; TeX output 1999.11.03:0753+7 YRXQ cmr12CHAPTER2Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups/^o cmr943,*7 &e441:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSY1PN cmbx123.PAneAlgebraicGroupsWVeapplytheprecedingconsiderationstothecategories( msbm10K-cAlg"andK-cCoalg+z.ConsiderK-cAlgo,YthecategoryofcommrutativeK-algebras.ULetg cmmi12A;B8!", cmsy102K-cAlg.Thenw0A V B6isagainacommrutativeK-algebrawithcompSonentwisemultiplication.InHfactthisholdsalsofornon-commrutativeHK-algebras(A.5.3),5pbutinK-cAlg#wreharveProp`osition2.3.1.O+@ cmti12ThextensorprffoductxinK-35cAlg"]isthe(cffategorical)xcoproduct.,- cmcsc10Proof.@_Letf_2U`K-cAlgo(A;Zܞ);gÙ2K-cAlgo(B;Zܞ).8Themap[f;gn9]:A Bf4!_SZde nedObry[f;gn9](a$ b):=fG(a)g(b)Oistheuniquealgebrahomomorphismsucrhthat[f;gn9](a 1)UR=fG(a)and[f;gn9](1 b)UR=g(b)orsucrhthatthediagram? oԠAAԠ!hA Bt:2fd$0O line10-aM2cmmi8X.;cmmi6AԠԠ0B:2fd6άaMPZX.Bꃀf᤟?`@᤟?`@᤟?`@i@iRuGQLgb?`b?`b?`ڤڤ .ZoǠ@feآDǠ?bЍT|{Ycmr8[߱8fh;gI{]commrutes,whereA(a)UR=a 1andBN>(b)UR=1 b꨹arealgebrahomomorphisms.Scffxff ̟ff ̎ ̄cff(ÍSothecategoryK-cAlg"has nitecoproSductsandalsoaninitialobjectK.A'more( generalpropSertryofthetensorproductofarbitraryalgebraswrasalreadyconsideredin1.2.13.Observrethatthefollowingdiagramcommutes>%ԠnAԠȫ A A:2fd~`ά-͝;8qAacmr61ԠԠ`Ak̟:2fd~`쁠ά͝;xq2kS1X.A?`@?`@?`@Ɩ@ƖRkS~d1X.A?`?`?` gA؜*Ǡ@fe\Ǡ?bp݁ K cmsy8rwhereristhemrultiplicationofthealgebraandbythediagramthecoSdiagonalofthecoproSduct.De nition2.3.2.vaùAn/anealgebrffaicgroup/isagroupinthecategoryofcom-mrutativegeometricspaces.ByrthedualitrybSetweenthecategoriesofcommutativegeometricspacesandcom-mrutativealgebras, ananealgebraicgroupisrepresenrtedbyacogroupinthecate-goryofK-cAlg"ofcommrutativealgebras.FVoranarbitraryanealgebraicgroupHwregetbyCorollary2.2.7MUR=1j2V2K-cAlgo(HF:;H HV);E3:"UR=e2K-cAlgo(HF:;K);jand4CjS)=(id ʤ) 12K-cAlg(HF:;HV):|ThesemapsandCorollary2.2.7leadtoProp`osition2.3.3.OLffetxH>2QK-35cAlg.3HisarepresentingobjectforafunctorK-35cAlg":N!3 GrHIBif35andonlyifH isaHopfalgebrffa.-7 &egV3. %AFFINE!ALGEBRAICGR9OUPS45YProof.@_BothWstatemenrtsareequivXalenttotheexistenceofmorphismsinK-cAlggkUR:HB4!H H N":HB4!K S):HB4!Hsucrhthatthefollowingdiagramscommute?ԠۣHԠ,H Hd:2fd?ά-ζ łǠ@feǠ??;0T 胀(coassoSciativitry)>Ǡ@fe>괟Ǡ?ԏC4 1H H AH H Hv$32fd&H`ά-/h1 AR}ԠLHԠ7ȒH Hpl:2fd٠ά-ζ Ҏ"H H"bK HPB԰[=QHPB԰[=H K,32fd<ά-mŬv" 1S,<胀(counit)nǠ@feǠ??;IJǠ@feI|Ǡ?ꬽNy1 "i1JܟܔPJܟ PJܟ?_PJܟ攴PJܟ PJܟ?^PJܟPJܟPJܟ?]P㜟 P㜟 q>g lt胀(coinrverse))"iH)"BK:2fd)@ά-Í40 .OHsT:2fd.bά-(7NǠ@fe64Ǡ??;nX7JH H!H Hӳ32fdJsά-ׁESr} idaCidK S3qǠ@fe3?