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1、1AUNIVERSALDIMENSIONFORMULAFORCOMPLEXSIMPLELIEALGEBRAS1J.M.LANDSBERGANDL.MANIVELAbstract.WepresentauniversalformulaforthedimensionoftheCartanpowersoftheadjointrepresentationofacomplexsimpleLiealgebra(i.e.,auniversalformulafortheHilbertfunctionsofhomogeneousc
2、omplexcontactmanifolds),aswellasseveralotheruniversalformulas.TheseformulasgeneralizeformulasofVogelandDeligneandaregivenintermsofrationalfunctionswhereboththenumeratoranddenominatordecomposeintoproductsoflinearfactorswithintegercoefficients.Wediscussconsequen
3、cesoftheformulasincludingarelationwithScorzavarieties.1.StatementofthemainresultVogel[17]definedatensorcategoryD′intendedtobeamodelforauniversalsimpleLiealgebra.Hismotivationcamefromknottheory,asD′wasdesignedtosurjectontothecategoryofVassilievinvariants.While
4、Vogel’sworkremainsunfinished(andunpublished),italreadyhasconsequencesforrepresentationtheory.LetgbeacomplexsimpleLiealgebra.VogelderivedauniversaldecompositionofS2ginto(possiblyvirtual)Casimireigenspaces,S2g=C⊕Y2(α)⊕Y2(β)⊕Y2(γ)whichturnsouttobeadecompositioni
5、ntoirreduciblemodules.Ifwelet2tdenotetheCasimireigenvalueoftheadjointrepresentation(withrespecttosomeinvariantquadraticform),thesemodulesrespectivelyhaveCasimireigenvalues4t−2α,4t−2β,4t−2γ,whichwemaytakeasthedefinitionsofα,β,γ.Vogelshowedthatt=α+β+γ.Hethenwen
6、tontofindCasimireigenspacesY3(α),Y3(β),Y3(γ)⊂S3gwitheigenvalues6t−6α,6t−6β,6t−6γ(whichagainturnouttobeirreducible),andcomputedtheirdimensionsthroughdifficultdiagrammaticcomputationsandthehelpofMaple[17]:(α−2t)(β−2t)(γ−2t)dimg=,αβγt(β−2t)(γ−2t)(β+t)(γ+t)(3α−2t)d
7、imY2(α)=−.α2βγ(α−β)(α−γ)t(α−2t)(β−2t)(γ−2t)(β+t)(γ+t)(t+β−α)(t+γ−α)(5α−2t)dimY3(α)=−,α3βγ(α−β)(α−γ)(2α−β)(2α−γ)arXiv:math/0401296v2[math.RT]1Feb2005andtheformulasforY2(β),Y2(γ)andY3(β),Y3(γ)areobtainedbypermutingα,β,γ.Theseformulassuggestacompletelydifferentp
8、erspectivefromtheusualdescriptionofthesimpleLiealgebrasbytheirrootsystemsandtheWeyldimensionformulathatcanbededucedforeachparticularsimpleLiealgebra.TheworkofVogelraisesmanyquestions.Inparticula
