; TeX output 1999.11.03:0813c7 YRXQ cmr12CHAPTER3Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups/^o cmr999d*7 &e1001:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYI.N cmbx126..ThebialgebracoSendLetg cmmi12!R:u!", cmsy10Do4!YCsand!n92 K cmsy80 :DUV20 4!'NCsbSediagramsinC5. }WVecallthediagram(DUV;!n9)w (D20#;!n920!n9(X+ Yp) BbL32fdЍά-'(X Y)뭍andG*$⍒R0( msbm10K⍒K K,{fdY?ά-‚.⍍hh=HɪǠ*FfeܟǠ?H 8*Ǡ*Ffe k\Ǡ?{`!n9(I)y!n9(I) B̞32fdKd ά-]'(I)commrute.WVedenotethesetofmonoidalnaturaltransformationsbryNat+Qx Gɹ(!n9;! B).Problem3.6.1.nRShorwthatNat+Qx Gɹ(!n9;! B)isafunctorinB.Theorem3.6.5.pLffet(DUV;!n9)beareconstructive,RmonoidaldiagraminVec.2`ThencoSendz(!n9)GisabialgebrffaandȄ:UR!Ë!J!' )coSendh(!)isamonoidalnaturffaltransforma-tion.If2LBRisabialgebrffaand@ι:-!!0 ! gBisamonoidalnaturffaltransformation,thenjtherffeisauniquehomomorphismofbialgebrasf:coend(!n9)!ߍBsuchjthatthef7 &e1021:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYdiagrffamE̴H@!HW! coSend ^"(!n9)ht{fd 0ά--0Hk}A@Xׁ @X @X @X @ԟt>@ԟt>RH! B椢Ǡ*FfeԟǠ?`T1 fs2cffommutes.Proof.@_ThemrultiplicationofcoSend!"(!n9)arisesfromthefollowingdiagramLNEH>x!n9(X) !(Yp)Hxe !n9(X) !(Yp) coSend ^"(!) coSend ^"(!)Y8t{fdά- A_L !n9(X+ Yp)v!n9(X+ Yp) coSend ^"(!)P32fdK(ά-mtn0H3S2Ǡ*Ffe3dǠ?H׺Ǡ*FfeǠ?HP԰չ=!n9(X) !(Yp) coSend ^"(!)`&ğׁ @0ğ @:ğ @Dğ @FD>@FD>R㕍FVoritheconstructionoftheunitwreconsiderthediagramD0u=-q(fIg;fid ʤg)togetherwith#!0D:D04!Vec/k, !0(I)=K,the#monoidalunitobjectinthemonoidalcategoryof.diagramsinVeciL.Then(K4!K K)=(!0 4!k!0 coSend j(!0)).istheunivrersalmap.8ThefollorwingdiagramtheninducedtheunitforcoSend!"(!n9)K]ō⍑j 8K⍒:K Kv4{fdAά-‚.⍍cxcx=c4h!n9(I)!n9(I) coSend ^"(!)}Ԟ32fd!wЍά-HnǠ*FfenǠ?HtǠ*FfȩǠ?HPN԰36=K coSend ^"(!n9)`t+ʬQttQtQt+QtrQ tQf4Qf4s8鍹ByusingtheunivrersalpropSertyonechecksthelawsforbialgebras.TheuWabSorvediagramsshowinparticularthatthenaturaltransformationhQ:!4!n! coSend ^"(!n9)ismonoidal.Scffxff ̟ff ̎ ̄cff&C;7 ,- cmcsc10+@ cmti12( msbm10#a6cmex8"u cmex10!q% cmsy6 K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8o cmr9N cmbx12Nff cmbx12XQ cmr12O line100