; TeX output 1998.02.24:1928={dRXQ cmr12CHAPTER3 sNff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups."N cmbx121.׈HopfAlgebrasDe nition3.1.1.vaùA@ cmti12leftHRHopfalgebrffa$g cmmi12HzisabialgebraHtogetherwithaleftantipffode S):URHB !", cmsy10\3!HV,;i.e.&^alinearmapSesucrhthatthefollowingdiagramcommutes:?I/)"QH)"$) msbm10K̟:2fd)@O line10-č2cmmi8 1HU<:2fd.bά-(70Ǡ@feǠ??<P@|{Ycmr83H HH H32fdJsά-ׁ'Sr}!K cmsy8 idSjǠ@fe?`6?<9rAHopf35algebrffaisaleftandrighrtHopfalgebra.t荍Problems3.1.2.sX1.YLetHv&bSeabialgebraandS\2bHom(HF:;HV).ThenS;isananrtipSodefforH(andHisaHopfalgebra)i S[=isatrwofsidedinrversefforidinthealgebra(Homy(HF:;HV);;n9)(see1.6.5).8InparticularSisuniquelydetermined.2.LetXHbSeaHopfalgebra.ThenSb/isananrtihomomorphismofalgebrasandcoalgebrasi.e.8S\inrvertstheorderofthemultiplicationandthecomultiplication".3.]LetHandK~,bSeHopfalgebrasandletfԢ:Hy! )KbSeahomomorphismofbialgebras.8ThenfGSH n=URSK;f,i.e.8f2iscompatiblewiththeanrtipSode.t荍De nition3.1.3.vaùBecause of3.1.2(3)evreryhomomorphismofbialgebrasbSe-trween[HopfalgebrasiscompatiblewiththeanrtipSodes. So[wede neahomomorphismoftHopfalgebrffasetobSeahomomorphismofbialgebras.ThecategoryofHopfalgebraswillbSedenotedbryK-Hopfalg1.Prop`osition3.1.4.OLffetHlbeabialgebrawithanalgebrageneratingsetX.KmLetS:7!H$w!HV2op}bffe%analgebrahomomorphismsuchthat#u cmex10PWS׹(x(1) \|)x(2)=7!n9"(x)forallxUR2X.fiThen35S isaleftantipffode35ofHV.- cmcsc10Proof.@_Assume;a;b2HܑsucrhthatPS׹(a(1) \|)a(2)mV=n9"(a)andPS׹(b(1) \|)b(2)mV=n9"(b).8Thenʍ-&0P;S׹((ab)(1) \|)(ab)(2)=URPS׹(a(1) \|b(1))a(2)b(2)ι=URPS׹(b(1))S(a(1))a(2)b(2)=URPS׹(b(1) \|)n9"(a)b(2)ι=UR"(a)"(b)="(ab):OSinceeevreryelementofHSGisa nitesumof niteproSductsofelementsinX,|forwhichtheequalitryholds,thisequalityextendstoallofHbyinduction.Y/cffxff ̟ff ̎ ̄cffExample3.1.5.oQ1.DfLet*V5bSeavrectorspaceandTƹ(Vp)thetensoralgebraoverVp.WVeVharveseenin1.5.10thatTƹ(Vp)isabialgebraandthatV'ƹgeneratesT(Vp)as/^,o cmr961>*{&e624w3.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPS.danTalgebra.w De neS: VS!Tƹ(Vp)2op 7ڹbryS׹(vn9):=vforallvx2Vp.w BytheunivrersalpropSertryofthetensoralgebrathismapextendstoanalgebrahomomorphismSӈ:Tƹ(Vp)URn!1T(V)2op.Sinces2(vn9)UR=vl> 1+1 vwrehave(S 1)s2(vn9)UR=vl>+vË=0=n9"(v),henceTƹ(Vp)isaHopfalgebrabrytheprecedingpropSosition.2. qLett-VbSeavrectorspaceandS׹(Vp)thesymmetricalgebraoverV(thatiscommrutative).4WVehaveseenin1.5.17thatS׹(Vp)isabialgebraandthatV:generatesS׹(Vp)Bwasanalgebra.@ODe neS:V9!S(Vp)bryS(vn9):=vforBwallvY2Vp.@OSNextendsto]analgebrahomomorphismS):URS׹(Vp)n!1S(Vp).5Since]s2(vn9)=v 1+1 vMwrehave(S 1)s2(vn9)9=vs+v?=0=n9"(v),hencepoS׹(Vp)isaHopfalgebrabrytheprecedingpropSosition."Example3.1.6.oQ(GroupSAlgebras)FVoreacrhalgebraAwecanformthegrffoupofunitsgU@(A)* :=fa2Aj9a212AgwiththemrultiplicationofAascompSositionofthey6group.ThenUisacorvXarianty6functorU۹:GK-Algf!/EGr?e.Thisfunctorleadstothefollorwinguniversalproblem.Let4GbSeagroup.;AnalgebraKGtogetherwithagrouphomomorphismyQ:G!"U@(KG)TKiscalleda(the)grffoupNalgebraofG,nifTKforevreryalgebraAandforeverygrouphomomorphismfQ:URGn!1U@(A)thereexistsauniquehomomorphismofalgebrasgË:URKGn!1A꨹sucrhthatthefollowingdiagramcommutesa HGHU@(KG)\{fd@ά- oH`Ç%flׁ @ƾl @оl @ھl @>@>RHiU@(A): Ǡ*Ffe=LǠ? g+ɍThegroupalgebraKGis(ifitexists)uniqueuptoisomorphism.ItisgeneratedaslanalgebrabrytheimageofG.vThemap2:GLy!iU@(KG)KGlڹisinjectiveandtheimageofGinKGisabasis.TheDAgroupalgebracanbSeconstructedasthefreevrectorspaceKGwithbasisGandxthealgebrastructureofKGisgivrenbyKG KG3g+" h7!gn9h2KGxandtheunitË:URK3 h7! e2KG.ThegroupalgebraKGisaHopfalgebra.ThecomrultiplicationisgivenbythediagramZȍj⍒0Gj⍒=SKGd{fd% ά-TDH`f tׁ @ t @ t @ t @Q>@Q>RH"њKG KGW"Ǡ*FfeTǠ?`<?{&e1.pHOPF!ALGEBRASGL63dwithfG(gn9):=g 6g&whicrhde nesagrouphomomorphismf:GA!U@(KG6 KG).ThecounitisgivrenbyCj⍒x&Gj⍒ꄻKG4̟{fd% ά-қH`7fRܟׁ @Rܟ @Rܟ @Rܟ @\>@\>RH""'K󞊟Ǡ*FfeѼǠ?]D<"wherefG(gn9)\=1forallg 2G.|OneshorwseasilybyusingtheuniversalpropSertyV,thatCiscoassoSciativreandhascounit".C~De neanalgebrahomomorphismSn:KG!n߹(KG)2op ŹbryDcj⍒Gj⍒[KGl{fd% ά-%LH`f|ׁ @| @| @| @">@">RH (KG)2op(*Ǡ*Ffe[\Ǡ?` S^withfG(gn9):=g21lwhicrhisagrouphomomorphismf:G!3U@((KG)2op).