommute,wecanwritetheexponentialasaproductoftheexponentialsofthesummands.QED

PAGE 79

5.4.TORALSUBALGEBRASANDCARTANSUBALGEBRAS.795.4ToralsubalgebrasandCartansubalgebras.ThestrategyisnowtoshowthatanytwoBSA'sofanarbitraryLiealgebraareconjugate.AnyCSAisnilpotent,hencesolvable,hencecontainedinaBSA.ThisreducestheproofoftheconjugacytheoremforCSA'stothatofBSA'sasweknowtheconjugacyofCSA'sinasolvablealgebra.SincetheradicaliscontainedinanyBSA,itisenoughtoprovethistheoremforsemi-simpleLiealgebras.SoforthissectiontheLiealgebragwillbeassumedtobesemi-simple.Sincegdoesnotconsistentirelyofadnilpotentelements,itcontainssomexwhichisnotadnilpotent,andthesemi-simplepartofxisanon-zeroadsemi-simpleelementofg.Asubalgebraconsistingentirelyofsemi-simpleelementsiscalledtoral,forexample,thelinethroughxs.Lemma7Anytoralsubalgebratisabelian.Proof.Theelementsadx;x2tcanbeeachbediagonalized.Wemustshowthatadxhasnoeigenvectorswithnon-zeroeigenvaluesint.Letybeaneigenvectorso[x;y]=ay.Thenadyx=)]TJ/F11 9.963 Tf 7.749 0 Td[(ayisazeroeigenvectorofady,whichisimpossibleunlessay=0,sinceadyannihilatesallitszeroeigenvectorsandisinvertibleonthesubspacespannedbytheeigenvectorscorrespondingtonon-zeroeigenvalues.QEDOneoftheconsequencesoftheconsiderationsinthissectionwillbe:Theorem11Asubalgebrahofasemi-simpleLiealgebragisaCSAifandonlyifitisamaximaltoralsubalgebra.Toprovethiswewanttodevelopsomeofthetheoryofroots.Soxamaximaltoralsubalgebrah.Decomposegintosimultaneouseigenspacesg=CghMghwhereCgh:=fx2gj[h;x]=08h2hgisthecentralizerofh,whererangesovernon-zerolinearfunctionsonhandgh:=fx2gj[h;x]=hx8h2hg:Ashwillbexedformostofthediscussion,wewilldropthehandwriteg=g0Mgwhereg0=Cgh.Wehave[g;g]g+byJacobisoadxisnilpotentifx2g;6=0If+6=0thenx;y=08x2g;y2g.

PAGE 80

80CHAPTER5.CONJUGACYOFCARTANSUBALGEBRAS.Thelastitemfollowsbychoosinganh2hwithh+h6=0.Then0=[h;x];y+x;[h;y]=h+hx;ysox;y=0.Thisimpliesthatg0isorthogonaltoalltheg;6=0andhencethenon-degeneracyofimpliesthatProposition14Therestrictionoftog0g0isnon-degenerate.Ournextintermediatestepistoprove:Proposition15h=g0.3ifhisamaximaltoralsubalgebra.Proceedaccordingtothefollowingsteps:x2g0xs2g0xn2g0:.4Indeed,x2g0,adx:h!0,andthenadxs;adxnalsomaph!0.x2g0;xsemisimplex2h:.5Indeed,suchanxcommuteswithallofh.Asthesumofcommutingsemi-simpletransformationsisagainsemisimple,weconcludethath+Cxisatoralsubalgebra.Bymaximalityitmustcoincidewithh.WenowshowthatLemma8TherestrictionoftheKillingformtohhisnon-degenerate.Sosupposethath;x=08x2h.Thismeansthath;x=08semi-simplex2g0.Supposethatn2g0isnilpotent.Sincehcommuteswithn,adhadnisagainnilpotent.Hencehastracezero.Henceh;n=0,andthereforeh;x=08x2g0.Henceh=0.QEDNextobservethatLemma9g0isanilpotentLiealgebra.Indeed,allsemi-simpleelementsofg0commutewithallofg0sincetheybelongtoh,andanilpotentelementisadnilpotentonallofgsocertainlyong0.Finallyanyx2g0canbewrittenasasumxs+xnofcommutingelementswhichareadnilpotentong0,hencexis.Thusg0consistsentirelyofadnilpotentelementsandhenceisnilpotentbyEngel'stheorem.QEDNowsupposethath2h;x;y2g0.Thenh;[x;y]=[h;x];y=;y=0andhence,bythenon-degeneracyofonh,weconcludethat

PAGE 81

5.5.ROOTS.81Lemma10h[g0;g0]=0:WenextproveLemma11g0isabelian.Supposethat[g0;g0]6=0.Sinceg0isnilpotent,ithasanon-zerocentercon-tainedin[g0;g0].Chooseanon-zeroelementz2[g0;g0]inthiscenter.Itcannotbesemi-simpleforthenitwouldlieinh.Soithasanon-zeronilpotentpart,n,whichalsomustlieinthecenterofg0,bytheBAtheoremweprovedinoursectiononlinearalgebra.Butthenadnadxisnilpotentforanyx2g0since[x;n]=0.Thisimpliesthatn;g0=0whichisimpossible.QEDCompletionofproofof.3.Weknowthatg0isabelian.Butthen,ifh6=g0,wewouldndanon-zeronilpotentelementing0whichcommuteswithallofg0proventobecommutative.Hencen;g0=0whichisimpossible.Thiscompletestheproofof5.3.QEDSowehavethedecompositiong=hM6=0gwhichshowsthatanymaximaltoralsubalgebrahisaCSA.Conversely,supposethathisaCSA.Foranyx=xs+xn2g;g0adxsg0adxsincexnisanadnilpotentelementcommutingwithadxs.Ifwechoosex2hminimalsothath=g0adx,weseethatwemayreplacexbyxsandwriteh=g0adxs.Butg0adxscontainssomemaximaltoralalgebracontainingxs,whichisthenaCartansubalgebracontainedinhandhencemustcoincidewithh.Thiscompletestheproofofthetheorem.QED5.5Roots.Wehaveprovedthattherestrictionoftohisnon-degenerate.Thisallowsustoassociatetoeverylinearfunctiononhtheuniqueelementt2hgivenbyh=t;h:Thesetof2h;6=0forwhichg6=0iscalledthesetofrootsandisdenotedby.Wehavespanshforotherwise9h6=0:h=082implyingthat[h;g]=08so[h;g]=0.2)]TJ/F11 9.963 Tf 20.479 0 Td[(2forotherwiseg?g.x2g;y2g)]TJ/F10 6.974 Tf 6.226 0 Td[(;2[x;y]=x;yt.Indeed,h;[x;y]=[h;x];y=t;hx;y=x;yt;h:

PAGE 82

82CHAPTER5.CONJUGACYOFCARTANSUBALGEBRAS.[g;g)]TJ/F10 6.974 Tf 6.226 0 Td[(]isonedimensionalwithbasist.Thisfollowsfromtheprecedingandthefactthatgcannotbeperpendiculartog)]TJ/F10 6.974 Tf 6.226 0 Td[(sinceotherwiseitwillbeorthogonaltoallofg.t=t;t6=0.Otherwise,choosingx2g;y2g)]TJ/F10 6.974 Tf 6.226 0 Td[(withx;y=1,weget[x;y]=t;[t;x]=[t;y]=0:Sox;y;tspanasolvablethreedimensionalalgebra.Actingasadong,itissuperdiagonizable,byLie'stheorem,andhenceadt,whichisinthecommutatoralgebraofthissubalgebraisnilpotent.Sinceitisadsemi-simplebydenitionofh,itmustlieinthecenter,whichisimpossible.Choosee2g;f2g)]TJ/F10 6.974 Tf 6.227 0 Td[(withe;f=2 t;t:Seth:=2 t;tt:Thene;f;hspanasubalgebraisomorphictosl.Callitsl.Weshallsoonseethatthisnotationisjustied,i.ethatgisonedimensionalandhencethatsliswelldened,independentofanychoices"ofe;fbutdependsonlyon.Considertheactionofslonthesubalgebram:=hLgnwheren2Z.Thezeroeigenvectorsofhconsistofhm.Oneofthesecorrespondstotheadjointrepresentationofslm.Theorthocomplementofh2hgivesdimh)]TJ/F8 9.963 Tf 8.444 0 Td[(1trivialrepresentationsofsl.Thismustexhaustalltheevenmaximalweightrepresentations,aswehaveaccountedforallthezeroweightsofslactingong.Inparticular,dimg=1andnointegermultipleofotherthan)]TJ/F11 9.963 Tf 7.749 0 Td[(isaroot.Nowconsiderthesubalgebrap:=hLgc;c2C.Thisisamoduleforsl.Henceallsuchc'smustbemultiplesof1=2.But1=2cannotoccur,sincethedoubleofarootisnotaroot.Hencethearetheonlymultiplesofwhichareroots.Nowconsider2;6=.Letk:=Mg+j:Eachnon-zerosummandisonedimensional,andkisanslmodule.Also+i6=0foranyi,andevaluationonhgivesh+2i.Allweightsdierbymultiplesof2andsokisirreducible.Letqbethemaximalintegersothat+q2,andrthemaximalintegersothat)]TJ/F11 9.963 Tf 10.337 0 Td[(r2.Thentheentirestring)]TJ/F11 9.963 Tf 9.962 0 Td[(r;)]TJ/F8 9.963 Tf 9.963 0 Td[(r)]TJ/F8 9.963 Tf 9.962 0 Td[(1;:::+q

PAGE 83

5.5.ROOTS.83areroots,andh)]TJ/F8 9.963 Tf 9.963 0 Td[(2r=)]TJ/F8 9.963 Tf 7.749 0 Td[(h+2qorh=r)]TJ/F11 9.963 Tf 9.963 0 Td[(q2Z:TheseintegersarecalledtheCartanintegers.Wecantransferthebilinearformfromhtohbydening;=t;t:Soh=t;h=2t;t t;t=2; ;:So2; ;=r)]TJ/F11 9.963 Tf 9.962 0 Td[(q2Z:Chooseabasis1;:::;`ofhconsistingofroots.Thisispossiblebecausetherootsspanh.Anyrootcanbewrittenuniquelyaslinearcombination=c11++c``wheretheciarecomplexnumbers.Weclaimthatinfacttheciarerationalnumbers.Indeed,takingthescalarproductrelativeto;ofthisequationwiththeigivesthe`equations;i=c11;i++c``;i:Multiplyingthei-thequationby2=i;igivesasetof`equationsforthe`coecientsciwhereallthecoecientsarerationalnumbersasarethelefthandsides.Solvingtheseequationsforthecishowsthattheciarerational.LetEbetherealvectorspacespannedbythe2.Then;restrictstoarealscalarproductonE.Also,forany6=02E,;:=:t;t:=tradt2=X2t2>0:Sothescalarproduct;onEispositivedenite.EisaEuclideanspace.Inthestringofroots,isqstepsdownfromthetop,soqstepsupfromthebottomisalsoaroot,so)]TJ/F8 9.963 Tf 9.963 0 Td[(r)]TJ/F11 9.963 Tf 9.963 0 Td[(q

PAGE 84

84CHAPTER5.CONJUGACYOFCARTANSUBALGEBRAS.isaroot,or)]TJ/F8 9.963 Tf 11.158 6.74 Td[(2; ;2:But)]TJ/F8 9.963 Tf 11.158 6.74 Td[(2; ;=swheresdenotesEuclideanreectioninthehyperplaneperpendicularto.Inotherwords,forevery2s:!:.6ThesubgroupoftheorthogonalgroupofEgeneratedbythesereectionsiscalledtheWeylgroupandisdenotedbyW.Wehavethusassociatedtoeverysemi-simpleLiealgebra,andtoeverychoiceofCartansubalgebraanitesubgroupoftheorthogonalgroupgeneratedbyreections.Thissubgroupisnite,becauseallthegeneratingreections,s,andhencethegrouptheygenerate,preservethenitesetofallroots,whichspanthespace.OncewewillhavecompletedtheproofoftheconjugacytheoremforCartansubalgebrasofasemi-simplealgebra,thenwewillknowthattheWeylgroupisdetermined,uptoisomorphism,bythesemi-simplealgebra,anddoesnotdependonthechoiceofCartansubalgebra.Wedeneh;i:=2; ;:Soh;i=h.7=r)]TJ/F11 9.963 Tf 9.962 0 Td[(q2Z.8ands=)-222(h;i:.9Sofar,wehavedenedthereectionspurelyintermsoftherootstructiononE,whichistherealsubspaceofhgeneratedbytheroots.Butinfact,s,andhencetheentireWeylgrouparisesasanautomorphismsofgwhichpreserveh.Indeed,weknowthate;f;hspanasubalgebraslisomorphictosl.Nowexpadeandexpad)]TJ/F11 9.963 Tf 7.749 0 Td[(fareelementsofEg.Consider:=expadeexpad)]TJ/F11 9.963 Tf 7.748 0 Td[(fexpade2Eg:.10WeclaimthatProposition16Theautomorphismpreserveshandonhitisgivenbyh=h)]TJ/F11 9.963 Tf 9.963 0 Td[(hh:.11Inparticular,thetransformationinducedbyonEiss.

PAGE 85

5.6.BASES.85Proof.Itsucestoprove.11.Ifh=0,thenbothadeandadfvanishonhsoh=hand.11istrue.Nowhandkerspanh.Soweneedonlycheck5.11forhwhereitsaysthath=)]TJ/F11 9.963 Tf 7.748 0 Td[(h.Butwehavealreadyveriedthisforthealgebrasl.QEDWecanalsoverify.11directly.Wehaveexpadeh=h)]TJ/F11 9.963 Tf 9.963 0 Td[(heforanyh2h.Now[f;e]=)]TJ/F11 9.963 Tf 7.749 0 Td[(hsoadf2e=[f;)]TJ/F11 9.963 Tf 7.749 0 Td[(h]=[h;f]=)]TJ/F8 9.963 Tf 7.749 0 Td[(2f:Soexp)]TJ/F8 9.963 Tf 9.409 0 Td[(adfexpadeh=id)]TJ/F8 9.963 Tf 9.963 0 Td[(adf+1 2adf2h)]TJ/F11 9.963 Tf 9.963 0 Td[(he=h)]TJ/F11 9.963 Tf 9.962 0 Td[(he)]TJ/F11 9.963 Tf 9.963 0 Td[(hf)]TJ/F11 9.963 Tf 9.962 0 Td[(hh+hf=h)]TJ/F11 9.963 Tf 9.962 0 Td[(hh)]TJ/F11 9.963 Tf 9.963 0 Td[(he:Ifwenowapplyexpadetothislastexpressionandusethefactthath=2,wegettherighthandsideof5.11.5.6Bases.isacalledaBaseifitisabasisofEso#=`=dimRE=dimChandevery2canbewrittenasP2k;k2Zwitheitherallthecoecientsk0orall0.Rootsareaccordinglycalledpositiveornegativeandwedenetheheightofarootbyht:=Xk:Givenabase,wegetpartialorderonEbydeningi)]TJ/F11 9.963 Tf 9.978 0 Td[(isasumofpositiverootsorzero.Wehave;0;;2.12sinceotherwise;>0ands=)]TJ/F8 9.963 Tf 11.158 6.74 Td[(2; ;isarootwiththecoecientof=1>0andthecoecientof<0,contradict-ingthedenitionwhichsaysthatrootsmusthaveallcoecientsnon-negativeornon-positive.Toconstructabase,choosea2E;j;6=082.Suchanelementiscalledregular.Theneveryroothaspositiveornegativescalarproductwith,dividingthesetofrootsintotwosubsets:=+[)]TJ/F11 9.963 Tf 6.725 -4.113 Td[(;)]TJ/F8 9.963 Tf 9.492 -4.113 Td[(=)]TJ/F8 9.963 Tf 7.749 0 Td[(+:

PAGE 86

86CHAPTER5.CONJUGACYOFCARTANSUBALGEBRAS.Aroot2+iscalleddecomposableif=1+2;1;22+,indecom-posableotherwise.Letconsistoftheindecomposableelementsof+.Theorem12isabase,andeverybaseisoftheformforsome.Proof.Every2+canbewrittenasanon-negativeintegercombinationofforotherwisechooseonethatcannotbesowrittenwith;assmallaspossible.Inparticular,isnotindecomposable.Write=1+2;i2+.Then62;;=;1+;2andhence;1<;and;2<;.Byourchoiceofthismeans1and2arenon-negativeintegercombinationsofelementsofandhencesois,contradiction.Now5.12holdsfor=forifnot,)]TJ/F11 9.963 Tf 8.364 0 Td[(isaroot,soeither)]TJ/F11 9.963 Tf 8.364 0 Td[(2+so=)]TJ/F11 9.963 Tf 9.962 0 Td[(+isdecomposableor)]TJ/F11 9.963 Tf 9.963 0 Td[(2+andisdecomposable.Thisimpliesthatislinearlyindependent:forsupposePc=0andletpbethepositivecoecientsand)]TJ/F11 9.963 Tf 7.748 0 Td[(qthenegativeones,soXp=Xqallcoecientspositive.Letbethiscommonvector.Then;=Ppq;0so=0whichisimpossibleunlessallthecoecientsvanish,sinceallscalarproductswitharestrictlypositive.SincetheelementsofspanEthisshowsthatisabasisofEandhenceabase.Nowletusshowthateverybaseisofthedesiredform:Foranybase,let+=+denotethesetofthoserootswhicharenon-negativeintegralcombinationsoftheelementsofandlet)]TJ/F8 9.963 Tf 9.492 -3.616 Td[(=)]TJ/F8 9.963 Tf 6.724 -3.616 Td[(denotetheoneswhicharenon-positiveintegralcombinationsofelementsof.Dene;2tobetheprojectionofontotheorthogonalcomplementofthespacespannedbytheotherelementsofthebase.Then;0=0;6=0;;=;>0so=Pr;r>0satises;>082hence++and)]TJ/F8 9.963 Tf 6.724 -4.113 Td[()]TJ/F8 9.963 Tf 6.725 -4.113 Td[(hence+=+and)]TJ/F8 9.963 Tf 6.725 -4.113 Td[(=)]TJ/F8 9.963 Tf 6.725 -4.113 Td[(:Sinceeveryelementof+canbewrittenasasumofelementsofwithnon-negativeintegercoecients,theonlyindecomposableelementscanbethe,sobutthentheymustbeequalsincetheyhavethesamecardinality`=dimE.QED

PAGE 87

5.7.WEYLCHAMBERS.875.7Weylchambers.DeneP:=?.ThenE)]TJ/F1 9.963 Tf 10.235 7.472 Td[(SPistheunionofWeylchamberseachcon-sistingofregular'swiththesame+.SotheWeylchambersareinonetoonecorrespondencewiththebases,andtheWeylgrouppermutesthem.Fixabase,.Ourgoalinthissectionistoprovethatthereectionss;2generatetheWeylgroup,W,andthatWactssimplytransitivelyontheWeylchambers.Eachs;2sends7!)]TJ/F11 9.963 Tf 20.478 0 Td[(.Butactingon=Pc,thereectionsdoesnotchangethecoecientofanyotherelementofthebase.If2+and6=,wemusthavec>0forsome6=inthebase.Thenthecoecientofintheexpansionofsispositive,andhenceallitscoecientsmustbenon-negative.Sos2+.Inshort,theonlyelementof+sentinto)]TJ/F8 9.963 Tf 10.056 -3.616 Td[(is.Soif:=1 2X2+thens=)]TJ/F11 9.963 Tf 9.963 0 Td[(:If2+;62,thenwecannothave;082forthen[wouldbelinearlyindependent.So)]TJ/F11 9.963 Tf 9.397 0 Td[(isarootforsome2,andsincewehavechangedonlyonecoecient,itmustbeapositiveroot.Henceany2canbewrittenas=1++pi2whereallthepartialsumsarepositiveroots.LetbeanyvectorinaEuclideanspace,andletsdenotereectioninthehyperplaneorthogonalto.LetRbeanyorthogonaltransformation.ThensR=RsR)]TJ/F7 6.974 Tf 6.227 0 Td[(1.13asfollowsimmediatelyfromthedenition.Let1;:::;i2;and,forshort,letuswritesi:=si.Lemma12Ifs1si)]TJ/F7 6.974 Tf 6.227 0 Td[(1i<0then9j
PAGE 88

88CHAPTER5.CONJUGACYOFCARTANSUBALGEBRAS.Keepingxedintheensuingdiscussion,wewillcalltheelmentsofsim-pleroots,andthecorrespondingreectionssimplereections.LetW0denotethesubgroupofWgeneratedbythesimplereections,s;2.EventuallywewillprovethatthisisallofW.Itnowfollowsthatifs2W0ands=thens=id.Indeed,ifs6=id,writesinaminimalfashionasaproductofsimplereections.Bywhatwehavejustproved,itmustsendsomesimplerootintoanegativeroot.SoW0permutestheWeylchamberswithoutxedpoints.WenowshowthatW0actstransitivelyontheWeylchambers:Let2Ebearegularelement.Weclaim9s2W0withs;>082:Indeed,chooses2W0sothats;isaslargeaspossible.Thens;ss;=s;s=s;)]TJ/F8 9.963 Tf 9.962 0 Td[(s;sos;082:Wecan'thaveequalityinthislastinequalitysincesisnotorthogonaltoanyroot.ThisprovesthatW0actstransitivelyonallWeylchambersandhenceonallbases.Wenextclaimthateveryrootbelongstoatleastonebase.Chooseanon-regular0?;but062P;6=.Thenchoosecloseenoughto0sothat;>0and;
PAGE 89

5.9.CONJUGACYOFBORELSUBALGEBRAS89Proof.Byinductionon`s.If`s=0thens=idandtheassertionisclearwiththeemptyproduct.Sowemayassumethatns>0,sossendssomepositiveroottoanegativeroot,andhencemustsendsomesimpleroottoanegativeroot.Solet2besuchthats2)]TJ/F8 9.963 Tf 6.725 -3.615 Td[(.Since2 C,wehave;0;82+andhence;s0.So0;s=s)]TJ/F7 6.974 Tf 6.227 0 Td[(1;=;0:So;=0sos=andhencess=.Butnss=ns)]TJ/F8 9.963 Tf 10.403 0 Td[(1sinces=)]TJ/F11 9.963 Tf 7.748 0 Td[(andspermutesalltheotherpositiveroots.So`ss=`s)]TJ/F8 9.963 Tf 10.479 0 Td[(1andwecanapplyinductiontoconcludethats=sssisaproductofsimplereectionswhichx.5.9ConjugacyofBorelsubalgebrasWeneedtoprovethisforsemi-simplealgebrassincetheradicaliscontainedineverymaximalsolvablesubalgebra.DeneastandardBorelsubalgebrarelativetoachoiceofCSAhandasystemofsimpleroots,tobeb:=hM2+g:DenethecorrespondingnilpotentLiealgebrabyn+:=M2+g:Sinceeachscanberealizedasexpeexp)]TJ/F11 9.963 Tf 7.749 0 Td[(fexpeeveryelementofWcanberealizedasanelementofEg.HenceallstandardBorelsubalgebrasrelativetoagivenCartansubalgebraareconjugate.NoticethatifxnormalizesaBorelsubalgebra,b,then[b+Cx;b+Cx]bandsob+Cxisasolvablesubalgebracontainingbandhencemustcoincidewithb:Ngb=b:Inparticular,ifx2bthenitssemi-simpleandnilpotentpartslieinb.Fromnowon,xastandardBSA,b.WewanttoprovethatanyotherBSA,b0isconjugatetob.WemayassumethatthetheoremisknownforLiealgebrasofsmallerdimension,orforb0withbb0ofgreaterdimension,sinceifdim

PAGE 90

90CHAPTER5.CONJUGACYOFCARTANSUBALGEBRAS.bb0=dimb,sothatb0b,wemusthaveb0=bbymaximality.Thereforewecanproceedbydownwardinductiononthedimensionoftheintersectionbb0.Supposebb06=0.Letn0bethesetofnilpotentelementsinbb0.Son0=n+b0.Also[bb0;bb0]n+b0=n0son0isanilpotentidealinbb0.Supposethatn06=0.Thensincegcontainsnosolvableideals,k:=Ngn06=g:Considertheactionofn0onb=bb0.ByEngel,thereexistsay62bb0with[x;y]2bb08x2n0.But[x;y]2[b;b]n+andso[x;y]2n0.Soy2k.Thusy2kb;y62bb0.Similarly,wecaninterchangetherolesofbandb0intheaboveargument,replacingn+bythenilpotentsubalgebra[b0;b0]ofb0,toconcludethatthereexistsay02kb0;y062bb0.Inotherwords,theinclusionskbbb0;kb0bb0arestrict.Bothbkandb0karesolvablesubalgebrasofk.Letc;c0beBSA'scontainingthem.Byinduction,thereisa2EkEgwithc0=c.Nowletb00beaBSAcontainingc.Wehaveb00bcbkbb0bwiththelastinclusionstrict.Sobyinductionthereisa2Egwithb00=b.Hencec0b:Thenbb0c0b0b0kbb0withthelastinclusionstrict.Sobyinductionwecanfurtherconjugateb0intob.Sowemustnowdealwiththecasethatn0=0,butwewillstillassumethatbb06=0.SinceanyBorelsubalgebracontainsboththesemi-simpleandnilpotentpartsofanyofitselements,weconcludethatbb0consistsentirelyofsemi-simpleelements,andsoisatoralsubalgebra,callitt.Ifx2b;t2t=bb0and[x;t]2t,thenwemusthave[x;t]=0,sinceallelementsof[b;b]arenilpotent.SoNbt=Cbt:LetcbeaCSAofCbt.SinceaCartansubalgebraisitsownnormalizer,wehavetc.SowehavetcCbt=NbtNbc:Lett2t;n2Nbc.Then[t;n]2candsuccessivebracketsbytwilleventuallyyield0,sincecisnilpotent.Thusadtkn=0forsomek,andsincetissemi-simple,[t;n]=0.Thusn2Cbtandhencen2csincecisitsownnormalizer

PAGE 91

5.9.CONJUGACYOFBORELSUBALGEBRAS91inCbt.ThuscisaCSAofb.WecannowapplytheconjugacytheoremforCSA'sofsolvablealgebrastoconjugatecintoh.Sowemayassumefromnowonthatth.Ift=h,thendecomposingb0intorootspacesunderh,wendthatthenon-zerorootspacesmustconsistentirelyofnegativeroots,andtheremustbeatleastonesuch,sinceb06=h.Butthenwecanndawhichconjugatesthisintoapositiveroot,preservingh,andthenb0bhaslargerdimensionandwecanfurtherconjugateintob.Sowemayassumethatthisstrict.Ifb0CgtthensincewealsohavehCgt,wecanndaBSA,b00ofCgtcontainingh,andconjugateb0tob00,sinceweareassumingthatt6=0andhenceCgt6=g.Sinceb00bhhasbiggerdimensionthanb0b,wecanfurtherconjugatetobbytheinductionhypothesis.Ifb06Cgtthenthereisacommonnon-zeroeigenvectorforadtinb0,callitx.Sothereisat02tsuchthat[t0;x]=c0x;c06=0.Settingt:=1 c0t0wehave[t;x]=x.Lettconsistofthoserootsforwhichtisapositiverationalnumber.Thens:=hM2tgisasolvablesubalgebraandsoliesinaBSA,callitb00.Sincetb00;x2b00weseethatb00b0hasstrictlylargerdimensionthanbb0.Alsob00bhasstrictlylargerdimensionthanbb0sincehbb00.Sowecanconjugateb0tob00andthenb00tob.Thisleavesonlythecasebb0=0whichwewillshowisimpossible.Lettbeamaximaltoralsubalgebraofb0.Wecannothavet=0,forthenb0wouldconsistentirelyofnilpotentelements,hencenilpotentbyEngel,andalsoself-normalizingasiseveryBSA.HenceitwouldbeaCSAwhichisimpossiblesinceeveryCSAinasemi-simpleLiealgebraistoral.SochooseaCSA,h00containingt,andthenastandardBSAcontainingh00.Bythepreceding,weknowthatb0isconjugatetob00and,inparticularhasthesamedimensionasb00.ButthedimensionofeachstandardBSArelativetoanyCartansubalgebraisstrictlygreaterthanhalfthedimensionofg,contradictingthehypothesisgbb0.QED

PAGE 92

92CHAPTER5.CONJUGACYOFCARTANSUBALGEBRAS.