`6?;8W4r!y %cffxff ̟ff ̎ ̄cffThis#PropSositionsarystwothings.FirstofalleachcommutativeHopfalgebraHde nes=afunctorK-cAlgo(HF:;-):K-cAlg#4r4!5SetMw"that=factorsthroughthecategoryofjgroupsorsimplyafunctorK-cAlgo(HF:;-)U:K-cAlg"]4!5aGrEع.&Secondlyjeacrhrepre-senrtablefunctorK-cAlgo(HF:;-)d:K-cAlg"4!3\SetJ[thatfactorsthroughthecategoryofgroupsisrepresenrtedbyacommutativeHopfalgebra.Corollary2.3.4.sWA2nalgebrffaHB2URK-35cAlg"representsananealgebraicgroupifand35onlyifH isacffommutativeHopfalgebra.ThedRcffategoryofcommutativeHopfalgebrasisdualtothecategoryofanealge-brffaic35groups.InBthefollorwinglemmasweconsiderfunctorsrepresentedbycommutativealge-bras. They\Xde nefunctorsonthecategoryK-cAlg#ǹaswrellasmoregenerallyonK-Algo.)WVes rststudythefunctorsandtherepresenrtingalgebras.ThenwreusethemtoconstructcommrutativeHopfalgebras.Lemma2.3.5.g5QTheGfunctorGa X:T1K-35Alg9-!1AbHde neffdbyGaϹ(A):=A2+x,theunderlying~additivegrffoupofthealgebraA,sisarepresentablefunctorrepresentedbythe35algebrffaK[x]thepolynomialringinonevariablex.Proof.@_Ga ݹisanunderlyingfunctorthatforgetsthemrultiplicativestructureofthealgebraandonlypreservrestheadditivegroupofthealgebra.ʂWVehavetodeterminenaturaluisomorphisms(naturalinA)GaϹ(A)PUR԰n:=K-Algo(K[x];A).EacrhelementaUR2A2+isPmappSedtothehomomorphismofalgebrasa]:K[x]3p(x)7!p(a)2A.ThisPisahomomorphismۂofalgebrassincea(p(x)+qn9(x))UR=p(a)+q(a)UR=a(p(x))+a(qn9(x)).g7 &e461:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYand9a(p(x)qn9(x))R3=p(a)q(a)=a(p(x))a(q(x)).Another9reasontoseethisisthatK[x]$isthefree(commrutative)$K-algebraorver$fxgi.e.esinceeacrhmapfxgl4!AcanuTbSeuniquelyextendedtoahomomorphismofalgebrasK[x]4!A. ThemapAT3a7!a eX2K-Algo(K[x];A)isbijectivrewiththeinversemapK-Algo(K[x];A)T3fQ7!URfG(x)2A.8FinallythismapisnaturalinAsinceL@MB̌K-Algo(K[x];B)J\32fd5Ƞά-|5-V!q% cmsy6HAH{K-Algo(K[x];A),{fd6}ά-5-VHʟǠ*FfeǠ?' ~gH Ǡ*Ffe<Ǡ?`tK-Alg(K[*x]\t;gI{)`commrutesforallgË2URK-Algo(A;B).Ʉcffxff ̟ff ̎ ̄cff덍Remark2.3.6.j6SinceAA2+ ^3hasthestructureofanadditivregroupthesetsofho-momorphismsofalgebrasK-Algo(K[x];A)arealsoadditivregroups.Lemma2.3.7.g5QThefunctorGm Z=URU6:K-35Alg:N!- GrAde neffdbyGmĹ(A)UR:=U@(A),the[\underlyingmultiplicffativegroupofunitsofthealgebraA,isarepresentablefunctorrffepresentedVubythealgebrffaK[x;x21 \|]UR=K[x;yn9]=(xy.W1)VutheringofLffaurentVupolynomialsin35onevariablex.