^_ThentheprecedingpropSositionshorwsthatKGisaHopfalgebra.XExample3.1.7.oQ(Universal#EnvelopingAlgebras)0hA0VLiesQalgebrffaconsistsofavrectorspace-%n eufm10gtogetherwithamultiplicationgI g3xI y7![x;yn9]2gsuchthatthefollorwinglawshold:EʍKY[x;x]UR=0;KY[Ox;[yn9;z]]xѫ+[yn9;[z;x]]+[z;[x;yn9]]UR=0(Jacobiidenrtity).AZhomomorphism=ofLiealgebrffasZf\:g-!AhisalinearmapfsucrhthatfG([x;yn9])=[fG(x);f(yn9)].8ThrusLiealgebrasformacategoryK-Lie.An*impSortanrtexampleistheLiealgebraassociatedwithanassociativrealgebra(withunit).8IfAisanalgebrathenthevrectorspaceAwiththeLiemultiplicationʒ[x;yn9]UR:=xyyx(1)u;isaLiealgebradenotedbryA2LGع.ThisLiealgebrade nesacovXariantfunctor-2LX:URK-Alg!nK-Lie.8Thisfunctorleadstothefollorwinguniversalproblem.LetugbSeaLiealgebra.ٯAnalgebraU@(g)togetherwithaLiealgebrahomomor-phism\E:g/!FU@(g)2L ۹iscalleda(the)universalgenvelopingalgebrffaofg,xZifforevreryalgebra{AAandforevreryLiealgebrahomomorphismfQ:URgn!1A2L there{AexistsauniquehomomorphismȾofalgebrasgË:URU@(g)n!1AȾsucrhthatthefollowingdiagramcommutesJȱ㍒Cgȱ㍒qU@(g)2L{fd!wЍά- X7H`Jfԟׁ @ǁԟ @сԟ @ہԟ @T>@T>RH#ӎA2LG:͂Ǡ*FfeǠ? 4gPTheGunivrersalenvelopingalgebraU@(g)is(ifitexists)uniqueuptoisomorphism.Itisgeneratedasanalgebrabrytheimageofg.8ThemapUR:gn!1U@(g)2L 2isinjective.@Š{&e644w3.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPS.dThe8univrersalenvelopingalgebracanbSeconstructedasU@(g)UR=Tƹ(g)=(x݁ yKy x[x;yn9])whereTƹ(g)UR=Kgg g:::Disthetensoralgebra.TheunivrersalenvelopingalgebraU@(g)isaHopfalgebra.8ThecomultiplicationisgivrenbythediagramGDЍH egHےU@(g){fd$ά-WH`-fItׁ @It @It @It @Տ>@Տ>RHRU@(g) U(g)"Ǡ*FfeTǠ?`znwith*fG(x)v}:=x 1+1 xwhicrhde nesaLiealgebrahomomorphismf|:v}g!(U@(g) U(g))2LGع.8ThecounitisgivrenbyLፍHٝgHa5U@(g)T{fd$ά-%LH`fׁ @ @ @ @^,>@^,>RH""KcZǠ*Ffe񖌟Ǡ?]DI "=withMvfG(x)=0forallx2g.aLOneshorwseasilybyusingtheuniversalpropSertyV,f*that{}iscoassoSciativreandhascounit". `De neanalgebrahomomorphismSm:U@(g)!n߹(U@(g))2op ŹbryG|HT5gHU@(g)T{fd$ά-ɟH`vfDׁ @ĒD @ΒD @ؒD @ğ>@ğ>RHz(U@(g))2opǠ*Ffe$Ǡ?`äS}withfG(x)UR:=xwhicrhisaLiealgebrahomomorphismfQ:gn!1(U@(g)2op)2LGع.%ThentheprecedingpropSositionshorwsthatU@(g)isaHopfalgebra.(Observre,[that79themeaningofUxinthisexampleandthepreviousexample(groupofNunits,funivrersalenvelopingalgebra)istotallydi erent,finthe rstcaseUcanbSeappliedtoanalgebraandgivresagroup,inthesecondcaseUعcanbSeappliedtoaLiealgebraandgivresanalgebra.)ؘProblems3.1.8.sX1. LetKbSea eld.Shorwthatanelementx2KGsatis es(x)UR=x x꨹and(x)UR=1ifandonlyifxUR=gË2G.2.BShorwthatthesetofprimitiveelementsPƹ(HV)UR=fx2Hj(x)=xֆ 1+1 xgofaHopfalgebraHareaLiesubalgebraofHV2L5..Prop`osition3.1.9.OLffet]HbeaHopfalgebrawithantipodeS.\Thefollowingareeffquivalent:1.fiSן22-=id.2.fiPS׹(a(2) \|)a(1)ι=URn9"(a)35foralla2HV.3.fiPa(2) \|S׹(a(1))UR=n9"(a)35forallaUR2HV.A,\{&e1.pHOPF!ALGEBRASGL65dProof.@_LetSן22-=id.8Then"ʍF/PTS׹(a(2) \|)a(1)㷹=URSן22r۹(PS׹(a(2) \|)a(1))=S׹(PS(a(1) \|)S22r۹(a(2)))=URS׹(PS(a(1) \|)a(2))=S׹(n9"(a))="(a)wbryusingProblem3.1.2.Conrverselyassumethat2.8holds.Thenʍ_oKS]Sן22r۹(a)a=URPS׹(a(1) \|S22r۹(a(2))UR=S׹(PS(a(2))a(1)a=URS׹(n9"(a))="(a):5ThrusSן22){andid7areinversesofSiwintheconvolutionalgebraHom0)(HF:;HV),ی> Hnu-:07,ׁ A7, A7, A7, AZl>AZl>U2KG(X)nG209(X)̶32fd_`ά- f(X)ŶcommruteforallXF2URC5.cProblems3.2.2.sX1)!IfasetZdtogetherwithamrultiplicationma]:Z}Z=W!ZisՊamonoid,Bthentheunitelemenrte2Z(isՊuniquelydetermined.Ifitisagroupthencalsotheinrverseci#+:ZT!cl{fd ά-).;1u"GGXgfG̞32fd_ά-č!mHǠ*FfeǠ?͞ABu1H\֊Ǡ*Ffe] Ǡ?]Da,Ǡ?1mi?-H:Ǡ*FfelǠ?(mi?-㕍commrutesifandonlyifMorѩC?h(-;m(m R1))UR=MorOC(-;m)(Mor5C(-;m) R1)UR=m̹-j(m̹-s1)_R=m̹-j(1m̹-)_R=MorOC(-;m)(1MorK Cʹ(-;m))_R=MorOC(-;m(1m))ifandonlyifm(m1)UR=m(1m)ifandonlyifthediagramKZ#GGGZmMGG۠ܟ{fdQά-).ۙm1GGdGϦ|32fd*Fά-čZmH^Ǡ*Ffe̟Ǡ?͞~ 1mHǠ*FfeLǠ?]DmzՍcommrutes.8InasimilarwayoneshowstheequivXalenceoftheotherdiagram(s).cffxff ̟ff ̎ ̄cffr:Problems3.2.4.sXLetC6bSeacategorywith niteproducts.'Shorwthatamor-phismfQ:URGn!1G20inCݹisahomomorphismofgroupsifandonlyifHZGGZIGI|{fd@ά-:mK̔G20xG20KG2032fdNά-.&m-:0HǠ*Ffe4̟Ǡ?`ffHڟǠ*Ffe Ǡ?`fecommrutes.De nition3.2.5.vaùA.Ncffogroup.`(comonoid)GinCᕹisagroup(monoid)inC52op R,?Ni.e.anaHobjectGUR2ObC=ObC52op togetheraHwithanaturaltransformationm(X)UR:G(X)G(X)URn!1G(X)whereG(X)UR=MorOCmr;cmmi6op&(XJg;G)=MorOC(G;X),sucrhthat(G(X);m(X))isagroup(monoid)foreacrhXF2URC5.Remark3.2.6.j6Let C7@bSeacategorywith nite(categorical)coproducts.  Anobject0GinCecarriesthestructuremӹ:G(-)ɨG(-)^!_3G(-)0ofacogroupinCeifandonly%*iftherearemorphisms:Gs!]G}qG,3":Gs!I,3and%*Sk:Gs!G%*sucrhthatthediagramsJ7GqGq9GqGqG[d32fdά- [q1bZBGbZ}8GqGO\{fd*ά-̍a\HZǠ*FfeǠ?` 1qHGXڟǠ*FfeG Ǡ?`;0ٚۄ8GqGٚI+qGPUR԰n9=GPUR԰n9=GqI32fdqά- "q1bZQ6GbZVGqG ܟ{fd_Zά-̍ 0Hg Ǡ*FfegA,Ǡ?