PAGE 93

Chapter6Thesimplenitedimensionalalgebras.InthischapterweclassifyallpossiblerootsystemsofsimpleLiealgebras.Aconsequence,asweshallsee,istheclassicationofthesimpleLiealgebrasthemselves.Theamazingresult-duetoKillingwithsomerepairworkbyElieCartan-isthatwithonlyveexceptions,therootsystemsoftheclassicalalgebrasthatwestudiedinChapterIIIexhaustallpossibilities.Thelogicalstructureofthischapterisasfollows:WerstshowthattherootsystemofasimpleLiealgebraisirreducibledenitionbelow.Wethendevelopsomepropertiesoftheoftherootstructureofanirreduciblerootsystem,inparticularwewillintroduceitsextendedCartanmatrix.WethenusethePerron-Frobeniustheoremtoclassifyallpossiblesuchmatrices.Fortheexpert,thismeansthatwerstclassifytheDynkindiagramsoftheanealgebrasofthesimpleLiealgebras.Surprisingly,thisissimplerandmoreecientthantheclassicationofthediagramsofthenitedimensionalsimpleLiealgebrasthemselves.Fromtheextendeddiagramsitisaneasymattertogetallpossiblebasesofirreduciblerootsystems.WethendevelopafewmorefactsaboutrootsystemswhichallowustoconcludethatanisomorphismofirreduciblerootsystemsimpliesanisomorphismofthecorrespondingLiealgebras.WepostponethetheproofoftheexistenceoftheexceptionalLiealgebrasuntilChapterVIII,whereweproveSerre'stheoremwhichgivesauniedpresentationofallthesimpleLiealgebrasintermsofgeneratorsandrelationsderiveddirectlyfromtheCartanintegersofthesimplerootsystem.Throughoutthischapterwewillbedealingwithsemi-simpleLiealgebrasoverthecomplexnumbers.93

PAGE 94

94CHAPTER6.THESIMPLEFINITEDIMENSIONALALGEBRAS.6.1SimpleLiealgebrasandirreduciblerootsys-tems.WechooseaCartansubalgebrahofasemi-simpleLiealgebrag,sowehavethecorrespondingsetofrootsandtherealEuclideanspaceEthattheyspan.Wesaythatisirreducibleifcannotbepartitionedintotwodisjointsubsets=1[2suchthateveryelementof1isorthogonaltoeveryelementof2.Proposition17Ifgissimplethenisirreducible.Proof.Supposethatisnotirreducible,sowehaveadecompositionasabove.If21and22then+;=;>0and+;=;>0whichmeansthat+cannotbelongtoeither1or2andsoisnotaroot.Thismeansthat[g;g]=0:Inotherwords,thesubalgebrag1ofggeneratedbyalltheg;21iscentralizedbyalltheg,sog1isapropersubalgebraofg,sinceifg1=gthiswouldsaythatghasanon-zerocenter,whichisnottrueforanysemi-simpleLiealgebra.Theaboveequationalsoimpliesthatthenormalizerofg1containsallthegwhererangesoveralltheroots.Buttheseggenerateg.Sog1isaproperidealing,contradictingtheassumptionthatgissimple.QEDLetuschooseabasefortherootsystemofasemi-simpleLiealgebra.Wesaythatisirreducibleifwecannotpartitionintotwonon-emptymutuallyorthogonalsetsasinthedenitionofirreducibilityofasabove.Proposition18isirreducibleifandonlyifisirreducible.Proof.Supposethatisnotirreducible,sohasadecompositionasabove.Thisinducesapartitionofwhichisnon-trivialunlessiswhollycontainedin1or2.If1say,thensinceEisspannedby,thismeansthatalltheelementsof2areorthogonaltoEwhichisimpossible.Soifisirreduciblesois.Conversely,supposethat=1[2isapartitionofintotwonon-emptymutuallyorthogonalsubsets.WehaveprovedthateveryrootisconjugatetoasimplerootbyanelementoftheWeylgroupWwhichisgeneratedbythesimplereections.Let1consistofthoserootswhichareconjugatetoanelementof1and2consistofthoserootswhichareconjugatetoanelementof2.Thereectionss;22commutewiththereectionss;21,andfurthermores=

PAGE 95

6.2.THEMAXIMALROOTANDTHEMINIMALROOT.95since;=0.Soanyelementof1isconjugatetoanelementof1byanelementofthesubgroupW1generatedbythes;21.Buteachsuchreectionaddsorsubtracts.So1isinthesubspaceE1ofEspannedby1andsoisorthogonaltoalltheelementsof2.Soif1isirreduciblesois.QEDWearenowintothebusinessofclassifyingirreduciblebases.6.2Themaximalrootandtheminimalroot.Supposethatisanirreduciblerootsystemandabase,soirreducible.Recallthatoncewehavechosen,everyrootisanintegercombinationoftheelementsofwithallcoecientsnon-negative,orwithallcoecientsnon-positive.Wewrite0intherstcase,and0inthesecondcase.ThisdenesapartialorderontheelementsofEbyifandonlyif)]TJ/F11 9.963 Tf 9.963 0 Td[(=X2k;.1wherethekarenon-negativeintegers.Thispartialorderwillproveveryim-portanttousinrepresentationtheory.Also,forany=Pk2+wedeneitsheightbyht=Xk:.2Proposition19Supposethatisanirreduciblerootsystemandabase.ThenThereexistsaunique2+whichismaximalrelativetotheordering.This=Pkwhereallthekarepositive.;0forall2and;>0foratleastone2.Proof.Choosea=Pkwhichismaximalrelativetotheordering.Atleastoneofthek>0.Weclaimthatallthek>0.Indeed,supposenot.Thispartitionsinto1,thesetofforwhichk>0and2,thesetofforwhichk=0.Nowthescalarproductofanytwodistinctsimplerootsis0.Recallthatthisfollowedfromthefactthatif1;2>0,thens21=1)-236(h1;2i2wouldbearootwhose1coecientispositiveandwhose2coecientisnegativewhichisimpossible.Inparticular,allthe1;20;121;222andso;20;8222:Theirreducibilityofimpliesthat1;26=0foratleastonepair121;222.Butthisscalarproductmustthenbenegative.So;2<0

PAGE 96

96CHAPTER6.THESIMPLEFINITEDIMENSIONALALGEBRAS.andhences2=)-222(h;2i2isarootwiths2)]TJ/F11 9.963 Tf 9.963 0 Td[(0contradictingthemaximalityof.Sowehaveprovedthatallthekarepositive.Furthermore,thissameargumentshowsthat;0forall2.SincetheelementsofformabasisofE,atleastoneofthescalarproductsmustnotvanish,andsobepositive.Wehaveestablishedthesecondandthirditemsinthepropositionforanymaximal.Wewillnowshowthatthismaximalweightisunique.Supposethereweretwo,and0.Write0=Pk0whereallthek0>0.Then;0>0since;0foralland>0foratleastone.Sinces0isaroot,thiswouldimplythat)]TJ/F11 9.963 Tf 9.871 0 Td[(0isaroot,unless=0.Butif)]TJ/F11 9.963 Tf 9.872 0 Td[(0isaroot,itiseitherpositiveornegative,contradictingthemaximalityofoneortheother.QEDLetuslabeltheelementsofas1;:::;`,andletusset0:=)]TJ/F11 9.963 Tf 7.748 0 Td[(sothat0istheminimalroot.Fromthesecondandthirditemsinthepropo-sitionweknowthat0+k11++k``=06.3andthath0;ii0foralliand<0forsomei.Letustakethelefthandsidecallitof.3andsuccessivelycomputeh;ii;i=0;1;:::;`.Weobtain0BBB@2h1;0ih`;0ih0;1i2h`;1i.........h0;`ih`)]TJ/F7 6.974 Tf 6.226 0 Td[(1;`i21CCCA0BBB@1k1...k`1CCCA=0:Thismeansthatifwewritethematrixontheleftofthisequationas2I)]TJ/F11 9.963 Tf 10.181 0 Td[(A,thenAisamatrixwith0onthediagonalandwhosei;jentryishj;ii.SoAanon-negativematrixwithintegerentrieswiththepropertiesifAij6=0thenAji6=0,ThediagonalentriesofAare0,Aisirreducibleinthesensethatwecannotpartitiontheindicesintotwonon-emptysubsetsIandJsuchthatAij=08i2I;j2JandAhasaneigenvectorofeigenvalue2withallitsentriespositive.

PAGE 97

6.3.GRAPHS.97WewillshowthatthePerron-Frobeniustheoremallowsustoclassifyallsuchmatrices.FromhereitisaneasymattertoclassifyallirreduciblerootsystemsandthenallsimpleLiealgebras.Forthisitisconvenienttointroducethelanguageofgraphtheory.6.3Graphs.Anundirectedgraph)-447(=N;EconsistsofasetNforusniteandasubsetEofthesetofsubsetsofNofcardinalitytwo.WecallelementsofNnodes"orvertices"andtheelementsofEedges".Ife=fi;jg2Ewesaythattheedge"ejoinstheverticesiandjorthatiandjareadjacent".Noticethatinthisdenitionouredgesareundirected":fi;jg=fj;ig,andwedonotallowself-loops.Anexampleofagraphisthecycle"A`with`+1vertices,soN=f0;1;2;:::;`gwith0adjacentto`andto1,with1adjacentto0andto2etc.TheadjacencymatrixAofagraph)-427(isthesymmetric0)]TJ/F8 9.963 Tf 10.583 0 Td[(1matrixwhoserowsandcolumnsareindexedbytheelementsofNandwhosei;j-thentryAij=1ifiisadjacenttojandzerootherwise.Forexample,theadjacencymatrixofthegraphA3is0BB@01011010010110101CCA:WecanthinkofAasfollows:LetVbethevectorspacewithbasisgivenbythenodes,sowecanthinkofthei-thcoordinateofavectorx2Vasassigningthevaluexitothenodei.Theny=Axassignstoithesumofthevaluesxjsummedoverallnodesjadjacenttoi.Apathoflengthrisasequenceofnodesxi1;xi2;:::;xirwhereeachnodeisadjacenttothenext.So,forexample,thenumberofpathsoflength2joiningitojisthei;j-thentryinA2andsimilarly,thenumberofpathsoflengthrjoiningitojisthei;j-thentryinAr.Thegraphissaidtobeconnectedifthereisapathofsomelengthjoiningeverypairofvertices.Intermsoftheadjacencymatrix,thismeansthatforeveryiandjthereissomersuchthatthei;jentryofArisnon-zero.Intermsofthetheoryofnon-negativematricesseebelowthissaysthatthematrixAisirreducible.Noticethatif1denotesthecolumnvectorallofwhoseentriesare1,then1isaneigenvectoroftheadjacencymatrixofA`,witheigenvalue2,andalltheentriesof1arepositive.InviewofthePerron-Frobeniustheoremtobestatedbelow,thisimpliesthat2isthemaximumeigenvalueofthismatrix.Wemodifythenotionoftheadjacencymatrixasfollows:Westartwithaconnectedgraph)-269(asbefore,butmodifyitsadjacencymatrixbyreplacingsomeoftheonesthatoccurbypositiveintegersaij.If,inthisreplacementaij>1,weredrawthegraphsothatthereisanarrowwithaijlinespointingtowards

PAGE 98

98CHAPTER6.THESIMPLEFINITEDIMENSIONALALGEBRAS.thenodei.Forexample,thegraphlabeledA1inTableA1correspondstothematrix0220whichclearlyhas11asanpositiveeigenvectorwitheigenvalue2.Similarly,diagramA2inTableA2correspondstothematrix0410whichhas21aseigenvectorwitheigenvalue2.Inthediagrams,thecoecientnexttoanodegivesthecoordinatesoftheeigenvectorwitheigenvalue2,anditisimmediatetocheckfromthediagramthatthisisindeedaneigenvectorwitheigenvalue2.Forexample,the2nexttoanodewithanarrowpointingtowarditinC`satises22=21+2etc.ItwillfollowfromthePerronFrobeniustheoremtobestatedandprovedbelow,thatthesearetheonlypossibleconnecteddiagramswithmaximaleigen-vectortwo.Allthegraphssofarhavezerosalongthediagonal.Ifwerelaxthiscondi-tion,andallowforanynon-negativeintegeronthediagonal,thentheonlynewpossibilitiesarethosegiveninFigure4.LetuscallamatrixsymmetrizableifAij6=0Aji6=0.ThemainresultofthischapterwillbetoshowthatthelistsintheFigures1-4exhaustallirre-duciblematriceswithnon-negativeintegermatrices,whicharesymmetrizableandhavemaximumeigenvalue2.6.4Perron-Frobenius.WesaythatarealmatrixTisnon-negativeorpositiveifalltheentriesofTarenon-negativeorpositive.WewriteT0orT>0.Wewillusethesedenitionsprimarilyforsquarennmatricesandforcolumnvectors=n1matrices.WeletQ:=fx2Rn:x0;x6=0gsoQisthenon-negativeorthant"excludingtheorigin.AlsoletC:=fx0:kxk=1g:SoCistheintersectionoftheorthantwiththeunitsphere.Anon-negativematrixsquareTiscalledprimitiveifthereisaksuchthatalltheentriesofTkarepositive.Itiscalledirreducibleifforanyi;jthereisak=ki;jsuchthatTkij>0.Forexample,asmentionedabove,theadjacencymatrixofaconnectedgraphisirreducible.

PAGE 99

6.4.PERRON-FROBENIUS.99 123456423E8 12343212E7 1232112E6 >12342F4 >123G2HH ...... HH112211D``4 > ...... < 1221C``2HH ...... >11222B``3 ............ `````111A`;`2< >11A1Figure6.1:A1.

PAGE 100

100CHAPTER6.THESIMPLEFINITEDIMENSIONALALGEBRAS. <12321E6< ... >1111D`+1;`2HH ...... < 11221A2`)]TJ/F7 6.974 Tf 6.227 0 Td[(1`3< ...... < 2221A2``2< 21A2Figure6.2:A2 < 121D4Figure6.3:A3

PAGE 101

6.4.PERRON-FROBENIUS.101L01 ..... 11L``1 > ..... 122LC``1 < ..... 111LB``1HH ...... 1122LD``2Figure6.4:Loopsallowed

PAGE 102

102CHAPTER6.THESIMPLEFINITEDIMENSIONALALGEBRAS.IfTisirreduciblethenI+Tisprimitive.InthissectionwewillassumethatTisnon-negativeandirreducible.Theorem13Perron-Frobenius.1.ThasapositiverealeigenvaluemaxsuchthatallothereigenvaluesofTsatisfyjjmax:2.Furthermoremaxhasalgebraicandgeometricmultiplicityone,andhasaneigenvectorxwithx>0.3.Anynon-negativeeigenvectorisamultipleofx.4.Moregenerally,ify0;y6=0isavectorandisanumbersuchthatTyytheny>0;andmaxwith=maxifandonlyifyisamultipleofx.5.If0ST;S6=TtheneveryeigenvalueofSsatisesjj
PAGE 103

6.4.PERRON-FROBENIUS.103applythePerronFrobeniustheoremtoA,andif)]TJ/F13 6.974 Tf 42.895 3.616 Td[(0isapropersubgraphsowehaveactuallydeletedsomerowsandcolumnsofAtoobtainA0,thenthemaximumeigenvalueofA0isstrictlylessthanthemaximumeigenvalueofA0isstrictlylessthanthemaximumeigenvalueofA.Similarly,ifanentryAijis>1,thematrixA0obtainedfromAbydecreasingthisentrywhilestillkeepingitpositivewillhaveastrictlysmallermaximaleigenvalue.WenowapplythistheoremtoconcludethatthediagramslistedinFiguresA1,A,andA3areallpossibleconnecteddiagramswithmaximaleigen-valuetwo.Adirectcheckshowsthatthevectorwhosecoordinateateachnodeistheintegerattachedtothatnodegiveninthegureisaneigenvectorwitheigen-value2.Perron-Frobeniusthenguarantees2isthemaximaleigenvalue.Butnowthatwehaveshownthatforeachofthesediagramsthemaximaleigenvalueistwo,anylarger"diagrammusthavemaximaleigenvaluestrictlygreaterthantwoandanysmaller"diagrammusthavemaximaleigenvaluestrictlylessthantwo.Togetstarted,thisargumentshowsthatA1istheonlydiagramforwhichthereisani;jforwhichbothaijandajiare>1.Indeed,ifAweresuchamatrix,bystrikingoutallbuttheiandjrowsandcolumns,wewouldobtainatwobytwomatrixwhoseodiagonalentriesareboth2.Iftherewerestrictinequality,themaximumeigenvalueofthismatrixwouldhavetobebiggerthan2andhencealsotheoriginaldiagrambyPerronFrobenius.SootherthanA1,wemayassumethatifaij>1thenaji=1.Sinceanydiagramwithsomeentryaij4mustcontainA2weseethatthisistheonlydiagramwiththispropertyandwithmaximumeigenvalue2.Sootherthanthiscase,allaij3.DiagramG2showsthatadiagramwithonlytwoverticesandatriplebondhasmaximumeigenvaluestrictlylessthan2,sinceitiscontainedinG2asasubdiagram.Soanydiagramwithatriplebondmusthaveatleastthreevertices.Butthenitmustcontain"eitherG2orD4.Butasbothofthesehavemaximaleigenvalue2,itcannotstrictlycontaineither.SoG2andD4.aretheonlypossibilitieswithatriplebond.SinceA`;`2isacyclewithmaximumeigenvalue2,nographcancontainacyclewithoutactuallybeingacycle,i.e.beingA`.Ontheotherhand,asimplechainwithonlysinglebondsiscontainedinA`,andsomusthavemaximumeigenvaluestrictlylessthan2,SootherthanA`,everycandidatemustcontainatleastonebranchpointoronedoublebond.Ifthegraphcontainstwodoublebonds,therearethreepossibilitiesastothemutualorientationofthearrows,theycouldpointtowardoneanotherasinC`,awayfromoneanotherasinD`+1orinthesamedirectionasinA2`.Butthenthesearetheonlypossibilitiesfordiagramswithtwodoublebonds,asnodiagramcanstrictlycontainanyofthem.Also,strikingooneendvertexofC`yieldsagraphwithoneextremevertexwithadoublebound,withthearrowpointingawayfromthevertex,and

PAGE 104

104CHAPTER6.THESIMPLEFINITEDIMENSIONALALGEBRAS.nobranchpoints.StrikingoutoneofthetwoverticesattheendoppositethedoublebondinB`yieldsagraphwithoneextremevertexwithwithadoubleboundandwiththearrowpointingtowardthisvertex.Soeitherdiagrammusthavemaximumeigenvalue<2.Thusiftherearenobranchpoints,theremustbeatleastonedoublebondandatleasttwoverticesoneithersideofthedoublebond.ThegraphwithexactlytwoverticesoneithersideisstrictlycontainedinF4andsoisexcluded.Sotheremustbeatleastthreeverticesononesideandtwoontheotherofthedoublebond.ButthenF4andE6exhaustthepossibilitiesforonedoublebondandnobranchpoints.Ifthereisadoublebondandabranchpointtheneitherthedoublebondpointstowardthebranch,asinA2`)]TJ/F7 6.974 Tf 6.227 0 Td[(1orawayfromthebranchasinB`.Butthentheseexhaustthepossibilitiesforadiagramcontainingbothadoublebondandabranchpoint.Iftherearetwobranchpoints,thediagrammustcontainD`andhencemustcoincidewithD`.Soweareleftwiththetaskofanalyzingthepossibilitiesfordiagramswithnodoublebondsandasinglebranchpoint.Letmdenotetheminimumnumberofverticesonsomelegofabranchexcludingthebranchpointitself.Ifm2,thenthediagramcontainsE6andhencemustcoincidewithE6.Sowemayassumethatm=1.Iftwobrancheshaveonlyonevertexemanating,thenthediagramisstrictlycontainedinD`andhenceexcluded.Soeachofthetwootherlegshaveatleasttwoormorevertices.Ifbothlegshavemorethantwoverticesonthem,thegraphmustcontain,andhencecoincidewithE7.Weareleftwiththesolepossibilitythatoneofthelegsemanatingfromthebranchpointhasonevertexandasecondleghastwovertices.ButtheneitherthegraphcontainsoriscontainedinE8soE8istheonlysuchpossibility.WehavecompletedtheproofthatthediagramslistedinA1,A2andA3aretheonlydiagramswithoutloopswithmaximumeigenvalue2.Ifweallowloops,aneasyextensionoftheaboveargumentshowsthattheonlynewdiagramsaretheonesinthetableLoopsallowed".6.5Classicationoftheirreducible.Noticethatifweremoveavertexlabeled1andthebondsemanatingfromitfromanyofthediagramsinA2orA3weobtainadiagramwhichcanalsobeobtainedbyremovingavertexlabeled1fromoneofthediagramsinA1.Inthediagramsoobtainedweignoretheremaininglabels.Indeed,removingtherighthandvertexlabeled1fromD4yieldsA2whichisobtainedfromA2byremovingavertex.Removingtheleftvertexmarked1givesG2,thediagramobtainedfromG2byremovingthevertexmarked1.RemovingavertexfromA2givesA1.Removingthevertexlabeled1fromA2`yieldsB2`,obtainedbyremovingoneoftheverticeslabeled1fromB`.

PAGE 105

6.6.CLASSIFICATIONOFTHEIRREDUCIBLEROOTSYSTEMS.105Removingavertexlabeled1fromA2`)]TJ/F7 6.974 Tf 6.226 0 Td[(1yieldsD2`orC2`,removingavertexlabeled1fromD`+1yieldsB`+1andremovingavertexlabeled1fromE6yieldsF4orC4.Thusallirreduciblecorrespondtographsobtainedbyremovingavertexlabeled1fromthetableA1.SowehaveclassiedallpossibleDynkindiagramsofallirreducible.TheyaregiveninthetablelabeledDynkindiagrams.6.6Classicationoftheirreduciblerootsystems.ItisusefultointroduceheresomenotationduetoBourbaki:AsubsetofaEuclideanspaceEiscalledarootsystemifthefollowingaxiomshold:isnite,spansEanddoesnotcontain0.If2thentheonlymultiplesofwhichareinare.If2thenthereectionsinthehyperplaneorthogonaltosendsintoitself.If;2thenh;i2Z,Recallthath;i:=2; ;sothatthereectionsisgivenbys=)-222(h;i:Wehaveshownthateachsemi-simpleLiealgebragivesrisetoarootsystem,andderivedpropertiesoftherootsystem.Ifwegobacktothevariousarguments,wewillndthatmostofthemapplytoageneral"rootsystemaccordingtotheabovedenition.TheoneplacewhereweusedLiealgebraargumentsdirectly,wasinshowingthatif6=isarootthenthecollectionofjsuchthat+jisarootformsanunbrokenchaingoingfrom)]TJ/F11 9.963 Tf 7.749 0 Td[(rtoqwherer)]TJ/F11 9.963 Tf 10.542 0 Td[(q=h;i.Forthisweusedtherepresentationtheoryofsl.Sowenowpausetogiveanalternativeproofofthisfactbasedsolelyonthataxiomsabove,andintheprocessderivesomeadditionalusefulinformationaboutroots.Foranytwonon-zerovectorsandinE,thecosineoftheanglebetweenthemisgivenbykkkkcos=;:Soh;i=2kk kkcos:Interchangingtheroleofandandmultiplyinggivesh;ih;i=4cos2:

PAGE 106

106CHAPTER6.THESIMPLEFINITEDIMENSIONALALGEBRAS. E8 E7 E6 >F4 >G2 ...... HHD``4 ...... < C``2 ...... >B``2 ......... A`;`1Figure6.5:Dynkindiagrams.

PAGE 107

6.6.CLASSIFICATIONOFTHEIRREDUCIBLEROOTSYSTEMS.107Therighthandsideisanon-negativeintegerbetween0and4.Soassumingthat6=andkkkkThepossibilitiesarelistedinthefollowingtable:h;ih;i0kk2=kk2 00=2undetermined11=31)]TJ/F8 9.963 Tf 7.749 0 Td[(1)]TJ/F8 9.963 Tf 7.749 0 Td[(12=3112=42)]TJ/F8 9.963 Tf 7.749 0 Td[(1)]TJ/F8 9.963 Tf 7.749 0 Td[(13=4213=63)]TJ/F8 9.963 Tf 7.749 0 Td[(1)]TJ/F8 9.963 Tf 7.749 0 Td[(35=63Proposition20If6=andif;>0then)]TJ/F11 9.963 Tf 9.149 0 Td[(isaroot.If;<0then+isaroot.Proof.Thesecondassertionfollowsfromtherstbyreplacingby)]TJ/F11 9.963 Tf 7.749 0 Td[(.Soweneedtoprovetherstassertion.Fromthetable,oneortheotherofh;iorh;iequalsone.Soeithers=)]TJ/F11 9.963 Tf 10.025 0 Td[(isarootors=)]TJ/F11 9.963 Tf 10.025 0 Td[(isaroot.Butrootsoccuralongwiththeirnegativessoineitherevent)]TJ/F11 9.963 Tf 10.393 0 Td[(isaroot.QEDProposition21Supposethat6=areroots.Letrbethelargestintegersuchthat)]TJ/F11 9.963 Tf 9.802 0 Td[(risaroot,andletqbethelargestintegersuchthat+qisaroot.Then+iisarootforall)]TJ/F11 9.963 Tf 7.748 0 Td[(riq.Furthermorer)]TJ/F11 9.963 Tf 10.136 0 Td[(q=h;isoinparticularjq)]TJ/F11 9.963 Tf 9.963 0 Td[(rj3.Proof.Supposenot.Thenwecanndapandanssuchthat)]TJ/F11 9.963 Tf 7.749 0 Td[(rp0.Nowsaddsamultipleoftoanyroot,andsopreservesthestringofroots)]TJ/F11 9.963 Tf 9.963 0 Td[(r;)]TJ/F8 9.963 Tf 9.962 0 Td[(r)]TJ/F8 9.963 Tf 9.963 0 Td[(1;:::;+q.Furthermores+i=)]TJ/F8 9.963 Tf 9.963 0 Td[(h;i+isosreversestheorderofthestring.Inparticulars+q=)]TJ/F11 9.963 Tf 9.963 0 Td[(r:Thelefthandsideis)]TJ/F8 9.963 Tf 7.898 0 Td[(h;i+qsor)]TJ/F11 9.963 Tf 7.897 0 Td[(q=h;iasstatedintheproposition.QEDWecannowapplyalltheprecedingdenitionsandargumentstoconcludethattheDynkindiagramsaboveclassifyalltheirreduciblebasesofrootsystems.

PAGE 108

108CHAPTER6.THESIMPLEFINITEDIMENSIONALALGEBRAS.Sinceeveryrootisconjugatetoasimpleroot,wecanusetheDynkindia-gramstoconcludethatinanirreduciblerootsystem,eitherallrootshavethesamelengthcasesA,D,Eortherearetworootlengths-theremainingcases.Furthermore,ifdenotesalongrootandashortroot,theratioskk2=kk2are2inthecasesB;C;andF4,and3forthecaseG2.Proposition22Inanirreduciblerootsystem,theWeylgroupWactsirre-duciblyonE.Inparticular,theW-orbitofanyrootspansE.Proof.LetE0beaproperinvariantsubspace.LetE00denoteitsorthogonalcomplement,soE=E0E00:Foranyroot,Ife2E0thense=e)-228(he;i2E0.Soeithere;=0foralle,andso2E00or2E0.Sincetherootsspan,theycan'tallbelongtothesamesubspace.Thiscontradictstheirreducibility.QEDProposition23Iftherearetwodistinctrootlengthsinanirreduciblerootsystem,thenallrootsofthesamelengthareconjugateundertheWeylgroup.Also,themaximalweightislong.Proof.Supposethatandhavethesamelength.WecanndaWeylgroupelementWsuchthatwisnotorthogonaltobytheprecedingproposition.Sowemayassumethath;i6=0.Sinceandhavethesamelength,bythetableabovewehaveh;i=1.Replacingby)]TJ/F11 9.963 Tf 7.748 0 Td[(=swemayassumethath;i=1.Thensss=ss)]TJ/F11 9.963 Tf 9.963 0 Td[(=s)]TJ/F11 9.963 Tf 7.749 0 Td[()]TJ/F11 9.963 Tf 9.963 0 Td[(+=s)]TJ/F11 9.963 Tf 7.749 0 Td[(=:QEDLetE;andE0;0betworootsystems.Wesaythatalinearmapf:E!E0isanisomorphismfromtherootsystemE;totherootsystemE0;0iffisalinearisomorphismofEontoE0withf=0andhf;fi=h;iforall;2.Theorem14Let=f1;:::;`gbeabaseof.SupposethatE0;0isasecondrootsystemwithbase0=f01;:::;0`gandthathi;ji=h0i;0ji;81i;j`:Thenthebijectioni7!0i

PAGE 109

6.7.THECLASSIFICATIONOFTHEPOSSIBLESIMPLELIEALGEBRAS.109extendstoauniqueisomorphismf:E;!E0;0.Inotherwords,theCartanmatrixAofdeterminesuptoisomorphism.Inparticular,TheDynkindiagramscharacterizeallpossibleirreduciblerootsystems.Proof.SinceisabasisofEand0isabasisofE0,themapi7!0iextendstoauniquelinearisomorphismofEontoE0.Theequalityinthetheoremimpliesthatfor;2wehavesff=f)-222(hf;fif=fs:SincetheWeylgroupsaregeneratedbythesesimplereections,thisimpliesthatthemapw7!fwf)]TJ/F7 6.974 Tf 6.227 0 Td[(1isanisomorphismofWontoW0.Every2isoftheformwwherew2Wandisasimpleroot.Thusf=fwf)]TJ/F7 6.974 Tf 6.226 0 Td[(1f20sof=0.Sinces=)-65(h;i,thenumberh;iisdeterminedbythereectionsactingon.Butthenthecorrespondingformulafor0togetherwiththefactthatsf=fsf)]TJ/F7 6.974 Tf 6.226 0 Td[(1impliesthathf;fi=h;i:QED6.7TheclassicationofthepossiblesimpleLiealgebras.Supposethatg;h;isapairconsistingofasemi-simpleLiealgebrag,andaCartansubalgebrah.ThisdeterminesthecorrespondingEuclideanspaceEandrootsystem.Supposewehaveasecondsuchpairg0;h0.WewouldliketoshowthatanisomorphismofE;withE0;0determinesaLiealgebraisomorphismofgwithg0.ThiswouldthenimplythattheDynkindiagramsclassifyallpossiblesimpleLiealgebras.WewouldstillbeleftwiththeproblemofshowingthattheexceptionalLiealgebrasexist.WewilldeferthisuntilChapterVIIIwhereweproveSerre'stheoremwithgivesadirectconstructionofallthesimpleLiealgebrasintermsofgeneratorsandrelationsdeterminedbytheCartanmatrix.Weneedafewpreliminaries.Proposition24Everypositiverootcanbewrittenasasumofsimplerootsi1+ikinsuchawaythateverypartialsumisagainaroot.