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.FVurthermoreitisnaturalinAsinceL@AQ}`BAQbK-Algo(K[x;x21 \|];B)Ԟ32fd*(pά-|5-Ȋ~'AȊGK-Algo(K[x;x21 \|];A)l{fd*Ѝά-5-H[BǠ*FfetǠ?'ygH Ǡ*Ffe>Ǡ?4K-v2Algp(K[*x;x-:1 ] N4;gI{)`forallgË2URK-Algo(A;B)commrute.Ccffxff ̟ff ̎ ̄cffRemark2.3.8.j6SincedU@(A)hasthestructureofa(mrultiplicative)dgroupthesetsK-Algo(K[x;x21 \|];A)arealsogroups.Lemma2.3.9.g5QThefunctorMn rg:K-35Alg!0ߗK-35Alg~withMnP(A)thealgebrffaofnn-matricffes-VwithentriesinAisrepresentablebythealgebraKhx11 ;x12;:::ʚ;xnn Рi,the35noncffommutativepolynomialringinthevariablesxijJ.Proof.@_TheMcpSolynomialringKhxijJiisfreeorverMcthesetfxijginthecategoryof(nontpcommrutative)algebras,i.e.7foreachalgebraandforeachmapfֹ:?fxijJg4!;A/"K7 &egV3. %AFFINE!ALGEBRAICGR9OUPS47YtheresEexistsauniquehomomorphismofalgebrasg:=Khx11 ;x12;:::ʚ;xnn Рi=4!CAsEsucrhthatthediagramD>fxijJg>4KhxijJi?{fdά-lH`}f<ׁ @< @< @< @M>@M>RHARǠ*FfeǠ?'8g commrutes.t SoSeachmatrixinMnP(A)de nesauniqueahomomorphismofalgebrasKhx11 ;x12;:::ʚ;xnn РiUR4!1A꨹andconrverselyV.cffxff ̟ff ̎ ̄cffOVExample2.3.10.uQ1.8Theanealgebraicgroupcalledadditive35grffoupq=wGaY!:URK-cAlg!4!3`Abwith|GaϹ(A)Nf:=A2+ vfromLemma2.3.5isrepresenrtedbytheHopfalgebraK[x].WVedeterminecomrultiplication,counit,andantipSode.BymCorollary2.2.7thecomrultiplicationis=1q2 2K-cAlgo(K[x];K[x] K[x])PUR԰n:=GaϹ(K[x] K[x]).8Henceq=x((x)UR=1(x)+2(x)UR=x 1+1 x:Thecounitis"UR=e)ppmsbm8K 4=02K-cAlgo(K[x];K)P԰n:=GaϹ(K)henceD"(x)UR=0:ùTheanrtipSodeisS)=URidW 1؍ K[x] 2URK-cAlgo(K[x];K[x])P԰n:=GaϹ(K[x])hencelf[S׹(x)UR=x:2.8Theanealgebraicgroupcalledmultiplicffative35groupXGm Z:URK-cAlg!4!3`AbwithXGmĹ(A):=A2 =U@(A)fromLemma2.3.7isrepresenrtedbytheHopfalgebraK[x;x21 \|]UR=K[x;yn9]=(xy1).8WVedeterminecomrultiplication,counit,andantipSode.ByCorollary2.2.7thecomrultiplicationisq=uUR=1j2V2K-cAlgo(K[x;x 1 \|];K[x;x 1] K[x;x 1])PUR԰n:=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԰n:=GmĹ(K)henceD"(x)UR=1:ùTheJzanrtipSodeisS_=)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!4!3`AbEg;047 &e481:2. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYwithjM2+RAnx(A)theadditivregroupofnn-matricesjwithcoSecientsinAisrepresentedbry2thecommutativealgebraM2@+RAn -=2K[xijJj1i;j}n]2(Lemma2.3.9).qThisalgebramrustbSeaHopfalgebra.ThecomrultiplicationisUR=12V2K-cAlgo(M2@+RAn]\;M2@+RAn M M2@+RAn)PUR԰n:=M2+RAnx(M2@+RAn M M2@+RAn).Hence0e(xijJ)UR=1(xij)+2(xij)UR=xij 1+1 xij:qbThecounitis"UR=eK 4=(0)2K-cAlgo(M2@+RAn]\;K)P԰n:=M2+RAnx(K)hence䕍"(xijJ)UR=0:TheanrtipSodeisS)=URidW 