`k1q"HZǠ*FfeǠ?`+H8Jb1H+ʬQHtQ HQH+Q HrQ*HQ-ܟQ-ܟsEhr{&eph2. %MONOIDS!ANDGR9OUPSINACA:TEGORYc69dZ@bZ4GbZտGI{fd K0ά-bZbZ&Gbܟ{fd K0ά-"gGqGYGqG32fd6@ά- бw1qSlбwSr}q1H*Ǡ*Ffe\Ǡ?`H*Ǡ*Ffe\ׁ 6` zrcommrutea`whererisdualtothemorphismde nedin1.3.6.ThemultiplicationsarerelatedbryX r۹=URMorOC(;X)UR=(X).Let!CoVbSeacategorywith nitecoproductsandletGandG20ZbecogroupsinC5.ThenElahomomorphismofgroupsf7й:G20 ו!hGisamorphismf:G \!/G20inElCsucrhthatthediagramIZVGZGG{fd@ά-̍ɆnK:G20KG20xG20532fdNά-.y+-:0H6Ǡ*FfeiǠ?`lffHzǠ*FfeǠ?` fcommrutes.8Ananalogousresultholdsforcomonoids.aRemark3.2.7.j6Obrviously/similarobservXationsandstatementscanbSemadeforotheralgebraicstructuresinacategoryC5.CSoonecaninrtroSducevectorspacesandcorvectorO spaces,&monoidsandcomonoids,ringsandcoringsinacategoryC5. fThestructures\8canbSedescribedbrymorphismsinCmifCisacategorywith nite(co-)proSducts.aProblems3.2.8.sXDetermine-thestructureofacorvector-spaceonavrectorspaceVfromthefactthatHomd1(V;Wƹ)isavrectorspaceforallvectorspacesWƹ.Prop`osition3.2.9.OLffetG2CSbeagroupwithmultiplicationab,unite,andinversePa21TinG(X).TThenthemorphismsm5:GG5!~G,XAu5:E@L!2G,andPS? :Gfk!G35arffegivenbyWtmUR=p1jp2;u=eE-;S)=idW 18O G|r:Proof.@_By[theYVonedaLemma1.8.2thesemorphismscanbSeconstructedfromtheXinaturaltransformationasfollorws.!UnderMorfC%(GG;GG)UR=GG(GG)PUR԰n9=G(GLG)G(GG)2 pUR !G(GG)UR=MorOC(GLG;G)xtheidenrtityxid CGG$=UR(p1;p2)ispmappSedtom9=p1p2.QUnderpMorʟC(E;E)9=E(E)9S2!G(E)9=MoroCc(E;G)ptheidenrtity;ofEŹismappSedtotheneutralelemenrtu;=eE-.+Under;MorqCj(G;G)=G(G)R|!nG(G)UR=MorOC(G;G)theidenrtityismappSedtoits-inrverseS)=URidW 18O G|r.4ӻcffxff ̟ff ̎ ̄cffCorollary3.2.10.zLffet_G2Cbe_acogroupwithmultiplicationaˋb,junit_e,andinversezxa21inG(X).<2ThenthemorphismsH:G?!GqqG,H"H:G?!I,HandzxS:Gfk!G35arffegivenby)xS'UR=1j2;"=eIM;S)=idW 18O G|r:Fza{&e704w3.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPS.d1P3.PAneAlgebraicGroupsWVeapplytheprecedingconsiderationstothecategoriesK-cAlg"andK-cCoalg+z.ConsiderK-cAlgo,XthecategoryofcommrutativeK-algebras.ULetA;B82K-cAlg.Then&$AC B*isagainacommrutativeK-algebrawithcompSonentwisemultiplication._Infactthisholdsalsofornon-commrutativeK-algebras(1.5.4),butinK-cAlg"wrehaveL1Prop`osition3.3.1.OThextensorprffoductxinK-35cAlg"]isthe(cffategorical)xcoproduct.Proof.@_Letf^2U_K-cAlgo(A;Zܞ);gØ2K-cAlgo(B;Zܞ).8Themap[f;gn9]:A Be !_QZde nedNbry[f;gn9](a$ b):=fG(a)g(b)Nistheuniquealgebrahomomorphismsucrhthat[f;gn9](a 1)UR=fG(a)and[f;gn9](1 b)UR=g(b)orsucrhthatthediagramA0}ԠAAԠ!hA Bt:2fd$0ά-aMX.AԠԠ0B:2fd6άaMPZX.Bꃀf᤟?`@᤟?`@᤟?`@i@iRuGQLgb?`b?`b?`ڤڤ .ZoǠ@feآDǠ?bЍT[߱8fh;gI{]+commrutes,whereA(a)UR=a 1andBN>(b)UR=1 b꨹arealgebrahomomorphisms.Scffxff ̟ff ̎ ̄cffڍSothecategoryK-cAlg"has nitecoproSductsandalsoaninitialobjectK.A'more( generalpropSertryofthetensorproductofarbitraryalgebraswrasalreadyconsideredin2.2.12.Observrethatthefollowingdiagramcommutes@IԠnAԠȫ A A:2fd~`ά-͝;8qAacmr61ԠԠ`Ak̟:2fd~`쁠ά͝;xq2kS1X.A?`@?`@?`@Ɩ@ƖRkS~d1X.A?`?`?` gA؜*Ǡ@fe\Ǡ?bp݁r+whereristhemrultiplicationofthealgebraandbythediagramthecoSdiagonalofthecoproSduct.L1De nition3.3.2.vaùAnanealgebrffaicgroup޹isacogroupinthecategoryofK-cAlg"ofcommrutativealgebras.FVoranarbitraryanealgebraicgroupHwregetbyCorollary3.2.10+MUR=1j2V2K-cAlgo(HF:;H HV);O:"UR=e2K-cAlgo(HF:;K);jand4CjS)=(id ʤ) 12K-cAlg(HF:;HV):ThesemapsandCorollary3.2.10leadtoProp`osition3.3.3.OLffetxH>2QK-35cAlg.3HisarepresentingobjectforafunctorK-35cAlg":N$!3 GrHIBif35andonlyifH isaHopfalgebrffa.Proof.@_BothWstatemenrtsareequivXalenttotheexistenceofmorphismsinK-cAlg+gkUR:HB\3!H H N":HB\3!K S):HB\3!HG{&egV3. %AFFINE!ALGEBRAICGR9OUPS71dsucrhthatthefollowingdiagramscommute?]ԠۣHԠ,H Hd:2fd?ά-ζ łǠ@feǠ??<0T 胀(coassoSciativitry)>Ǡ@fe>괟Ǡ?ԐC4 1H H AH H Hv$32fd&H`ά-0h1 BԠLHԠ7ȓH Hpl:2fd٠ά-ζ Ҏ"H H"bK HPB԰[=QHPB԰[=H K,32fd<ά-nŬv" 1S,<胀(counit)nǠ@feǠ??<IJǠ@feI|Ǡ?ꬾNy1 "j1JܟܔPJܟ PJܟ?_PJܟ攴PJܟ PJܟ?^PJܟPJܟPJܟ?]P㜟 P㜟 q?ۊlt胀(coinrverse))"iH)"BK:2fd)@ά-č40 .OHsT:2fd.bά-(7NǠ@fe64Ǡ??<nX7KH H!H Hӳ32fdJsά-ׁESr} idaCidK S3qǠ@fe3?`6?<8W4r۹ %cffxff ̟ff ̎ ̄cffThis#PropSositionsarystwothings.FirstofalleachcommutativeHopfalgebraHde nes=afunctorK-cAlgo(HF:;-):K-cAlg#4r%M!5SetMw"that=factorsthroughthecategoryofjgroupsorsimplyafunctorK-cAlgo(HF:;-)U:K-cAlg"\% !5aGrEֹ.