PAGE 110

110CHAPTER6.THESIMPLEFINITEDIMENSIONALALGEBRAS.Proof.Byinductiononsaytheheightitisenoughtoprovethatforeverypositiverootwhichisnotsimple,thereisasimplerootsuchthat)]TJ/F11 9.963 Tf 10.01 0 Td[(isaroot.Wecannothave;0forall2forthiswouldimplythatthesetfg[isindependentbythesamemethodthatweusedtoprovethatwasindependent.So;>0forsome2andso)]TJ/F11 9.963 Tf 10.066 0 Td[(isaroot.Sinceisnotsimple,itsheightisatleasttwo,andsosubtractingwillnotbezerooranegativeroot,hencepositive.QEDProposition25Letg;hbeasemi-simpleLiealgebrawithachoiceofCartansubalgebra,Letbethecorrespondingrootsystem,andletbeabase.ThengisgeneratedasaLiealgebrabythesubspacesg;g)]TJ/F10 6.974 Tf 6.227 0 Td[(;2.Fromtherepresentationtheoryofslweknowthat[g;g]=g+if+isaroot.Thusfromtheprecedingproposition,wecansuccessivelyobtainallthegforpositivebybracketingtheg;2.Similarlywecangetallthegfornegativefromtheg)]TJ/F10 6.974 Tf 6.226 0 Td[(.Sowecangetalltherootspaces.But[g;g)]TJ/F10 6.974 Tf 6.227 0 Td[(]=Chsowecangetallofh.Thedecompositiong=hM2gthenshowsthatwehavegeneratedallofg.Hereisthebigtheorem:Theorem15Letg;handg0;h0besimpleLiealgebraswithchoicesofCartansubalgebras,andlet;0bethecorrespondingrootsystems.Supposethereisanisomorphismf:E;!E0;0whichisanisometryofEuclideanspaces.Extendftoanisomorphismofh!h0viacomplexication.Letf:h!h0denotethecorrespondingisomorphismontheCartansubalgebrasobtainedbyidentifyinghandh0withtheirdualsusingtheKillingform.Fixabaseofand0of0.Choose06=x2g;2and06=x002g00.Extendftoalinearmapf:hM2g!h0M020g0byfx=x00:Thenfextendstoauniqueisomorphismofg!g0.Proof.Theuniquenessiseasy.Givenxthereisauniquey2g)]TJ/F10 6.974 Tf 6.227 0 Td[(forwhich[x;y]=hsof,ifitexists,isdeterminedontheyandhenceonallofgsincethexandygenerategbetheprecedingproposition.

PAGE 111

6.7.THECLASSIFICATIONOFTHEPOSSIBLESIMPLELIEALGEBRAS.111Toprovetheexistence,wewillconstructthegraphofthisisomorphism.Thatis,wewillconstructasubalgebrakofgg0whoseprojectionsontotherstandontothesecondfactorareisomorphisms:Usethexandyasabove,withthecorrespondingelementsx00andy00ing0.Let x:=xx002gg0andsimilarlydene y:=yy00;and h:=hh00:Letbetheuniquemaximalrootofg,andchoosex2g.Makeasimilarchoiceofx02g00where0isthemaximalrootofg0.Set x:=xx0:Letmgg0bethesubspacespannedbyallthead yi1ad yim x:Theelementad yi1ad yim xbelongstog)]TJ/F41 6.974 Tf 6.227 5.23 Td[(Pijg00)]TJ/F41 6.974 Tf 6.227 5.23 Td[(P0ijsomgg00isonedimensional:Inparticularmisapropersubspaceofgg0.Letkdenotethesubalgebraofgg0generatedbythe xthe yandthe h.Weclaimthat[k;m]m:Indeed,itisenoughtoprovethatmisinvariantundertheadjointactionofthegeneratorsofk.Forthead ythisfollowsfromthedenition.Forthead hweusethefactthat[h;y]=)]TJ/F11 9.963 Tf 7.749 0 Td[(hytomovethead hpastallthead yatthecostofintroducingsomescalarmultiple,whilead h x=h;ix+h0;0ix00=h;i xbecausefisanisomorphismofrootsystems.Finally[x1;y2]=0if16=22since1)]TJ/F11 9.963 Tf 10.303 0 Td[(2isnotaroot.Ontheotherhand[x;y]=h.Sowecanmovethead xpastthead yattheexpenseofintroducinganad heverytime=.Now+isnotaroot,sinceisthemaximalroot.So[x;x]=0.Thusad x x=0,andwehaveprovedthat[k;m]m.Butsincemisapropersubspaceofgg0,thisimpliesthatkisapropersubalgebra,sinceotherwisemwouldbeaproperideal,andtheonlyproperidealsingg0aregandg0.

PAGE 112

112CHAPTER6.THESIMPLEFINITEDIMENSIONALALGEBRAS.Nowthesubalgebrakcannotcontainanyelementoftheformz0;z6=0,foritifdid,itwouldhavetocontainalloftheelementsoftheformu0sincewecouldrepeatedlyapplyadx'suntilwereachedthemaximalrootspaceandthengetallofg0,whichwouldmeanthatkwouldalsocontainallof0g0andhenceallofgg0whichweknownottobethecase.Similarlykcannotcontainanyelementoftheform0z0.Sotheprojectionsofkontogandontog0arelinearisomorphisms.ByconstructiontheyareLiealgebrahomomorphisms.Hencetheinverseoftheprojectionofkontogfollowedbytheprojectionofkontog0isaLiealgebraisomorphismofgontog0.Byconstructionitsendsxtox00andhtoh0andsoisanextensionoff.QED

PAGE 113

Chapter7Cyclichighestweightmodules.Inthischapter,gwilldenoteasemi-simpleLiealgebraforwhichwehavechosenaCartansubalgebra,handabasefortheroots=+[)]TJ/F8 9.963 Tf 10.046 -3.615 Td[(ofg.Wewillbeinterestedindescribingitsnitedimensionalirreduciblerepre-sentations.IfWisanitedimensionalmoduleforg,thenhhasatleastonesimultaneouseigenvector;thatisthereisa2handaw6=02Wsuchthathw=hw8h2h:.1Thelinearfunctioniscalledaweightandthevectorviscalledaweightvector.Ifx2g,hxw=[h;x]w+xhw=+hxw:Thisshowsthatthespaceofallvectorswsatisfyinganequationofthetype.1forvaryingspansaninvariantsubspace.IfWisirreducible,thentheweightvectorsthosesatisfyinganequationofthetype.1mustspanallofW.Furthermore,sinceWisnitedimensional,theremustbeavectorvandalinearfunctionsuchthathv=hv8h2h;ev=0;82+:.2Usingirreducibilityagain,weconcludethatW=Ugv:Themoduleiscyclicgeneratedbyv.Infactwecanbemoreprecise:Leth1;:::;h`bethebasisofhcorrespondingtothechoiceofsimpleroots,letei2gi;fi2g)]TJ/F10 6.974 Tf 6.227 0 Td[(iwhere1;:::;mareallthepositiveroots.Wecanchoosethemsothateacheandfgeneratealittlesl.Theng=n)]TJ/F14 9.963 Tf 8.939 1.495 Td[(hn+;113

PAGE 114

114CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.wheree1;:::;emisabasisofn+,whereh1;:::;h`isabasisofh,andf1;:::;fmisabasisofn)]TJ/F8 9.963 Tf 6.725 1.494 Td[(.ThePoincare-Birkho-Witttheoremsaysthatmonomialsoftheformfi11fimmhj11hj``ek11ekmmformabasisofUg.Herewehavechosentoplaceallthee'stotheextremeright,withtheh'sinthemiddleandthef'stotheleft.Itnowfollowsthattheelementsfi11fimmvspanW.Everysuchelement,ifnon-zero,isaweightvectorwithweight)]TJ/F8 9.963 Tf 9.962 0 Td[(i11++imm:Recallthatmeansthat)]TJ/F11 9.963 Tf 9.963 0 Td[(=Xkii;i>0;wherethekiarenon-negativeintegers.WehaveshownthateveryweightofWsatises:Sowemakethedenition:Acyclichighestweightmoduleforgisamodulenotnecessarilynitedimensionalwhichhasavectorv+suchthatx+v+=0;8x+2n+;hv+=hv+8h2handV=Ugv+:InanysuchcyclichighestweightmoduleeverysubmoduleisadirectsumofitsweightspacesbyvanderMonde.TheweightspacesVallsatisfyandwehaveV=MV:Anypropersubmodulecannotcontainthehighestweightvector,andsothesumoftwopropersubmodulesisagainapropersubmodule.HenceanysuchVhasauniquemaximalsubmoduleandhenceauniqueirreduciblequotient.Thequotientofanyhighestweightmodulebyaninvariantsubmodule,ifnotzero,isagainacyclichighestweightmodulewiththesamehighestweight.7.1Vermamodules.Thereisabiggest"cyclichighestweightmodule,associatedwithany2hcalledtheVermamodule.Itisdenedasfollows:Letussetb:=hn+:

PAGE 115

7.2.WHENISDIMIRR<1?115Givenany2hletCdenotetheonedimensionalvectorspaceCwithbasisz+andwiththeactionofbgivenbyh+X0xz+:=hz+:SoitisaleftUbmodule.BythePoincareBirkhoWitttheorem,UgisafreerightUbmodulewithbasisffi11fim`g,andsowecanformtheVermamoduleVerm:=UgUbCwhichisacyclicmodulewithhighestweightvectorv+:=1z+.Furthermore,anytwoirreduciblecyclichighestweightmoduleswiththesamehighestweightareisomorphic.Indeed,ifVandWaretwosuchwithhighestweightvectorv+;u+,considerVWwhichhasv+;u+asamaximalweightvectorwithweight,andhenceZ:=Ugv+;u+iscyclicandofhighestweight.Projectionsontotherstandsecondfactorsgivenon-zerohomomorphismswhichmustbesurjective.ButZhasauniqueirreduciblequotient.Hencethesemustinduceisomorphismsonthisquotient,VandWareisomorphic.Hence,uptoisomorphism,thereisauniqueirreduciblecyclichighestweightmodulewithhighestweight.WecallitIrr:Inshort,wehaveconstructedalargest"highestweightmoduleVermandasmallest"highestweightmoduleIrr.7.2WhenisdimIrr<1?IfIrrisnitedimensional,thenitisnitedimensionalasamoduleoveranysubalgebra,inparticularoveranysubalgebraisomorphictosl.Appliedtothesubalgebrasligeneratedbyei;hi;fiweconcludethathi2Z:Suchaweightisiscalledintegral.Furthermoretherepresentationtheoryofslsaysthatthemaximalweightforanynitedimensionalrepresentationmustsatisfyhi=h;ii0sothatliesintheclosureofthefundamentalWeylchamber.Suchaweightiscalleddominant.SoanecessaryconditionforIrrtobenitedimensionalisthatbedominantintegral.Wenowshowthatconversely,Irrisnitedimensionalwheneverisdominantintegral.ForthiswerecallthatintheuniversalenvelopingalgebraUgwehave1.[ej;fk+1i]=0,ifi6=j

PAGE 116

116CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.2.[hj;fk+1i]=)]TJ/F8 9.963 Tf 7.749 0 Td[(k+1ihjfk+1i3.[ei;fk+1i]=)]TJ/F8 9.963 Tf 7.749 0 Td[(k+1fkik1)]TJ/F11 9.963 Tf 9.963 0 Td[(hiwherethersttwoequationsareconsequencesofthefactthatadisaderivationand[ei;fj]=0ifi6=jsincei)]TJ/F11 9.963 Tf 9.963 0 Td[(jisnotarootand[hj;fj]=)]TJ/F11 9.963 Tf 7.748 0 Td[(jhifj:ThelastisathefactaboutslwhichwehaveprovedinChapterII.Noticethatitfollowsfrom1.thatejfkiv+=0forallkandalli6=jandfrom3.thateifhi+1iv+=0sothatfhi+1iv+isamaximalweightvector.Ifitwerenon-zero,thecyclicmoduleitgenerateswouldbeapropersubmoduleofIrrcontradictingtheirreducibility.Hencefhi+1iv+=0:Soforeachithesubspacespannedbyv+;fiv+;;fhiiv+isanitedimen-sionalslimodule.InparticularIrrcontainssomenitedimensionalslimodules.LetV0denotethesumofallsuch.IfWisanitedimensionalslimodule,theneWisagainnitedimensional,thussotheirsum,whichisanitedimensionalslimodule.HenceV0isg-stable,henceallofIrr.Inparticular,theeiandthefiactaslocallynilpotentoperatorsonIrr.Sotheoperatorsi:=expeiexp)]TJ/F11 9.963 Tf 7.749 0 Td[(fiexpeiarewelldenedandiIrr=IrrsisodimIrrw=dimIrr8w2W.3whereWdenotestheWeylgroup.Theseareallnitedimensionalsubspaces:IndeedtheirdimensionisatmostthecorrespondingdimensionintheVermamoduleVerm,sinceIrrisaquotientspaceofVerm.ButVermhasabasisconsistingofthosefk11fkmmv+.Thenumberofsuchelementsisthenumberofwaysofwriting)]TJ/F11 9.963 Tf 9.962 0 Td[(=k11+kmm:SodimVermisthenumberofm-tupletsofnon-negativeintegersk1;:::;kmsuchthattheaboveequationholds.Thisnumberisclearlynite,andisknownasPK)]TJ/F11 9.963 Tf 9.515 0 Td[(,theKostantpartitionfunctionof)]TJ/F11 9.963 Tf 9.515 0 Td[(,whichwillplayacentralroleinwhatfollows.NoweveryelementofEisconjugateunderWtoanelementoftheclosureofthefundamentalWeylchamber,i.e.toasatisfying;i0

PAGE 117

7.3.THEVALUEOFTHECASIMIR.117i.e.toathatisdominant.Weclaimthatthereareonlynitelymanydominantweightswhichare,whichwillcompletetheproofofnitedimensionality.Indeed,thesumoftwodominantweightsisdominant,so+isdominant.Ontheotherhand,)]TJ/F11 9.963 Tf 9.963 0 Td[(=Pkiiwiththeki0.So;)]TJ/F8 9.963 Tf 9.963 0 Td[(;u=+;)]TJ/F11 9.963 Tf 9.963 0 Td[(=Xki+;i0:Soliesintheintersectionoftheballofradiusp ;withthediscretesetofweightswhichisnite.Werecordaconsequenceof.3whichisusefulunderveryspecialcircum-stances.Supposewearegivenanitedimensionalrepresentationofgwiththepropertythateachweightspaceisonedimensionalandallweightsareconju-gateunderW.Thenthisrepresentationmustbeirreducible.Forexample,takeg=sln+1andconsidertherepresentationofgon^kCn+1;1kn.Intermsofthestandardbasise1;:::;en+1ofCn+1theelementsei1^^eikareweightvectorswithweightsLi1++Lik,WherehconsistsofalldiagonaltracelessmatricesandLiisthelinearfunctionwhichassignstoeachdiagonalmatrixitsi-thentry.TheseweightspacesareallonedimensionalandconjugateundertheWeylgroup.Hencetheserepresentationsareirreduciblewithhighestweight!i:=L1++Lkintermsoftheusualchoiceofbase,h1;:::;hnwherehjisthediagonalmatrixwith1inthej-thposition,)]TJ/F8 9.963 Tf 7.748 0 Td[(1inthej+1-stpositionandzeroselsewhere.Noticethat!ihj=ijsothatthe!iformabasisoftheweightlattice"consistingofthose2hwhichtakeintegralvaluesonh1;:::;hn.7.3ThevalueoftheCasimir.RecallthatourbasisofUgconsistsoftheelementsfi11fimmhj11hj``ek11ekmm:TheelementsofUharethentheoneswithnoeorfcomponentintheirexpression.SowehaveavectorspacedirectsumdecompositionUg=UhUgn++n)]TJ/F11 9.963 Tf 6.725 1.495 Td[(Ug;wheren+andn)]TJ/F8 9.963 Tf 10.739 1.495 Td[(arethecorrespondingnilpotentsubalgebras.Letdenoteprojectionontotherstfactorinthisdecomposition.Nowsupposez2Zg,thecenteroftheuniversalenvelopingalgebra.Inparticular,z2Ugh.Theeigenvaluesofthemonomialaboveundertheactionofh2haremXs=1ks)]TJ/F11 9.963 Tf 9.962 0 Td[(issh:

PAGE 118

118CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.Soanymonomialintheexpressionforzcannothaveffactorsalone.Wehaveprovedthatz)]TJ/F11 9.963 Tf 9.962 0 Td[(z2Ugn+;8z2Zg:.4Forany2h,theelementz2Zgactsasascalar,callitzontheVermamoduleassociatedto.Inparticular,ifisadominantintegralweight,itactsbythissamescalarontheirreduciblenitedimensionalmoduleassociatedto.Ontheotherhand,thelinearmap:h!Cextendstoahomomorphism,whichwewillalsodenotebyofUh=Sh!C.Explicitly,ifwethinkofelementsofUh=Shaspolynomialsonh,thenP=PforP2Sh.Sincen+v=0ifvisthemaximalweightvector,weconcludefrom.4thatz=z8z2Zg:.5WewanttoapplythisformulatothesecondorderCasimirelementassociatedtotheKillingform.Soletk1;:::;k`2hbethedualbasistoh1;:::;h`relativeto,i.e.hi;kj=ij:Letx2gbeabasisi.e.non-zeroelementandz2g)]TJ/F10 6.974 Tf 6.227 0 Td[(bethedualbasiselementtoxundertheKillingform,sothesecondorderCasimirelementisCas=Xhiki+Xxz:wherethesecondsumontherightisoverallroots.Wemightchoosethex=eforpositiveroots,andthenthecorrespondingzissomemultipleofthef.And,forpresentpurposeswemightevenchoosef=zforpositive.TheproblemisthatthezforpositiveintheaboveexpressionforCasarewrittentotheright,andwemustmovethemtotheleft.SowewriteCas=Xihiki+X>0[x;z]+X>0zx+X<0xz:ThisexpressionforCashasallthen+elementsmovedtotheright;inpartic-ular,allofthesummandsinthelasttwosumsannihilatev.HenceCas=Xihiki+X>0[x;z]andCas=Xihiki+X>0[x;z]:Foranyh2hwehaveh;[x;z]=[h;x];z=hx;z=hso[x;z]=t

PAGE 119

7.3.THEVALUEOFTHECASIMIR.119wheret2hisuniquelydeterminedbyt;h=h8h2h:Let;denotethebilinearformonhobtainedfromtheidenticationofhwithhgivenby.ThenX>0[x;z]=X>0t=X>0;=2;.6where:=1 2X>0:Ontheotherhand,lettheconstantsaibedenedbyh=Xiaihi;h8h2h:InotherwordscorrespondstoPaihiundertheisomorphismofhwithhso;=Xi;jaiajhi;hj:Sincehi;kj=ijwehaveki=ai:Combinedwithhi=Pjajhj;hithisgives;=Xihiki:.7Combinedwith7.6thisyieldsCas=+;+)]TJ/F8 9.963 Tf 9.963 0 Td[(;:.8Wenowusethisinnocuouslookingformulatoprovethefollowing:WeletL=LghRdenotethelatticeofintegrallinearformsonh,i.e.L=f2hj2; ;2Z82g:.9Liscalledtheweightlatticeofg.For;2Lrecallthatif)]TJ/F11 9.963 Tf 9.963 0 Td[(isasumofpositiveroots.Then

PAGE 120

120CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.Proposition26AnycyclichighestweightmoduleZ;2Lhasacomposi-tionserieswhosequotientsareirreduciblemodules,Irrwheresatises+;+=+;+:.10Infact,ifd=XdimZwherethesumisoverallsatisfying.10thenthereareatmostdstepsinthecompositionseries.Remark.Thereareonlynitelymany2Lsatisfying.10sincethesetofallsatisfying.10iscompactandLisdiscrete.Eachweightisofnitemultiplicity.Thereforedisnite.Proofbyinductionond.Werstshowthatifd=1thenZisirreducible.Indeed,ifnot,anypropersubmoduleW,beingthesumofitsweightspaces,musthaveahighestweightvectorwithhighestweight,say.ButthenCas=CassinceWisasubmoduleofZandCastakesontheconstantvalueCasonZ.Thusandbothsatisfy.10contradictingtheassumptiond=1.Ingeneral,supposethatZisnotirreducible,sohasasubmodule,WandquotientmoduleZ=W.Eachoftheseisacyclichighestweightmodule,andwehaveacorrespondingcompositionseriesoneachfactor.Inparticular,d=dW+dZ=Wsothatthed'sarestrictlysmallerforthesubmoduleandthequotientmodule.Hencewecanapplyinduction.QEDForeach2Lweintroduceaformalsymbol,ewhichwewanttothinkofasanexponential"andsothesymbolsaremultipliedaccordingtotheruleee=e+:.11ThecharacterofamoduleNisdenedaschN=XdimNe:Inallcaseswewillconsidercyclichighestweightmodulesandthelikeallthesedimensionswillbenite,sothecoecientsarewelldened,butinthecaseofVermamodulesforexampletheremaybeinnitelymanytermsintheformalsum.Logically,suchaformalsumisnothingotherthanafunctiononLgivingthecoecient"ofeache.InthecasethatNisnitedimensional,theabovesumisnite.Iff=Xfeandg=Xgearetwonitesums,thentheirproductusingtherule.11correspondstoconvolution:XfeXge=Xf?ge

PAGE 121

7.4.THEWEYLCHARACTERFORMULA.121wheref?g:=X+=fg:SoweletZnLdenotethesetofZvaluedfunctionsonLwhichvanishoutsideaniteset.Itisacommutativeringunderconvolution,andwewilloscillateinnotationbetweenwritinganelementofZnLasanexponentialsum"thinkingofitasafunctionofnitesupport.SincewealsowanttoconsiderinnitesumssuchasthecharactersofVermamodules,weenlargethespaceZnLbydeningZgenLtoconsistofZval-uedfunctionswhosesupportsarecontainedinniteunionsofsetsoftheform)]TJ/F1 9.963 Tf 10.194 7.472 Td[(P0k.TheconvolutionoftwofunctionsbelongingtoZgenLiswelldened,andbelongstoZgenL.SoZgenLisagainaring.ItnowfollowsfromProp.26thatchZ=XchIrrwherethesumisoverthenitelymanytermsinthecompositionseries.Inparticular,wecanapplythistoZ=Verm,theVermamodule.Letusordertheisatisfying.10insuchawaythatijij.ThenforeachofthenitelymanyioccurringwegetacorrespondingformulaforchVermiandsowegetcollectionofequationschVermj=XaijchIrriwhereaii=1andijinthesum.WecaninvertthisuppertriangularmatrixandthereforeconcludethatthereisaformulaoftheformchIrr=XbchVerm.12wherethesumisoversatisfying.10withcoecientsbthatweshallsoondetermine.Butwedoknowthatb=1.7.4TheWeylcharacterformula.WewillnowproveProposition27Thenon-zerocoecientsin7.12occuronlywhen=w+)]TJ/F11 9.963 Tf 9.963 0 Td[(wherew2W,theWeylgroupofg,andthenb=)]TJ/F8 9.963 Tf 7.748 0 Td[(1w:Here)]TJ/F8 9.963 Tf 7.749 0 Td[(1w:=detw:

PAGE 122

122CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.Wewillprovethisbyprovingsomecombinatorialfactsaboutmultiplicationofsumsofexponentials.Werecallournotation:For2h,Irrdenotestheuniqueirreduciblemoduleofhighestweight,,andVermdenotestheVermamoduleofhighestweight,andmoregenerally,Zdenotesanarbitrarycyclicmoduleofhighestweight.Also:=1 2X2+isonehalfthesumofthepositiveroots.Leti;i=1;:::;dimhbethebasisoftheweightlattice,Ldualtothebase.Soihj=hi;ji=ij:Sincesii=)]TJ/F11 9.963 Tf 7.749 0 Td[(iwhilekeepingalltheotherpositiverootspositive,wesawthatthisimpliedthatsi=)]TJ/F11 9.963 Tf 9.962 0 Td[(iandthereforeh;ii=1;i=1;:::;`:=dimh:Inotherwords=1 2X2+=1++`:.13TheKostantpartitionfunction,PKisdenedasthenumberofsetsofnon-negativeintegers,ksuchthat=X2+k:Thevalueiszeroifcannotbeexpressedasasumofpositiveroots.ForanymoduleNandany2h,Ndenotestheweightspaceofweight.Forexample,intheVermamodule,Verm,theonlynon-zeroweightspacesaretheoneswhere=)]TJ/F1 9.963 Tf 10.801 7.472 Td[(P2+kandthemultiplicityofthisweightspace,i.e.thedimensionofVermisthenumberofwaysofexpressinginthisfashion,i.e.dimVerm=PK)]TJ/F11 9.963 Tf 9.962 0 Td[(:.14IntermsofthecharacternotationintroducedintheprecedingsectionwecanwritethisaschVerm=XPK)]TJ/F11 9.963 Tf 9.962 0 Td[(e:TobeconsistentwithHumphreys'notation,denetheKostantfunctionpbyp=PK)]TJ/F11 9.963 Tf 7.748 0 Td[(andtheninsuccinctlanguagechVerm=p)]TJ/F11 9.963 Tf 14.944 0 Td[(:.15

PAGE 123

7.4.THEWEYLCHARACTERFORMULA.123Observethatiff=Xfethenfe=Xfe+=Xf)]TJ/F11 9.963 Tf 9.962 0 Td[(e:Wecanexpressthisbysayingthatfe=f)]TJ/F11 9.963 Tf 14.944 0 Td[(:Thus,forexample,chVerm=p)]TJ/F11 9.963 Tf 14.944 0 Td[(=pe:Alsoobservethatiff=1 1)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F11 9.963 Tf 7.749 0 Td[(:=1+e)]TJ/F11 9.963 Tf 7.749 0 Td[(+e)]TJ/F8 9.963 Tf 7.749 0 Td[(2+then)]TJ/F11 9.963 Tf 9.963 0 Td[(e)]TJ/F11 9.963 Tf 7.749 0 Td[(f=1andY2+f=pbythedenitionoftheKostantfunction.Denethefunctionqbyq:=Y2+e=2)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F11 9.963 Tf 7.748 0 Td[(=2=eY)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F11 9.963 Tf 7.748 0 Td[(sincee=Q2+e=2.Noticethatwq=)]TJ/F8 9.963 Tf 7.749 0 Td[(1wq:Itisenoughtocheckthisonfundamentalreections,buttheyhavethepropertythattheymakeexactlyonepositiverootnegative,hencechangethesignofq.Wehaveqp=e:.16Indeed,qpe)]TJ/F11 9.963 Tf 7.749 0 Td[(=hY)]TJ/F11 9.963 Tf 9.963 0 Td[(e)]TJ/F11 9.963 Tf 7.749 0 Td[(iepe)]TJ/F11 9.963 Tf 7.749 0 Td[(=hY)]TJ/F11 9.963 Tf 9.963 0 Td[(e)]TJ/F11 9.963 Tf 7.749 0 Td[(ip=Y)]TJ/F11 9.963 Tf 9.963 0 Td[(e)]TJ/F11 9.963 Tf 7.749 0 Td[(Yf=1:Therefore,qchVerm=qpe=ee=e+:

