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1、iSemi-SimpleLieAlgebrasandTheirRepresentationsRobertN.CahnLawrenceBerkeleyLaboratoryUniversityofCaliforniaBerkeley,California1984THEBENJAMIN/CUMMINGSPUBLISHINGCOMPANYAdvancedBookProgramMenloPark,CaliforniaReading,MassachusettsLondonAmsterdamDonMills,On
2、tarioSydneyiiPrefaceiiiPrefaceParticlephysicshasbeenrevolutionizedbythedevelopmentofanewparadigm",thatofgaugetheories.TheSU(2)xU(1)theoryofelectroweakin-teractionsandthecolorSU(3)theoryofstronginteractionsprovidethepresentexplanationofthreeofthefourprevi
3、ouslydistinctforces.FornearlytenyearsphysicistshavesoughttounifytheSU(3)xSU(2)xU(1)theoryintoasinglegroup.ThishasledtostudiesoftherepresentationsofSU(5),O(10),andE.6E ortstounderstandthereplicationoffermionsingenerationshaveprompteddiscussionsofevenlargerg
4、roups.Thepresentvolumeisintendedtomeettheneedofparticlephysicistsforabookwhichisaccessibletonon-mathematicians.Thefocusisonthesemi-simpleLiealgebras,andespeciallyontheirrepresentationssinceitisthey,andnotjustthealgebrasthemselves,whichareofgreatestinterest
5、tothephysicist.Ifthegaugetheoryparadigmiseventuallysuccessfulindescribingthefundamentalparticles,thensomerepresentationwillencompassallthoseparticles.ThesourcesofthisbookaretheclassicalexpositionofJacobsoninhisLieAlgebrasandthreegreatpapersofE.B.Dynkin.Ali
6、stingofthereferencesisgivenintheBibliography.Inaddition,attheendofeachchapter,referencesivPrefacearegiven,withtheauthors'namesincapitalletterscorrespondingtothelistinginthebibliography.Thereaderisexpectedtobefamiliarwiththerotationgroupasitarisesinquantumm
7、echanics.Areviewofthismaterialbeginsthebook.AfamiliaritywithSU(3)isextremelyusefulandthisisreviewedaswell.Thestructureofsemi-simpleLiealgebrasisdeveloped,mostlyheuristically,inChaptersI-VII,culminatingwiththeintroductionofDynkindiagrams.TheclassicalLiealge
8、brasarepresentedinChapterVIIandtheexceptionalonesinChapterIX.Propertiesofrepresentationsareexploredinthenexttwochapters.TheWeylgroupisdevelopedinChapterXIIandexploitedinChapterXIVintheproofofWeyl'sdimensionfo