&Secondlyjeacrhrepre-senrtablefunctorK-cAlgo(HF:;-)d:K-cAlg"$!3ZSetJ[thatfactorsthroughthecategoryofgroupsisrepresenrtedbyacommutativeHopfalgebra.s&Corollary3.3.4.sWA2nalgebrffaH2K-35cAlg$fisananealgebraicgroupifandonly35ifH isacffommutativeHopfalgebra.ThedQcffategoryofcommutativeHopfalgebrasisdualtothecategoryofanealge-brffaic35groups.InBthefollorwinglemmasweconsiderfunctorsrepresentedbycommutativealge-bras. They\Xde nefunctorsonthecategoryK-cAlg#ǹaswrellasmoregenerallyonK-Algo.)WVes rststudythefunctorsandtherepresenrtingalgebras.ThenwreusethemtoconstructcommrutativeHopfalgebras.Lemma3.3.5.g5QTheGfunctorGa X:T1K-35Alg9- !1AbHde neffdbyGaϹ(A):=A2+x, theunderlying~additivegrffoupofthealgebraA,sisarepresentablefunctorrepresentedbythe35algebrffaK[x]thepolynomialringinonevariablex.Proof.@_Ga ܹis anunderlyingfunctorthatforgetsthemrultiplicative structureofthealgebraandonlypreservrestheadditivegroupofthealgebra.ʃWVehavetodeterminenaturaluisomorphisms(naturalinA)GaϹ(A)PUR԰n9=K-Algo(K[x];A).EacrhelementaUR2A2+isOmappSedtothehomomorphismofalgebrasa]:K[x]3p(x)7!p(a)2A.ThisOisahomomorphismۂofalgebrassincea(p(x)+qn9(x))UR=p(a)+q(a)UR=a(p(x))+a(qn9(x))and9a(p(x)qn9(x))R2=p(a)q(a)=a(p(x))a(q(x)).Another9reasontoseethisisthatK[x]$isthefree(commrutative)$K-algebraorver$fxgi.e.esinceeacrhmapfxgl!AcanuSbSeuniquelyextendedtoahomomorphismofalgebrasK[x]!A. ThemapHI{&e724w3.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPS.dAS3a7!a eW2K-Algo(K[x];A)isbijectivrewiththeinversemapK-Algo(K[x];A)S3fQ7!URfG(x)2A.8FinallythismapisnaturalinAsinceQWAMB̌K-Algo(K[x];B)J\32fd5Ƞά-|6-VHAH{K-Algo(K[x];A),{fd6}ά-6-VHʟǠ*FfeǠ?' ~gH Ǡ*Ffe<Ǡ?`tK-Alg(K[*x]\t;gI{)commrutesforallgË2URK-Algo(A;B).Ʉcffxff ̟ff ̎ ̄cff鍍Remark3.3.6.j6SinceAA2+ ^2hasthestructureofanadditivregroupthesetsofho-momorphismsofalgebrasK-Algo(K[x];A)arealsoadditivregroups.Lemma3.3.7.g5QThefunctorGm Z=URU6:K-35Alg:N!- GrAde neffdbyGmĹ(A)UR:=U@(A),the[]underlyingmultiplicffativegroupofunitsofthealgebraA,isarepresentablefunctorrffepresentedVvbythealgebrffaK[x;x21 \|]UR=K[x;yn9]=(xy.W1)VvtheringofLffaurentVvpolynomialsin35onevariablex.Proof.@_WVe0harvetodeterminenaturalisomorphisms(naturalinA)GmĹ(A)P԰=K-Algo(K[x;x21 \|];A).tEacrhSelementa2GmĹ(A)ismappSedtothehomomorphismofalgebrasa ,:=((K[x;x21 \|]3x7!a2A).Thisde nesauniquehomomorphismofalgebrassinceeacrhhomomorphismofalgebrasffromK[x;x21 \|]_r=K[x;yn9]=(xyT1)towAiscompletelydeterminedbrytheimagesofxandofy butinadditiontheimagesharvetosatisfyfG(x)f(yn9)UR=1,-@i.e.fG(x)mrustbSeinvertibleandfG(yn9)mustbSetheinversetofG(x).8Thismappingisbijectivre.FVurthermoreitisnaturalinAsinceQWAAQ}`BAQbK-Algo(K[x;x21 \|];B)Ԟ32fd*(pά-|6-Ȋ~'AȊGK-Algo(K[x;x21 \|];A)l{fd*Ѝά-6-H[BǠ*FfetǠ?'ygH Ǡ*Ffe>Ǡ?4K-v2Algp(K[*x;x-:1 ] N4;gI{)forallgË2URK-Algo(A;B)commrute.Ccffxff ̟ff ̎ ̄cffRemark3.3.8.j6SincedU@(A)hasthestructureofa(mrultiplicative)dgroupthesetsK-Algo(K[x;x21 \|];A)arealsogroups.Lemma3.3.9.g5QThefunctorMn rf:K-35Alg }!0ߕK-35Alg}withMnP(A)thealgebrffaofnn-matricffes-WwithentriesinAisrepresentablebythealgebraKhx11 ;x12;:::ʚ;xnn Рi,the35noncffommutativepolynomialringinthevariablesxijJ.Proof.@_TheMcpSolynomialringKhxijJiisfreeorverMcthesetfxijginthecategoryof(nontocommrutative)algebras,i.e.7foreachalgebraandforeachmapfչ:?fxijJgYa!9AtheresEexistsauniquehomomorphismofalgebrasg:=Khx11 ;x12;:::ʚ;xnn Рi=We!AAsEsucrhI{&egV3. %AFFINE!ALGEBRAICGR9OUPS73dthatthediagramE\>fxijJg>4KhxijJi?{fdά-lH`}f<ׁ @< @< @< @M>@M>RHARǠ*FfeǠ?'8g&commrutes.t SoSeachmatrixinMnP(A)de nesauniqueahomomorphismofalgebrasKhx11 ;x12;:::ʚ;xnn РiURn!1A꨹andconrverselyV.cffxff ̟ff ̎ ̄cffߍExample3.3.10.uQ1.8Theanealgebraicgroupcalledadditive35grffoup퍒wGaY!:URK-cAlg!$ L!3`Abwith|GaϹ(A)Nf:=A2+ ufromLemma3.3.5isrepresenrtedbytheHopfalgebraK[x].WVedeterminecomoltiplication,counit,andanrtipSode.ByCorollary3.2.10thecomrultiplicationis:=1,l2 2K-cAlgo(K[x];K[x] K[x])PUR԰n9=GaϹ(K[x] K[x]).8Hencex((x)UR=1(x)+2(x)UR=x 1+1 x:Thecounitis"UR=e*ppmsbm8K 4=02K-cAlgo(K[x];K)P԰n9=GaϹ(K)henceD"(x)UR=0:iTheanrtipSodeisS)=URidW 1؍ K[x] 2URK-cAlgo(K[x];K[x])P԰n9=GaϹ(K[x])hence0f[S׹(x)UR=x:2.8Theanealgebraicgroupcalledmultiplicffative35groupXGm Z:URK-cAlg!$ L!3`AbwithXGmĹ(A):=A2 =U@(A)fromLemma3.3.7isrepresenrtedbytheHopfalgebraK[x;x21 \|]UR=K[x;yn9]=(xy1).8WVedeterminecomoltiplication,counit,andanrtipSode.ByCorollary3.2.10thecomrultiplicationisuUR=1j2V2K-cAlgo(K[x;x 1 \|];K[x;x 1] K[x;x 1])PUR԰n9=GmĹ(K[x;x 1] K[x;x 1]):Hence(x)UR=1(x)2(x)UR=x x:&Thecounitis"UR=eK 4=12K-cAlgo(K[x;x21 \|];K)PUR԰n9=GmĹ(K)henceD"(x)UR=1:iTheJzanrtipSodeisS_=)idWv1эvK[x;x1 ]6t2)K-cAlgo(K[x;x21 \|];K[x;x21])P)԰=GaϹ(K[x;x21])|hencebS׹(x)UR=x 1 \|:&3.8Theanealgebraicgroupcalledadditive35matrixgrffoupɸM +ڍn qʹ:URK-cAlg!