automorphism of special symmetry space

《automorphism of special symmetry space》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、MATH311:COMPLEXANALYSIS—AUTOMORPHISMGROUPSLECTURE1.IntroductionRatherthanstudyindividualexamplesofconformalmappingsoneatatime,wenowwanttostudyfamiliesofconformalmappings.Ensemblesofconformalmappingsnaturallycarrygroupstructures.2.AutomorphismsofthePlaneTheautomorphismgroupofth

2、ecomplexplaneisAut(C)={analyticbijectionsf:C−→C}.Anyautomorphismoftheplanemustbeconformal,foriff￿(z)=0forsomezthenftakesthevaluef(z)withmultiplicityn>1,andsobytheLocalMappingTheoremitisn-to-1nearz,impossiblesincefisanautomorphism.Byaproblemonthemidterm,weknowtheformofsuchautom

3、orphisms:theyaref(z)=az+b,a,b∈C,a￿=0.Thisdescriptionofsuchfunctionsoneatatimelosestrackofthegroupstructure.Iff(z)=az+bandg(z)=a￿z+b￿then(f◦g)(z)=aa￿z+(ab￿+b),f−1(z)=a−1z−a−1b.Buttheseformulasarenotveryilluminating.Forabetterpictureoftheautomor-phismgroup,representeachautomorph

4、ismbya2-by-2complexmatrix,￿￿ab(1)f(z)=ax+b←→.01Thenthematrixcalculations￿￿￿￿￿￿aba￿b￿aa￿ab￿+b=,010101￿￿−1￿￿aba−1−a−1b=0101naturallyencodetheformulasforcomposingandinvertingautomorphismsoftheplane.Withthisinmind,definetheparabolicgroupof2-by-2complexmatrices,￿￿￿￿abP=:a,b∈C,a￿=0.0

5、1Thenthecorrespondence(1)isanaturalgroupisomorphism,Aut(C)∼=P.12MATH311:COMPLEXANALYSIS—AUTOMORPHISMGROUPSLECTURETwosubgroupsoftheparabolicsubgroupareitsLevicomponent￿￿￿￿a0M=:a∈C,a￿=0,01describingthedilationsf(z)=ax,anditsunipotentradical￿￿￿￿1bN=:b∈C,01describingthetranslation

6、sf(z)=z+b.Proposition2.1.TheparabolicgrouptakestheformP=MN=NM.Also,MnormalizesN,meaningthatm−1nm∈Nforallm∈Mandn∈N.Proof.Toestablishthefirststatement,simplycompute:￿￿￿￿￿￿￿￿￿￿aba01a−1b1ba0==.0101010101Similarlyforthesecondstatement,￿￿￿￿￿￿￿￿a−101ba01a−1b=.01010101￿Thegeometriccont

7、entoftheproposition’sfirststatementisthatanyaffinemapisthecompositionofatranslationandadilationandisalsothecompositionofadilationandatranslation.Thecontentofthesecondstatementisthatadilationfollowedbyatranslationfollowedbythereciprocaldilationisagainatranslation.(Idonotfindthislas

8、tresultquicklyobviousgeometrically.)Insumsofar,consideringthe