资源描述：
《Topics in Representation Theory- The Adjoint Representation》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、TopicsinRepresentationTheory:TheAdjointRepresentation1TheAdjointRepresentationBesidestheleftandrightactionsofGonitself,thereistheconjugationactionc(g):h→ghg−1Unliketheleftandrightactionswhicharetransitive,thisactionhasfixedpoints,includingtheidentity.Definition1(AdjointRepresentation)
2、.Thedifferentialoftheconjugationaction,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−1sinceonecanthinkoftheLiealgebraintermsofmatricesinfinitesimallyclosetotheunitmatrixandcarryovertheconjugationactiontothem.GivenanyLiegrouprepresentationπ:G→GL(V)takingthedifferentialgivesarepresentationdπ:g→End(V)definedbyddπ(X)v=(π(exp(tX))v)
5、t=0dt1forv∈V.Usingourpreviousformulafor
6、thederivativeofthedifferentialoftheexponentialmap,wefindfortheadjointrepresentationAd(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.Borelseemstohavebeenthefirsttousethisterminology,butnowsayshecan’trememberwhatinspiredhimtouseit[1].Definit
11、ion2(KillingForm).T