$ L!3`AbEg;J{&e744w3.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPS.dwithjM2+RAnx(A)theadditivregroupofn n-matricesjwithcoSecientsinAisrepresentedbry2thecommutativealgebraM2@+RAn -=2K[xijJj1i;j}n]2(Lemma3.3.9).qThisalgebramrustbSeaHopfalgebra.ThecomrultiplicationisUR=12V2K-cAlgo(M2@+RAn]\;M2@+RAn M M2@+RAn)PUR԰n9=M2+RAnx(M2@+RAn M M2@+RAn).HenceIIe(xijJ)UR=1(xij)+2(xij)UR=xij 1+1 xij:{Thecounitis"UR=eK 4=(0)2K-cAlgo(M2@+RAn]\;K)P԰n9=M2+RAnx(K)hence/"(xijJ)UR=0:/TheanrtipSodeisS)=URidW 1 ,썑 Mi@"+Ln 2URK-cAlgo(M2@+RAn]\;M2@+RAn)PUR԰n9=M2+RAnx(M2@+RAn)hencev}wS׹(xijJ)UR=xij:4. ThematrixalgebraMnP(A)alsohasanoncommrutativemultiplication,thema-trixRmrultiplication, S=(id ʤ)21@2K-cCoalg+z(HF:;HV).4These>maps,theYVonedaLemmaandRemark3.2.9leadtotheproSofoftheproposition.)=@cffxff ̟ff ̎ ̄cff?TRemark3.4.4.j6In:particulartherepresenrtingobject(HF:;r;u;;";S׹):ofaformalgroup GisacoScommrutative HopfalgebraandevrerysuchHopfalgebrarepresentsaformalgroup.HencethecategoryofformalgroupsisequivXalenrttothecategoryofcoScommrutativeHopfalgebras.Corollary3.4.5.sWA=cffoalgebrahH}2K-35cCoalg0representsaformalgroupifandonly35ifH isacffocommutative35Hopfalgebrffa.The6cffategoryofcocommutativeHopfalgebrasisequivalenttothecategoryofformalgrffoups.Corollary3.4.6.sWThe35followingcffategories35areequivalent: 1. #The35cffategoryofcommutative,cocommutativeHopfalgebras. 2. #The35cffategoryofcommutativeformalgroups. 3. #The35dualofthecffategory35ofcffommutativeanealgebraicgroups.Example3.4.7.5U1.8GroupalgebrasKGareformalgroups. 2. #UnivrersalenvelopingalgebrasU@(g)ofLiealgebrasgareformalgroups. 3. #ThetensoralgebraTƹ(Vp)andthesymmetricalgebraS׹(V)areformalgroups. 4. #LetCbSeacocommrutativecoalgebraandGbeagroup. Thenthegroup #KG(Cܞ)=K-cCoalg+z(C5;KG)Yisisomorphictothesetoffamilies(h2RAgjg2G)of #decompSositionsBoftheunitofCܞ2ߝinrtoasumoforthogonalidempotenrtsh2RAg*P2URCܞ2 #thatareloScally nite.2#TVo HseethisemrbSedK-cCoalg+z(C5;KG) xHom$(C;KG) HandemrbSed #Hom8(C5;KG))inrtotheset(Cܞ2)2G IofG-familiesofelementsinthealgebraCܞ2M뼠{&er5.pINTEGRALS!ANDF9OURIERTRANSFORMSeĿ77d #bryrh;7!(h2RAg)withh(c)=PßgI{2G&h2RAg(c)gn9. T?Thelinearmaphisahomomor-C #phismީofcoalgebrasi (hP h)=hީand"h="ީi PTh(c(1) \|)P h(c(2))= #P.3h(c)(1) / |h(c)(2)0and)"(h(c))UR="(c)forallc2Cǹi P~h2RAg(c(1) \|)g |h2yl(c(2))l= #P.3h2RAg(c)gM ߫gandGP1h2RAg(c)UR="(c)Gi Ph2RAg(c(1) \|)h2yl(c(2))UR=gI{lvh2RAg(c)GandP1h2RAg*P=UR" #i _h2RAg{}h2yl=gI{lvh2RAg 4andP h2RAg=1C t.FVurthermorethefamiliesmrustbSelocally # nite,i.e.8foreacrhcUR2CFonly nitelymanyofthemgivenon-zerovXalues. 5. #LetCϵbSeacocommrutativecoalgebraandK[x]betheHopfalgebrawith(x)UR= #x܆ 1+1 x(thesymmetricalgebraoftheonedimensionalvrectorspaceZ΍ #Kx). WVe5emrbSedasbeforeK-cCoalg+z(C5;K[x])-Hom(C;K[x])=(Cܞ2)2f02@cmbx8Nq0*g@, #thesetofloScally niteN0-familiesinCܞ2 bryh(c)=P*o1 U_oi=0!h2RAi(c)x2idڹ.$Themap$ #hisahomomorphismofcoalgebrasi (h(c))k=PBh2RAi(c)(x+~ 1+1 x)2iE= #P.3h2RAi(c)G܍i v~ldܟG x2l' x2ill=UR(h h)(c)=Ph2RAi(c(1) \|)h2RAj(c(2))x2ij x2jWandM"(Ph2RAi(c)x2idڹ)UR=" #"(c)i h2RAijh2RAjV=URG܍Ti+j v~iKGofK2G isthemap"v?:KG! KrestrictedtoG,hence"(x)v?=1=hx;1>KG iforallxUR2G.8TheanrtipSodeoffQ2URK2G @isgivrenbyS׹(fG)(x)UR=hx;S(fG)i=f(x21 \|).TheDelemenrtsofthedualbasis(x2jx}2G)Dwithhx;yn92.=i}=x;y$consideredDasmaps-JfromGtoKformabasisofK2G.8Theysatisfytheconditions=tHx yn9 =URx;y "hx and!~X!x2G4x V=1>KG sincehz;x2yn92.=iUR=hz;x2ihz;yn92.=iUR=zV;x zV;y=x;y "hhz;x2iandhz;Px2G"=x2iUR=1=hz;1>KG i.MHencesthedualbasis(x2jxY2G)sisadecompSositionoftheunitinrtoasetofminimalorthogonalidempSotenrtsandthealgebraofK2G @hasthestructureQ$K G t=URx2G/Kx PV԰.==K:::K:InparticularK2G @iscommrutativeandsemisimple.ThediagonalofK2G @is=tf/(x )UR=XSyI{2Gyn9  (yn9 1 ʵx) V=XyI{;zV2G;yz=x3MUyn9  z Q7<{&er5.pINTEGRALS!ANDF9OURIERTRANSFORMSeĿ81dsincer덍>)hz3 u;(x2)iUR=hzu;x2i=x;zVu19=zV1 HEx;u"=PyI{2G#GRyI{;z /yI{1 ;;x;uGϹ=URPyI{2G!GThz;yn92.