; TeX output 2002.06.17:15107 VN cmbx12MathematischesInstitutSS2002derUniversitXatM`unchenSet9Prof.Dr.B.Pareigis;KrNff cmbx12Problemffsetfor&QuantumffGroupsandNoncommutativeGeometry, XQ cmr12(33)%(LinearJyAlgebra)FVorg cmmi12U9M!", cmsy10iVde neU@2 K cmsy8? UŹ:=ff@h2Vp2\tjfG(U@)=0g.XRFVorZVp2%〹de neZܞ2? Nh:=URfvË2VpjZܞ(vn9)=0g.8Shorwthatthefollowinghold:)((a)=ѬU6URV=)LU=U@2??Թ;)' (b)=ѬZ1URVp2 GanddimZ<1꨹=)Z=Zܞ2??;*uD(c)=ѬfUPVpjdimDVN8=U<1gP԰)=ufZVp2\tjdimDZ<1g1underthemaps=ѬU67!URU@2? HandZ17!Zܞ2?.ʍ (34)%Let?V="u cmex10L*:h1 U_:h2cmmi8i|{Ycmr8=1!{( msbm10KxibSeanin nite-dimensionalvrectorspace.6:Findanelement%gË2UR(VG Vp)2thatisnotinV2  V2l(UR(VG V)2). (35)%FVor$morphismsf:Ij!zMewandg& :Ij!zNewinamonoidalcategorywrede ne%(f T18:N?X!VM N@):=(f 1IM)(I)21@ŹandI(1 glq:M?X!VM N@):=%(1 gn9)(I)21 \|.8ShorwthatthediagramMN㚠M Nd32fd'8O line10- Ġf 1bYIbYDM4{fd6"Ѝά-i9fH낟Ǡ*FfeǠ?'6gHaǠ*Ffe4Ǡ?q-F1 g썑%〹commrutes. (36)%Let{GbSea nitegroupandK2G t:=URK[G]2;thedualofthegroupalgebra.Shorw%thatNK2G misaHopfalgebraandthateacrhmoSdulestructureK[G]_ M?Y !WM%〹translates89My andconrverselyV.gShow%thatthisde nesamonoidalequivXalenceofcategories.%DescribSe88thegroupvXaluedfunctorK-+@ cmti12c Alg f(K2G;)intermsofsetsandtheir%groupstructure.=}Duedate:8TVuesdary,25.06.2002,16:15inLectureHallE41*;7 +@ cmti12( msbm10"u cmex10 K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8Nff cmbx12N cmbx12XQ cmr12O line10 a