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1、FiniteDimensionalAlgebrasandQuantumGroupsBangmingDengJieDuBrianParshallJian-panWangDepartmentofMathematics,BeijingNormalUniversity,Beijing,ChinaE-mailaddress:dengbm@bnu.edu.cnSchoolofMathematics,UniversityofNewSouthWales,SydneyNSW2052,AustraliaE-mailaddress:j.du@unsw.edu.auDepartm
2、entofMathematics,UniversityofVirginia,Charlottesville,VA22904-4137,USAE-mailaddress:bjp8w@virginia.eduDepartmentofMathematics,EastChinaNormalUniversity,Shang-hai200062,ChinaE-mailaddress:jpwang@ecnu.edu.cn2000MathematicsSubjectClassification.Primary17B37,20G05,20C08;Secondary16G20,
3、20C33Keywordsandphrases.Quivers,finitedimensionalalgebras,Hallalgebras,generallineargroups,Heckealgebras,cells,quantumgroupsThesecondauthorwassupportedinpartbyARC..ThethirdauthorwassupportedinpartbyNSFGrant#000000.Abstract.ContentsChapter0.GettingStarted10.1.Cartanmatricesandtheirt
4、worealizations10.2.Freealgebrasandpresentationswithgeneratorsandrelations40.3.Examples:therealizationproblem80.4.Countingoverfinitefields:Gaussianpolynomials110.5.Canonicalbases:thematrixconstruction14Part1.QuiversandTheirRepresentations21Chapter1.Representationsofquivers231.1.Quive
5、rsandtheirrepresentations231.2.Eulerformsandrootsystems261.3.Bernstein-Gelfand-Ponomarevreflectionfunctors311.4.Gabriel’stheorem341.5.Representationvarieties,degenerationandgenericextensions38Chapter2.AlgebraswithFrobeniusmorphisms432.1.Fq-structuresonvectorspaces432.2.Algebraswith
6、FrobeniusmorphismsandFrobeniustwists452.3.F-stableA-modules492.4.AconstructionofindecomposableF-stablemodules512.5.Almostsplitsequences532.6.Frobeniusfoldingofalmostsplitsequences57Chapter3.Quiverswithautomorphisms633.1.Quiverswithautomorphismsandvaluedquivers633.2.Modulatedquiver
7、sassociatedtoquiverswithautomorphisms673.3.ExtendedDynkinquiverswithautomorphisms723.4.Finitedimensionalalgebrasoverafinitefield773.5.Auslander-Reitenquiverswithautomorphisms78Part2.SomeQuantizedAlgebras87Chapter4.CoxetergroupsandHeckealgebras894.1.Coxetergroups894.2.Heckealgebras95
8、4.3.SemisimplicityofHeckealgebras