=ihu;(yn921 ʵx)2iUR=hz3 u;PyI{2G!yn92 (yn921x)2i:bLeto]a752KG.Thenade nesamap/eaѓ:GP!K2K2G brya=Px2G$5:e$ua*(x)x.FVorӒarbitraryfQ2URK2G @anda2KGthisgivresm:y?tha;fGiUR=f(X!x2GeUVaW(x)x)=Xvfx2Gjeaթ(x)f(x): ThecounitofK2G @isgivrenby"(x2)UR=x;eswheree2Gistheunitelemenrt.TheanrtipSodeis,asaborve,S׹(x2)UR=(x21 \|)2.WVeconsiderHpF=K2G sasthefunctionalgebraonthe nitegroupGandKGasthedualspaceofHB=URK2G @henceasthesetofdistributionsonHV.Then~HUQR\H:=URXvfx2GxUR2HV  =KG(2)Cލisa(trwosided)inrtegralonHsincePUx2G$(yn9xUR=Px2G#,x="(y)Px2G"=x=Px2G#,yx.WVewriteA甆ZfG(x)dxUR:=hUQR;fi=Xvfx2Gf(x):덑WVei.harveseenthatthereisadecompSositionoftheunit1l2K2G ƹinrtoasetofprimitivreorthogonalidempSotentsfx2jx|2Ggsuchthateveryelementf{2|K2G hasauniquerepresenrtationf86=7PfG(x)x2. SinceUQRyn92 t=7Px2G#.hx;yn92.=iwegetUQRfGyn92 t=RP x2G=hx;fGyn92.=iUR=Pf(x)yn92.=(x)=f(yn9)henceقfQ=URX(甆ZfG(x)yn9 .=(x)dx)yn9 :~nProblems3.5.11.zDescribSe thegroupvXaluedfunctorK-cAlgo(K2G;)intermsofsetsandtheirgroupstructure.VʍDe nitionandRemark3.5.12._LetKbSeanalgebraiclyclosed eldandletGbSea niteabeliangroup(replacingRaborve).+AssumethatthecrharacteristicofKdoSesnotdividetheorderofG.DLetH=9K2G.WVeidenrtifyK2G ѹ=9Hom(KG;K)alongthelinearexpansionofmapsasinExample3.5.10.0Let=usconsiderthesetx?^G):=s@f:G! K2j꨹grouphomomorphismtB.g.mSince=K2Ais]܍anabSeliangroup,thesetx^Gisanabeliangroupbrypoinrtwisemultiplication.Thegroupx^Giscalledthecharffacter35group꨹ofG.ObrviouslygthecharactergroupisamultiplicativesubsetofK2G $=tHom(KG;K).Actually$itisasubgroupofK-cAlgo(KG;K)URHom(KG;K)$sincetheelemenrtsUR2xT^GexpandeWtoalgebrahomomorphisms:.?(ab)=(P xHxP y yn9)=P x y (xyn9)=(a)(b)and(1)=(e)=1.~]Conrverselyanalgebrahomomorphismf 2K-cAlgo(KG;K)qrestrictstoacrharacterfQ:URGn!1K2.Thusx^qG =K-cAlgo(KG;K),thesetofrationalpSoinrtsoftheanealgebraicgrouprepresentedbyKG.RJ{&e824w3.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPS.dThereisamoregeneralobservXationbSehindthisremark.Lemma3.5.13.mQLffetiHbea nitedimensionalHopfalgebra.ThenthesetGr(HV2Z)of35grffouplikeelementsofHV2 isequaltoK-Alg(HF:;K).Proof.@_Infactf:PH=Wg!Kisanalgebrahomomorphismi hf uf;a bi=hf;aihf;biUR=hf;abi=h(fG);a bi꨹and1UR=hf;1i="(fG).}݄cffxff ̟ff ̎ ̄cffHencethereisaHopfalgebrahomomorphism'UR:Kx^G !K2G @bry3.1.6.]HProp`osition3.5.14.TheHopfalgebrffahomomorphism'6:Kx^G {! K2G ̩isbijec-tive.Proof.@_WVegivretheproSofbyseverallemmas.Lemma3.5.15.mQA2nysetofgrffouplikeelementsinaHopfalgebraH@islinearlyindepffendent.Proof.@_AssumethereisalinearlydepSendenrtsetfx0;x1;:::ʚ;xnPgofgrouplikreelemenrtsHinHV.ChoSosesuchasetwithnminimal.ObviouslynUR1Hsinceallelementsarenonzero.8Thrusx0V=URP*n U_i=1 AS idxiOandfx1;:::ʚ;xnPglinearlyindepSendent.8WVegetVHX ㇍Z,Hi;jij id jf xi xj\=URx0j x0V=(x0)=X ㇍ i idxi xi:#[Sinceall i,6=UR0andthexia |xj:arelinearlyindepSendenrtwegetnUR=1and 1V=UR1sothatx0V=URx1,aconrtradiction.5ӄcffxff ̟ff ̎ ̄cffCorollary3.5.16.z(Deffdekind'svLemma)A2nysetofcharactersinK2G `islinearlyindepffendent.]HThrusd3'$5:Kx^G hl!K2G ˹isinjectivre.Nowweprovethatthemap'$5:Kx^G hl!K2G ˹issurjectivre.Lemma3.5.17.mQ(Pontryagintduality)Theevaluationx!^GQGb!1K2 4isanon-deffgenerate35bilinearmapofabeliangroups.Proof.@_FirstwreobservethatHom (CnP;K2)P԰׹=7Cn OUforacyclicgroupofordernsinceKhasaprimitivren-throSotofunity(char(K)6u tjGj).SincezthedirectproSductandthedirectsumcoincideinAbwrecanusethefunda-menrtaltheoremfor niteabSeliangroupsGPUR԰n9=Cnq1 :::t0Cnt togetHom|(G;K2)PUR԰n9=Gfor anryabSeliangroupGwithchar(K)6u tjGj.sThusx^GPۢ԰=Gandύ1^31x^Gۢ=G.sInparticular(x)?=1Eforallx?2GEi ?=1.H+ByEthesymmetryofthesituationwregetthatthe]܍bilinearformh:;:iUR:xT^GDGn!1K2isnon-degenerate.+Jcffxff ̟ff ̎ ̄cffThrusjx^G DjUR=jGjhencedim(Kx^G D)=dim(K2G).8ThisprorvesPropSosition3.5.14.cffxff ̟ff ̎ ̄cffS\{&er5.pINTEGRALS!ANDF9OURIERTRANSFORMSeĿ83dDe nition3.5.18.}!ùLet$dHbSeaHopfalgebra.A$K-moduleMeHthatisarighrtHV-moSduleUbryUR:M1 HB\3!M9andUarightHV-comoSdulebyȄ:URM6!M1 H|iscalledaHopf35moffduleifthediagram?"⍍ԠM HԠzH :2fdzά-(73ԠԠ M H :2fdzά-mSqLM H H H8M H H H㌞32fdנά-n][1 I{ 1Ǡ@feǠ? L SjǠ@fe?`6x9 r㍹commrutes,ui.e.+vifhs2(mh)UR=Pm(M") hh(1) Xpm(1) \|h(2)holdsforallh2Handallm2M@.NObservreythatHfisanHopfmoSduleoveritself.FVurthermoreeachmoSduleoftheformV %'HisaHopfmoSdulebrytheinducedstructure.TMoregenerallythereisafunctorVecz3URV7!VG HB2Hopf-Mo`d@1ƹ-DnHV.Prop`osition3.5.19.Thetwofunctors-2coH+ֹ:URHopf-Mo`d@1-DdHB!gVec.and- F ?HB:Vec3URV7!VG HB2Hopf-Mo`d@1-DdH arffe35inverseequivalencesofeachother.Proof.@_De nenaturalisomorphismsn~w/ h:URM@ coH HB3m h7!mh2Mwithinrversemap`O  1]:URM63m7!Xm(M") hS׹(m(1) \|) m(2)2URM@ coH Haand. , :URV3vË7!v 12(VG HV) coH;withinrversemap(VG HV) coH83URv h7!vn9"(h)2V:ObrviouslyLthesehomomorphismsarenaturaltransformationsinMeandVp.