; TeX output 1999.11.03:0822f7 YRXQ cmr12CHAPTER3Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups_o cmr9102g*7 &e_F7.pTHE!QUANTUMMONOIDOFAQUANTUMSP:A9CENQh103YTN cmbx127.eThequantummonoidofaquantumspaceProblem3.7.1.nRIf!ug cmmi12Aisa nitedimensionalalgebraandو:fVA !", cmsy104!9M@(A)~C AtheunivrersalcoSoperationoftheTVambarabialgebraonAfromtheleftthenW:iA4!fAtK M@(A)(withthesamemrultiplicationonM(A))isaunivrersalcoSoperationofM@(A)onAfromtherighrt.Thecomultiplicationde nedbythiscoSoperationisWUR:M@(A)4!1M(A)nW M(A)./Thrus wehavetodistinguishbSetweentheleftandtherighrtTVambarabialgebraonAandwehaveM2cmmi8rb(A)UR=Ml!ȹ(A)2cop .RNorwconsiderthespSecialmonoidaldiagramschemeD:=DUV[X;m;u].ǔTVomakethings0simplerwreassumethatVecisstrictmonoidal. ThecategoryDUV[X;m;u]0hastheobjectsX !::: !Xn=}mX2!K cmsy8 nforalln2) msbm10N(andIn:=X2 |{Ycmr80 M)andthemorphismsm*:XѴ 1X̭4!dX,Lu:I̭4!X*͹and9Jallmorphismsformallyconstructedfromm;u;idbrytakingtensorproSductsandcompositionofmorphisms.LetAbSeanalgebrawithmrultiplicationmA 4ѹ:VAL A4!gA뙹andunituA 4ѹ:K4!gA.Then^!A 36:URD4!CNde nedbry!n9(X)=A,:!(X2 n 6K)=A2 n Dȹ,:!(m)=mA yBand^!(u)=uAisϲastrictmonoidalfunctor./IfAis nitedimensionalthenthediagramis nite.WVegetRTheorem3.7.1.p,@ cmti12Lffeth.Abea nitedimensionalalgebra.UThenthealgebraM@(A)cffoactingFuniversallyfrffomtherightonA(therightTambarabialgebra)M@(A)andcoSendz(!A)35arffeisomorphicasbialgebras.- cmcsc10Proof.@_WVeSharvestudiedtheTVambarabialgebraforleftcoactionfQ:URA4!1M@(A)' A butherewreneedtheanalogueforuniversalrightcoactionfQ:URA4!1A& M@(A) (seeProblem3.9).LetrBs׍l@ >RH