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1、ADVANCESINMATHEMATICS33,144-160(1979)RootSystemsofHyperbolicTypeROBERTV.MOODYDepartmentofMathematics,UniversityofSaskatchewan,Saskatoon,CanadaThispaperbeginswithasurveyoftheknownresultsaboutrootsystems.Therefollowsthedefinitionofhyperbolicrootsystems,andtheexplicitdescrip
2、-tionoftheimaginaryrootsofsuchasystemasthepointsofintersectionofalatticeandacone.Finallythereisaproofoftheconjugacyofbasesinasym-metrizablerootsystembytheextendedWeylgroup.1.INTRODUCTIONTheconceptofarootsystemaroseoutoftheKilling-Cartanclassificationofthesemi-simplefinite
3、dimensionalLiealgebrasoverthecomplexnumbers.ApartfromtheirfundamentalnatureinrelationtotheLiealgebras,therootsystemsarefascinatingcombinatorialstrujturesandasubjectofstudyintheirownright.Thegeneralizationofthefiniterootsystems(whicharethosewhichappearedintheclassicaltheor
4、y)toinfiniteones,appearedindependentlyin[12]and[16],inbothcasesasaresultofexploringtheLiealgebrasarisingoutofanaturalgeneralizationofapresentationforthefinitedimensionalsemi-simpleLiealgebras.Ofthese,theEuclidean(oraffine)rootsystemsespeciallyhaveturnedupinanunexpectedvar
5、ietyofways;forexample,inthestudyofp-adicChevalleygroups[5,11],inMacdonald'sidentitieswhichareintricatelytiedupwithDedekind's2-function[15],andmostrecentlyintherepresentationtheoryofgraphs[8,9].Whatevertheirprospectivepositionmaybeinmathematics,therootsystemsareremarkablye
6、lusivewhenitcomestodeterminingtheirinternalstructureexplicitly.Probablythebasicproblemisthatwehavecomeatthembyaninductiveprocess,buildinguptherootsfromabaseandCartanmatrixwhicharegivenapriori.However,asintheclassicalcasetherootsystemisinvariantbylargegroupW,theWeylgroup,w
7、hichindicatestheabundantinternalsymmetryofthesystemandtheweaknessofbeingtiedtoaparticularbase.Quitedifferentfromtheclassicalcaseisthefactthat,ingeneral,thetranslatesofthebasebytheWeylgroup(realroots)donotcomeclosetocoveringthetherootsystem.Theremainingroots(imaginaryroots
8、)arebothqualitativelyandquantitativelyunknownexceptinspecialcases.Noonereadingthispapercanfailto