^lFVur-thermore 7isahomomorphismofHV-moSdules.8 21Ziswrell-de nedsince'iʍRs2(Pm(M") hS׹(m(1)))@7=URPm(M") hS׹(m(3) \|) m(1)S׹(m(2))@7(sinceM+isaHopfmoSdule)@7=URPm(M") hS׹(m(2) \|) n9"(m(1))@7=URPm(M") hS׹(m(1) \|) 1(泍hencexP#m(M") hS׹(m(1) \|)Gg2M@2coH$a.FVurthermorex 21isahomomorphismofcomoSdulessincenʍ%s2 21 p (m)T=URs2(Pm(M") hS׹(m(1) \|) m(2))UR=Pm(M") hS׹(m(1)) m(2)$ m(3)T=URP 21 p (m(M") h) m(1)ι=UR( 21 1)s2(m):Finally 7and 21Zareinrversetoeacrhotherby`+D{  1 p (m)UR= (XUVm(M") hS׹(m(1) \|) m(2))UR=Xm(M") hS׹(m(1))m(2)ι=maandnʍ' 21 p (m h)n6=UR 21 p (mh)=Pm(M") hh(1) \|S׹(m(1)h(2)) m(2) \|h(3)n6=URPmh(1) \|S׹(h(2)) h(3)G$(brys2(m)UR=m 1){=URm h:Tm{&e844w3.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPS.dThrus 7and 21ZaremutuallyinversehomomorphismsofHopfmoSdules.The4qimageof isin(VyT HV)2coHbrys2(vK 1)=vK (1)=(vK 1) 1.:Hy&!Hhisanalgebraanrtihomomorphism,άthedualHV2 isanHV-moSduleinfourdi erenrtways:⍍ʍL1xh(fQ*URa);gn9i:=ha;gn9fGi;ޯXh(aUR(fG);gn9i:=ha;fGgn9i;L1xh(fQ+URa);gn9i:=ha;S׹(fG)gn9i;ޯXh(aUR)fG);gn9i:=ha;gn9S׹(fG)i:(3)zIfHis nitedimensionalthenHV2 aisaHopfalgebra.Theequalitryh(f*a);gn9i=ha;gn9fGiUR=Pha(1) \|;gn9iha(2);fGi꨹implies د(fQ*URa)=Xa(1) \|ha(2);fGi:(4)iAnalogouslywrehaveiخ(aUR(fG)=Xha(1) \|;fia(2):(5)CProp`osition3.5.20.LffetKH9bea nitedimensionalHopfalgebra.%ThenHV2 #isaright35HopfmoffduleoverHV.TProof.@_HV2 zisK aleftHV2Z-moSdulebryleftmultiplicationhenceby3.5.1arightHV-comoSdule,Nbrys2(a)UR=Pidb2RAia% bidڹ.mLet,Nf;gË2URHanda;bUR2HV2Z.mThe,N(left)mrultiplicationofHV2 satis es)r?abUR=Xb(H) ha;b(1) \|i:NWVeusetherighrtHV-moSdulestructureiP(aUR)fG)=Xa(1) \|hS׹(f);a(2)i:-onHV2 =URHom(HF:;K).NorwwechecktheHopfmoSdulepropertryV..ڍ 1. #If35H is nitedimensionalthentherffeexistsauniqueDiracs2-function. 2. #If35H isin nitedimensionalthentherffeexistsnoDiracs2-function.Proof.@_1. GSinceHLj3_f7!(f*UQR _)2HV2 4$isanisomorphismthereisaF2_HU^sucrhthat(/*XUQR Y)X=":Then(fG*XUQR Y)X=(fW*(*UQR Y))=(fW*")="(fG)"="(fG)(9q*?UQR @)gwhicrhimpliesf9q=?"(f)s2.FVurthermorewrehavehUQR;s2i?=hUQR;1HDs2i=h(Ȅ*URUQR US);1HDiUR="(1H)=1K.2.8is[Swreedler]exerciseV.4.؄cffxff ̟ff ̎ ̄cffvLemma3.5.26.mQLffetkHYbea nitedimensionalHopfalgebra.ThenUQR)2HV2 isaleft35inteffgrali 𩍍gva(XUVUQRUW*(1))\{ S׹(UQR*(2)\}))UR=(XUVUQRUW*(1) S׹(UQR*(2)\}))a(10)i y|&X|S׹(a)UQR*(1)% UQR *(2)\w=URXUQR*(1), aUQR*(2)(11)i UݟX3f(1) \|hUQR;f(2)iUR=hUQR;fGi1HD:(12)W{&er5.pINTEGRALS!ANDF9OURIERTRANSFORMSeĿ87dProof.@_LetꨟUQRQbSealeftinrtegral.8Thenr =XQa(1) \|UQR\}*(1)!c S׹(UQR*(2)\})S(a(2) \|)UR=X(aUQR)(1)$ S׹((aUQR)(2) \|)UR="(a)(XUVUQRUW*(1))\{ S׹(UQR*(2)\}));forallaUR2HV.8Hence$ٔ?:(PUQR*(1)$ S׹(UQR*(2)\}))a=URP"(a(1) \|)(UQR*(1)% S׹(UQR*(2)\}))a(2)y=URPa(1) \|UQR\}*(1)!c S׹(UQR*(2)\})S(a(2) \|)a(3)=URPa(1) \|UQR\}*(1)!c S׹(UQR*(2)\})"(a(2) \|)UR=a(PUQR*(1)$ S׹(UQR*(2)\})):',Conrverselya(PUQRUV*(1)!"(S׹(UQR*(2)')))C=(PUQRUV*(1)"(S׹(UQR*(2)')a))="(a)(PUQRUV*(1)"(S׹(UQR*(2)'))),^henceꨟUQR?=URPUQR*(1)%""(S׹(UQR*(2)'))isaleftinrtegral.SinceSisbijectivrethefollowingholds{QcPrS׹(a)UQR*(1)# UQR*(2)\w=URPS(a)UQR*(1) S21 S(S(UQR*(2)'))ye=URPUQR*(1)%" Sן21 S(S׹(UQR*(2)')S(a))UR=PUQR*(1) aUQR*(2)#:TheconrversefollowseasilyV.IfꨟUQR?2URInrtWşly(HV)isaleftinrtegralthenPUha;f(1) \|ihUQR;f(2)iUR=haUQR ;fGi=ha;1HDihUQR;fGi.Conrverselyif2HV2 !with(12)isgivrenthenha;fGiݹ=P@ha;f(1) \|ih;f(2)iݹ=ha;1HDih;fGi꨹henceaUR="(a). cffxff ̟ff ̎ ̄cffqIfGisa nitegroupthen s2(x)UR=z( 0ifx6=e;ɍ 1ifx=e:(13)"InfactsinceCOisleftinrvXariantwegetfG(x)s2(x)UR=f(e)s2(x)forallxUR2GandfQ2URK2G.SincejG/HV2 ܹ=KGisabasis,wregets2(x)=0ifx6=e.oFVurthermoreUQRjs2(x)dx=P x2G =s2(x)UR=1implies(e)UR=1.8SowrehaveȄ=URe2.RIf%Gisa niteAbSeliangroupwreget= P:2P ^G"Ϡforsome _2K. YTherevXaluationgivres1ZF=hUQR;s2i= P:x2G;2P ^G4?Eh;xi.ANorwletZF2wbG .Then푟P>2P ^G"h;xi=CP 2P ^Gh;xi=h;xiP2P ^G h;xi.Since'Wforeacrhx2Gnfeg'Wthereisasuchthat>h;xiUR6=1andwreget၍SX-K2P ^G`h;xiUR=jGje;x :#,HenceꨟPUx2G;2P ^G6`h;xiUR=jGj= 21Zand$UȄ=URjGj 1 zX- \z2P ^G :(14)$rLetYHbSe nitedimensionalfortherestofthissection."qInCorollary5.23wrehaveseen&thatthemapHM3f7!(UQR (fG)2HV2 Xis&anisomorphism.zZThismapwillbSecalledtheFourier35trffansform.XQ{&e884w3.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPS.dUTheorem3.5.27.wXThe35FouriertrffansformHB3URfQ7!V~e*f K2HV2 is35bijectivewithmVse*Jf=UR(UQR US(fG)=XhUQR*(1)\};fiUQR*(2)(15)mThe35inverseFouriertrffansformisde nedbyne]Caݖ=URXSן 1 S((1) \|)ha;(2)i:(16)Sincffe35thesemapsareinversesofeachotherthefollowingformulashold$ppGN hV)Re*f;gn9iUR=甆ZUTfG(x)g(x)dxUha;V*e*biUR=甆ZUTMXSן21 S(a)(x)b(x)dx*N fQ=URPSן21 S((1) \|)hV)Re*f;(2)iUaUR=PhUQR*(1)';)ea*iUQR*(2)#:(17)(`Proof.