; TeX output 2002.02.27:16017 VN cmbx12MathematischesInstitutSS2002derUniversitXatM`unchenSet4Prof.Dr.B.Pareigis;KrNff cmbx12Problemffsetfor&QuantumffGroupsandNoncommutativeGeometry, XQ cmr12(13)%Determineexplicitlythedualcoalgebrag cmmi12A2 K cmsy8ofAUR:=( msbm10K!", cmsy10hxi=(x2|{Ycmr83). (14)%DeterminePanddescribSethecoendomorphismbialgebraofAfromproblem12.%(Hinrt:Determine\3 rstasetofalgebrageneratorsofM@(A). dThendescribSethe%relations.) (15)%Let{lAbSea nitedimensionalK-algebrawithunivrersalbialgebraAURn!1Bbz tA.%Shorw0\i)=ѬthateA22cmmi8op !"rB2op * A2opHisunivrersal(whereA2ophasthemrultiplication=Ѭro:URA An!1A An!1A);-ii)=ѬthatAPUR԰n9=A2op ŹimpliesBPX԰ ?=B2op h˹(asbialgebras);*uDiii)=Ѭthat/forcommrutative/algebrasAthealgebraB5satis esBP԰g=QB2opRbut=ѬthatBneednotbSecommrutative.*iv)=ѬFind aanisomorphismBPX԰ ?=B2op forthebialgebraBX=URKha;bi=(a22;abބ+ba).=Ѭ(compareproblem14). (16)%ConsiderthealgebraKܞ[]=(22)thesocalledalgebraofdualnrumbSersovera% eld5Kܞ.ConsiderthealgebraBwith(noncommruting)generatorsa;b;c;d5and%relations:eI5(acW =URacac ad=acad+adacbcUR=acbc=bcacITbdW =URacbd+adbc=bcad+bdac bcbc=bcbd+bdbc=0I5(acW =UR1 ad=0%ShorwthatBtogetherwiththecoSoperationȄ:URA!BE A%〹withs2()UR=bc 1+bd isthecoSendomorphismbialgebraofA.=}Duedate:8TVuesdary,14.05.2002,16:15inLectureHallE41*;7 ( msbm10 K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8Nff cmbx12N cmbx12XQ cmr12 x