; TeX output 2002.07.02:14217 VN cmbx12MathematischesInstitutSS2002derUniversitXatM`unchenŸSet11Prof.Dr.B.Pareigis;KrNff cmbx12Problemffsetfor&QuantumffGroupsandNoncommutativeGeometry. XQ cmr12(41)%LetT(g cmmi12M@2 K cmsy8;ev /)bSealeftdualforM@.Shorwthatthereisa+@ cmti12uniquemorphism%db6G:URIF!", cmsy10``!M M@2 됹satisfyingtheconditions ʍx(M26|{Ycmr8dbr 1p ^!M M@2 M21 evp6 !@M@)UR=12cmmi8Mm߹(M@22 V:1 dbp ҳ( ! M@2 M M@22 V:ev" 1p Ua!!TM@2)UR=1M"!q% cmsy6 D:!1Z%〹(Uniquenessofthedualbasis.) (42)%Let BbSethebialgebra( msbm10Khx;yn9i=I,7ZwhereI isgeneratedbryx22;xyO+yn9x,7Zwiththe%diagonal#(yn9)UR=yL y,K(x)=x 1+yL x,and#thecounit"(yn9)UR=1;"(x)=0.1Achain35cffomplexhastheform^؍M6=UR(:::'0x#@qAacmr63ЍuK!'ʟM2'0 m@q2ЍVj!M1'0 m@q1ЍVj!M0)oH%with@n1@n=UR0.1Shorw!CthatthecategoryK-C5ompn:ofchaincomplexesisequivXalenttothe%categoryB-C5omoffd!ofB-comoSdules.1Use|thefollorwingconstruction.IfMȹisachaincomplexthende neaB-%comoSduleǔonM6=URi2)ppmsbm8N֥Mi,nwiththestructuremapȄ:M6!B cM@,Θs2(m):=%yn92i m+xyn92i1ch @idڹ(m)(forallmlM2Mi]and(forallilM2N(resp.a`s2(m):=1 m%〹form12M0.QConrverselyifM;c:1MXq!B ^M3ٹisaB-comoSdulethenwe%de nenK-moSdulesMi,:=URfm2M@j9m20#2M[s2(m)=yn92ip ]m+xyn92i1 m209]gnand%K-linear7&maps@i,:URMiӷ!) Mi1x|bry7&@idڹ(m):=m20_fors2(m)=yn92i <m+xyn92i1 m209.%Checrkthatthisde nesanequivXalenceofcategories.1(Hinrt:Letm2M2B-C5omoffd`.Sinceyn92i;xyn92i!formabasisofBZwrehave%s2(m)}="u cmex10P(ixyn92i mi¹+Peimxyn92i m20RAidڹ.|WVeapplytothistheequation(1 s2)%=%( 1)]ڹandcomparecoSecienrtstogetU€s2(midڹ)UR=yn9 i} mi+xyn9 i1Z7 m 0ڍi1AV; (m 0ڍidڹ)UR=yn9 i} m 0ڍi%〹forqalliUR2N01(withqm20RA1ι=0).[Consequenrtlyforeachmi,2URMithereisexactly%one@(midڹ)UR=m20RAi12M+sucrhthatzb&&s2(midڹ)UR=yn9 i} mi+xyn9 i1Z7 @(mi):*7 ,o cmr92V%〹Sinceas2(m20RAi1AV) =yn92i1K m20RAi1foralli2Nwreseethat@(midڹ)2Mi1AV.Sowre%harveTde ned@:Mi_ yG!t+Mi1AV.FVurthermorewreseefromthisequationthat%@22g (midڹ)?=0LJforalli?2N. ]HenceLJwrehaveobtainedachaincomplexfrom%(M;s2).1IfNFwreapply({ 1)s2(m)=mNFthenwegetm=Pmi withNFmic2Mihence%MҢ=Li2N$Midڹ. This[togetherwiththeinrverse[constructionleadstotheM%requiredequivXalence.)ʍ (43)%Findk=anexampleofanobjectM!inamonoidalcategoryCrthathasaleftdual%butnorighrtdual. (44))((a)=ѬInOthecategoryofN-gradedvrectorspacesdetermineallobjectsMܹthat=Ѭharvealeftdual.)' (b)=ѬIn1thecategoryofcrhaincomplexesK-C5omp~determineallobjectsMrֹthat=Ѭharvealeftdual.*uD(c)=ѬInthecategoryofcoScrhaincomplexesK-C5offcomp&ӹdetermineallobjectsM=Ѭthatharvealeftdual.=}Duedate:8TVuesdary,09.07.2002,16:15inLectureHallE41;7  ,o cmr9+@ cmti12)ppmsbm8( msbm10"u cmex10!q% cmsy6 K cmsy8!", cmsy102cmmi8g cmmi12Aacmr6|{Ycmr8Nff cmbx12N cmbx12XQ cmr12G