@_WVe:usetheisomorphismsH ' !G7HV2 de nedbryV!b*f+ :=VI|e* *fR= *(UQR +(fG)=U^P hUQR*(1)';fGiUQR*(2)͹andHV2  7!qHde nedbryba j:=UR(a*s2)=P(1) \|ha;(2)i.8Becauseof~}ha;V*b*biUR=ha;(b*s2)i=hab;s2i(18))and~hV)Re*f;gn9iUR=h(UQR US(fG);gn9i=hUQR;fGgn9i(19)wregetforallaUR2HV2 andfQ2H9 ha;UUObVvb*Lf _pi*=URhaV)Rb*f;s2i=Pha;(1) \|ihV)Rb*f;(2)iUR=Pha;(1) \|ihUQR;fG(2)i(bryLemma5.26)*=URPha;S׹(fG)(1) \|ihUQR;(2)iUR=ha;S׹(fG)ihUQR;s2iUR=ha;S(fG)i:%;This_&givresUU5bVb*If =S׹(fG).[SotheinversemapofH"|!>HV2 withVxb*fh=(UQR (fG)=VDe*fis_&HV2-P!HwithSן21 S(+ba+)&=P>Sן21((1) \|)ha;(2)i&=TQea '.Thenthegivreninversionformulasareclear.WVenoteforlateruseha;V*e*biUR=ha;Sן21 S(V*b*b)i=hSן21(a);V*b*bi=hSן21(a)b;s2i.2v-cffxff ̟ff ̎ ̄cff-XIfGisa nitegroupandHB=URK2G @thenmVsje*Jf=URXvfx2GfG(x)x:"U Since.(s2)UR=Px2G#,x215 <) 1+x2o2wherethex2V2K2G ƹarethedualbasistothex2G,wreget,w8e a7`=URXvfx2Gha;x ix :!1If )Gisa niteAbSeliangroupthenthegroupsGandwSbG\areisomorphicsotheFVouriertransforminducesalinearautomorphisme- *:URK2G t u!K2G @andwrehavemeaw߹=URjGj 1 zX- \z2P ^Gha;i 1Yà{&er5.pINTEGRALS!ANDF9OURIERTRANSFORMSeĿ89dBysubstitutingtheformrulasfortheintegralandtheDiracs2-function(2)and(14)wregetLVSe*Q_;f[4=URPx2G#,fG(x)x;W)eߖaQ=URjGj21 \zP'2P ^G.a()21 \|;㓍Q_;fQ=URjGj21 \zP'2P ^GV0Ae*.f5+4()21 \|;ߖaUR=Px2G#SWe#,a)-(x)x:(20)ThisimpliesVzAe*xDf*()UR=Xvfx2GfG(x)(x)=甆ZURf(x)(x)dx(21)Jwithinrversetransformee?aj(x)UR=jGj 1 zX- \z2P ^G (a) 1 \|(x):(22)"[pCorollary3.5.28.zThe`FouriertrffansformsoftheleftinvariantintegralsinHand35HV2 arffeVq*_e*qz=UR"ǟ 1s2HV !andQX3ePUQR\۹=12HF::(23)Proof.@_WVePharvehV%e*q;fGi.=hUQR;s2fGi=hUQR;ǟ21 C(fG)s2i="ǟ21(fG)hUQR;s2i="ǟ21(fG)_henceV'e* k=UR"ǟ21 C.8FVromnPe꨹1 =(UQR US(1)=UQR?wregetSeUQR=UR1.cffxff ̟ff ̎ ̄cff(Prop`osition3.5.29.De ne35aconrvolutionmultiplication35onHV2 byWxfchab;fGiUR:=Xha;Sן 1 S((1) \|)fGihb;(2)i:XThen35thefollowingtrffansformationruleholdsforf;gË2URHV:fcV 8%W!5GrJLjde nedbrySL(n)(A),Lthe8groupofn߮n-matrices8withcoSecienrtsinthecommutativealgebraAandwithdeterminanrt1,YGisrepresentedbythealgebraOUV(SL(n))%=SL(n)=K[xijJ]=(detQ(xij)1)i.e. ߍb~SLqz+(n)(A)PUR԰n9=K-cAlgo(K[xijJ]=(detQ(xij)1);A):SincebSL(n)(A)bhasagroupstructurebrythemultiplicationofmatrices,1therepresent-ingcommrutativealgebrahasaHopfalgebrastructurewiththediagonal=12hence:z(xikl)UR=Xxij xjvk ;\e{&e924w3.pHOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPS.dthecounit"(xijJ)UR=ij 'andtheanrtipSodeS׹(xij)UR=adjӹ(X)ij 'whereadj(X)istheadjoinrtmatrixofXFչ=UR(xijJ).8WVelearvethevreri cationofthesefactstothereader.fvWVenconsiderSLU(n)URMn=A2n-:2 Aasnasubspaceofthen22-dimensionalanespace.&䍍Example3.6.5.oQLetKMq(2)o=Kq qʍWa$dbcc$Xd*Zq2q==IןqI-detZq`qʍi=a20N920Zb20N920jvc20N920~̰d20N920$qz:(28)cInparticularwrehave(detQq溹)\1=detqB detQqjand"(detQq)=1.DThequanrtumdetermi-nanrtisagrouplikeelement(see3.5.13).Norwwede neanalgebra5-SLq(2)UR:=Mq(2)=(detQqb1):]{&e6.pQUANTUM!GR9OUPS^93dThealgebraSLq(2)represenrtsthefunctorrHZSLWq[(2)(A)UR=fqʍXa20\b20 cc20Βd20$͟q0w2Mq(2)(A)jdetQq渟qʍa203CVb20 JIc202Jd20;qG/=1g:YnThereisasurjectivrehomomorphismofalgebrasMq(2)[!-SLq(2)andSLq(2)isasubfunctorofMq(2).LetIBXJg;Y岹bSecommrutingquantummatricessatisfyingdetq/(X)=1=detӟqѹ(Yp).SincedetKZqX(X)detQq渹(Yp) ==det~q$|(XYp)forcommrutingquantummatriceswegetdetQq溹(XYp)=1,henceO{SL(q&(2)O{isaquanrtumsubmonoidofMq(2)andSLq(2)isabialgebrawithdiagonal"qʍ Vacb ccWd!Yq-F=URqʍ *ab cUd"pq- qʍ a9 b 9cd!q,Y;ands"qʍ Vacb ccWd!Yq-F=URqʍ *1 0 *0 1!ꢟq,:;TVoshorwthatSLq(2)hasanantipSodewe rstde neahomomorphismofalgebrasT:URMq(2)n!1Mq(2)2op ŹbryrpTğqʍ wa0)b Ucd"q.ɹ:=URqʍMd5qn9b *qn921 ʵc$ Mq(2)>xMq(2)[t]t{fd@ά->>MGq(2)񥔟{fdЍά-H0A`ğׁ @ğ @ğ @ğ @D>@D>RH#rǠ*FfeVǠ?H`Մׁ Մ Մ Մ >> 5NwithtUt?7!deteq&aqʍ#a207b20$c206d20?.qH<1V:ThrusGLɭq^(2)(A)isasubsetofMq(2)(A).ObservethatfpMq(2)URn!1GLq(2)isnotsurjectivre.Since lthequanrtumdeterminantpreservesproSductsandtheproductofinrvertibleelemenrtsAisagaininvertiblewegetGL(q(2)isaquantumsubmonoidofMq(2),hencetUR:GLq(2)n!1GLq(2) GLq(2)hwithqʍ Vacb ccWd!Yq-F=URqʍ *ab cUd"pq+ qʍ \a"b Ncdq+Uعand(t)UR=t t.fpWVeconstructtheanrtipSodeforGLq(2).8Wede neT:URMq(2)[t]n!1Mq(2)[t]2op Źbry {Tğqʍ wa0)b Ucd"q.ɹ:=URtqʍyd49qn9b Vqn921 ʵc; 1aH1şqnqandwTƹ(t):=detq< qʍ#ba3ob#c3