分次雙代數(shù)與量子群余模范疇中的Hopf代數(shù).pdf

1、中國科學(xué)技術(shù)大學(xué)博士學(xué)位論文分次雙代數(shù)與量子群余模范疇中的Hopf代數(shù)姓名：孫建華申請學(xué)位級別：博士專業(yè)：基礎(chǔ)數(shù)學(xué)指導(dǎo)教師：李尚志2002.3.1AbstractHopfalgebraisanimportantresearchfieldinalgebraThelastfifteenyearshaveseenanumberofsignificantadvancesinHopfal—gebrasThebestknownistheintrod

2、uctionofquantumgroups，whichareHopfalgebrasthataroseinmathematicalphysicsandnOWhaveconnectionstOmanybranchesofmathematicsSomenewconstructionsfromquantumgroupsappeared，forexample，quasi—triangularHopfalgebras，coquasit“angul

3、arHopfalgebrasandHopfalgebrasinabraidedcategor5r_Theyarestudied協(xié)rmanymathe—maticianssuchasSMajid，VGDrinfeldandSMontgomeryTheaimofthisPhDdessertationistodefineandstudyanabeliangroupGgraded(A，r)一Hopfalgebra，whichisageneral

4、iza—tionoftheGgradedAHopfalgebra[M]，andthenmakeresearchesonbathHopfalgebraandHopfmoduleinacomodulecategory√卅月，where日isacoquasitriangularHopfalgebraFirst，byusing“twisted”methodswegiveanmnberofnewconstructionsofagraded(A，r

5、)一HopfalgebraItturnsoutthatthegradeddualspaceofalocalfiniteGgraded(A，T)一Hopfalgebraisagraded(A。，7_)一Hopfalgebra，andthataGgraded(A，A)一bialge—bra(AHopfalgebra)canbeembeddedinausualbialgebra(Hopfalgebra)byusingadirectandsim

6、plewaySecond，weresearchintoconstructionsofHopfalgebrasinacomodulecategoryMHwhere(H，O)isacoquasitriangularHopfalgebraWeobtainthatifBisaHopfalgebrainM爿thentheantipodeSfromBto(BFopisacomodulealgebrahomomorphism，andSfrom(B一)

7、?！眛OBisacomodulecoalgebrahomomorphismTheseresultsasabovecanbeappliedtO(k，C，I，)()一HopfalgebrassuchasRingel—Hallalgebra，compositionalgebraandthepositivepartofDrinfeld—JimboquantizedenvelopingalgebraFinally，westudyHopfmodul

8、esinM8WegainthatifBisaHopfalgebrainM“thensmashprodoctB鞋日。叩isaleftH。op—comodulealgebra，smashcoproductB#日。0pisaleftH?！抟籱odulecoalgebraandthengivetheconditionsofaleftB—Hopfmodulein朋”beingausualleftbiproductB★Hc。p—Hopfmodule