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《Topics in Representation Theory- The Adjoint Representation》由會員上傳分享，免費在線閱讀，更多相關(guān)內(nèi)容在學(xué)術(shù)論文-天天文庫。
1、TopicsinRepresentationTheory:TheAdjointRepresentation1TheAdjointRepresentationBesidestheleftandrightactionsofGonitself,thereistheconjugationactionc(g):h→ghg?1Unliketheleftandrightactionswhicharetransitive,thisactionhas?xedpoints,includingtheidentity.De?nition1(AdjointRepresentation)
2、.Thedi?erentialoftheconjugationaction,evaluatedattheidentity,iscalledtheadjointactionAd(g)=c?(g)(e):TeG→TeGIdentifyinggwithTeGandinvokingthechainruletoshowthatAd(g1)?Ad(g2)=Ad(g1g2)thisgivesahomomorphismAd(g):G→GL(g)calledtheadjointrepresentation.So,foranyLiegroup,wehaveadistinguish
3、edrepresentationwithdimensionofthegroup,givenbylineartransformationsontheLiealgebra.LaterwewillseethatthereisaninnerproductontheLiealgebrawithrespecttowhichthesetransformationsareorthogonal.Forthematrixgroupcase,theadjointrepresentationisjusttheconjugationactiononmatricesAd(g)(y)=gY
4、g?1sinceonecanthinkoftheLiealgebraintermsofmatricesin?nitesimallyclosetotheunitmatrixandcarryovertheconjugationactiontothem.GivenanyLiegrouprepresentationπ:G→GL(V)takingthedi?erentialgivesarepresentationdπ:g→End(V)de?nedbyddπ(X)v=(π(exp(tX))v)
5、t=0dt1forv∈V.Usingourpreviousformulafor
6、thederivativeofthedi?erentialoftheexponentialmap,we?ndfortheadjointrepresentationAd(g)thattheassociatedLiealgebrarepresentationisgivenbyddad(X)(Y)=(c(exp(tX))?(Y))
7、t=0=(Ad(exp(tX))(Y))
8、t=0=[X,Y]dtdtForthespecialcaseofmatrixgroupswecancheckthiseasilysinceexpandingthematrixexponential
9、givesetXYe?tX=Y+t[X,Y]+O(t2)SoassociatedtoAd(G),theadjointrepresentationoftheLiegroupGong,takingthederivativewehavead(g),aLiealgebrarepresentationofgonitselfad(g):X∈g→ad(X)=[X,·]∈End(g)Animportantpropertyoftheadjointrepresentationisthatthereisaninvari-antbilinearformong.Thisiscalled
10、the“Killingform”,afterthemathematicianWilhelmKilling(1847-1823).KillingwasresponsibleformanyimportantideasinthetheoryofLiealgebrasandtheirrepresentations,butnotfortheKillingform.Borelseemstohavebeenthe?rsttousethisterminology,butnowsayshecan’trememberwhatinspiredhimtouseit[1].De?nit
11、ion2(KillingForm).T