PAGE 124

124CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.Letusnowmultiplybothsidesof.12byqandusetheprecedingequation.WeobtainqchIrr=Xbe+wherethesumisoverallsatisfying.10,andthebarecoecientswemustdetermine.NowchIrrisinvariantundertheWeylgroupW,andqtransformsby)]TJ/F8 9.963 Tf 7.749 0 Td[(1w.Henceifweapplyw2Wtotheprecedingequationweobtain)]TJ/F8 9.963 Tf 7.749 0 Td[(1wqchIrr=Xbew+:Thisshowsthatthesetof+withnon-zerocoecientsisstableunderWandthecoecientstransformbythesignrepresentationforeachWorbit.Inparticular,eachelementoftheform=w+)]TJ/F11 9.963 Tf 8.407 0 Td[(has)]TJ/F8 9.963 Tf 7.749 0 Td[(1wasitscoecient.WecanthuswriteqchV=Xw2W)]TJ/F8 9.963 Tf 7.749 0 Td[(1wew++RwhereRisasumoftermscorrespondingto+whicharenotoftheformw+.WeclaimthattherearenosuchtermsandhenceR=0.Indeed,ifthereweresuchaterm,thetransformationpropertiesunderWwoulddemandthattherebesuchatermwith+intheclosureoftheWeylchamber,i.e.+2:=LDwhereD=Dg=f2Ej;082+gandE=hRdenotesthespaceofreallinearcombinationsoftheroots.Butweclaimthat;+;+=+;+;&+2==:Indeed,write=)]TJ/F11 9.963 Tf 9.963 0 Td[(;=Pk;k0so0=+;+)]TJ/F8 9.963 Tf 9.963 0 Td[(+;+=+;+)]TJ/F8 9.963 Tf 9.963 0 Td[(+)]TJ/F11 9.963 Tf 9.963 0 Td[(;+)]TJ/F11 9.963 Tf 9.962 0 Td[(=+;+;++;since+20since+2andinfactliesintheinteriorofD.Butthelastinequalityisstrictunless=0.Hence=0.Wewillhaveoccasiontousethistypeofargumentseveraltimesagaininthefuture.InanyeventwehavederivedthefundamentalformulaqchIrr=Xw2W)]TJ/F8 9.963 Tf 7.749 0 Td[(1wew+:.17

PAGE 125

7.5.THEWEYLDIMENSIONFORMULA.125Noticethatifwetake=0andsothetrivialrepresentationwithcharacter1forV,.17becomesq=X)]TJ/F8 9.963 Tf 7.749 0 Td[(1wewandthisispreciselythedenominatorintheWeylcharacterformula:WCFchIrr=Pw2W)]TJ/F8 9.963 Tf 7.749 0 Td[(1wew+ Pw2W)]TJ/F8 9.963 Tf 7.748 0 Td[(1wew.187.5TheWeyldimensionformula.Foranyweight,wedeneA:=Xw2W)]TJ/F8 9.963 Tf 7.749 0 Td[(1wew:ThenwecanwritetheWeylcharacterformulaaschIrr=A+ A:ForanyweightdenethehomomorphismfromtheringZnLintotheringofformalpowerseriesinonevariabletbytheformulae=e;tandextendlinearly.ThelefthandsideoftheWeylcharacterformulabelongstoZnL,andhencesodoestherighthandsidewhichisaquotientoftwoelementsofZnL.ThereforeforanywehavechIrr=A+ A:A=A.19foranypairofweights.Indeed,A=Xw)]TJ/F8 9.963 Tf 7.749 0 Td[(1we;wt=Xw)]TJ/F8 9.963 Tf 7.749 0 Td[(1wew)]TJ/F6 4.981 Tf 5.397 0 Td[(1;t=X)]TJ/F8 9.963 Tf 7.749 0 Td[(1wew;t=A:

PAGE 126

126CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.Inparticular,A=A=q=Ye=2)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F11 9.963 Tf 7.748 0 Td[(=2=Y2+e;t=2)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F7 6.974 Tf 6.227 0 Td[(;t=2=Y;t#++termsofhigherdegreeint:HencechIrr=A+ A=Q+; Q;+termsofpositivedegreeint:Nowconsiderthecompositehomomorphism:rstapplyandthensett=0.Thishastheeectofreplacingeveryebytheconstant1.HenceappliedtothelefthandsideoftheWeylcharacterformulathisgivesthedimensionoftherepresentationIrr.ThepreviousequationshowsthatwhenthiscompositehomomorphismisappliedtotherighthandsideoftheWeylcharacterformula,wegettherighthandsideoftheWeyldimensionformula:dimIrr=Q2++; Q2+;:.207.6TheKostantmultiplicityformula.Letusmultiplythefundamentalequation.17bype)]TJ/F11 9.963 Tf 7.748 0 Td[(andusethefact.16thatqpe)]TJ/F11 9.963 Tf 7.749 0 Td[(=1toobtainchIrr=Xw2W)]TJ/F8 9.963 Tf 7.748 0 Td[(1wpe)]TJ/F11 9.963 Tf 7.749 0 Td[(ew+:Butpe)]TJ/F11 9.963 Tf 7.749 0 Td[(ew+=p)]TJ/F11 9.963 Tf 14.944 0 Td[(w++or,inmorepedestrianterms,thelefthandsideofthisequationhas,asitscoecientofethevaluep+)]TJ/F11 9.963 Tf 9.963 0 Td[(w+:Ontheotherhand,bydenition,chIrr=XdimIrre:WethusobtainKostant'sformulaforthemultiplicityofaweightintheirreduciblemodulewithhighestweight:KMFdimIrr=Xw2W)]TJ/F8 9.963 Tf 7.748 0 Td[(1wp+)]TJ/F11 9.963 Tf 9.963 0 Td[(w+:.21

PAGE 127

7.7.STEINBERG'SFORMULA.127ItwillbeconvenienttointroducesomenotationwhichsimpliestheappearanceoftheKostantmultiplicityformula:Forw2Wand2LorinEforthatmatterdenew:=w+)]TJ/F11 9.963 Tf 9.963 0 Td[(:.22ThisdenesanotheractionofWonEwheretheoriginoftheorthogonaltransformationswhasbeenshiftedfrom0to)]TJ/F11 9.963 Tf 7.749 0 Td[(".ThenwecanrewritetheKostantmultiplicityformulaasdimIrr=Xw2W)]TJ/F8 9.963 Tf 7.749 0 Td[(1wPKw)]TJ/F11 9.963 Tf 9.962 0 Td[(.23oraschIrr=Xw2WX)]TJ/F8 9.963 Tf 7.749 0 Td[(1wPKw)]TJ/F11 9.963 Tf 9.962 0 Td[(e;.24wherePKistheoriginalKostantpartitionfunction.Forthepurposesofthenextsectionitwillbeusefultorecordthefollowinglemma:Lemma14Ifisadominantweightande6=w2Wthenwisnotdominant.Proof.Ifisdominant,soliesintheclosureofthepositiveWeylchamber,then+liesintheinteriorofthepositiveWeylchamber.Henceifw6=e,thenw+hi<0forsomei,andsow=w+)]TJ/F11 9.963 Tf 9.531 0 Td[(isnotdominant.QED7.7Steinberg'sformula.Supposethat0and00aredominantintegralweights.DecomposeIrr0Irr00intoirreducibles,andletn=n;000denotethemultiplicityofIrrinthisdecompositionintoirreducibleswithn=0ifIrrdoesnotappearasasummandinthedecomposition.Inparticular,n=0ifisnotadominantweightsinceIrrisinnitedimensionalinthiscase,socannotappearasasummandinthedecomposition.Intermsofcharacters,wehavechIrr0chIrr00=XnchIrr:Steinberg'sformulaisaformulaforn.Toderiveit,usetheWeylcharacterformulachIrr00=A00+ A;chIrr=A+ AintheaboveformulatoobtainchIrr0A00+=XnA+:

PAGE 128

128CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.UsetheKostantmultiplicityformula.24for0:chIrr0=Xw2WX)]TJ/F8 9.963 Tf 7.749 0 Td[(1wPKw0)]TJ/F11 9.963 Tf 9.962 0 Td[(eandthedenitionA00+=Xu2W)]TJ/F8 9.963 Tf 7.748 0 Td[(1ueu00+andthesimilarexpressionforA+togetXXu;w2W)]TJ/F8 9.963 Tf 7.749 0 Td[(1uwPKw0)]TJ/F11 9.963 Tf 9.962 0 Td[(eu00++=XXwn)]TJ/F8 9.963 Tf 7.749 0 Td[(1wew+:Letusmakeachangeofvariablesontherighthandside,writing=wsotherighthandsidebecomesXXw)]TJ/F8 9.963 Tf 7.748 0 Td[(1wnw)]TJ/F7 6.974 Tf 6.227 0 Td[(1e+:Ifisadominantweight,thenbyLemma14w)]TJ/F7 6.974 Tf 6.227 0 Td[(1isnotdominantifw)]TJ/F7 6.974 Tf 6.227 0 Td[(16=e.Sonw)]TJ/F7 6.974 Tf 6.227 0 Td[(1=0ifw6=1andsothecoecientofe+ispreciselynwhenisdominant.Onthelefthandsidelet=)]TJ/F11 9.963 Tf 9.963 0 Td[(u00toobtainX;u;w)]TJ/F8 9.963 Tf 7.748 0 Td[(1uwPKw0+u00)]TJ/F11 9.963 Tf 9.963 0 Td[(e+:Comparingcoecientsfordominantgivesn=Xu;w)]TJ/F8 9.963 Tf 7.749 0 Td[(1uwPKw0+u00)]TJ/F11 9.963 Tf 9.963 0 Td[(:.257.8TheFreudenthal-deVriesformula.Wereturntothestudyofasemi-simpleLiealgebragandgetarenementoftheWeyldimensionformulabylookingatthenextordertermintheexpansionweusedtoderivetheWeyldimensionformulafromtheWeylcharacterformula.Bydenition,theKillingformrestrictedtotheCartansubalgebrahisgivenbyh;h0=Xhh0

PAGE 129

7.8.THEFREUDENTHAL-DEVRIESFORMULA.129wherethesumisoverallroots.If;2hwitht;ttheelementsofHcorrespondingtothemundertheKillingform,wehave;=t;t=Xttso;=X;;:.26ForeachintheweightlatticeLwehaveletedenotetheformalexponential"soZfinListhespacespannedbytheeandwehavedenedthehomomorphism:Zfin!C[[t]];e7!e:t:LetNandDbetheimagesunderoftheWeylnumeratoranddenominator.SoN=A+=+Aby.19andA=q=Y2+e=2)]TJ/F11 9.963 Tf 9.963 0 Td[(e)]TJ/F10 6.974 Tf 6.226 0 Td[(=2.27andthereforeNt=Y>0e+;t=2)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F7 6.974 Tf 6.227 0 Td[(+;t=2=Y+;t[1+1 24+;2t2+]withasimilarformulaforD.ThenN=D!d=thedimensionoftherepresentationast!0istheusualproofthatwereproducedaboveoftheWeyldimensionformula.StickingthisintoN=DgivesN D=d1+1 24X>0[+;2)]TJ/F8 9.963 Tf 9.963 0 Td[(;2]t2+!:Foranyweightwehave;=P;2by.26,wherethesumisoverallrootssoN D=d1+1 48[+;+)]TJ/F8 9.963 Tf 9.963 0 Td[(;]t2+;andwerecognizethecoecientof1 48t2intheaboveexpressionasCas,thescalargivingthevalueoftheCasimirassociatedtotheKillingformintherepresentationwithhighestweight.Ontheotherhand,theimageunderofthecharacteroftheirreduciblerepresentationwithhighestweightisXe;t=X+;t+1 2;2t2+

PAGE 130

130CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.wherethesumisoverallweightsintheirreduciblerepresentationcountedwithmultiplicity.ComparingcoecientsgivesX;2=1 24dCas:Appliedtotheadjointrepresentationthelefthandsidebecomes;by.26,whiledisthedimensionoftheLiealgebra.Ontheotherhand,Cas=1sincetradCas=dimgbythedenitionofCas.Soweget;=1 24dimg.28foranysemisimpleLiealgebrag.AnalgebrawhichisthedirectsumacommutativeLieandasemi-simpleLiealgebraiscalledreductive.ThepreviousresultofFreudenthalanddeVrieshasbeengeneralizedbyKostantfromasemi-simpleLiealgebratoallreductiveLiealgebras:Supposethatgismerelyreductive,andthatwehavechosenasymmtricbilinearformongwhichisinvariantundertheadjointrepresentation,anddenotetheassociatedCasimirelementbyCasg.Weclaimthat.28generalizesto1 24tradCasg=;:.29NoticethatifgissemisimpleandwetakeoursymmetricbilinearformtobetheKillingform;.29becomes.28.Toprove.29observethatbothsidesdecomposeintosumsaswedecomposegintoassumofitscenteranditssimpleideals,sincethismustbeanorthogonaldecompositionforourinvariantscalarproduct.Thecontributionofthecenteriszeroonbothsides,sowearereducedtoproving.29forasimplealgebra.Thenoursymmetricbiinearform;mustbeascalarmultipleoftheKillingform:;=c2;forsomenon-zeroscalarc.Ifz1;:::;zNisanorthonormalbasisofgfor;thenz1=c;:::;zN=cisanorthonormalbasisfor;.ThusCasg=1 c2Cas:SotradCasg=1 c2tradCas=1 c21 24dimg:Butonhwehavethedualrelation;=1 c2;:Combiningthelasttwoequationsshowsthat.29becomes.28.Noticethatthesameproofshowsthatwecangeneralize.8asCas=+;+)]TJ/F8 9.963 Tf 9.963 0 Td[(;.30validforanyreductiveLiealgebraequippedwithasymmetricbilinearforminvariantundertheadjointrepresentation.

PAGE 131

7.9.FUNDAMENTALREPRESENTATIONS.1317.9Fundamentalrepresentations.Welet!idenotetheweightwhichsatises!ihj=ijsothatthe!iformanintegralbasisofLandaredominant.Wecallthesethebasicweights.IfV;andW;aretwonitedimensionalirreduciblerepresentationswithhighestweightsand,thenVW;containstheirreduciblerepresentationwithhighestweight+,andhighestweightvectorvw,thetensorproductofthehighestweightvectorsinVandW.Tak-ingthishighest"componentinthetensorproductisknownastheCartanproductofthetwoirreduciblerepresentations.LetVi;ibetheirreduciblerepresentationscorrespondingtothebasicweight!i.TheneverynitedimensionalirreduciblerepresentationofgcanbeobtainedbyCartanproductsfromthese,andforthatreasontheyarecalledthefundamentalrepresentations.ForthecaseofAn=sln+1wehavealreadyveriedthatthefundamentalrepresentationsare^kVwhereV=Cn+1andwherethebasicweightsare!i=L1++LiWenowsketchtheresultsfortheotherclassicalsimplealgebras,leavingthedetailsasanexerciseintheuseoftheWeyldimensionformula.ForCn=spnitisimmediatetocheckthatthesesameexpressionsgivethebasicweights.HoweverwhileV=C2n=^1Visirreducible,thehigherorderexteriorpowersarenot:Indeed,thesymplecticform2^2Vispreserved,andhencesoisthethemap^jV!^j)]TJ/F7 6.974 Tf 6.227 0 Td[(2Vgivenbycontractionby.Itiseasytocheckthattheimageofthismapissurjectiveforj=2;:::;n.thekernelisthusaninvariantsubspaceofdimension2nj)]TJ/F1 9.963 Tf 9.962 14.048 Td[(2n2j)]TJ/F8 9.963 Tf 9.963 0 Td[(2andanotcompletelytrivialapplicationoftheWeyldimensionformulawillshowthattheseareindeedthedimensionsoftheirreduciblerepresentationswithhighestweight!j.ThusthesekernelsarethefundamentalrepresentationsofCn.Herearesomeofthedetails:Wehave=!1++!n=Xn)]TJ/F11 9.963 Tf 9.962 0 Td[(i+1Li:ThemostgeneraldominantweightisoftheformXki!i=a1L1++anLn

PAGE 132

132CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.wherea1=k1++kn;a2=k2++kn;an=knwherethekiarenon-negativeintegers.Sowecanequallywelluseanydecreasingsequencea1a2an0ofintegerstoparameterizetheirreduciblerepresentations.Wehave;Li)]TJ/F11 9.963 Tf 9.963 0 Td[(Lj=j)]TJ/F11 9.963 Tf 9.963 0 Td[(i;;Li+Lj=2n+2)]TJ/F11 9.963 Tf 9.962 0 Td[(i)]TJ/F11 9.963 Tf 9.962 0 Td[(j:MultiplyingthesealltogethergivesthedenominatorintheWeyldimensionformula.SimilarlythenumeratorbecomesYi2.Inapplyingtheprecedingformula,allofthetermswith2
PAGE 133

7.10.EQUALRANKSUBGROUPS.133andthetermsr1+1;r2+1contributeafactorn+1 n)]TJ/F8 9.963 Tf 9.963 0 Td[(1:Inmultiplyingallofthesetermstogetherthereisahugecancellationandwhatisleftforthedimensionofthisfundamentalrepresentationisn+1n)]TJ/F8 9.963 Tf 9.962 0 Td[(2 2:Noticethatthisequals2n2)]TJ/F8 9.963 Tf 9.963 0 Td[(1=dim^2V)]TJ/F8 9.963 Tf 9.963 0 Td[(1:Moregenerallythisdimensionargumentwillshowthatthefundamentalrepre-sentationsarethekernelsofthecontractionmapsi:^k!V^k)]TJ/F7 6.974 Tf 6.227 0 Td[(2Vwhereisthesymplecticform.ForBnitiseasytocheckthat!i:=L1++Liin)]TJ/F8 9.963 Tf 10.232 0 Td[(1;and!n=1 2L1++LnarethebasicweightsandtheWeyldimensionformulagivesthevalue2n+1jforjn)]TJ/F8 9.963 Tf 10.538 0 Td[(1asthedimensionsoftheirreducibleswiththeseweight,sothattheyare^jV;j=1;:::n)]TJ/F8 9.963 Tf 9.214 0 Td[(1whilethedimensionoftheirreduciblecorrespondingto!nis2n.Thisisthespinrepresentationwhichwewillstudylater.Finally,forDn=onthebasicweightsare!j=L1++Lj;jn)]TJ/F8 9.963 Tf 9.962 0 Td[(2;and!n)]TJ/F7 6.974 Tf 6.226 0 Td[(1:=1 2L1++Ln)]TJ/F7 6.974 Tf 6.226 0 Td[(1+Lnand!n:=1 2L1++Ln)]TJ/F7 6.974 Tf 6.226 0 Td[(1)]TJ/F11 9.963 Tf 9.962 0 Td[(Ln:TheWeyldimensionformulashowsthatthetherstn)]TJ/F8 9.963 Tf 9.933 0 Td[(2fundamentalrepre-sentationsareinfacttherepresentationon^jV;j=1;:::;n)]TJ/F8 9.963 Tf 10.283 0 Td[(2whilethelasttwohavedimension2n)]TJ/F7 6.974 Tf 6.227 0 Td[(1.Thesearethehalfspinrepresentationswhichwewillalsostudylater.7.10Equalranksubgroups.InthissectionwepresentageneralizationoftheWeylcharacterformuladuetoRamond-Gross-Kostant-Sternberg.ItdependsonaninterpretationoftheWeyldenominatorintermsofthespinrepresentationoftheorthogonalgroupOg=h,andsoonsomeresultswhichwewillproveinChapterIX.Butitslogicalplaceisinthischapter.Sowewillquotetheresultsthatwewillneed.YoumightprefertoreadthissectionafterChapterIX.

PAGE 134

134CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.Letpbeanevendimensionalspacewithasymmetricbilinearsuchthatp=p+p)]TJ/F8 9.963 Tf -194.078 -18.871 Td[(isadirectsumdecompositionofpintotwoisotropicsubspaces.Inotherwordsp+andp)]TJ/F8 9.963 Tf 9.855 1.494 Td[(areeachhalfthedimensionofp,andthescalarproductofanytwovectorsinp+vanishes,asdoesthescalarproductofanytwoelementsofp)]TJ/F8 9.963 Tf 6.725 1.495 Td[(.Forexample,wemighttakep=n+n)]TJ/F8 9.963 Tf 10.336 1.495 Td[(andthesymmetricbilinearformtobetheKillingform.Thenp=nissuchadesireddecomposition.Thesymmetricbilinearformthenputspintoduality,i.e.wemayidentifyp)]TJ/F8 9.963 Tf 10.373 1.494 Td[(withp+andviceversa.SupposethatwehaveacommutativeLiealgebrahactingonpasinnitesimalisometries,soastopreserveeachp,thatthee+iareweightvectorscorrespondingtoweightsiandthatthee)]TJ/F10 6.974 Tf 0 -7.039 Td[(iformthedualbasis,correspondingtothenegativeoftheseweights)]TJ/F11 9.963 Tf 7.748 0 Td[(i.Inparticular,wehaveaLiealgebrahomomorphismfromhtoop,andthetwospinrepresentationsofopgivetworepresentationsofh.Byabuseoflanguage,letusdenotethesetworepresentationsbySpin+andSpin)]TJ/F10 6.974 Tf 6.227 0 Td[(.Wecanalsoconsiderthecharactersoftheserepresentationsofh.Accordingtoequation9.22tobeprovedinChapterIXwehavechSpin+)]TJ/F8 9.963 Tf 9.963 0 Td[(chSpin)]TJ/F10 6.974 Tf 5.895 0.997 Td[(=Yje1 2j)]TJ/F11 9.963 Tf 9.962 0 Td[(e)]TJ/F8 9.963 Tf 8.944 6.74 Td[(1 2j:InthecasethathistheCartansubalgebraofasemi-simpleLiealgebraandandp=nwerecognizethisexpressionastheWeyldenominator.Nowletgbeasemi-simpleLiealgebraandrgareductivesubalgebraofthesamerank.ThismeansthatwecanchooseaCartansubalgebraofgwhichisalsoaCartansubalgebraofr.Therootsofrformasubsetoftherootsofg.TheWeylgroupWgactssimplytransitivelyontheWeylchambersofgeachofwhichiscontainedinaWeylchamberforr.Wechooseapositiverootsystemforg,whichthendeterminesapositiverootsystemforr,andthepositiveWeylchamberforgiscontainedinthepositiveWeylchamberforr.LetCWgdenotethesetofthoseelementsoftheWeylgroupofgwhichmapthepositiveWeylchamberofgintothepositiveWeylchamberforr.Bythesimpletransi-tivityoftheWeylgroupactionsonchambers,weknowthatelementsofCformcosetrepresentativesforthesubgroupWrWg.Inparticular,thenumberofelementsofCisthesameastheindexofWrinWg.Letgandrdenotehalfthesumofthepositiverootsofgandrrespectively.Foranydominantweightofgtheweight+gliesintheinteriorofthepositiveWeylchamberforg.Henceforeachc2C,theelementc+gliesintheinteriorforrandhencec:=c+g)]TJ/F11 9.963 Tf 9.963 0 Td[(r

PAGE 135

7.10.EQUALRANKSUBGROUPS.135isadominantweightforr,andeachoftheseisdistinct.LetVdenotetheirreduciblerepresentationofgwithhighestweight.Wecanconsideritasarepresentationofthesubalgebrar.AlsotheKillingformormoregenerallyanyadinvariantsymmetricbilinearformonginducesaninvariantformonr.Letpdenotetheorthogonalcomplementofring.Wethusgetahomomorphismofrintotheorthogonalalgebraog=r,whichisanevendimensionalorthogonalalgebra,andhencehastwospinrepresentations.TospecifywhichofthesetwospinrepresentationsweshalldenotebyS+andwhichbyS)]TJ/F8 9.963 Tf 6.724 1.494 Td[(,wenotethatthereisaonedimensionalweightspacewithweightg)]TJ/F11 9.963 Tf 10.047 0 Td[(r,andweletS+denotethespinrepresentationwhichcontainsthatonedimensionalspace.ThespacesSareog=rmodules,andviathehomomor-phismr!og=rwecanconsiderthemasrmodules.Finally,foranydominantintegralweightofrweletUdenotetheirre-duciblemoduleofrwithhighestweight.WithallthisnotationwecannowstateTheorem16[G-K-R-S]IntherepresentationringRrwehaveVS+)]TJ/F11 9.963 Tf 9.963 0 Td[(VS)]TJ/F8 9.963 Tf 9.492 1.494 Td[(=Xc2C)]TJ/F8 9.963 Tf 7.748 0 Td[(1cUc:.31Proof.Tosaythattheaboveequationholdsintherepresentationringofrmeansthatwhenwetakethesignedsumsofthecharactersoftherepresentationsoccurringonbothsideswegetequality.Inthespecialcasethatr=h,wehaveobservedthat.31isjusttheWeylcharacterformula:IrrS+g=h)]TJ/F11 9.963 Tf 9.962 0 Td[(S)]TJ/F52 6.974 Tf 6.227 0 Td[(g=h=Xw2Wg)]TJ/F8 9.963 Tf 7.749 0 Td[(1wew+g:ThegeneralcasefollowsfromthisspecialcasebydividingbothsidesofthisequationbyS+r=h)]TJ/F11 9.963 Tf 9.894 0 Td[(S)]TJ/F52 6.974 Tf 6.227 0 Td[(r=h.Thelefthandsidebecomesthecharacterofthelefthandsideof.31becausetheweightsthatgointothisquotientvia.22areexactlythoserootsofgwhicharenotrootsofr.Therighthandsidebecomesthecharacteroftherighthandsideof.22byreorganizingthesumandusingtheWeylcharacterformulaforr.QED

PAGE 136

136CHAPTER7.CYCLICHIGHESTWEIGHTMODULES.

