弱Hopf代數(shù)的結(jié)構(gòu)及其箭圖表示.pdf

1、浙江大學(xué)理學(xué)院博士學(xué)位論文弱Hopf代數(shù)的結(jié)構(gòu)及其箭圖表示姓名：穆尼爾申請(qǐng)學(xué)位級(jí)別：博士專業(yè)：基礎(chǔ)數(shù)學(xué)指導(dǎo)教師：李方20070601AbstractweakHopfalgebraHwithweakantipodeTiscalledasemilattice廂adedweakHopfalgebraifH=o^kisasemilatticegradedo∈yLet≈rbeacoalgebrathenthesetofitsgrouplikeel

2、ements|sgivenbythesetG(kr)=扣∈打JZX(u)=d圓d，E(n)=1)Itcaneasilybeverifiedthat“i3acoalgebraInfactitisthedualversionofthepathalgebrastructureoilafinitequiverasconsideredbyWchinandSMontgomeryin117I_F0ranarbitraryquiverrthepathc

3、oalgebrakFispointedsinceaⅡ0fjtssimplesubcoalgebrasareonedimensionalThen七ro箋k(G(kr))isthekvectorspacewithbasisa(kr)IfthepathcoalgebrakFisabialgebraoverkthenG(kFl望F0isasemigroupundermultiDlicationand讓七risaHopfalgebraoverkt

4、henG(kF)isAgroup(bylgtl)Thevec斷space燈】isakFobicomoduIewiththeleftandfightcomodulestructuremaps6L(a)=t(8)on，and5R(a)=ao3(n)，respectivelyforalla∈捌r卜Thecotensorproductofkrlwithitseffisthekeruelofthemaphol一1@屯：kFl0南r1_七rlokr

5、00krl，wherekrxistherigh￡andleftkrocomoduleswiththestructuremaps屯and6nrespectively(8∞『23|)Oul“researchworkmainlyconsistsoffivepartswhich啪described∞followsfI)Inf231CCibil8andMRossodefinedthenotionofHopfquiverandclassifiedt

6、heP叮adedHopfalgebrastructuresusingsuchquiversWiththequestionofgeneralizationofHopfquivertheauthorsdefinethenotiouofweakHopfquiverDle￡3311withramificationdataofsomecli肋rdmortoldSWealsodescribethe盯adedstructuresofweakHopfm

7、udnl髑overtheCliffordmonoidalgebrak8whichisasemilattice阜adedweakHopfalgebraWeest出lishastrongconditionbetweenthepathcoalgebraendowedwiththestructureofsemilatticegradedweakHopfalgebraandtheweakHopfquiverofsomeCliffordmonoid

8、withrespecttosomeramificationdata(s∞Theorem335)HopfmuduleearenaturalrepresentationsofHopfalgebrasTheyfo珊allabeliancategoryU(H)by(20i，providedwiththetensorproductHopfbimodulesoveragroupalgebraprovidesustheclassificationof

9、patheoalgebraswhichadmitgradedHopfalgebrastructures，intheformofcategoryt；(kG)【2S]WeakHopfhimudnlesaretherepresentationsoftheweakHopfalgebrasWe址HopfbimodulesoveraCliffordmonoidalgebrakSca丑beconsideredtoclassifythepathcoal

10、gebrawhichadmitsasemiiattice審adedweakHopfalgebrastructureThugweakHopfkSbimoduiesoxetherepresentationsofthesemitattice盯adedweakHopfalgebrasMRoesoin1821established蛆equivalenceofB(日)asamonoidalcategorytothecategoryofmodules

11、overtheDrinfel’ddoubleofHC，CibiisandMRussoin123lhasestablishedaonetoonecorrespondencebetweentheobjectsofHopfbimodulesB(kG)andacompletellstof日adedHopfalgebrastructuresonthepathcoalgebrakFcorrespondingtotheHopfquiverrWegiv

12、eanequivalericebetweenthecategoryofCliflurdmonoidalgebrasandthecategoryof女z百一bimoduies(seeTbeorem339)Weuisoestablisha11equivalencebetweenthecategoryofweakHopfkSbimodulesB(ks)andacompletelistofsemilattice燈adedweakHopfalge