量子群、扭Hopf代數(shù)與Ringel-Hall代數(shù).pdf

1、中國科學技術大學博士學位論文量子群、扭Hopf代數(shù)與RingelHall代數(shù)姓名：李立斌申請學位級別：博士專業(yè)：基礎數(shù)學指導教師：李尚志章璞2000.2.1QuantumGroups，TwistedHopfAlgebrasandRingel—HallAlgebrasLiLibinAbstractQuantumgrouporDrinfeld—Jimboquantizedenvelopingalgebradevelopinginthemid

2、oftheeightiesisoneoftheimportantbranchesofalgebraSincequantumgrouphasthefundamentalalgebrastructurebehindmathematicalphysics，andthetightrelationwithotherbranchesofmathematics，ithasdevelopedrapidlyduringthelastfifteenyear

3、sHowever，RingelHallalgebradevelopinginthebeginingoftheninetiesprovidesavalidapproachtoquantumgroupintermsoftherepresentationtheoryofalgebraRoughlyspeaking，withtheRigel—Green—Lustingisomorphism，quantizedenvelopingalgebra，

4、definedbythegeneratorsandquantumSerrerelatons，canbeinterpretedasRingel—Hallalgebrawithabasisofthemodulesoverfinite—dimensionalalgebraoverafinitefieldandwiththeHallmultiplicationWiththerapiddevelopmentoftherepresentationt

5、heoryofalgebrasincetheseventies，thismethodhaspresentanewviewpointandwaytostudyquantizedenvelopingalgebraTheaimofthisPhDdissertationistostudythepropertiesoftheide—ajsofquantumgroup乩(s，(2))andthestructuresofRingelHallalgeb

6、raintheframeoftwistedHopfalgebraandGreencategoryFirstlyusingthewell—knownrepresentationtheory，andthestructuresoflocallyfinitesubal—gebraandthepropertiesofannihilatorpolynomialsofsimplemodules，weprovethateverynon—zeroidea

7、lof砜(sf(2))canbegeneratedbyasumoftwohighestweightvectorsItturnsoutthatourresultsgiveUSacompletelistofallprime，primitiveandmaximalidealsSecondly，weintroducethetheoryoftwistedHopfalgebrasuchasthepropertiesofantipode，theCOn

8、—structionandextension；andthestructureofselfdualtwistedHopfalgebraFinially，fromtheviewpointoftwistedHopfalgebra，westudyinfurtherstepthestructuresofRingel—Hallalgebra，compositionalgebra，thepositivepartofDrinfeld—Jimboquan