PAGE 137

Chapter8Serre'stheorem.WehaveclassiedallthepossibilitiesforanirreducibleCartanmatrixviatheclassicationofthepossibleDynkindiagrams.Thefourmajorseriesinourclas-sicationcorrespondtotheclassicalsimplealgebrasweintroducedinChapterIII.Theremainingvecasesalsocorrespondtosimplealgebras-theexcep-tionalalgebras".Eachdeservesadiscussiononitsown.HoweveratheoremofSerreguaranteesthatstartingwithanyCartanmatrix,thereisacorrespondingsemi-simpleLiealgebra.AnyrootsystemgivesrisetoaCartanmatrix.Soevenbeforestudyingeachofthesimplealgebrasindetail,weknowinadvancethattheyexist,providedthatweknowthatthecorrespondingrootsystemexists.WepresentSerre'stheoreminthischapter.Attheendofthechapterweshowthateachoftheexceptionalrootsystemsexists.ThisthenprovestheexistenceoftheexceptionalsimpleLiealgebras.8.1TheSerrerelations.Recallthatifandareroots,h;i:=2; ;andthestringofrootsoftheform+jisunbrokenandextendsfrom)]TJ/F11 9.963 Tf 9.963 0 Td[(rto+qwherer)]TJ/F11 9.963 Tf 9.963 0 Td[(q=h;i:Inparticular,if;2sothat)]TJ/F11 9.963 Tf 9.963 0 Td[(isnotaroot,thestringis;+;:::;+qwhereq=h;i:Thusadeh;i+1e=0;137

PAGE 138

138CHAPTER8.SERRE'STHEOREM.fore2g;e2gbutadeke6=0for0kh;i;ife6=0;e6=0.Soif=f1;:::;`gwemaychooseei2gi;fi2g)]TJ/F10 6.974 Tf 6.227 0 Td[(isothate1;:::;;e`;f1;:::;f`generatethealgebraand[hi;hj]=0;1i;j;`.1[ei;fi]=hi.2[ei;fj]=0i6=j.3[hi;ej]=hj;iiej.4[hi;fj]=hj;iifi.5adeihj;ii+1ej=0i6=j.6adfihj;ii+1fj=0i6=j:.7Serre'stheoremsaysthatthisisapresentationofasemi-simpleLiealgebra.Inparticular,theCartanmatrixgivesapresentationofasimpleLiealgebra,showingthatforeveryDynkindiagramthereexistsauniquesimpleLiealgebra.8.2Therstverelations.LetfbethefreeLiealgebraon3`generators,X1;:::;X`;Y1;:::;Y`;Z1;:::;Z`.Ifgisasemi-simpleLiealgebrawithgeneratorsandrelations.1{8.7,wehaveauniquehomomorphismf!gwhereXi!ei;Yi!fi;Zi!hi.Wewanttoconsideranintermediatealgebra,m,wherewemakeuseofallbutthelasttwosetsofrelations.SoletIbetheidealinfgeneratedbytheelements[Zi;Zj];[Xi;Yj])]TJ/F11 9.963 Tf 9.963 0 Td[(ijZi;[Zi;Xj])-222(hj;iiXj;[Zi;Yj]+hj;iiYj:Weletm:=f=IanddenotetheimageofXiinmbyxietc.WewillrstexhibitmasLiesubalgebraofthealgebraofendomorphismsofavectorspace.Thiswillallowustoconcludethatthexi;yjandzkarelinearlyindependentandfromthisdeducethestructureofm.Wewillthenndthatthereisahomomorphismofmontoourdesiredsemi-simpleLiealgebrasendingx7!e;y7!f;z7!h.Soconsideravectorspacewithbasisv1;:::;v`andletAbethetensoralgebraoverthisvectorspace.Wedropthetensorproductsignsinthealgebra,sowritevi1vi2vit:=vi1vit

PAGE 139

8.2.THEFIRSTFIVERELATIONS.139foranynitesequenceofintegerswithvaluesfrom1to`.WemakeAintoanfmoduleasfollows:WelettheZiactasderivationsofA,determinedbyitsactionsongeneratorsbyZi1=0;Zjvi=hi;jivj:Soifwedenecij:=hi;jiwehaveZjvi1vit=)]TJ/F8 9.963 Tf 7.749 0 Td[(ci1j++citjvi1vit:TheactionoftheZiisdiagonalinthisbasis,sotheiractionscommute.WelettheYiactbyleftmultiplicationbyvi.SoYjvi1vit:=vjvi1vitandhence[Zi;Yj]=)]TJ/F11 9.963 Tf 7.748 0 Td[(cjiYj=hj;iiYjasdesired.WenowwanttodenetheactionoftheXisothattherelationsanalogousto.2and.3hold.SinceZi1=0theserelationswillholdwhenappliedtotheelement1ifwesetXj1=08jandXjvi=08i;j:SupposewedeneXjvpvq=)]TJ/F11 9.963 Tf 7.749 0 Td[(jpcqjvq:ThenZiXjvpvq=jpcqjcqivq=)]TJ/F11 9.963 Tf 7.749 0 Td[(cqiXjvpvqwhileXjZivpvq=jpcqjcpi+cqivq=)]TJ/F8 9.963 Tf 7.749 0 Td[(cpi+cqiXjvjvq:Thus[Zi;Xj]vpvq=cjiXjvpvqasdesired.Ingeneral,deneXjvp1vpt:=vp1Xjvp2vpt)]TJ/F11 9.963 Tf 9.691 0 Td[(p1jcp2j++cptjvp2vpt.8fort2.WeclaimthatZiXjvp1vpt=)]TJ/F8 9.963 Tf 7.749 0 Td[(cp1i++cpti)]TJ/F11 9.963 Tf 9.963 0 Td[(cjiXjvp1vpt:Indeed,wehaveveriedthisforthecaset=2.Byinduction,wemayassumethatXjvp2vptisaneigenvectorofZiwitheigenvaluecp2i++cpti)]TJET

PAGE 140

140CHAPTER8.SERRE'STHEOREM.cji.Multiplyingthisontheleftbyvp1producesthersttermontherightof.8.Ontheotherhand,thismultiplicationproducesaneigenvectorofZiwitheigenvaluecp1i++cpti)]TJ/F11 9.963 Tf 9.689 0 Td[(cji.Asforthesecondtermontherightof.8,ifj6=p1itdoesnotappear.Ifj=p1thencp1i++cpti)]TJ/F11 9.963 Tf 8.139 0 Td[(cji=cp2i++cpti.Soineithercase,therighthandsideof.8isaneigenvectorofZiwitheigenvaluecp1i++cpti)]TJ/F11 9.963 Tf 9.963 0 Td[(cji.Butthen[Zi;Xj]=hj;iiXjasdesired.WehavedenedanactionoffonAwhosekernelcontainsI,hencedescendstoanactionofmonA.Let:m!EndAdenotethisaction.Supposethatz:=a1z1++a`z`forsomecomplexnumbersa1;:::;a`andthatz=0.Theoperatorzhaseigenvalues)]TJ/F1 9.963 Tf 9.409 9.464 Td[(XajcijwhenactingonthesubspaceVofA.Allofthesemustbezero.ButtheCartanmatrixisnon-singular.Hencealltheai=0.Thisshowsthatthespacespannedbytheziisinfact`-dimensionalandspansan`-dimensionalabeliansubalgebraofm.Callthissubalgebraz.Nowconsiderthe3`-dimensionalsubspaceoffspannedbytheXi;YiandZi;i=1;:::;`.Wewishtoshowthatitprojectsontoa3`dimensionalsubspaceofmunderthenaturalpassagetothequotientf!m=f=i.Theimageofthissubspaceisspannedbyxi;yiandzi.Sincexi6=0andyi6=0weknowthatxi6=0andyi6=0.SupposewehadalinearrelationoftheformXaixi+Xbiyi+z=0:Choosesomez02zsuchthatiz06=0andiz06=jz0foranyi6=j.Thisispossiblesincetheiarealllinearlyindependent.Bracketingtheaboveequationbyz0givesXz0aixi)]TJ/F1 9.963 Tf 9.963 9.465 Td[(Xiz0biyi=0bytherelations.4and.5.Repeatedbracketingbyz0andusingthevanderMondeorinductionargumentshowsthatai=0;bi=0andhencethatz=0.Wehaveprovedthattheelementsxi;yj;zkinmarelinearlyindependent.Theelement[xi1;[xi2;[[xit)]TJ/F6 4.981 Tf 5.396 0 Td[(1;xit]]]]isaneigenvectorofziwitheigenvalueci1i++citi:Foranypairofelementsandofzorofhrecallthat

PAGE 141

8.2.THEFIRSTFIVERELATIONS.141denotesthefactthat)]TJ/F11 9.963 Tf 9.304 0 Td[(=Pkiiwherethekiareallnon-negativeintegers.Forany2zletmdenotethesetofallm2msatisfying[z;m]=zm8z2z:Thenwehaveshownthatthesubalgebraxofmgeneratedbyx1;:::;x`iscontainedinm+:=M0m:Similarly,thesubalgebrayofmgeneratedbytheyiliesinm)]TJ/F8 9.963 Tf 9.492 1.494 Td[(:=M0m:Inparticular,thevectorspacesumy+z+xisdirectsincezm0.Weclaimthatthisisinfactallofm.Firstofall,observethatitisasubalgebra.Indeed,[yi;xj]=)]TJ/F11 9.963 Tf 7.749 0 Td[(ijziliesinthissubspace,andhence[yi;[xj1;[[xjt)]TJ/F7 6.974 Tf 6.227 0 Td[(1;xjt]]2xfort2:Thusthesubspacey+z+xisclosedunderadyiandhenceunderanyproductoftheseoperators.Similarlyforadxi.Sincethesegeneratethealgebramweseethaty+z+x=mandhencex=m+andy=m)]TJ/F11 9.963 Tf 6.725 1.495 Td[(:Wehaveshownthatm=m)]TJ/F14 9.963 Tf 8.939 1.494 Td[(zm+wherezisanabeliansubalgebraofdimension`,wherethesubalgebram+isgeneratedbyx1;:::;x`,wherethesubalgebram)]TJ/F8 9.963 Tf 9.053 1.494 Td[(isgeneratedbyy1;:::;y`,andwherethe3`elementsx1;:::;x`;y1:::;y`;z1;:::;z`arelinearlyindependent.ThereisafurtherpropertyofmwhichwewanttouseinthenextsectionintheproofofSerre'stheorem.Foralli6=jbetween1and`denetheelementsxijandyijbyxij:=adxi)]TJ/F10 6.974 Tf 6.227 0 Td[(cji+1xj;yij:=adyi)]TJ/F10 6.974 Tf 6.227 0 Td[(cji+1yj:Conditions.6and.7amounttosettingtheseelements,andhencetheidealthattheygenerateequaltozero.Weclaimthatforallkandalli6=jbetween1and`wehaveadxkyij=0.9andadykxij=0:.10

PAGE 142

142CHAPTER8.SERRE'STHEOREM.Bysymmetry,itisenoughtoprovetherstoftheseequations.Ifk6=ithen[xk;yi]=0by.3andhenceadxkyij=adyi)]TJ/F10 6.974 Tf 6.227 0 Td[(cji+1[xk;yj]=adyi)]TJ/F10 6.974 Tf 6.226 0 Td[(cji+1kjhjby.2and.3.Ifk6=jthisiszero.Ifk=jwecanwritethisasadyi)]TJ/F10 6.974 Tf 6.226 0 Td[(cjiadyihj=adyi)]TJ/F10 6.974 Tf 6.227 0 Td[(cjicijyi:Ifcij=0thereisnothingtoprove.Ifcij6=0thencji6=0andinfactisstrictlynegativesincetheanglesbetweenallelementsofabaseareobtuse.Butthenadyi)]TJ/F10 6.974 Tf 6.227 0 Td[(cjiyi=0:Itremainstoconsiderthecasewherek=i.Thealgebrageneratedbyxi;y;ziisisomorphictoslwith[xi;yi]=zi;[zi;xi]=2xi;[zi;yi]=)]TJ/F8 9.963 Tf 7.748 0 Td[(2yi.Wehaveadecompositionofmintoweightspacesforallofz,inparticularintoweightspacesforthislittlesl.Now[xi;yj]=0from.3soyjisamaximalweightvectorforthisslwithweight)]TJ/F11 9.963 Tf 7.748 0 Td[(cjiand.9isjustastandardpropertyofamaximalweightmoduleforslwithnon-negativeintegermaximalweight.8.3ProofofSerre'stheorem.Letkbetheidealofmgeneratedbythexijandyijasdenedabove.Wewishtoshowthatg:=m=kisasemi-simpleLiealgebrawithCartansubalgebrah=z=kandrootsystem.Forthispurpose,letinowdenotetheidealinm+generatedbythexijandjbetheidealinm)]TJ/F8 9.963 Tf 10.046 1.495 Td[(generatedbytheyijsothati+jk:Weclaimthatjisanidealofm.Indeed,eachyijisaweightvectorforz,and[z;m)]TJ/F8 9.963 Tf 6.725 1.495 Td[(]m)]TJ/F8 9.963 Tf 6.725 1.495 Td[(,hence[z;j]j.Ontheotherhand,weknowthat[xk;m)]TJ/F8 9.963 Tf 6.724 1.494 Td[(]m)]TJ/F8 9.963 Tf 8.957 1.494 Td[(+zand[xk;yij]=0by.9.SoadxkjjbyJacobi.Sincethexkgeneratem+Jacobithenimpliesthat[m+;j]jaswell,hencejisanidealofm.Similarly,iisanidealofm.Hencei+jisanidealofm,andsinceitcontainsthegeneratorsofk,itmustcoincidewithk,i.e.k=i+j:Inparticular,zk=f0gandsozprojectsisomorphicallyontoan`-dimensionalabeliansubalgebraofg=m=k.Furthermore,sincejm+=f0gandim)]TJ/F8 9.963 Tf 9.493 1.495 Td[(=f0gwehaveg=n)]TJ/F14 9.963 Tf 8.939 1.494 Td[(hn+.11asavectorspacewheren)]TJ/F8 9.963 Tf 9.492 1.495 Td[(=m)]TJ/F11 9.963 Tf 6.724 1.495 Td[(=j;andn+=m+=i;

PAGE 143

8.3.PROOFOFSERRE'STHEOREM.143andn+isaasumofweightspacesofh,summedover0whilen)]TJ/F8 9.963 Tf 9.848 1.494 Td[(isasumofweightspacesofhwith0.Wehavetoseewhichweightspacessurvivethepassagetothequotient.Theslgeneratedbyxi;yi;ziisnotsentintozerobytheprojectionofmontogsinceziisnotsentintozero.Sinceslissimple,thismeansthattheprojectionmapisanisomorphismwhenrestrictedtothissl.Letusdenotetheimagesofxi;yi;zibyei;fi;hi.Thusgisgeneratedbythe3`elementse1;:::;e`;f1;:::;f`;h1;:::;h`andalltheaxioms.1-.7aresatised.Wemustshowthatgisnitedimensional,semi-simple,andhasasitsrootsystem.Firstobservethatadeiactsnilpotentlyoneachofthegeneratorsofthealgebrag,andhenceactslocallynilpotentlyonallofg.Similarlyforadfi.Hencetheautomorphismi:=expadeiad)]TJ/F11 9.963 Tf 9.963 0 Td[(fiexpadeiiswelldenedonallofg.SoifsidenotesthereectionintheWeylgroupWcorrespondingtoi,wehaveig=gsi:Noticethateachofthemisnitedimensional,sincethedimensionofmfor0isatmostthenumberofwaystowriteasasumofsuccessivei,eachsuchsumcorrespondingtotheelement[xi1;[xi2;[;xit]].Inparticularmk=f0gfork>1.Similarlyfor0.Soitfollowsthateachofthegisnitedimensional,thatdimgw=dimg8w2Wandthatgk=0fork6=)]TJ/F8 9.963 Tf 7.748 0 Td[(1;0;1:Furthermore,giisonedimensional,andsinceeveryrootisconjugatetoasimpleroot,weconcludethatdimg=182:Wenowshowthatg=f0gfor6=0;62:Indeed,supposethatg6=f0g.Weknowthatisnotamultipleofforany2,sinceweknowthistobetrueforsimpleroots,andthedimensionsofthegareinvariantundertheWeylgroup,eachrootbeingconjugatetoasimpleroot.So?doesnotcoincidewithanyhyperplaneorthogonaltoanyroot.Sowecannda2?suchthat;6=0forallroots.Wemayndaw2WwhichmapsintothepositiveWeylchamberforsothati;0andhencei;w>0fori=1;:::;`.Nowdimgw=dimg

PAGE 144

144CHAPTER8.SERRE'STHEOREM.andforthelattertobenon-zero,wemusthavew=Xkiiwiththecoecientsallnon-negativeornon-positiveintegers.But0=;=w;w=Xkii;withi;>08i.Hencealltheki=0.Sodimg=`+Card:Weconcludetheproofifweshowthatgissemi-simple,i.e.containsnoabelianideals.Sosupposethataisanabelianideal.Sinceaisanideal,itisstableunderhandhencedecomposesintoweightspaces.Ifga6=f0g,thengaandhence[g)]TJ/F10 6.974 Tf 6.226 0 Td[(;g]aandhencetheentireslgeneratedbygandg)]TJ/F10 6.974 Tf 6.227 0 Td[(iscontainedinawhichisimpossiblesinceaisabelianandslissimple.Soah.Butthenamustbeannihilatedbyalltheroots,whichimpliesthata=f0gsincetherootsspanh.QED8.4Theexistenceoftheexceptionalrootsys-tems.Theideaoftheconstructionisasfollows.ForeachDynkindiagramwewillchosealatticeLinaEuclideanspaceV,andthenletconsistofallvectorsinthislatticehavingallthesamelength,orhavingoneoftwoprescribedlengths.Wethencheckthat21;2 1;12Z81;22:Thisimpliesthatreectionthroughthehyperplaneorthogonalto1preservesL,andsincereectionspreservelengththatthesereectionspreserve.Thiswillshowthatisarootsystemandthencalculationshowsthatitisofthedesiredtype.G2.LetVbetheplaneinR3consistingofallvectors0@xyz1Awithx+y+z=0:LetLbetheintersectionofthethreedimensionalstandardlatticeZ2withV.LetL1;L2;L3denotethestandardbasisofR3.LetconsistofallvectorsinLofsquaredlength2or6.SoconsistsofthesixshortvectorsLi)]TJ/F11 9.963 Tf 9.963 0 Td[(Lji
PAGE 145

8.4.THEEXISTENCEOFTHEEXCEPTIONALROOTSYSTEMS.145andthesixlongvectorsLi)]TJ/F11 9.963 Tf 9.963 0 Td[(Lj)]TJ/F11 9.963 Tf 9.962 0 Td[(Lki=1;2;3j6=i;j6=k:Wemaychoosethebasetoconsistof=L1)]TJ/F11 9.963 Tf 9.963 0 Td[(L2;2=)]TJ/F8 9.963 Tf 7.749 0 Td[(2L1+L2+L3:

PAGE 146

146CHAPTER8.SERRE'STHEOREM.F4.LetV=R4andL=Z4+Z1 2L1+L2+L3+L4.LetconsistofallvectorsofLofsquaredlength1or2.Soconsistsofthe24longrootsLiLji
PAGE 147

Chapter9Cliordalgebrasandspinrepresentations.9.1Denitionandbasicproperties9.1.1Denition.Letpbeavectorspacewithasymmetricbilinearform;.TheCliordalgebraassociatedtothisdataisthealgebraCp:=Tp=IwhereTpdenotesthetensoralgebraTp=kpppandwhereIdenotestheidealinTpgeneratedbyallelementsoftheformy1y2+y2y1)]TJ/F8 9.963 Tf 9.962 0 Td[(2y1;y21;y1;y22pand1istheunitelementofthetensoralgebra.ThespacepinjectsasasubspaceofCpandgeneratesCpasanalgebra.AlinearmapfofptoanassociativealgebraAwithunit1AiscalledaCliordmapiffy1fy2+fy2fy1=2y1;y2A;8y1;y22porwhatamountstothesamethingbypolarizationsincewearenotoveraeldofcharacteristic2iffy2=y;yA8y2p:AnyCliordmapgivesrisetoauniquealgebrahomomorphismofCptoAwhoserestrictiontopisf.TheCliordalgebraisuniversal"withrespecttothisproperty.Ifthebilinearformisidenticallyzero,thenCp=^p,theexterioralgebra.Butwewillbeinterestedintheoppositeextreme,thecasewherethebilinearformisnon-degenerate.147

PAGE 148

148CHAPTER9.CLIFFORDALGEBRASANDSPINREPRESENTATIONS.9.1.2Gradation.TheidealIdeningtheCliordalgebraisnotZhomogeneousunlessthebilinearformisidenticallyzerosinceitsgeneratorsy1y2+y2y1)]TJ/F8 9.963 Tf 9.314 0 Td[(2y1;y21aremixed",beingasumoftermsofdegreetwoanddegreezeroinTp.Butthesetermsarebotheven.SotheZ=2Zgradationispreserveduponpassingtothequotient.Inotherwords,CpisaZ=2Zgradedalgebra:Cp=C0pC1pwheretheelementsofC0pconsistofsumsofproductsofelementsofpwithanevennumberoffactorsandC1pconsistofsumsoftermseachaproductofelementsofpwithanoddnumberoffactors.Theusualrulesformultiplicationofagradedalgebraobtain:C0pC0pC0p;C0pC1pC1p;C1pC1pC0p:9.1.3^pasaCpmodule.Letpbeavectorspacewithanon-degeneratesymmetricbilinearform.Theexterioralgebra,^pinheritsabilinearformwhichwecontinuetodenoteby;.Herethespaces^kpand^`pareorthogonalifk6=`whilex1^^xk;y1^^yk=detxi;yj:Forv2pletv2End^pdenoteexteriormultiplicationbyvandvbethetransposeofvrelativetothisbiinearformon^p.Sovisinteriormultiplicationbytheelementofpcorrespondingtovunderthemapp!pinducedby;p.Themapp!End^p;v7!v+visaCliordmap,i.e.satisesv+v2=v;vpidandsoextendstoahomomorphismofCp!End^pmaking^pintoaCpmodule.WeletxydenotetheproductofxandyinCp.9.1.4Chevalley'slinearidenticationofCpwith^p.ConsiderthelinearmapCp!^p;x7!x1

PAGE 149

9.1.DEFINITIONANDBASICPROPERTIES149where12^0pundertheidenticationof^0pwiththegroundeld.Theelementx1ontheextremerightmeanstheimageof1undertheactionofx2Cp.Forelementsv1;:::;vk2pthismapsendsv17!v1v1v27!v1^v2+v1;v2v1v2v37!v1^v2^v3+v1;v2v3)]TJ/F8 9.963 Tf 9.962 0 Td[(v1;v3v2+v2;v3v1v1v2v3v47!v1^v2^v3^v4+v2;v3v1^v4)]TJ/F8 9.963 Tf 9.962 0 Td[(v2;v4v1^v3+v3;v4v1^v2+v1;v2v3^v4)]TJ/F8 9.963 Tf 9.963 0 Td[(v1;v3v1^v4+v1;v4v2^v3+v1;v4v2;v3)]TJ/F8 9.963 Tf 9.962 0 Td[(v1;v3v2;v4+v1;v2v3;v4......Ifthev'sformanorthonormal"basisofpthentheproductsvi1vik;i1
PAGE 150

150CHAPTER9.CLIFFORDALGEBRASANDSPINREPRESENTATIONS.Forsmallvaluesofkwehavek )]TJ/F8 9.963 Tf 7.749 0 Td[(11 2kk)]TJ/F7 6.974 Tf 6.227 0 Td[(1 0 11 12 )]TJ/F8 9.963 Tf 7.749 0 Td[(13 )]TJ/F8 9.963 Tf 7.749 0 Td[(14 15 16 )]TJ/F8 9.963 Tf 7.749 0 Td[(1:Wewillusesubscriptstodenotethehomogeneouscomponentsofelementsof^p.Noticethatifu2^2pthenau=)]TJ/F11 9.963 Tf 7.749 0 Td[(ubytheabovetable,whileau2=au2=u2.Sinceu2isevenandhencehasonlyevenhomogeneouscomponentsandsincethemaximumdegreeofthehomogeneouscomponentofu2is4,weconcludethatu2=u20+u248u2^2p:.2Forthesamereasonv2=v20+v248v2^3p:.3Wealsoclaimthefollowing:ww00=aw;w0=)]TJ/F8 9.963 Tf 7.749 0 Td[(11 2kk)]TJ/F7 6.974 Tf 6.226 0 Td[(1w;w08w;w02^kp:.4Indeed,itissucienttoverifythisforw;w0belongingtoabasisof^p,saythebasisgivenbyallelementsoftheform.1,inwhichcasebothsidesof.4vanishunlessw=w0.Ifw=w0=v1^^vksaythenww0=v1vkv1^^vk=)]TJ/F8 9.963 Tf 7.749 0 Td[(11 2kk)]TJ/F7 6.974 Tf 6.227 0 Td[(1v1;v1vk;vk=)]TJ/F8 9.963 Tf 7.749 0 Td[(11 2kk)]TJ/F7 6.974 Tf 6.227 0 Td[(1w;wproving.4.Asspecialcasesthatwewilluselateron,observethatuu00=)]TJ/F8 9.963 Tf 7.748 0 Td[(u;u08u;u02^2p.5andvv00=)]TJ/F8 9.963 Tf 7.748 0 Td[(v;v08v;v02^3p:.69.1.6Commutatorbyanelementofp.Foranyy2pconsiderthelinearmapw7![y;w]=yw)]TJ/F8 9.963 Tf 9.962 0 Td[()]TJ/F8 9.963 Tf 7.749 0 Td[(1kwyforw2^kp

PAGE 151

9.1.DEFINITIONANDBASICPROPERTIES151whichisanticommutatorintheCliordmultiplicationbyy.Weclaimthat[y;w]=2yw:.7Inparticular,[y;],whichisautomaticallyaderivationfortheCliordmulti-plication,isalsoaderivationfortheexteriormultiplication.Alternatively,thisequationsaysthaty,whichisaderivationfortheexterioralgebramultipli-cation,isalsoaderivationfortheCliordmultiplication.Toprove.7writewy=ayaw:Thenyw=y^w+yw;wy=ay^aw+ayaw=w^y+ayaw:Wemayassumethatw2^kp.Theny^w)]TJ/F8 9.963 Tf 9.963 0 Td[()]TJ/F8 9.963 Tf 7.749 0 Td[(1kw^y=0;sowemustshowthatayaw=)]TJ/F8 9.963 Tf 7.749 0 Td[(1k)]TJ/F7 6.974 Tf 6.227 0 Td[(1yw:Forthiswemayassumethaty6=0andwemaywritew=u^z+z0;whereyu=1andyz=yz0=0.Infact,wemayassumethatzandz0aresumsofproductsoflinearelementsallofwhichareorthogonaltoy.Thenyaz=yaz0=0soyaw=)]TJ/F8 9.963 Tf 7.748 0 Td[(1k)]TJ/F7 6.974 Tf 6.226 0 Td[(1azsincezhasdegreeonelessthanwandhenceayaw=)]TJ/F8 9.963 Tf 7.749 0 Td[(1k)]TJ/F7 6.974 Tf 6.227 0 Td[(1z=)]TJ/F8 9.963 Tf 7.749 0 Td[(1k)]TJ/F7 6.974 Tf 6.227 0 Td[(1yw:QED9.1.7Commutatorbyanelementof^2p.Supposethatu2^2p:Thenfory2pwehave[u;y]=)]TJ/F8 9.963 Tf 7.748 0 Td[([y;u]=)]TJ/F8 9.963 Tf 7.748 0 Td[(2yu:.8Inparticular,ifu=yi^yjwhereyi;yj2pwehave[u;y]=2yj;yyi)]TJ/F8 9.963 Tf 9.962 0 Td[(2yi;yyj8y2p:.9Ifyi;yj=0thisisaninnitesimalrotation"intheplanespannedbyyiandyj.Sinceyi^yj;i
PAGE 152

152CHAPTER9.CLIFFORDALGEBRASANDSPINREPRESENTATIONS.givesanisomorphismof^2pwiththeorthogonalalgebraop.Thisidenti-cationdiersbyafactoroftwofromtheidenticationthatwehadbeenusingearlier.Noweachelementofopinfactanylineartransformationonpinducesaderivationof^p.Weclaimthatundertheaboveidenticationof^2pwithop,thederivationcorrespondingtou2^2pisCliordcommutationbyu.Insymbols,ifudenotesthisinducedderivation,weclaimthatuw=[u;w]=uw)]TJ/F11 9.963 Tf 9.963 0 Td[(wu8w2^p:.10Toverifythis,itisenoughtocheckitonbasiselementsoftheform.1,andhencebythederivationpropertyforeachvj,wherethisreducesto9.8.WecannowbemoreexplicitaboutthedegreefourcomponentoftheCliordsquareofanelementof^2p,i.e.theelementu24occurringontherightof.2.Weclaimthatforanythreeelementsy;y0;y002p1 2y00y0yu2=y^y0;uy00u+y0^y00;uyu+y00^y;uy0u:.11Toprovethisobservethatyu2=yuu+uyuy0yu2=y0yuu)]TJ/F11 9.963 Tf 9.963 0 Td[(yuy0u+y0uyu+uy0yu=2y^y0;uu+y0u^yu1 2y00y0yu2=y^y0;uy00u+y00y0u^yu)]TJ/F11 9.963 Tf 9.963 0 Td[(y0u^y00yu=y^y0;uy00u+y0^y00;uyu+y00^y;uy0uasrequired.Wecanalsobeexplicitaboutthedegreezerocomponentofu2.Indeed,itfollowsfrom.9thatifu=yi^yj;i
PAGE 153

9.2.ORTHOGONALACTIONOFALIEALGEBRA.1539.2OrthogonalactionofaLiealgebra.LetrbeaLiealgebra.Supposethatwehavearepresentationofractingasinnitesimalorthogonaltransformationsofpwhichmeans,inviewoftheidenticationof^2pwithopthatwehaveamap:r!^2psuchthatxy=)]TJ/F8 9.963 Tf 7.749 0 Td[(2yx.13wherexydenotestheactionofx2rony2p.9.2.1Expressionforintermsofdualbases.Itwillbeusefulforustowriteequation.13intermsofabasis.Solety1;:::;ynbeabasisofpandletz1;:::;znbethedualbasisrelativeto;=;p.Weclaimthatx=)]TJ/F8 9.963 Tf 8.945 6.739 Td[(1 4Xjyj^xzj:.14Indeed,itsucestoverify.13foreachoftheelementszi.Nowzi0@)]TJ/F8 9.963 Tf 8.945 6.74 Td[(1 4Xjyj^xzj1A=)]TJ/F8 9.963 Tf 8.944 6.74 Td[(1 4xzi+1 4Xjzi;xzjyj:Butzi;xzj=)]TJ/F8 9.963 Tf 7.749 0 Td[(xzi;zjsincexactsasaninnitesimalorthogonaltransformationrelativeto;.Sowecanwritethesumas1 4Xjzi;xzjyj=)]TJ/F8 9.963 Tf 8.945 6.74 Td[(1 4Xjxzi;zjyj=)]TJ/F8 9.963 Tf 8.944 6.74 Td[(1 4xziyieldingzi0@)]TJ/F8 9.963 Tf 8.944 6.74 Td[(1 4Xjyj^xzj1A=)]TJ/F8 9.963 Tf 8.944 6.74 Td[(1 2xziwhichis9.13.9.2.2TheadjointactionofareductiveLiealgebra.Forfutureusewerecordhereaspecialcaseof9.14:Supposethatp=r=gisareductiveLiealgebrawithaninvariantsymmetricbilinearform,andtheactionistheadjointaction,i.e.xy=[x;y].LethbeaCartansubalgebraofg

PAGE 154

154CHAPTER9.CLIFFORDALGEBRASANDSPINREPRESENTATIONS.andletdenotethesetofrootsandsupposethatwehavechosenrootvectorse;e)]TJ/F10 6.974 Tf 6.226 0 Td[(;2sothate;e)]TJ/F10 6.974 Tf 6.227 0 Td[(=1:Leth1;:::;hsbeabasisofhandk1;:::ksthedualbasis.Let:g!^2gbethemapwhenappliedtothisadjointaction.Then.14becomesx=1 40@sXi=1hi^[ki;x]+X2e)]TJ/F10 6.974 Tf 6.227 0 Td[(^[e;x]1A:.15Incasex=h2hthisformulasimplies.The[ki;h]=0,andinthesecondsumwehavee)]TJ/F10 6.974 Tf 6.227 0 Td[(^[e;h]=)]TJ/F11 9.963 Tf 7.749 0 Td[(he)]TJ/F10 6.974 Tf 6.226 0 Td[(^ewhichisinvariantundertheinterchangeofand)]TJ/F11 9.963 Tf 7.748 0 Td[(.Soletusmakeachoice+ofpositiveroots.Thenwecanwrite.15ash=)]TJ/F8 9.963 Tf 8.944 6.74 Td[(1 2X2+he)]TJ/F10 6.974 Tf 6.227 0 Td[(^e;h2h:.16Nowe)]TJ/F10 6.974 Tf 6.227 0 Td[(^e=)]TJ/F8 9.963 Tf 7.748 0 Td[(1+e)]TJ/F10 6.974 Tf 6.227 0 Td[(e:Soif:=1 2X2+.17isonehalfthesumofthepositiverootswehaveh=h)]TJ/F8 9.963 Tf 11.158 6.739 Td[(1 2X2+he)]TJ/F10 6.974 Tf 6.227 0 Td[(e;h2h:.18Inthisequation,themultiplicationontherightisintheCliordalgebra.9.3Thespinrepresentations.Ifp=p1p2isadirectsumdecompositionofavectorspacepwithasymmetricbilinearformintotwoorthogonalsubspacesthenitfollowsfromthedenitionoftheCliordalgebrathatCp=Cp1Cp2