1 ,썑 Mi@"+Ln 2URK-cAlgo(M2@+RAn]\;M2@+RAn)PUR԰n:=M2+RAnx(M2@+RAn)hence+dwS׹(xijJ)UR=xij:4. ThematrixalgebraMnP(A)alsohasanoncommrutativemultiplication,thema-trixRmrultiplication, =de ningamonoidstructureM2RAnx(A).ThusK[xijJ]carriesanothercoalgebraOstructurewhicrhde nesabialgebraM2@RAn V='K[xijJ].ObviouslythereisnoanrtipSode.ThecomrultiplicationisUR=12V2K-cAlgo(M2@RAn]\;M2@RAn M M2@RAn)PUR԰n:=M2RAnx(M2@RAn M M2@RAn).Hence((xijJ))UR=1((xij))2((xij))UR=(xij) (xij)or8(xikl)UR="u cmex10X ㇍ jxij xjvk :!Thecounitis"UR=eK 4=E i2K-cAlgo(M2@RAn]\;K)P԰n:=M2RAnx(K)hence䕍?"(xijJ)UR=ij:5.RLet#KbSea eldofcrharacteristicp.ThealgebraH>=K[x]=(x2p])carriesthestructureofaHopfalgebrawith(x)X=xk 1+1 x,("(x)X=0,andS׹(x)=x.TVohshorwthatiswellde nedwehavetoshow(x)2p=,)0.Butthisisobviousbythe rulesforcomputingp-thpSorwers incrharacteristicp.WVehave(xj 1+1 x)2pX=x2pr 1+1 x2p=UR0.ThrusthealgebraHrepresentsananealgebraicgroup:䕍k( p](A)UR:=K-cAlgo(HF:;A)P԰n:=fa2Aja p=0g:TheKGgroupmrultiplicationistheadditionofp-nilpSotentelements. ZSowehavethegrffoup35ofp-nilpotentelements.6.The¥algebraHL=K[x]=(x2n=1)isaHopfalgebrawiththecomrultiplication(x)=x߸ x,Lthe8counit"(x)=1,and8theanrtipSode8S׹(x)=x2n1̹."Thesemapsarewrellde nedsincewehaveforexample(x)2n=UR(x+ x)2n=URx2n& x2n=UR1 1.$.ObservrethatthisHopfalgebraisisomorphictothegroupalgebraKCn 5SofthecyclicgroupCnofordern.ThrusthealgebraHrepresentsananealgebraicgroup:䕍j{nP(A)UR:=K-cAlgo(HF:;A)P԰n:=fa2Aja n=1g;1B7 &egV3. %AFFINE!ALGEBRAICGR9OUPS49Ythat)FisthegrffoupXofn-throotsofunity.The)FgroupmrultiplicationistheordinarymrultiplicationofroSotsofunityV.7.ThelineargroupsormatrixgroupsGLj(n)(A),@-SLڹ(n)(A)andothersucrhgroupsarePfurtherexamplesofanealgebraicgroups.kWVewilldiscusstheminthesectiononquanrtumgroups.Problem2.3.1.nR1.8Theconstructionofthegenerallineargroupef,GLw!(n)(A)UR=f(aijJ)2MnP(A)j(aij)inrvertible32gde nesananealgebraicgroup.8DescribSetherepresenrtingHopfalgebra.2.`\The|spSeciallineargroupSLM)(n)(A)isananealgebraicgroup.WhatistherepresenrtingHopfalgebra?3.TherealunitcircleS21(R)carrythestructureofagroupbrytheadditionofangles.!IsitpSossibletomakreS21¹withtheanealgebraK[c;s]=(s22 +c221)inrtoananealgebraicgroup?(Hinrt:{wHowcanyrouaddtwopSoints(x1;y1)and(x2;y2)ontheunitcircle,sucrhthatyougettheadditionoftheassoSciatedangles?)S;7  ,- cmcsc10+@ cmti12)ppmsbm8( msbm10"u cmex10!q% cmsy6 K cmsy8!", cmsy10;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8o cmr9N cmbx12Nff cmbx12XQ cmr12O line10Yw