; TeX output 2002.06.07:15267 VN cmbx12MathematischesInstitutSS2002derUniversitXatM`unchenSet8Prof.Dr.B.Pareigis;KrNff cmbx12Problemffsetfor&QuantumffGroupsandNoncommutativeGeometry, XQ cmr12(29)%Determine{thestructureofacorvector{spaceonavrectorspaceg cmmi12Vfromthefact%thatHomd1(V;Wƹ)isavrectorspaceforallvectorspacesWƹ.ʍ (30)%Therealunitcircle!", cmsy10Sb2|{Ycmr81f(( msbm10R)carriesthestructureofagroupbrytheadditionof%angles.Is*itpSossibletomakreSb21 SwiththeanealgebraK[c;s]=(s22n+jc221)%inrtoananealgebraicgroup?(Hint:PHowcanyouaddtwopSoints(x1;y1)%andJ(x2;y2)ontheunitcircle,jsucrhthatyougettheadditionoftheassoSciated%angles?)%FindagroupstructureonthetorusT ͹. (31)%Let]VbSeavrectorspace.ShowthatthereisauniversalvectorspaceEand%homomorphism:E~ VA[!V<(sucrhthatforeachvectorspaceZjandeach%homomorphism?f:cZ Va!}Vthereisauniquehomomorphismg*:Z@Z! E%〹sucrhthat@KCE^ VV932fdL O line10-gۍWH`Df?$ׁ @?$ @?$ @?$ @녤>@녤>RYZF VӒǠ*FfeğǠ?q-"2cmmi8gI{ K cmsy8 1R%〹commrutes).ݐWVe!callEդand¹:E VO2h!Va!+@ cmti12veffctorespaceactinguniversally%onVp. (32)%Let7Eιand|:E6 Vtw!fV'bSe7avrectorspaceactinguniversallyonVp. Show%that E!hasauniquelydeterminedstructureofanalgebrasucrhthatVzbSecomes%aleftE-moSdule.=}Duedate:8TVuesdary,18.06.2002,16:15inLectureHallE41*;7 +@ cmti12( msbm10 K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8Nff cmbx12N cmbx12XQ cmr12O line10 _