; TeX output 2002.06.25:09147 VN cmbx12MathematischesInstitutSS2002derUniversitXatM`unchenŸSet10Prof.Dr.B.Pareigis;KrNff cmbx12Problemffsetfor&QuantumffGroupsandNoncommutativeGeometry/-n XQ cmr12(37)%WVeڋharveseenthatinrepresentationtheoryandincorepresentationtheoryof%quanrtum;groupsg cmmi12H)suchas( msbm10KG,PU@(+%n eufm10g),SL2cmmi8q(2),Uq(slC(2));theordinarytensor%proSduct(inK-!", cmsy10M0.@ cmti12offd 9)oftrwo(co-)reprensentationsisinacanonicalwayagain%a(co-)reprensenrtation.o6FVortwoHV-moSdulesM=andNdescribSethemodule%structureonM N+if)((a)=ѬHB=URKG:8gn9(m n)=::: forgË2G;)' (b)=ѬHB=URU@(g):8gn9(m n)=::: forgË2g;*uD(c)=ѬHB=URUq(slC(2)):JN(i)[bE(m n)UR=:::,GH(ii)[bFƹ(m n)UR=:::,D6(iii)[bKܞ(m n)UR=:::=ѬfortheelemenrtsE;FS;K12URUq(slC(2)).s (38)%LetGbSeamonoid.1(a)Thecffategory-sofG-familiesofveffctorspacesM2G t=UR"u cmex10QgI{ K cmsy82G!Vechasfamilies>%ofvrectorspaces(Vgjg2)WG)asobjectsandfamiliesoflinearmaps(fg U:Vg%' !8yWgjgv2lG)Sasmorphisms.tThecompSositionis(fgjgv2lG)L(hgjgv2lG)=-J%(fg]_hgjgË2URG).ShorwthatM2G #isamonoidalcategorywiththetensorproSductFT(VgjgË2URG) (WgjgË2URG):=(h;k62G;hk|{Ycmr8=g/Vh Wk#jg2G):%〹WheredounitandassoSciativitrylawsofGentertheproSof8?1(b)?7A?!vrectorspaceVۧtogetherwithafamilyofsubspaces(Vg=?VpjgSx2G)%isUcalledG-grffaded,ifUV¹=URgI{2GFVgSholds.'oLet(V;(VgjgË2G))and(Wr;(WgjgË2%G))SbSeG-gradedvrectorspaces.=Alinearmapf(:X)V$!fOWiscalledG-grffaded,%iffG(Vg)URWgforallgË2G.(TheG-gradedvrectorspacesandG-gradedlinearӍ%mapslformthecategoryM2[G]EKofG-grffadedvectorspaces.IShorwlthatM2[G]EKis%amonoidalcategorywiththetensorproSductV rSWƹ,Yspacethen(VgjgRj21G)isaG-familyofvectorspaces.4FVoraG-graded%vrectorgspace(V;(VgjgË2URG))constructtheKG-comoSduleVwiththestructure%map:Q :VzK!qV} EKG,Ns2(vn9):=vO~ gfor:all(homogeneouselemenrts)vL2Vg%〹andRBforallgs2G.oConrverselyRBlet(V;x۹:V!OV 1KG)bSeaKG-comodule.%Then>constructthevrectorspaceVpwithdengraded(homogenous)compSonents%Vg*P:=URfvË2Vpjs2(vn9)=v gg).e (39)%Letv(DUV;!n9)bSeadiagraminVeffc .KLetD̹beamonoidalcategoryand!bea%monoidalfunctor.8Then(DUV;!n9)iscalledamonoidal35diagrffam.1Let(DUV;!n9)bSeamonoidaldiagramVeffc .LetAF2Veffcbeanalgebra.A%naturalhtransformation'Nm:!1!$! tBnishcalledmonoidalmonoidalifthe%diagramsMHyH@!n9(X) !(Yp)Hq!n9(X) !(Yp) BE B{fdoˀO line10-`'(X) '(Y)HbzǠ*FfecǠ?'YH63Ǡ*Ffe6g,Ǡ?G; mI0&!n9(X+ Yp)w>!n9(X+ Yp) BbL32fdЍά-'(X Y)8鍑%〹andCw`⍒R0K⍒K K,{fdY?ά-‚.⍍hh=HɪǠ*FfeܟǠ?H 8*Ǡ*Ffe k\Ǡ?{`!n9(I)y!n9(I) B̞32fdKd ά-]'(I)鍑%〹commrute.1WVedenotethesetofmonoidalnaturaltransformationsbryNatFx  (!n9;!< 0B).%ShorwthatNat+Qx Gɹ(!n9;! B)isafunctorinB. (40)%Shorwthattheadjoint actionH> Q6HB3URh a7!Ph(1) \|aS׹(h(2))2H3makresH%〹anHV-moSdulealgebra.!Duedate:8TVuesdary,02.07.2002,16:15inLectureHallE41;7  /o cmr9.@ cmti12+%n eufm10)ppmsbm8( msbm10"u cmex10 K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8Nff cmbx12N cmbx12XQ cmr12O line10[