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1、RepresentationringsandtheirbiFrobeniusalgebrastructuresByZhihuaWangSupervisor：AssociateSupervisor：ProfLibinLiProfYinhuoZhangSchoolofMathematicalScienceYangzhouUniversityJune，2015Submittedintotalfulfilmentoftherequirement

2、sforthedegreeofPhDinmathematicsAbstractAsaninvariantofmonoidalcategories，therepresentationringsplayamoreandmoreimportantroleinthestudyofmonoidalcategoriesandhavearousedgreatattentionInthisthesis，wemainlydealwithsomestruc

3、turesandpropertiesoftherepresentationringofafinitedimensionalHopfalgebraoveranalgebraicallyclosedfieldEspeciallywestudytheFrobeniuspropertyoftherepresentationringi13theframeworkofabilinearformAsanapplicationofthispropert

4、yweinvestigatetheFrobeniusalgebraandtheFrobeniuscoalgebrastruc—turesonaquotientoftherepresentationring，whichleadstotheconstructionofbi—FrobeniusalgebraintermsoftherepresentationringFirstly，weintroducethenotionofaquantumt

5、raceontherepresentationcategoryofafinitedimensionalHopfalgebra溉usethisnotiontoanswerthequestionraisedbyCibils：whendoesthetrivialmoduleappearasadirectsum—mandofthetensorproductofanytwoindecomposablemodulesWethengivesomech

6、aracterizationsoffinitedimensionalindecomposablemodulesofquantumtracezerobyvirtueofahnostsplitsequencesandshalldemonstratetheroleoftheobjectsofquantumtracezerointhestructureoftherepresentationringSecondl弘accordingtothedi

7、mensionsofmorphismspaces，wedefineanassociativeandnondegenerateZbilinearformontherepresentationringofafinitedimensionalHopfalgebraWeinvestigatepropertiesandstructuresoftherepresentationringintheframeworkofthebilinearform，

8、includingthestudyoftheFrobeniusproperties，thestudyofsomeonesidedideals，thedescriptionofthenilpotentradicalandidempotentsoftherepresentationringThirdly，westudythestablerepresentationring(therepresentationringofthestableca

9、tegory)ofafinitedimensionalHopfalgebraWeshowthatthestablerepresentationringisisomorphictoaquotientoftherepresentationringmodulotheidealgeneratedbytheisomorphismclassesofprojectivemodulesBecauseofthisisomorphism，thebiline

10、arformontherepresentationringinducesabilinearformonthestablerepresentationringTheinducedbilinearformiSnotnon—degenerateingeneral，wegivesomeequivalentconditionsforthenon—degeneracyoftheinducedbilinearformWecheckthatthesta

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