PAGE 155

9.3.THESPINREPRESENTATIONS.155wherethemultiplicationonthetensorproductistakeninthesenseofsuperal-gebras,thatisa1a2b1b2:=a1b1a2b2ifeithera2orb1areeven,buta1a2b1b2:=)]TJ/F11 9.963 Tf 7.749 0 Td[(a1b1a2b2ifbotha2andb1areodd.Itcostsasigntomoveoneoddsymbolpastanother.9.3.1Theevendimensionalcase.Supposethatpisevendimensional.Ifthemetricissplitwhichisalwaysthecaseifthemetricisnon-degenerateandweareoverthecomplexnumbersthenpisadirectsumoftwodimensionalmutuallyorthogonalsplitspaces,Wi,soletusexaminerstthecaseofatwodimensionalsplitspacep,spannedby;with;=;=0;;=1 2.LetTbeaonedimensionalspacewithbasistandconsiderthelinearmapofp!End^Tdeterminedby7!t;7!twheretdenotesexteriormultiplicationbytandtdenotesinteriormulti-plicationbyt,thedualelementtotinT.ThisisaCliordmapsincet2=0=t2;tt+tt=id:ThisthereforeextendstoamapofCp!End^T.Explicitly,ifweuse12^0T;t2^1Tasabasisof^Tthismapisgivenby17!10017!01007!00107!1000:Thisshowsthatthemapisanisomorphism.Ifnowp=W1Wmisadirectsumoftwodimensionalsplitspaces,andwewriteT=T1Tm

PAGE 156

156CHAPTER9.CLIFFORDALGEBRASANDSPINREPRESENTATIONS.wheretheCWi=End^Tiasabove,thensince^T=^T1^TmweseethatCp=End^T:Inparticular,Cpisisomorphictothefull2m2mmatrixalgebraandhencehasauniqueuptoisomorphismirreduciblemodule.OnemodelofthisisS=^T:WecanwriteS=S+S)]TJ/F8 9.963 Tf -193.981 -15.775 Td[(asasupervectorspace,wherewechoosethestandardZ2gradingon^TtodeterminethegradingonSifmiseven,butusetheoppositegradingforreasonswhichwillbecomeapparentinamomentifmisodd.Theevenpart,C0pofCpactsirreduciblyoneachofS.Since^2ptogetherwiththeconstantsgeneratesC0pweseethattheactionof^2poneachofSisirreducible.Since^2punderCliordcommutationisisomorphictoopthetwomodulesSgiveirreduciblemodulesfortheevenorthogonalalgebraop.Thesearethehalfspinrepresentationsoftheevenorthogonalalgebras.WecanidentifyS=S+S)]TJ/F8 9.963 Tf 10.773 1.495 Td[(asaleftidealinCpasfollows:Supposethatwewritep=p+p)]TJ/F8 9.963 Tf -194.077 -15.774 Td[(whereparecomplementaryisotropicsubspaces.Chooseabasise+1;:::;e+mofp+andlete+:=e+1e+m=e+1^^e+m2^mp+:Wehavey+e+=0;8y+2p+andhence^p++e+=0:Inotherwords^p+e+consistsofallscalarmultiplesofe+.Since^p)]TJ/F14 9.963 Tf 8.938 1.494 Td[(^p+!Cp;w)]TJ/F14 9.963 Tf 8.939 1.494 Td[(w+7!w)]TJ/F11 9.963 Tf 6.725 1.494 Td[(w+isalinearbijection,weseethatCpe+=^p)]TJ/F11 9.963 Tf 6.725 1.494 Td[(e+:Thismeansthattheleftidealgeneratedbye+inCphasdimension2m,andhencemustbeisomorphicasaleftCpmoduletoS.Inparticularitisaminimalleftideal.

PAGE 157

9.3.THESPINREPRESENTATIONS.157Lete)]TJ/F7 6.974 Tf 0 -6.918 Td[(1;:::e)]TJ/F10 6.974 Tf 0 -6.079 Td[(mbeabasisofp)]TJ/F8 9.963 Tf 10.676 1.494 Td[(andforanysubsetJ=fi1;:::;ijg;i1
PAGE 158

158CHAPTER9.CLIFFORDALGEBRASANDSPINREPRESENTATIONS.itself,saythediagonalmatricesintheblockdecompositionofopgivenbythedecompositionC2k=CkCkintotwoisotropicsubspaces.Inthiscasetheiisjustthei-thdiagonalentryand.22yieldsthestandardformulaforthedierenceofthecharactersofthespinrepresentationsoftheevenorthogonalalgebras.AsecondveryimportantcaseiswherewetakehtobetheCartansubalgebraofasemi-simpleLiealgebrag,andtakep:=n+n)]TJ/F8 9.963 Tf -195.462 -20.652 Td[(relativetoachoiceofpositiveroots.Thenthejarejustthepositiveroots,andweseethattherighthandsideof.22isjusttheWeyldenominator,thedenominatoroccurringintheWeylcharacterformula.ThismeansthatwecanwritetheWeylcharacterformulaaschIrrS+)]TJ/F8 9.963 Tf 9.962 0 Td[(chIrrS)]TJ/F8 9.963 Tf 6.725 1.494 Td[(=Xw2W)]TJ/F8 9.963 Tf 7.749 0 Td[(1wewwherew:=w+:IfweletUdenotetheonedimensionalmoduleforhgivenbytheweightwecandropthecharactersfromtheprecedingequationandsimplywritetheWeylcharacterformulaasanequationinvirtualrepresentationsofh:IrrS+)]TJ/F8 9.963 Tf 9.963 0 Td[(IrrS)]TJ/F8 9.963 Tf 9.492 1.494 Td[(=Xw2W)]TJ/F8 9.963 Tf 7.748 0 Td[(1wUw:.23ThereadercannowgobacktotheprecedingchapterandtoTheorem16wherethisversionoftheWeylcharacterformulahasbeengeneralizedfromtheCartansubalgebratothecaseofareductivesubalgebraofequalrank.InthenextchapterweshallseethemeaningofthisgeneralizationintermsoftheKostantDiracoperator.9.3.2Theodddimensionalcase.Sinceeveryodddimensionalspacewithanon-singularbilinearformcanbewrittenasasumofaonedimensionalspaceandanevendimensionalspacebothnon-degenerate,weneedonlylookattheCliordalgebraofaonedimensionalspacewithabasiselementxsuchthatx;x=1sinceweareoverthecomplexnumbers.ThisCliordalgebraistwodimensional,spannedby1andxwithx2=1,theelementxbeingodd.ThisalgebraclearlyhasitselfasacanonicalmoduleunderleftmultiplicationandisirreducibleasaZ=2Zmodule.WemaycallthisthespinrepresentationofCliordalgebraofaonedimensionalspace.UndertheevenpartoftheCliordalgebrai.e.underthescalarsitsplitsintotwoisomorphiconedimensionalspacescorrespondingtothebasis1;xof

PAGE 159

9.3.THESPINREPRESENTATIONS.159theCliordalgebra.Relativetothisbasis1;xwehavetheleftmultiplicationrepresentationgivenby17!1001;x7!0110:LetususeCCtodenotetheCliordalgebraoftheonedimensionalor-thogonalvectorspacejustdescribed,andSCitscanonicalmodule.Thenifq=pCisanorthogonaldecompositionofanodddimensionalvectorspaceintoadirectsumofanevendimensionalspaceandaonedimensionalspacebothnon-degeneratewehaveCq=CpCC=EndSqwhereSq:=SpSCalltensorproductsbeingtakeninthesenseofsuperalgebra.Wehaveadecom-positionSq=S+qS)]TJ/F8 9.963 Tf 6.725 1.494 Td[(qasasupervectorspacewhereS+q=S+pxS)]TJ/F8 9.963 Tf 6.725 1.494 Td[(p;S)]TJ/F8 9.963 Tf 6.725 1.494 Td[(q=S)]TJ/F8 9.963 Tf 6.725 1.494 Td[(pxS+p:ThesetwospacesareequivalentandirreducibleasC0qmodules.SincetheevenpartoftheCliordalgebraisgeneratedby^2qtogetherwiththescalars,weseethateitherofthesespacesisamodelfortheirreduciblespinrepresenta-tionofoqinthisodddimensionalcase.Considerthedecompositionp=p+p)]TJ/F8 9.963 Tf 9.587 1.494 Td[(thatweusedtoconstructamodelforSpasbeingtheleftidealinCpgeneratedby^mp+wherem=dimp+.Wehave^Cp)]TJ/F8 9.963 Tf 6.725 1.495 Td[(=^C^p)]TJ/F11 9.963 Tf 6.725 1.495 Td[(;andProposition28TheleftidealintheCliordalgebrageneratedby^mp+isamodelforthespinrepresentation.Noticethatthisdescriptionisvalidforboththeevenandtheodddimensionalcase.9.3.3SpinadandV.Wewanttoconsiderthefollowingsituation:gisasimpleLiealgebraandwetake;tobetheKillingform.Wehave:g!^2gCg

PAGE 160

160CHAPTER9.CLIFFORDALGEBRASANDSPINREPRESENTATIONS.whichisthemapassociatedtotheadjointrepresentationofg.LethbeaCartansubalgebraandthecollectionofroots.Wechooserootvectorse;2sothate;e)]TJ/F10 6.974 Tf 6.227 0 Td[(=1:Thenitfollowsfrom.14thatx=1 40@Xhi^[ki;x]g+X2e)]TJ/F10 6.974 Tf 6.227 0 Td[(^[e;x]g1A.24wherethebracketsaretheLiebracketsofg,wherethehirangeoverabasisofhandthekioveradualbasis.Thisequationsimpliesinthespecialcaseswherex=h2handinthecasewherex=e;2+relativetoachoice,+ofpositiveroots.Inthecasethatx=h2hwehaveseenthat[ki;h]=0andtheequationsimpliestoh=h)]TJ/F8 9.963 Tf 11.158 6.74 Td[(1 2X2+he)]TJ/F10 6.974 Tf 6.226 0 Td[(e.25where=1 2X2+isonehalfthesumofthenpositiveroots.Weclaimthatfor2+wehavee=Xx0e0.26wherethesumisoverpairs0;0suchthateither1.0=0;0=andx02hor2.02;02+and0+0=;andx02g0:Toseethis,rstobservethatthisrstsumontherightof.24givesXkihi^eandsoallthesesummandsareoftheform1.Foreachsummande)]TJ/F10 6.974 Tf 6.227 0 Td[(^[e;e]ofthesecondsum,wemayassumethateither+=0orthat+2forotherwise[e;e]=0.If+=0,so=)]TJ/F11 9.963 Tf 7.749 0 Td[(6=0,wehave[e;e]2hwhichisorthogonaltoe)]TJ/F10 6.974 Tf 6.226 0 Td[(since6=0.Soe)]TJ/F10 6.974 Tf 6.227 0 Td[(^[e;e]=)]TJ/F8 9.963 Tf 7.749 0 Td[([e;e]eagainhastheform1.

PAGE 161

9.3.THESPINREPRESENTATIONS.161If+=6=0isaroot,thene)]TJ/F10 6.974 Tf 6.226 0 Td[(;e=0since6=.If2+thene)]TJ/F10 6.974 Tf 6.226 0 Td[(^[e;e]=e)]TJ/F10 6.974 Tf 6.226 0 Td[(y;whereyisamultipleofesothissummandisoftheform2.Ifisanegativeroot,themustbeanegativerootso)]TJ/F11 9.963 Tf 7.749 0 Td[(isapositiveroot,andwecanswitchtheorderofthefactorsintheprecedingexpressionattheexpenseofintroducingasign.Soagainthisisoftheform2,completingtheproofof.26.Letn+bethesubalgebraofggeneratedbythepositiverootvectorsandsimilarlyn)]TJ/F8 9.963 Tf 10.046 1.494 Td[(thesubalgebrageneratedbythenegativerootvectorssog=n+b)]TJ/F11 9.963 Tf 6.725 1.494 Td[(;b)]TJ/F8 9.963 Tf 9.492 1.494 Td[(:=n)]TJ/F14 9.963 Tf 8.939 1.494 Td[(hisanhstabledecompositionofgintoadirectsumofthenilradicalanditsoppositeBorelsubalgebra.LetNbethenumberofpositiverootsandlet06=n2^Nn+:Clearlyyn=08y2n+:Henceby.26wehaven+n=0whileby.25hn=hn8h2h:ThisimpliesthatthecyclicmoduleUgnisamodelfortheirreduciblerepresentationVofgwithhighestweight.Leftmultiplicationbyx;x2ggivestheactionofgonthismodule.Furthermore,ifnc6=0forsomec2Cgthennchasthesameproperty:n+nc=0;hnc=hnc;8h2h:Thuseverync6=0alsogeneratesagmoduleisomorphictoV.Nowthemap^n+^b)]TJ/F14 9.963 Tf 9.492 1.494 Td[(!Cg;xb!xbisalinearisomorphismandrightCliordmultiplicationof^Nn+by^n+isjust^Nn+,alltheelementsofof^+n+yielding0.SowehavethevectorspaceisomorphismnCg=^Nn+^b)]TJ/F11 9.963 Tf 6.725 1.495 Td[(:Inotherwords,UgnCg

PAGE 162

162CHAPTER9.CLIFFORDALGEBRASANDSPINREPRESENTATIONS.isadirectsumofirreduciblemodulesallisomorphictoVwithmultiplicityequaltodim^b)]TJ/F8 9.963 Tf 9.492 1.494 Td[(=2s+Nwheres=dimhandN=dimn)]TJ/F8 9.963 Tf 10.508 1.495 Td[(=dimn+.LetuscomputethedimensionofVusingtheWeyldimensionformulawhichassertsthatforanyirreduciblenitedimensionalrepresentationVwithhighestweightwehavedimV=Q2++; Q2+;:Ifweplugin=weseethateachfactorinthenumeratoristwicethecorrespondingfactorinthedenominatorsodimV=2N:.27ButthendimUgnCg=2s+2N=dimCg:ThisimpliesthatCg=UgnCg=Ugn^b)]TJ/F8 9.963 Tf 6.725 1.495 Td[(;.28provingthatCgisprimaryoftypeVwithmultiplicity2s+Nasarepresen-tationofgundertheleftmultiplicationactionofg.Thisimpliesthatanysubmoduleforthisaction,inparticularanyleftidealofCg,isprimaryoftypeV.SincewehaverealizedthespinrepresentationofCgasaleftidealinCgwehaveprovedtheimportantTheorem17SpinadisprimaryoftypeV.Oneconsequenceofthistheoremisthefollowing:Proposition29TheweightsofVare)]TJ/F11 9.963 Tf 9.963 0 Td[(J.29whereJrangesoversubsetsofthepositiverootsandJ=X2JJeachoccurringwithmultiplicityequaltothenumberofsubsetsJyieldingthesamevalueofJ.Indeed,.21givestheweightsofSpinad,butseveraloftheJareequalduetothetrivialactionofadhonitself.Howeverthiscontributiontothemultiplicityofeachweightoccurringin.21isthesame,andhenceisequaltothemultiplicityofVinSpinad.SoeachweightvectorofVmustbeoftheform.29eachoccurringwiththemultiplicitygivenintheproposition.

PAGE 163

Chapter10TheKostantDiracoperatorLetpbeavectorspacewithanon-degeneratesymmetricbilinearform.WehavetheCliordalgebraCpandtheidenticationofop=^2pinsideCp.10.1Antisymmetrictrilinearforms.Letbeanantisymmetrictrilinearformonp.Thendenesanantisymmetricmapb=b:pp!pbytheformulaby;y0;y00=y;y0;y008y;y0;y002p:Thisbilinearmapleaves;invariant"inthesensethatby;y0;y00=y;by0;y00:Conversely,anyantisymmetricmapb:pp!psatisfyingthisconditiondenesanantisymmetricform.Finallyeitherofthesetwoobjectsdenesanelementv2^3pby)]TJ/F8 9.963 Tf 7.748 0 Td[(2v;y^y0^y00=by;y0;y00=y;y0;y00:.1Wecanwritethisrelationinseveralalternativeways:Since)]TJ/F8 9.963 Tf 7.749 0 Td[(2v;y^y0^y00=)]TJ/F8 9.963 Tf 7.749 0 Td[(2y0yv;y00=2yy0v;y00wehaveby;y0=2yy0v:.2Also,yv2^2pandsoisidentiedwithanelementofopbycommutatorintheClifordalgebra:adyvy0=[yv;y0]=)]TJ/F8 9.963 Tf 7.749 0 Td[(2y0yv163

PAGE 164

164CHAPTER10.THEKOSTANTDIRACOPERATORsoadyvy0=[yv;y0]=by;y0:.310.2JacobiandCliord.Givenanantisymmetricbilinearmapb:pp!pwemaydeneJacb:ppp!pbyJacby;y0;y00=bby;y0;y00+bby0;y00;y+bby00;y;y0sothatthevanishingofJacbistheusualJacobiidentity.ItiseasytocheckthatJacbisantisymmetricandthatifbsatisesby;y0;y00=y;by0;y00thenthefourformy;y0;y00;y0007!Jacby;y0;y00;y000isantisymmetric.Weclaimthatifv2^3pasintheprecedingsubsection,theny00y0yv2=1 2Jacby;y0;y00:.4Toprovethisobservethatyv2=yvv)]TJ/F11 9.963 Tf 9.962 0 Td[(vyvy0yv2=y0yvv+yvy0v)]TJ/F8 9.963 Tf 9.963 0 Td[(y0vyv+vy0yvy00y0yv2=)]TJ/F8 9.963 Tf 7.749 0 Td[(y0yvy00v+y00vy0yv+y00yvy0v+yvy00y0v)]TJ/F8 9.963 Tf 9.963 0 Td[(y00y0vyv)]TJ/F8 9.963 Tf 9.963 0 Td[(y0vy00yv=[y00v;y0yv]+[y0v;yy00v]+[yv;y0y00v]=1 2Jacby;y0;y00by.2and.3.Equation10.4describesthedegreefourcomponentofv2intermsofJacb.Wecanbeexplicitaboutthedegreezerocomponentofv2.Weclaimthatv20=1 24trnXj=1[y!jbyj;byj;y];j:=yj;yj:.5Indeed,by.6weknowthatv20=)]TJ/F8 9.963 Tf 7.749 0 Td[(v;vandsinceyi^yj^yk;i
PAGE 165

10.3.ORTHOGONALEXTENSIONOFALIEALGEBRA.165formanorthonormal"basisof^3pwehave)]TJ/F8 9.963 Tf 7.749 0 Td[(v;v=)]TJ/F1 9.963 Tf 24.577 9.464 Td[(X1i
PAGE 166

166CHAPTER10.THEKOSTANTDIRACOPERATORSowehaveprovedx;[y;y0]rr=xy;y0p:.7Thishasthefollowingsignicance:SupposethatwewanttomakerpintoaLiealgebrawithaninvariantsymmetricbilinearform;suchthatrandpareorthogonalunder;,therestrictionof;toris;randtherestrictionof;topis;p,and[r;p]pandthebracketofanelementofrwithanelementofpisgivenby[x;y]=xy.Thenthercomponentof[y;y0]mustbegivenby[y;y0]r:ThustodeneaLiealgebrastructureonrpwemustspecifythepcomponentofthebracketoftwoelementsofp.Thisamountstospecifyingav2^3paswehaveseen,andtheconditionthattheJacobiidentityholdforx;y;y0withx2randy;y02pamountstotheconditionthatv2^3pbeinvariantundertheactionofr.Itthenfollowsthatifwetrytodene[;]=[;]vby[y;y0]=[y;y0]r+2yy0vthen[z;z0];z00=z;[z0;z00]foranythreeelementsofg:=rp,andtheJacobiidentityissatisedifatleastoneoftheseelementsbelongstor.Furthermore,foranyx2rwehave[[y;y0];y00];x=[[y;y0];y00];xr=[y;y0];[y00;x]pby.7=[x;[y;y0]];y00p=[[x;y];y0];y00p+[y;[x;y0]];y00p=[x;y];[y0;y00]p+[x;y0];[y00;y]p=x;[y;[y0;y00]r+x;[y0;[y00;y]]ror[[y;y0];y00]+[[y0;y00];y]+[[y00;y];y0];x=0:Inotherwords,thercomponentoftheJacobiidentityholdsforthreeelementsofp.SowhatremainstobecheckedisthepcomponentoftheJacobiidentityforthreeelementsofp.ThisisthesumJacby;y0;y00+[y;y0]ry00+[y0;y00]ry+[y00;y]ry0:

PAGE 167

10.4.THEVALUEOF[V2+CASR]0.167Letuschooseanorthonormal"basisfxig;i=1;:::;rofrandwrite[y;y0]r=Xii[y;y0];xixi;i:=xi;xir=1so[y;y0]ry00=Xii[y;y0]r;xixiy00:Thenby.11and.4weseethattheJacobiidentityisyy0y00)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(v2+Casr=0whereCasr:=Xiix2i2Urdoesnotdependonthechoiceofbasis,and:Ur!Cpistheextensionofthehomomorphism:r!Cp:Inparticular,wehaveprovedthatvdenesanextensionoftheLiealgebrastructuresatisfyingourconditionifandonlyifv2+Casr2C.8i.e.hasnocomponentofdegreefour.Supposethatthisconditionholds.WethenhavedenedaLiealgebrastructureong=rp:WeletPrandPpdenoteprojectionsontotherstandsecondcomponentsofourdecomposition.OurLiebracketong,denotedsimplyby[;]satises[x;x0]=[x;x0]r;x;x02r.9[x;y]=xy;x2r;y2p.10Pr[y;y0]=[y;y0]r=)]TJ/F8 9.963 Tf 7.749 0 Td[(2yy^y0y;y02p.11Pp[y;y0]=by;y0=2yy0v;y;y02p:.12Fromnowonwewillassumethatweareoverthecomplexnumbersorthatweareovertherealsandthesymmetricbilinearformsarepositivedenite.Thisisnotforanymathematicalreasonsbutbecausetheformulasbecomeabitcomplicatedifweputinallthesigns.Weleavethegeneralcasetothereader.10.4Thevalueof[v2+Casr]0.Condition.8saysthatthedegreefourcomponentofv2+Casrvanishes.Assumethatthisholds,sowehaveconstructedaLiealgebra.Wewillnowcomputethedegreezerocomponentofv2+Casr.Theanswerwillbegiveninequation.13below.

PAGE 168

168CHAPTER10.THEKOSTANTDIRACOPERATORWecanwrite.5asv20=1 24trPpnXj=1adgyjPpadgyjPp=1 24trPpnXj=1adgyjPpadgyjinviewof.12whereadgdenotestheadjointactiononallofg.Ontheotherhand,wehavefrom.12thatCasr0=1 8trXiadxi2Pp=1 8rXi=1nXj=1[xi;[xi;yj]];yj:Wecanrewritethissumas1 8r;nXi=1;j=1[xi;yj];[yj;xi]whichequals1 8Pr;n;ni=1;j=1;k=1[xi;yj];ykyk;[yj;xi].Butthisequals1 8r;n;nXi=1;j=1;k=1xi;[yj;yk][yk;yj];xi=1 8Xj=1;k=1Pr[yj;yk];Pr[yk;yj]=1 8n;nXj=1;k=1Pr[yj;yk];[yk;yj]=1 8n;nXj=1;k=1[yj;Pr[yj;yk]];yk=1 8trPpnXj=1adgyjPradgyjPp:InotherwordsCasr0=1 8trPpnXj=1adgyjPradgyjPp=1 8trnXj=1adgyjPradgyjPp:Multiplyingthisequationby1=3andaddingittotheaboveexpressionforv20gives1 3Casr0+v20=1 24trnXj=1adgyj2Pp:

PAGE 169

10.5.KOSTANT'SDIRACOPERATOR.169WecanwriteCasr0=1 8r;nXi=1;j=1[xi;yj];[yj;xi]=1 8r;nXi=1;j=1xi;[yj;[yj;xi]]=1 8trPrnXj=1adgyj2Pr=1 8trnXj=1adgyj2Pr:Multiplyingby1=3andaddingtotheprecedingequationgives2 3Casr0+v20=1 24trnXj=1adgyj2:OntheotherhandCasr0=1 8r;nXi=1;j=1[xi;[xi;yj]];yj=1 8tradpCasr=1 8tradgCasr)]TJ/F8 9.963 Tf 9.963 0 Td[(tradrCasr:Multiplyingby1=3andaddingtotheprecedingequation,andusingthefactthatCasg=Casr+Py2jgivesCasr+v2=1 24tradgCasg)]TJ/F8 9.963 Tf 9.963 0 Td[(tradrCasr.13when.8holds.SupposenowthattheLiealgebrarisreductiveandthattheLiealgebragwecreatedoutofrandpusingav2^3psatisfying.8isalsoreductive.Using.29forgandforrintherighthandsideof.13yieldsCasr+v2=g;g)]TJ/F8 9.963 Tf 9.963 0 Td[(r;r:.1410.5Kostant'sDiracOperator.SupposethatwehaveconstructedourLiealgebrag=r+pfromav2^3psatisfying.8.Wearegoingtodene6K2UgCpasfollows:Lety1;:::;ynbeanorthonormalbasisofp.Then6K:=Xiyiyi+1v:.15Ontheleftofthetensorproductsigntheyi2pisconsideredasanelementofUgviathecanonicalinjectionofpinUpUgandontherightofthetensorproductsignitliesinCpviathecanonicalinjectionofpintoCp.

PAGE 170

170CHAPTER10.THEKOSTANTDIRACOPERATORWehaveahomomorphismUr!Ug,inparticularaLiealgebrainjectionr!Ug.WealsohaveaLiealgebrahomomorphism:r!Cp.Inparticular,wehavethediagonalLiealgebramapdiag:r!UgCp;diagx=x1+1xandthisextendstoanalgebramapdiag:Ur!UgCp:Forexample,diagCasr=Xixi1+1xi2wherex1;:::;xrisanorthonormalbasisofr.InotherwordsdiagCasr=rXi=1x2i1+2rXi=1xixi+rXi=11xi2:.16Weclaimthat6K2=Casg1)]TJ/F8 9.963 Tf 9.963 0 Td[(diagCasr+1 24tradgCasg)]TJ/F8 9.963 Tf 9.963 0 Td[(tradrCasr11:.17Toprovethis,letuswrite.15as6K=6K0+6K00:So6K002=1v2andhence6K002+rXj=11xi2=1 24tradgCasg)]TJ/F8 9.963 Tf 9.963 0 Td[(tradrCasr11by.13.Wehave6K02=Xijyiyjyiyj=Xiy2i1+Xi6=jyiyjyiyj=Xiy2i1+Xi
PAGE 171

10.5.KOSTANT'SDIRACOPERATOR.171wherewehaveusedthedecompositionof[yi;yj]intoitsrandpcomponentstogettothelastexpression.Wecanwritethemiddleterminthelastexpressionas)]TJ/F8 9.963 Tf 7.748 0 Td[(2Xi
PAGE 172

172CHAPTER10.THEKOSTANTDIRACOPERATORSupposethatisthehighestweightofanitedimensionalirreduciblerep-resentationVofgsothatwegetasurjectivehomomorphismUg!EndVandhenceacorrespondinghomomorphismUgCp!EndVCp:Alsoletdiagdenotethecompositionofthishomomorphismwithdiag:Ur!UgCp:ThenfromthevalueoftheCasimir.8and.18weget6K2=+g;+g)]TJ/F8 9.963 Tf 9.962 0 Td[(r;r11)]TJ/F8 9.963 Tf 9.962 0 Td[(diagCasr:.1910.6EigenvaluesoftheDiracoperator.Weconsiderthesituationwhereg=rpisaLiealgebrawithinvariantsymmetricbilinearform,whererhasthesamerankasg,andwherewehavechosenacommonCartansubalgebrahrg:Welet`denotethedimensionofh,i.e.thecommonrankofrandg.Welet=gdenotethesetofrootsofg,letW=WgdenotetheWeylgroupofg,andletWrdenotetheWeylgroupofrsothatWrWandweletcdenotetheindexofWrinW.Achoiceofpositiveroots+forgamountstoachoiceofaBorelsubalgebrabofgandthenbrisaBorelsubalgebraofr,whichpicksoutasystemofpositiveroots+rforrandthen+r+:ThecorrespondingWeylchambersareD=Dg=f2hRj;082+gandDr=f2hRj;082+rgsoDDrandwehavechosenacross-sectionCofWrinWasC=fw2WjwDDrg;

PAGE 173

10.6.EIGENVALUESOFTHEDIRACOPERATOR.173soW=WrC;Dr=[w2CwD:WeletL=LghRdenotethelatticeofgintegrallinearformsonh,i.e.L=f2hj2; ;2Z82g:Welet=g=1 2X2+andr=1 2X2+r:WesetLr=thelatticespannedbyLandr;and:=LD;r:=LrDr:ForanyrmoduleZwelet\050Zdenoteitssetofweights,andweshallassumethat\050ZLr:ForsucharepresentationdenemZ:=max2\050Z+r;+r:.20Forany2rweletZdenotetheirreduciblemodulewithhighestweight.Proposition30Let)]TJ/F7 6.974 Tf 6.227 -1.495 Td[(maxZ:=f2\050Zj+r;+r=mZg:Let2)]TJ/F7 6.974 Tf 6.227 -1.494 Td[(maxZ.Then1.2r.2.Ifz6=0isaweightvectorwithweightthenzisahighestweightvector,andhencethesubmoduleUrzisirreducibleandequivalenttoZ.3.LetYmax:=X2)]TJ/F6 4.981 Tf 4.926 -0.996 Td[(maxZZandY:=UrYmax:ThenmZ)]TJ/F8 9.963 Tf 9.943 0 Td[(r;risthemaximaleigenvalueofCasronZandYisthecorrespondingeigenspace.

