lectures on lie groups - d. milicic

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1、LecturesonLieGroupsDraganMiliˇci´cContentsChapter1.Basicdifferentialgeometry11.Differentiablemanifolds12.Quotients43.Foliations114.Integrationonmanifolds19Chapter2.Liegroups231.Liegroups232.LiealgebraofaLiegroup433.HaarmeasuresonLiegroups72Chapter3.CompactLiegroups771.CompactLiegroups77Chapter4.Bas

2、icLiealgebratheory971.Solvable,nilpotentandsemisimpleLiealgebras972.Liealgebrasandfieldextensions1043.Cartan’scriterion1094.SemisimpleLiealgebras1135.Cartansubalgebras125Chapter5.StructureofsemisimpleLiealgebras1371.Rootsystems1372.RootsystemofasemisimpleLiealgebra145iiiCHAPTER1Basicdifferentialgeo

3、metry1.Differentiablemanifolds1.1.Differentiablemanifoldsanddifferentiablemaps.LetMbeatopo-logicalspace.AchartonMisatriplec=(U,ϕ,p)consistingofanopensubsetU⊂M,anintegerp∈andahomeomorphismϕofUontoanopensetin+
p.TheopensetUiscalledthedomainofthechartc,andtheintegerpisthedimensionofthechartc.Thecharts

4、c=(U,ϕ,p)andc0=(U0,ϕ0,p0)onMarecompatibleifeitherU∩U0=∅orU∩U06=∅andϕ0◦ϕ−1‘:ϕ(U∩U0)−→ϕ0(U∩U0)isaC∞-diffeomorphism.AfamilyAofchartsonMisanatlasofMifthedomainsofchartsformacoveringofMandallanytwochartsinAarecompatible.AtlasesAandBofMarecompatibleiftheirunionisanatlasonM.Thisisobviouslyanequivalencere

5、lationonthesetofallatlasesonM.Eachequivalenceclassofatlasescontainsthelargestelementwhichisequaltotheunionofallatlasesinthisclass.Suchatlasiscalledsaturated.AdifferentiablemanifoldMisahausdorfftopologicalspacewithasaturatedatlas.Clearly,adifferentiablemanifoldisalocallycompactspace.Itisalsolocallyco

6、nnected.Therefore,itsconnectedcomponentsareopenandclosedsubsets.LetMbeadifferentiablemanifold.Achartc=(U,ϕ,p)isachartaroundm∈Mifm∈U.Wesaythatitiscenteredatmifϕ(m)=0.Ifc=(U,ϕ,p)andc0=(U0,ϕ0,p0)aretwochartsaroundm,thenp=p0.There-fore,allchartsaroundmhavethesamedimension.Therefore,wecallpthedimen-sio

7、nofMatthepointmanddenoteitbydimmM.Thefunctionm7−→dimmMislocallyconstantonM.Therefore,itisconstantonconnectedcomponentsofM.IfdimmM=pforallm∈M,wesaythatMisanp-dimensionalmanifold.LetMandNbetwodifferentiablemanifolds.Acont