PAGE 174

174CHAPTER10.THEKOSTANTDIRACOPERATORProof.Werstshowthat2)]TJ/F7 6.974 Tf 6.227 -1.495 Td[(max+r2r:Supposenot,sothereexistsaw6=1;w2Wrsuchthatw+wr2r:Butwchangesthesignofsomeofthepositiverootsthenumberofsuchchangesbeingequalthelengthofwintermsofthegeneratingreections,andsor)]TJ/F11 9.963 Tf 7.782 0 Td[(wrisanon-trivialsumofpositiveroots.Thereforew+wr;r)]TJ/F11 9.963 Tf 9.963 0 Td[(wr0;r)]TJ/F11 9.963 Tf 9.962 0 Td[(wr;r)]TJ/F11 9.963 Tf 9.962 0 Td[(wr>0andw+r=w+wr+r)]TJ/F11 9.963 Tf 9.962 0 Td[(wrsatisesw+r;w+r>w+wr;w+wr=+r;+=mZcontradictingthedenitionofmZ.Nowsupposethatzisaweightvectorwithweightwhichisnotahighestweightvector.ThentherewillbesomeirreduciblecomponentofZcontainingzandhavingsomeweight0suchthat0)]TJ/F11 9.963 Tf 9.962 0 Td[(isanontrivialsumofpositiveroots.Wehave0+r=0)]TJ/F11 9.963 Tf 9.963 0 Td[(++rsobythesameargumentweconcludethat0+r;0+r>mZsince+r2r,andagainthisisimpossible.Hencezisahighestweightvectorimplyingthat2r.Thisproves1and2.WehavealreadyveriedthattheeigenvalueoftheCasimirCasronanyZis+r;+r)]TJ/F8 9.963 Tf 9.963 0 Td[(r;r:Thisproves3.ConsidertheirreduciblerepresentationVofgcorrespondingto=g.Bythesamearguments,anyweight6=ofVlyinginDmustsatisfy;<;andhenceanyweightofVsatisfying;=;mustbeoftheform=wforauniquew2W:Butw=)]TJ/F1 9.963 Tf 12.88 9.465 Td[(X2Jw=)]TJ/F11 9.963 Tf 9.963 0 Td[(JwhereJw:=w)]TJ/F8 9.963 Tf 7.749 0 Td[(++:

PAGE 175

10.6.EIGENVALUESOFTHEDIRACOPERATOR.175WeknowthatalltheweightsofVareoftheform)]TJ/F11 9.963 Tf 10.019 0 Td[(JasJrangesoverallsubsetsof+.So;)]TJ/F11 9.963 Tf 9.963 0 Td[(J;)]TJ/F11 9.963 Tf 9.963 0 Td[(J10.21wherewehavestrictinequalityunlessJ=Jwforsomew2W.Nowlet2,letVbethecorrespondingirreduciblemodulewithhighestweightandletbeaweightofV.Asusual,letJdenoteasubsetofthepositiveroots,J+.WeclaimthatProposition31Wehave+;++)]TJ/F11 9.963 Tf 9.963 0 Td[(J;+)]TJ/F11 9.963 Tf 9.963 0 Td[(J0.22withstrictinequalityunlessthereexistsaw2Wsuchthat=w;andJ=Jwinwhichcasethewisunique.Proof.Choosewsuchthatw)]TJ/F7 6.974 Tf 6.227 0 Td[(1+)]TJ/F11 9.963 Tf 9.963 0 Td[(J2:Sincew)]TJ/F7 6.974 Tf 6.227 0 Td[(1isaweightofV,)]TJ/F11 9.963 Tf 8.932 0 Td[(w)]TJ/F7 6.974 Tf 6.227 0 Td[(1isasumpossiblyemptyofpositiveroots.Alsow)]TJ/F7 6.974 Tf 6.226 0 Td[(1)]TJ/F11 9.963 Tf 9.987 0 Td[(JisaweightofVandhence)]TJ/F11 9.963 Tf 9.986 0 Td[(w)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F11 9.963 Tf 9.986 0 Td[(Jisasumpossiblyemptyofpositiveroots.Since+=)]TJ/F11 9.963 Tf 9.962 0 Td[(w)]TJ/F7 6.974 Tf 6.227 0 Td[(1+)]TJ/F11 9.963 Tf 9.963 0 Td[(w)]TJ/F7 6.974 Tf 6.227 0 Td[(1)]TJ/F11 9.963 Tf 9.963 0 Td[(J+w)]TJ/F7 6.974 Tf 6.227 0 Td[(1+)]TJ/F11 9.963 Tf 9.963 0 Td[(J;weconcludethat+;+w)]TJ/F7 6.974 Tf 6.227 0 Td[(1+)]TJ/F11 9.963 Tf 9.752 0 Td[(J;w)]TJ/F7 6.974 Tf 6.227 0 Td[(1+)]TJ/F11 9.963 Tf 9.752 0 Td[(J=+)]TJ/F11 9.963 Tf 9.752 0 Td[(J;+)]TJ/F11 9.963 Tf 9.752 0 Td[(Jwithstrictinequalityunless)]TJ/F11 9.963 Tf 10.176 0 Td[(w)]TJ/F7 6.974 Tf 6.227 0 Td[(1=0=)]TJ/F11 9.963 Tf 10.175 0 Td[(w)]TJ/F7 6.974 Tf 6.226 0 Td[(1)]TJ/F11 9.963 Tf 10.176 0 Td[(J,andthislastequalityimpliesthatJ=Jw.QEDWehavethespinrepresentationSpinwhere:r!Cp.CallthismoduleS.ConsiderVSasarmodule.Then,lettingdenoteaweightofV,wehave\050VS=f=+p)]TJ/F11 9.963 Tf 9.963 0 Td[(Jg.23wherep=1 2XJ2+p;+p:=+=+r:Inotherwords,paretherootsofgwhicharenotrootsofr,or,putanotherway,theyaretheweightsofpconsideredasarmodule.Ourequalrank

PAGE 176

176CHAPTER10.THEKOSTANTDIRACOPERATORassumptionsaysthat0doesnotoccurasoneoftheseweights.FortheweightsofVStheform.23gives+r=+)]TJ/F11 9.963 Tf 9.962 0 Td[(J;J+p:SoifwesetZ=VSasarmodule,.22saysthat+;+mZ:ButwemaytakeJ=;asoneofourweightsshowingthatmZ=+g;+g:.24Todetermine)]TJ/F7 6.974 Tf 69.514 -1.494 Td[(maxZasinProp.30weagainuseProp.31and.23:A=+p)]TJ/F11 9.963 Tf 10.252 0 Td[(Jbelongsto)]TJ/F7 6.974 Tf 55.026 -1.494 Td[(maxZifandonlyif=wandJ=Jw.Buttheng)]TJ/F11 9.963 Tf 9.963 0 Td[(J=wg:Sinceg=r+pweseefromtheform10.23that+r=w+g.25wherewisunique,andJw+p:WeclaimthatthisconditionisthesameastheconditionwDDrdeningourcross-section,C.Indeed,w2Cifandonlyif;wg>0;82+r.But;w=w)]TJ/F7 6.974 Tf 6.227 0 Td[(1;>0ifandonlyif2w+.SinceJw=w)]TJ/F8 9.963 Tf 7.749 0 Td[(++,weseethatJw+pisequivalenttotheconditionw2C.Nowfor2)]TJ/F7 6.974 Tf 6.227 -1.494 Td[(maxZwehave=w+)]TJ/F11 9.963 Tf 9.962 0 Td[(r=:w.26where=wandsohasmultiplicityoneinV.Furthermore,weclaimthattheweightp)]TJ/F11 9.963 Tf 10.149 0 Td[(JwhasmultiplicityoneinS.Indeed,considertherepresentationZrSofr.Ithastheweight=r+pasahighestweight,andinfact,alloftheweightsofVgoccuramongitsweights.Hence,ondimensionalgrounds,sayfromtheWeylcharacterformula,weconcludethatitcoincides,asarep-resentationofr,withtherestrictionoftherepresentationVgtor.Butsinceg)]TJ/F11 9.963 Tf 10.544 0 Td[(Jw=wghasmultiplicityoneinVg,weconcludethatp)]TJ/F11 9.963 Tf 10.544 0 Td[(JwhasmultiplicityoneinS.WehaveprovedthateachofthewhavemultiplicityoneinVSwithcorrespondingweightvectorszw:=vweJw)]TJ/F11 9.963 Tf 10.564 2.48 Td[(e+:

PAGE 177

10.6.EIGENVALUESOFTHEDIRACOPERATOR.177SoeachofthesubmodulesZw:=Urzw.27occurswithmultiplicityoneinVS.Thelengthofw2CintermsofthesimplereectionsofWdeterminedbyisthenumberofpositiverootschangedintonegativeroots,i.e.thecardinalityofJw.ThiscardinalityisthesignofdetwandalsodetermineswhethereJ)]TJ/F11 9.963 Tf 6.725 2.463 Td[(e+belongstoS+ortoS)]TJ/F8 9.963 Tf 6.724 1.495 Td[(.FromProp.31andequation.24weknowthatthemaximumeigenvalueofCasronVSis+g;+g)]TJ/F8 9.963 Tf 9.962 0 Td[(r;r:Now6K2EndVScommuteswiththeactionofrwithVS+!VS)]TJ/F14 9.963 Tf -120.292 -13.45 Td[(6K:VS)]TJ/F14 9.963 Tf 9.492 1.494 Td[(!VS+:Furthermore,by10.19,thekernelof6K2istheeigenspaceofCasrcorrespond-ingtotheeigenvalue+;+)]TJ/F8 9.963 Tf 9.962 0 Td[(r;r.ThusKer6K2=Xw2CZw:EachofthesemoduleslieseitherinVS+orVS)]TJ/F8 9.963 Tf 6.725 1.494 Td[(,oneortheotherbutnotboth.HenceKer6K2=Ker6KandsoKer6KjVS+=Xw2C;detw=1Zw.28andKer6KjVS)]TJ/F8 9.963 Tf 9.658 5.651 Td[(=Xw2C;detw=)]TJ/F7 6.974 Tf 6.227 0 Td[(1Zw.29LetK:=Xw2C;detw=1Zw:.30Itfollowsfrom.28that6KinducesaninjectionofVS+=K+!VS)]TJ/F8 9.963 Tf -221.268 -19.792 Td[(whichwecanfollowbytheprojectionVS)]TJ/F14 9.963 Tf 9.492 1.494 Td[(!VS)]TJ/F8 9.963 Tf 6.724 1.494 Td[(=K)]TJ/F11 9.963 Tf 6.725 1.494 Td[(:Hence6Kinducesabijection~6K:VS+=K+!VS)]TJ/F8 9.963 Tf 6.725 1.494 Td[(=K)]TJ/F11 9.963 Tf 6.725 1.494 Td[(:.31

PAGE 178

178CHAPTER10.THEKOSTANTDIRACOPERATORInshort,wehaveprovedthatthesequence0!K+!VS+!VS)]TJ/F14 9.963 Tf 9.492 1.494 Td[(!K)]TJ/F14 9.963 Tf 9.492 1.494 Td[(!0.32isexactinaveryprecisesense,wherethemiddlemapistheKostantDiracoperator:eachsummandofK+occursexactlyonceinVS+andsimilarlyforK)]TJ/F8 9.963 Tf 6.725 1.494 Td[(.ThisgivesamuchmoreprecisestatementofTheorem16andacompletelydierentproof.10.7Thegeometricindextheorem.LetrbetherepresentationofGonthespaceFGofsmoothoronL2GofL2functionsonGcomingfromrightmultiplication.Thus[rgf]a=fag:Then6KactsonFGSoronL2GSandcentralizestheactionofdiagr.IfUisamoduleforR,wemayconsiderFGSUorL2GSU,and6K1commuteswithdiagr1andwiththeactionofRonU,i.ewith11.IfRisconnected,thisimpliesthat6Kcommuteswiththediagonalactionof~R,theuniversalcoverofR,onFSUorL2GSUgivenbyk7!rkSpinkk;k2RwhereSpin:~R!SpinpisthegrouphomomorphismcorrespondingtotheLiealgebrahomomorphism.IfG=Risaspinmanifold,theinvariantsunderthisRactioncorrespondtosmoothorL2sectionsofSUwhereSisthespinbundleofG=RandUisthevectorbundleonG=RcorrespondingtoU.Thus6Kdescendsbyrestrictiontoadierentialoperator6@onG=RandweshallcomputeitsG-indexforirreducibleU.Thekeyresult,duetoLandweber,assertsthatifUbelongstoamultipletcomingfromanirreducibleVofG,thenthisindexis,uptoasign,equaltoV.IfUdoesnotbelongtoamultiplet,thenthisindexiszero.WebeginwithsomepreliminaryresultsduetoBott.10.7.1TheindexofequivariantFredholmmaps.LetEandFbeHilbertspaceswhichareunitarymodulesforthecompactLiegroupG.SupposethatE=dMnEn;F=dMnFnarecompleteddirectsumdecompositionsintosubspaceswhichareG-invariantandnitedimensional,andthatT:E!F

PAGE 179

10.7.THEGEOMETRICINDEXTHEOREM.179isaFredholmmapnitedimensionalkernelandcokernelsuchthatTEnFn:WewriteIndexGT=KerT)]TJ/F8 9.963 Tf 9.962 0 Td[(CokerTasanelementofRG,theringofvirtualrepresentationsofG.ThusRGisthespaceofnitelinearcombinationsPaV;a2ZasVrangesovertheirreduciblerepresentationsofG.Here,andinwhatfollows,weareregard-inganynitedimensionalrepresentationofGasanelementofRGbyitsdecompositionintoirreducibles,andsimilarlythedierenceofanytwonitedimensionalrepresentationsisanelementofRG.IfwedenotetherestrictionofTtoEnbyTn,thenIndexGT=XIndexGTnwhereallbutanitenumberoftermsontherightvanish.Foreachnwehavetheexactsequence0!KerTn!En!Fn!CokerTn!0:ThusIndexGTn=En)]TJ/F11 9.963 Tf 9.962 0 Td[(FnaselementsofRG.ThereforewecanwriteIndexGT=XEn)]TJ/F11 9.963 Tf 9.963 0 Td[(Fn.33inRG,whereallbutanitenumberoftermsontherightvanish.WeshallrefertothisasBott'sequation.10.7.2InducedrepresentationsandBott'stheorem.LetRbeaclosedsubgroupofG.GivenanyR-actiononavectorspaceU,weconsidertheassociatedvectorbundleGRVoverthehomogeneousspaceG=R.ThesectionsofthisbundlearethenequivariantU-valuedfunctionsonGsatisfyingsgk=k)]TJ/F7 6.974 Tf 6.227 0 Td[(1sgforallk2R.ApplyingthePeter-Weyltheorem,wecandecomposethespaceofL2mapsfromGtoUintoasumovertheirreduciblerepresentationsVofG,L2GU=dMVVU;withrespecttotheGGRactionlr.TheR-equivarianceconditionisequivalenttorequiringthatthefunctionsbeinvariantunderthediagonalR-actionk7!rkk.RestrictingthePeter-WeyldecompositionabovetotheRinvariantsubspace,weobtainL2GRU=cLVVUR=cLVHomRV;U:.34

PAGE 180

180CHAPTER10.THEKOSTANTDIRACOPERATORTheLiegroupGactsonthespaceofsectionsbylg,theleftactionofGonfunctions,whichispreservedbythisconstruction.ThespaceL2GHUisthusaninnitedimensionalrepresentationofG.TheintertwiningnumberoftworepresentationsgivesusaninnerproducthV;WiG=dimCHomGV;WonRG,withrespecttowhichtheirreduciblerepresentationsofGformanorthonormalbasis.TakingtheformalcompletionofRG,wedene^RGtobethespaceofpossiblyinniteformalsumsPaV.TheintertwiningnumberthenextendstoapairingRG^RG!Z.IfRisasubgroupofG,everyrepresentationofGautomaticallyrestrictstoarepresentationofR.Thisgivesusapullbackmapi:RG!RR,correspondingtotheinclusioni:R,!G.ThemapU7!L2GHUdis-cussedaboveassignstoeachR-representationaninducedinnitedimensionalG-representation.Expressedintermsofourrepresentationringnotation,thisinductionmapbecomesthehomomorphismi:RR!^RGgivenbyiU=XhiV;UiRV;theformaladjointtothepullbacki.ThisisthecontentoftheFrobeniusreciprocitytheorem.AhomogeneousdierentialoperatoronG=RisadierentialoperatorD:\050E!\050FbetweentwohomogeneousvectorbundlesEandFthatcommuteswiththeleftactionofGonsections.Iftheoperatoriselliptic,thenitskernelandcokernelarebothnitedimensionalrepresentationsofG,andthusitsG-indexisavirtualrepresentationinRG.Inthiscase,theindextakesaparticularlyelegantform.Theorem18BottIfD:\050GHU0!\050GHU1isanelliptichomoge-neousdierentialoperator,thentheG-equivariantindexofDisgivenbyIndexGD=iU0)]TJ/F11 9.963 Tf 9.963 0 Td[(U1;whereiU0)]TJ/F11 9.963 Tf 9.962 0 Td[(U1isaniteelementin^RG,i.e.belongstoRG.Inparticular,notethattheindexofahomogeneousdierentialoperatordependsonlyonthevectorbundlesinvolvedandnotontheoperatoritself!Toprovethetheorem,justuseBott'sformula.33,wherethesubscriptnisreplacedbylabelingthetheG-irreducibles.10.7.3Landweber'sindextheorem.SupposethatGissemi-simpleandsimplyconnectedandRisareductivesub-groupofmaximalrank.SupposefurtherthatG=Risaspinmanifold,thenwecancomposethespinrepresentationS=S+S)]TJ/F8 9.963 Tf 10.757 1.495 Td[(ofSpinpwiththelifted

PAGE 181

10.7.THEGEOMETRICINDEXTHEOREM.181mapSpin:~R!Spinptoobtainahomogeneousvectorbundle,thespinbun-dleSoverG=R.ForanyrepresentationofRonUtheKostantDiracoperatordescendstoanoperator6@U:\050SU!\050SU:ThisoperatorhasthesamesymbolastheDiracoperatorarisingfromtheLevi-CivitaconnectiononG=RtwistedbyU,andhasthesameindexbyBott'stheorem.ForthepreciserelationbetweenthisDiracoperatorcomingfrom6KandthediracoperatorcomingfromtheLevi-civitaconnectionwerefertoLandweber'sthesis.ThefollowingtheoremofLandwebergivesanexpressionfortheindexofthisKostantDiracoperator.Inparticular,ifweconsiderG=T,whereTisamaximaltoruswhichisalwaysaspinmanifold,thistheorembecomesaversionoftheBorel-Weil-BotttheoremexpressedintermsofspinorsandtheDiracoperator,insteadofinitscustomaryforminvolvingholomorphicsectionsandDolbeaultcohomology.Theorem19LandweberLetG=Rbeaspinmanifold,andletUbeanirreduciblerepresentationUofRwithhighestweight.TheG-equivariantindexoftheDiracoperator6@UisthevirtualG-representationIndexG6@U=)]TJ/F8 9.963 Tf 7.749 0 Td[(1dimp=2)]TJ/F8 9.963 Tf 7.748 0 Td[(1wVw+H)]TJ/F10 6.974 Tf 6.227 0 Td[(G.35ifthereexistsanelementw2WGintheWeylgroupofGsuchthattheweightw+H)]TJ/F11 9.963 Tf 9.963 0 Td[(GisdominantforG.Ifnosuchwexists,thenIndexG6@U=0.Proof.ForanyirreduciblerepresentationVofGwithhighestweightwehaveVS+)]TJ/F11 9.963 Tf 9.963 0 Td[(S)]TJ/F8 9.963 Tf 6.725 1.495 Td[(=Xw2C)]TJ/F8 9.963 Tf 7.748 0 Td[(1wUwby[GKRS].HenceHomRVS+)]TJ/F11 9.963 Tf 9.962 0 Td[(S)]TJ/F8 9.963 Tf 6.725 1.494 Td[(;U=0if6=wforsomew2CwhileHomRVS+)]TJ/F11 9.963 Tf 9.963 0 Td[(S)]TJ/F8 9.963 Tf 6.725 1.495 Td[(;U=)]TJ/F8 9.963 Tf 7.748 0 Td[(1wif=w.But,by.33andTheorem18wehaveIndexG6@U=dMVVS+)]TJ/F11 9.963 Tf 9.963 0 Td[(S)]TJ/F8 9.963 Tf 6.725 1.494 Td[(UR=dMHomRVS+)]TJ/F11 9.963 Tf 9.963 0 Td[(S)]TJ/F8 9.963 Tf 6.725 1.495 Td[(;U:NowS+)]TJ/F11 9.963 Tf 9.723 0 Td[(S)]TJ/F8 9.963 Tf 6.725 1.494 Td[(=S+)]TJ/F11 9.963 Tf 9.724 0 Td[(S)]TJ/F8 9.963 Tf 9.926 1.494 Td[(ifdimp=0mod4whileS+)]TJ/F11 9.963 Tf 9.724 0 Td[(S)]TJ/F8 9.963 Tf 6.725 1.494 Td[(=S)]TJ/F14 9.963 Tf 8.699 1.494 Td[()]TJ/F11 9.963 Tf 9.724 0 Td[(S+ifdimp=2mod4:HenceIndexG6@U=)]TJ/F8 9.963 Tf 7.749 0 Td[(1dimp=2dMHomRVS+)]TJ/F11 9.963 Tf 9.963 0 Td[(S)]TJ/F8 9.963 Tf 6.725 1.494 Td[(;U:.36

PAGE 182

182CHAPTER10.THEKOSTANTDIRACOPERATORTherighthandsideof.36vanishesifdoesnotbelongtoamultiplet,i.eisnotoftheformw=w+g)]TJ/F11 9.963 Tf 9.963 0 Td[(rforsome.Theconditionw=canthusbewrittenasw)]TJ/F7 6.974 Tf 6.227 0 Td[(1+r)]TJ/F11 9.963 Tf 9.963 0 Td[(g=:Ifthisequationdoeshold,thenwegettheformulainthetheoremwithwreplacedbyw)]TJ/F7 6.974 Tf 6.227 0 Td[(1whichhasthesamedeterminant.QEDIngeneral,ifG=Risnotaspinmanifold,theninordertoobtainasimilarresultwemustinsteadconsidertheoperator6@U:)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(L2GSUr!)]TJ/F11 9.963 Tf 4.566 -8.07 Td[(L2GSUrviewedasanoperatoronG,restrictedtothespaceofSU-valuedfunctionsonGthatareinvariantunderthediagonalr-action%Z=diagZ+Z,whereisther-actiononU.NotethatifSUisinducedbyarepresentationoftheLiegroupR,thenthisoperatordescendstoawell-denedoperatoronG=Rasbefore.Ingeneral,theG-equivariantindexofthisoperator6@Uisonceagaingivenby.35.Toprovethis,wenotethatBott'sidentity.33andhistheoremcontinuetoholdfortheinductionmapi:Rr!^RgusingtherepresentationringsfortheLiealgebrasinsteadoftheLiegroups.WorkingintheLiealgebracontext,wenolongerneedconcernourselveswiththetopologicalobstructionsoccurringintheglobalLiegrouppicture.TherestoftheproofofTheorem19continuesunchanged.

PAGE 183

Chapter11ThecenterofUg.Thepurposeofthischapteristostudythecenteroftheuniversalenvelopingalgebraofasemi-simpleLiealgebrag.WehavealreadymadeuseofthesecondorderCasimirelement.11.1TheHarish-Chandraisomorphism.LetusreturntothesituationandnotationofSection7.3.Wehavethemonomialbasisfi11fimmhj11hj``ek11ekmmofUg,thedecompositionUg=UhUgn++n)]TJ/F11 9.963 Tf 6.725 1.494 Td[(Ugandtheprojection:Ug!Uhontotherstfactorofthisdecomposition.Thisprojectionrestrictstoaprojec-tion,alsodenotedby:Zg!Uh:Theprojection:Zg!Uhisabitawkward.HoweverHarish-ChandrashowedthatbymakingaslightmodicationinwegetanisomorphismofZgontotheringofWeylgroupinvariantsofUh=Sh.Harish-Chandra'smodicationisasfollows:Asusual,let:=1 2X>0:Recallthatforeachi,thereectionsisendsi7!)]TJ/F11 9.963 Tf 23.063 0 Td[(iandpermutestheremainingpositiveroots.Hencesi=)]TJ/F11 9.963 Tf 9.963 0 Td[(i183

PAGE 184

184CHAPTER11.THECENTEROFUG.Butbydenition,si=)-222(h;iiiandsoh;ii=1foralli=1;:::;m.So=!1++!m;i.e.isthesumofthefundamentalweights.11.1.1StatementDene:h!Uh;h=h)]TJ/F11 9.963 Tf 9.963 0 Td[(h:.1ThisisalinearmapfromhtothecommutativealgebraUhandhence,bytheuniversalpropertyofUh,extendstoanalgebrahomomorphismofUhtoitselfwhichisclearlyanautomorphism.Wewillcontinuetodenotethisautomorphismby.SetH:=:ThenHarish-Chandra'stheoremassertsthatH:Zg!UhWandisanisomorphismofalgebras.11.1.2Exampleofsl.Toseewhatisgoingonlet'slookatthissimplestcase.TheCasimirofdegreetwois1 2h2+ef+fe;ascanbeseenfromthedenition.Oritcanbecheckeddirectlythatthiselementisinthecenter.Itisnotwritteninourstandardformwhichrequiresthatthefbeontheleft.Butef=fe+[e;f]=fe+h.Sothewayofwritingthiselementintermsoftheabovebasisis1 2h2+h+2fe;andapplyingtoityields1 2h2+h:Thereisonlyonepositiverootanditsvalueonhis2,soh=1.Thussends1 2h2+hinto1 2h)]TJ/F8 9.963 Tf 9.963 0 Td[(12+h)]TJ/F8 9.963 Tf 9.963 0 Td[(1=1 2h2)]TJ/F11 9.963 Tf 9.963 0 Td[(h+1 2+h)]TJ/F8 9.963 Tf 9.963 0 Td[(1=1 2h2)]TJ/F8 9.963 Tf 9.962 0 Td[(1:TheWeylgroupinthiscaseisjusttheidentitytogetherwiththereectionh7!)]TJ/F11 9.963 Tf 20.479 0 Td[(h,andtheexpressionontherightisclearlyWeylgroupinvariant.

PAGE 185

11.1.THEHARISH-CHANDRAISOMORPHISM.18511.1.3UsingVermamodulestoprovethatH:Zg!UhW.Any2hcanbethoughtofasalinearmapofhintothecommutativealgebra,CandhenceextendstoanalgebrahomomorphismofUhtoC.IfweregardanelementofUh=Shasapolynomialfunctiononh,thenthishomomorphismisjustevaluationat.Fromourdenitions,)]TJ/F11 9.963 Tf 9.962 0 Td[(z=Hz8z2Zg:LetusconsidertheVermamoduleVerm)]TJ/F11 9.963 Tf 9.619 0 Td[(wherewedenoteitshighestweightvectorbyv+.Foranyz2Zg,wehavehzv+=zhv+=)]TJ/F11 9.963 Tf 9.796 0 Td[(hzv+andeizv+=zeiv+=0.Sozv+isahighestweightvectorwithweight)]TJ/F11 9.963 Tf 10.326 0 Td[(andhencemustbesomemultipleofv+.Callthismultiple'z.Thenzfi11fimmv+=fi11fimm'zv+;sozactsasscalarmultiplicationby'zonallofVerm)]TJ/F11 9.963 Tf 10.102 0 Td[(.Toseewhatthisscalaris,observethatsincez)]TJ/F11 9.963 Tf 9.505 0 Td[(z2Ugn+,weseethatzhasthesameactiononv+asdoeszwhichismultiplicationby)]TJ/F11 9.963 Tf 9.966 0 Td[(z=Hz.Inotherwords,'z=Hz=Hz:NoticethatinthisargumentweonlyusedthefactthatVerm)]TJ/F11 9.963 Tf 8.043 0 Td[(isacyclichighestweightmodule:IfVisanycyclichighestweightmodulewithhighestweight)]TJ/F11 9.963 Tf 10.434 0 Td[(thenzactsasmultiplicationby'z=Hz=Hz.Wewillusethisobservationinamoment.GettingbacktothecaseofVerm)]TJ/F11 9.963 Tf 11.035 0 Td[(,forasimpleroot=iletm=mi:=h;iiandsupposethatmisaninteger.Theelementfmiv+2Verm)]TJ/F11 9.963 Tf 9.963 0 Td[(where=)]TJ/F11 9.963 Tf 9.962 0 Td[()]TJ/F11 9.963 Tf 9.963 0 Td[(m=si)]TJ/F11 9.963 Tf 9.962 0 Td[(:Nowfromthepointofviewoftheslgeneratedbyei;fi,thevectorv+isamaximalweightvectorwithweightm)]TJ/F8 9.963 Tf 10.864 0 Td[(1.Henceeifmiv+=0.Since[ej;fi]=0;i6=jwehaveejfmiv+=0aswell.Sofmiv+6=0isamaximalweightvectorwithweightsi)]TJ/F11 9.963 Tf 9.574 0 Td[(.Callthehighestweightmoduleitgenerates,M.ThenfromMweseethat'siz='z:HencewehaveprovedthatHzw=Hz8w2W

PAGE 186

186CHAPTER11.THECENTEROFUG.ifisdominantintegral.Buttwopolynomialswhichagreeonalldominantintegralweightsmustagreeeverywhere.WehaveshownthattheimageofliesinShW.Furthermore,wehavez1z2)]TJ/F11 9.963 Tf 9.963 0 Td[(z1z2=z1z2)]TJ/F11 9.963 Tf 9.962 0 Td[(z2+z2z1)]TJ/F11 9.963 Tf 9.963 0 Td[(z22Ugn+:Soz1z2=z1z2:Thissaysthatisanalgebrahomomorphism,andsinceH=whereisanautomorphism,weconcludethatHisahomomorphismofalgebras.Equallywell,wecanarguedirectlyfromthefactthatz2Zgactsasmultiplicationby'z=HzonVerm)]TJ/F11 9.963 Tf 11.002 0 Td[(thatHisanalgebrahomomorphism.11.1.4Outlineofproofofbijectivity.TocompletetheproofofHarish-Chandra'stheoremwemustprovethatHisabijection.Forthiswewillintroducesomeintermediatespacesandho-momorphisms.LetYg:=Sggdenotethesubspacexedbytheadjointrepresentationextendedtothesymmetricalgebrabyderivations.Thisisasubalgebra,andtheltrationonSginducesaltrationonYg.Weshallproduceanisomorphismf:Yg!ShW:WealsohavealinearspaceisomorphismofUg!SggivenbythesymmetricembeddingofelementsofSkgintoUg,andletsbetherestrictionofthistoZg.Weshallseethats:Zg!Ygisanisomorphism.Finally,deneSkg=S0gS1gSkgsoastogetaltrationonSg.ThisinducesaltrationonShSg.Weshallshowthatforanyz2UkgZgwehavefszHzmodSk)]TJ/F7 6.974 Tf 6.227 0 Td[(1g:ThisprovesinductivelythatHisanisomorphismsincesandfare.Also,sincedoesnotchangethehighestordercomponentofanelementinSh,itwillbeenoughtoprovethatforz2UkgZgwehavefszzmodSk)]TJ/F7 6.974 Tf 6.226 0 Td[(1g:.2Wenowproceedtothedetails.

PAGE 187

11.1.THEHARISH-CHANDRAISOMORPHISM.18711.1.5RestrictionfromSggtoShW.Werstdiscusspolynomialsong|thatiselementsofSg.Letbeanitedimensionalrepresentationofofg,andconsiderthesymmetricfunctionFofdegreekonggivenbyX1;:::;Xk7!XtrX1Xkwherethesumisoverallpermutations.ForanyY2g,bydenition,YFX1;:::;Xk=F[Y;X1];X2;:::;Xk++FX1;:::;Xk)]TJ/F7 6.974 Tf 6.227 0 Td[(1;[Y;Xk]:AppliedtotheaboveFX1;:::;Xk=trX1XkwegettrYX1Xk)]TJ/F8 9.963 Tf 9.963 0 Td[(trX1YXk+trX1XX2Xk)-222(=trYX1Xk)]TJ/F8 9.963 Tf 9.963 0 Td[(trX1XkY=0:Inotherwords,thefunctionX7!trXnbelongstoSgg.Nowsincehisasubspaceofg,therestrictionmapinducesahomomorphism,r:Sg!Sh:IfF2Sgg,then,asafunctionongitisinvariantundertheautomorphismi:=expadeiexpad)]TJ/F11 9.963 Tf 7.749 0 Td[(fiexpadeiandhencer:Sgg!ShW:IfF2Sggissuchthatitsrestrictiontohvanishes,thenitsvalueatanyelementwhichisconjugatetoanelementofhunderEgthesubgroupofautomorphismsofggeneratedbytheimustalsovanish.Buttheseincludeadenseseting,soF,beingcontinuous,mustvanisheverywhere.SotherestrictionofrtoSggisinjective.Toprovethatitissurjective,itisenoughtoprovethatShWisspannedbyallfunctionsoftheformX7!trXkasrangesoverallnitedimensionalrepresentationsandkrangesoverallnon-negativeintegers.NowthepowersofanysetofspanningelementsofhspanSh.SowecanwriteanyelementofShWasalinearcombinationoftheAkwhereAdenotesaveragingoverW.Soitisenoughtoshowthatforanydominantweight,wecanexpresskintermsoftrk.LetEdenotethenitesetofalldominantweights.Letdenotethenitedimensionalrepresentationwithhighestweight.ThentrXk)-272(AkXisacombinationofAXkwhere2E.HencebyinductiononthenitesetEwegetthedesiredresult.Inshort,wehaveprovedthatr:Sgg!ShWisbijective.

PAGE 188

188CHAPTER11.THECENTEROFUG.11.1.6FromSggtoShW.NowwetransferallthisinformationfromSgtoSg:UsetheKillingformtoidentifygwithgandhencegetanisomorphism:Sg!Sg:Similarly,let:Sh!ShbetheisomorphisminducedbytherestrictionoftheKillingformtoh,whichweknowtobenon-degenerate.NoticethatcommuteswiththeactionoftheWeylgroup.WecanwriteSg=Sh+JwhereJistheidealinSggeneratedbyn+andn)]TJ/F8 9.963 Tf 6.724 1.494 Td[(.Letj:Sg!Shdenotethehomomorphismobtainedbyquotientingoutbythisideal.WeclaimthatthediagramSg)334()222()222()334(!Sgj??y??yrSh0)334()222()222()334(!Shcommutes.Indeed,sinceallmapsarealgebrahomomorphisms,itisenoughtocheckthisongenerators,thatisonelementsofg.IfX2g,thenhh;rXi=hh;Xi=h;XwherethescalarproductontherightistheKillingform.Butsincehisorthog-onalundertheKillingformton++n)]TJ/F8 9.963 Tf 6.725 1.494 Td[(,wehaveh;X=h;jX=hh;jXi:QEDUponrestrictiontothegandW-invariants,wehaveprovedthattherighthandcolumnisabijection,andhencesoisthelefthandcolumn,sinceisaW-modulemorphism.RecallingthatwehavedenedYg:=Sgg,wehaveshownthattherestrictionofjtoYgisanisomorphism,callitf:f:Yg!ShW:11.1.7Completionoftheproof.NowwehaveacanonicallinearbijectionofSgwithUgwhichmapsSg3X1Xr7!1 r!X2rX1Xr;

PAGE 189

11.2.CHEVALLEY'STHEOREM.189wherethemultiplicationontheleftisinSgandthemultiplicationontherightisinUgandwhererdenotesthepermutationgrouponrletters.Thismapisagmodulemorphism.Inparticularthismapinducesabijections:Zg!Yg:Ourproofwillbecompleteonceweprove.2.Thisisacalculation:writeuAJB:=fAhJeBforourusualmonomialbasis,wherethemultiplicationontherightisintheuniversalenvelopingalgebra.LetusalsowritepAJB:=fAhJeB=fihj11hj``ek2SgwherenowthepowersandmultiplicationareinSg.TheimageofuAJBunderthecanonicalisomorphismwithofUgwithSgwillnotbepAJBingeneral,butwilldierfrompAJBbyatermoflowerltrationdegree.Nowtheprojection:Ug!UhcomingfromthedecompositionUg=Uhn)]TJ/F11 9.963 Tf 6.724 1.495 Td[(Ug+Ugn+sendsuAJB7!0unlessA=0=Bandistheidentityonu0J0.Similarly,jpAJB=0unlessA=0=Bandjp0J0=p0J0=hJ:Thesetwofactscompletetheproofof.2.QED11.2Chevalley'stheorem.HarishChandra'stheoremsaysthatthecenteroftheuniversalenvelopingalge-braofasemi-simpleLiegroupisisomorphictotheringofWeylgroupinvariantsinthepolyinomialalgebraSh.Chevalley'stheoremassertsthatthisringisinfactapolynomialringin`generatorswhere`=dimh.ToproveCheval-ley'stheoremweneedtocallonsomefactsfromeldtheoryandfromtherepresentationtheoryofnitegroups.11.2.1Transcendencedegrees.AeldextensionL:Kisnitelygeneratedifthereareelements1;:::;nofLsothatL=K1;:::;n.Inotherwords,everyelementofLcanbewrittenasarationalexpressioninthe1;:::;n.Elementst1;:::;tkofLarecalledalgebraicallyindependentoverKifthereisnonon-trivialpolynomialpwithcoecientsinKsuchthatpt1;:::;tk=0:

PAGE 190

190CHAPTER11.THECENTEROFUG.Lemma15IfL:Kisnitelygenerated,thenthereexistsanintermediateeldMsuchthatM=K1;:::;rwherethe1;:::;rareindependenttranscen-dentalelementsandL:Misaniteextensioni.e.LhasnitedimensionoverMasavectorspace.Proof.WeareassumingthatL=K1;:::;q.Ifalltheiarealgebraic,thenL:Kisaniteextension.Otherwiseoneoftheiistranscendental.Callthis1.IfL:K1isaniteextensionwearedone.OtherwiseoneoftheremainingiistranscendentaloverK1.Callit2.So1;2areindependent.Proceed.Lemma16Ifthereisanothercollection1:::ssothatL:K1;:::;sisnitethenr=s.ThiscommonnumberiscalledthetranscendencedegreeofLoverK.Proof.Ifs=0,theneveryelementofLisalgebraic,contradictingtheassump-tionthatthe1;:::;rareindependent,unlessr=0.Sowemayassumethats>0.SinceL:Misnite,thereisapolynomialpsuchthatp1;1;:::;r=0:Thispolynomialmustcontainatleastone,since1istranscendental.Renum-berifnecessarysothat1occursinp.Then1isalgebraicoverK1;2;:::;randL:K1;2;:::;risnite.Continuingthiswaywecansuccessivelyre-placesbysuntilweconcludethatL:K1;:::;risnite.Ifs>rthenthesarenotalgebraicallyindependent.sosrandsimilarlyrs.Noticethatif1;:::;narealgebraicallyindependentthenK1;:::;nisisomorphictotheeldofrationalfunctionsinnindeterminatesKt1;:::;tnsinceK1;:::;n=K1;:::;n)]TJ/F7 6.974 Tf 6.227 0 Td[(1nbyclearingdenominators.11.2.2Symmetricpolynomials.ThesymmetricgroupSnactsonKt1;:::;tnbypermutingthevariables.Thexedeld,F,containsallthesymmetricpolynomials,inparticulartheele-mentarysymmetricpolynomialss1;:::;snwheresristhesumofallpossibledistinctproductstakenratatime.Usingthegeneraltheoryofeldextensions,wecanconcludethatProposition32F=Ks1;:::;sn.ThestrategyoftheproofistorstshowthatthedimensionoftheextensionKt1;:::;tn:Ks1;:::;snisn!andthenabasictheoreminGaloistheorywhichweshallrecallsaysthatthedimensionofK:FequalstheorderofthegroupG=Snwhichisn!.SinceKs1;:::;snFthiswillimplytheproposition.ConsidertheextensionsKt1;:::;tnKs1;:::;sn;tnKs1;:::;sn:

PAGE 191

11.2.CHEVALLEY'STHEOREM.191Wehavetheequationftn=0whereft:=tn)]TJ/F11 9.963 Tf 9.963 0 Td[(s1tn)]TJ/F7 6.974 Tf 6.226 0 Td[(1+tn)]TJ/F7 6.974 Tf 6.226 0 Td[(2s2+)]TJ/F8 9.963 Tf 7.749 0 Td[(1nsn:ThisshowsthatthedimensionoftheeldextensionKs1;:::;sn;tn:Ks1;:::;snisn.Ifwelets01:::s0n)]TJ/F7 6.974 Tf 6.227 0 Td[(1denotetheelementarysymmetricfunctionsinn)]TJ/F8 9.963 Tf 9.552 0 Td[(1variables,wehavesj=tns0j)]TJ/F7 6.974 Tf 6.227 0 Td[(1+s0jsoKs1;:::;sn;tn=Ktn;s01;:::;s0n)]TJ/F7 6.974 Tf 6.227 0 Td[(1:ByinductionwemayassumethatdimKt1;:::;tn:Ks1;:::;sn;tn=dimKtnt1;:::;tn)]TJ/F7 6.974 Tf 6.227 0 Td[(1:Ktns01;:::;s0n)]TJ/F7 6.974 Tf 6.226 0 Td[(1n)]TJ/F8 9.963 Tf 9.962 0 Td[(1!provingthatdimKt1;:::;tn:Ks1;:::;snn!:AfundamentaltheoremofGaloistheorysaysTheorem20LetGbeanitesubgroupofthegroupofautomorphismsoftheeldLovertheeldK,andletFbethexedeld.Thendim[L:F]=#G:Thistheorem,whoseproofwewillrecallinthenextsection,thencompletestheproofoftheproposition.Thepropositionimpliesthateverysymmetricpolynomialisarationalfunctionoftheelementarysymmetricfunctions.Infact,everysymmetricpolynomialisapolynomialintheelementarysym-metricfunctions,givingastrongerresult.Thisisprovedasfollows:putthelexicographicorderonthesetofn)]TJ/F8 9.963 Tf 7.748 0 Td[(tuplesofintegers,andthereforonthesetofmonomials;soxi11xinnisgreaterthanxj11xjnninthisorderingifi1>j1ofi1=j1andi2>j2oretc.Anypolynomialhasaleadingmonomial"thegreatestmonomialrelativetothislexicographicorder.Theleadingmonomialoftheproductofpolynomialsistheproductoftheirleadingmonomials.Weshallproveourcontentionbyinductionontheorderoftheleadingmonomial.Noticethatifpisasymmetricpolynomial,thentheexponentsi1;:::;inofitsleadingtermmustsatisfyi1i2in;forotherwisethemonomialobtainedbyswitchingtwoadjacentexponentswhichoccurswiththesamecoecientinthesymmetricpolynomial,pwouldbestrictlyhigherinourlexicographicorder.Supposethatthecoecientofthisleadingmonomialisa.Thenq=asi1)]TJ/F10 6.974 Tf 6.227 0 Td[(i21si2)]TJ/F10 6.974 Tf 6.226 0 Td[(i32sin)]TJ/F6 4.981 Tf 5.396 0 Td[(1)]TJ/F10 6.974 Tf 6.227 0 Td[(inn)]TJ/F7 6.974 Tf 6.227 0 Td[(1sinnhasthesameleadingmonomialwiththesamecoecient.Hencep)]TJ/F11 9.963 Tf 10.429 0 Td[(qhasasmallerleadingmonomial.QED

PAGE 192

192CHAPTER11.THECENTEROFUG.11.2.3Fixedelds.Wenowturntotheproofofthetheoremoftheprevioussection.Lemma17EverydistinctsetofmonomorphismsofaeldKintoaeldLarelinearlyindependentoverL.Let1;:::;nbedistinctmonomorphismsofK!L.Theassertionisthattherecannotexista1;:::;an2Lsuchthata11x++annx08x2Kunlessalltheai=0.Assumethecontrary,sothatsuchanequationholds,andwemayassumethatnoneoftheai=0.Lookingatallsuchpossibleequations,wemaypickonewhichinvolvesthefewestnumberofterms,andwemayassumethatthisistheequationwearestudying.Inotherwords,nosuchequationholdswithfewerterms.Since16=n,thereexistsay2Ksuchthat1y6=nyandinparticulary6=0.Substitutingyxforxgivesa11yx++annyx=0soa11y1x++annynx=0andmultiplyingouroriginalequationby1yandsubtractinggivesa21y)]TJ/F11 9.963 Tf 9.963 0 Td[(2y2x++anny)]TJ/F11 9.963 Tf 9.963 0 Td[(1ynx=0whichisanon-trivialequationwithfewerterms.Contradiction.Letn=#G,andlettheelementsofGbeg1=1;:::;gn.SupposethatdimL:F=mn.Letx1;:::;xn;xn+1belinearlyindependentoverF,andndy1;:::;yn+12Lnotallzerosolvingthenequationsgjx1y1++gjxn+1yn+1=0;j=1;:::;n:

PAGE 193

11.2.CHEVALLEY'STHEOREM.193Chooseasolutionwithfewestpossiblenon-zeroy0sandrelabelsothattherstarethenon-vanishingones,sotheequationsnowreadgjx1y1++gjxryr=0;j=1;:::;n;andnosuchequationsholdwithlessthanry0s.Applyingg2Gtotheprecedingequationgivesggjx1gy1++ggjxrgyr=0:ButggjrunsoveralltheelementsofG,andsogy1;:::;gyrisasolutionofouroriginalequations.Inotherwordswehavegjx1y1++gjxryr=0andgjx1gy1++gjxrgyr=0:Multiplyingtherstequationsbygy1,thesecondbyy1andsubtracting,givesgjx2[y2gy1)]TJ/F11 9.963 Tf 9.963 0 Td[(gy2y1]++gjxr[yrgy1)]TJ/F11 9.963 Tf 9.962 0 Td[(gyry1]=0;asystemwithfewery0s.Thiscannothappenunlessthecoecientsvanish,i.e.yigy1=y1gyioryiy)]TJ/F7 6.974 Tf 6.227 0 Td[(11=gyiy)]TJ/F7 6.974 Tf 6.227 0 Td[(118g2G:Thismeansthatyiy)]TJ/F7 6.974 Tf 6.226 0 Td[(112F:Settingzi=yi=y1andk=y1,wegettheequationx1kz1++xrkzr=0astherstofoursystemofequations.Dividingbykgivesalinearrelationamongx1;:::;xrcontradictingtheassumptionthattheyareindependent.11.2.4Invariantsofnitegroups.LetGbeanitegroupactingonavectorspace.ItactiononthesymmetricalgebraSVwhichisthesameasthealgebraofpolynomialfunctionsonVbygfv=fg)]TJ/F7 6.974 Tf 6.226 0 Td[(1v:LetR=SVGbetheringofinvariants.LetS=SVandLbetheeldofquotientsofS,sothatL=Kt1;:::;tnwheren=dimV.Fromthetheoremonxedelds,weknowthatthedimensionofLasanextensionofLGisequaltothenumberofelementsinG,inparticularnite.SoLGhastranscendencedegreenoverthegroundeld.

PAGE 194

194CHAPTER11.THECENTEROFUG.ClearlytheeldoffractionsofRiscontainedinLG.Weclaimthattheycoincide.Indeed,supposethatp;q2S;p=q2LG.Multiplythenumeratoranddenominatorbygptheproducttakenoverallg2G;g6=1.Thenewnu-meratorisGinvariant.Thereforesoisthedenominator,andwehaveexpressedp=qasthequotientoftwoelementsofR.IfthenitegroupGactsonavectorspace,thenaveragingoverthegroup,i.e.themapE3f7!f]:=1 #GXgfisaprojectionontothesubspaceofinvariantelements:A:f7!f]E!EG:Inparticular,ifEisnitedimensional,dimEG=trA:.3WemayapplytheaveragingoperatortoourinnitedimensionalsituationwhereS=SVandR=SGinwhichcasewehavetheadditionalobviousfactthatpq]=p]q8p2S;q2R:LetR+RdenotethesubringofRconsistingofelementswithconstanttermzero.LetI:=SR+sothatIisanidealinS.BytheHilbertbasistheoremwhoseproofwerecallinthenextsectiontheidealIisnitelygenerated,andhence,fromanysetofgenerators,wemaychooseanitesetofgenerators.Theorem21Letf1;:::;frbehomogeneouselementsofR+whichgenerateIasanidealofS.Thenf1;:::;frtogetherwith1generateRasaKalgebra.Inparticular,RisanitelygeneratedKalgebra.Proof.Wemustshowthatanyf2Rcanbeexpressedasapolynomialinthef1;:::;fr,andsinceeveryfisasumofitshomogeneouscomponents,itisenoughtodothisforhomogeneousfandweproceedbyinductiononitsdegree.Thestatementisobviousfordegreezero.Forpositivedegree,f2RIsof=s1f1++srfr;si2Iandsincef;f1;:::;frarehomogeneous,wemayassumethesiarehomogeneousofdegreedegf)]TJ/F8 9.963 Tf 9.509 0 Td[(degfisinceallothercontributionsmustcancel.NowapplyAtogetf=s]1f1++s]rfr:Thes]ilieinRandhavelowerhomogeneousdegreethanf,andhencecanbeexpressedaspolynomialsinf1;:::;fr.Hencesocanf.

PAGE 195

11.2.CHEVALLEY'STHEOREM.19511.2.5TheHilbertbasistheorem.AcommutativeringiscalledNoetherianifanyofthefollowingequivalentcon-ditionsholds:1.IfI1I2isanascendingchainofidealsthenthereisaksuchthatIk=Ik+1=Ik+2=::::2.Everynon-emptysetofidealshasamaximalelementwithrespecttoinclusion.3.Everyidealisnitelygenerated.TheHilbertbasistheoremassertsthatifRisaNoetherianring,thensoisthepolynomialringR[X].Inparticular,allidealsinK[X1;:::;Xn]arenitelygenerated.LetIbeanidealinRXandforanypositiveintegerkletLkIRbedenedbyLkI:=fak2Rj9ak)]TJ/F7 6.974 Tf 6.227 0 Td[(1;:::;a12RwithkX0ajXj2Ig:Foreachk,LkIisanidealinR.MultiplyingbyXshowsthatLkILk+1I:Hencetheseidealsstabilize.IfIJandLkI=LkJforallk,weclaimthatthisimpliesthatI=J.Indeed,supposenot,andchooseapolynomialofsmallestdegreebelongingtoJbutnottoI,saythisdegreeisk.ItsleadingcoecientbelongstoLkJandcannotbelongtoLkIbecauseotherwisewecouldndapolynomialofsmallerdegreebelongingtoJandnottoI.ProofoftheHilbertbasistheorem.LetI0I1beanascendingchainofidealsinR[X].ConsiderthesetofidealsLpIq.Wecanchooseamaximalmember.SoforkpwehaveLkIj=LkIq8jq:Foreachofthenitelymanyvaluesj=1;:::;p)]TJ/F8 9.963 Tf 9.963 0 Td[(1,theascendingchainsLiI0LiI1stabilizes.SowecanndalargeenoughrbiggerthanthenitelymanylargevaluesneededtostabilizethevariouschainssothatLiIj=LiIr8jr;8i:ThisshowsthatIj=Ir8jr.

PAGE 196

196CHAPTER11.THECENTEROFUG.11.2.6ProofofChevalley'stheorem.ThissaysthatifK=RandWisanitesubgroupofOVgeneratedbyreections,thenitsringofinvariantsisapolynomialringinn-generators,wheren=dimV.WithoutlossofgeneralitywemayassumethatWactseectively,i.e.nonon-zerovectorisxedbyallofW.Letf1;:::;frbeaminimalsetofhomogeneousgenerators.Supposewecouldprovethattheyarealgebraicallyindependent.SincethetranscendencedegreeofthequotienteldofRisn=dimV,weconcludethatr=n.Sothewholepointistoprovethataminimalsetofhomogeneousgeneratorsmustbealgebraicallyindependent-thattherecannotexistanon-zeropolynomialh=hy1;:::;yrsothathf1;:::;fr=0:.4Sowewanttogetasmallersetofgeneratorsassumingthatsucharelationexists.Letd1:=degf1;:::;dr:=degfr:Foranynon-zeromonomialaye11yfrroccurringinhthetermafr11ferrwegetbysubstitutingf'sfory'shasdegreed=e1d1+erdrandhencewemaythrowawayallmonomialsinhwhichdonotsatisfythisequation.Nowsethi:=@h @yif1;;frsothathi2Rishomogeneousofdegreed)]TJ/F11 9.963 Tf 10.88 0 Td[(di,andletJbetheidealinRgeneratedbythehi.Renumberf1;:::;frsothath1;:::;hmisaminimalgeneratingsetforJ.Thismeansthathi=mXj=1gijhj;gij2Rfori>mifm
PAGE 197

11.2.CHEVALLEY'STHEOREM.197andsubstitutetheaboveexpressionsforhi;i>mtogetmXi=1hi0@@fi @xk+rXj=m+1gji@fj @xk1Ak=1;:::;n:Setpi:=@fi @xk+rXj=m+1gji@fj @xki=1;:::;msothateachpiishomogeneouswithdegpi=di)]TJ/F8 9.963 Tf 9.962 0 Td[(1andwehavetheequationh1p1+hmpm=0:.5Wewillprovethatthisimpliesthatp12I:.6Assumingthisforthemoment,thismeansthat@f1 @xk+rXj=m+1gj1@fj @xk=sumri=1fiqiwhereqi2S.MultiplytheseequationsbyxkandsumoverkandapplyEuler'sformulaforhomogeneouspolynomialsXxk@f @xk=degff:Wegetd1f1+Xdjgj1fj=Xfiriwithdegr1>0ifitisnotzero.Onceagain,thelefthandsideishomogeneousofdegreed1sowecanthrowawayalltermsontherightwhicharenotofthisdegreebecauseofcancellation.Butthismeansthatwethrowawaytheterminvolvingf1,andwehaveexpressedf1intermsoff2;:::;fr,contradictingourchoiceoff1;:::;frasaminimalgeneratingset.SotheproofofChevalley'stheoremreducedtoprovingthat.5implies.6,andforthiswemustusethefactthatWisgeneratedbyreections,whichwehavenotyetused.ThedesiredimplicationisaconsequenceofthefollowingProposition33Leth1;:::;hm2Rbehomogeneouswithh1notintheidealofRgeneratedbyh2;:::;hm.Supposethat.5holdswithhomogeneousele-mentspi2S.Then.6holds.

PAGE 198

198CHAPTER11.THECENTEROFUG.Noticethath1cannotlieintheidealofSgeneratedh2;:::hmbecausewecanapplytheaveragingoperatortotheequationh1=k2h2++kmhmki2Stoarrangethatthesameequationholdswithkireplacedbyk]i2R.Weprovethepropositionbyinductiononthedegreeofp1.Thismustbepositive,sincep16=0constantwouldputh1intheidealgeneratedbytheremaininghi.LetsbeareectioninWandHitshyperplaneofxedvectors.Thenspi)]TJ/F11 9.963 Tf 9.963 0 Td[(pi=0onH:Let`beanon-zerolinearfunctionwhosezerosetisthishyperplane.Withnolossofgenerality,wemayassumethatthelastvariable,xn,occurswithnon-zerocoecientin`relativetosomechoiceoforthogonalcoordinates.Infact,byrotation,wecanarrangetemporarilythat`=xn.Expandingoutthepolynomialspi)]TJ/F11 9.963 Tf 10.451 0 Td[(piinpowersoftherotatedvariables,weseethatsgi)]TJ/F11 9.963 Tf 10.451 0 Td[(gimusthavenotermswhicharepowersofx1;:::;xn)]TJ/F7 6.974 Tf 6.227 0 Td[(1alone.Putinvariantly,weseethatspi)]TJ/F11 9.963 Tf 9.963 0 Td[(pi=`riwhereriishomogeneousofdegreeonelessthatthatofpi.Applystoequation.5andsubtracttoget`h1r1+hmrm=0:Since`6=0wemaydivideby`togetanequationoftheform.5withp1replacedbyr1oflowerdegree.Sor12Ibyinduction.Sosp1)]TJ/F11 9.963 Tf 9.962 0 Td[(p12I:NowWstabilizesR+andhenceIandwehaveshownthateachw2Wactstriviallyonthequotientofp1inthisquotientspaceS=I.Thusp]1=Ap1p1modI.Sop12Isincep]12I.QED