; TeX output 1999.10.27:2107K7 YRXQ cmr12CHAPTER3Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups/^o cmr975L*7 &e761:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYN cmbx122.MonoidalCategoriesFVorourfurtherinrvestigationsweneedageneralizedversionofthetensorproSductthatwrearegoingtointroSduceinthissection.iThiswillgiveusthepSossibilitytostudymoregeneralvrersionsofthenotionofalgebrasandrepresentations.rDe nition3.2.1.vaùA@ cmti12monoidal35cffategory(ortensor35category)consistsofacategory!!", cmsy10C5,acorvXariantfunctor UR:C]C4!wfC5,calledthetensor35prffoduct,anobjectg cmmi12IF2URC5,calledtheunit,naturalisomorphisms ;ʍjVD (A;B;Cܞ)UR:(A B) C14!A (BE Cܞ);jVD(A)UR:I+ A4!1A;jVD(A)UR:A IF4!A;calledeassoffciativity,Bleftfunitandrightunit,sucrhthatthefollowingdiagramscommute:A9K((A B) Cܞ) Dx(A (BE Cܞ)) Dol:2fd3O line10-̯qі2cmmi8 |{Ycmr8(A;Bd;C)"K cmsy8 1E[A ((BE Cܞ) DS)=̟:2fd3ά-̯ (A;Bd C;D| (A Bd;C;DInparticulartheJ8* msbm10K-moSdulesformamonoidalcategoryV.WThisisourmostimportanrtexampleofamonoidalcategoryV.2.Let9BǹbSeabialgebraandB-Mo`d#-bethecategoryofleftB-modules.WVede nethestructureofaB-moSduleonthetensorproductM N6=URM +ppmsbm8K cN+bry~ mBE M Ns6 1X. ;cmmi6M 1X.N:0!3B B M Ns61X.BX r 1X.N c!/B M B Nh6X.M X.NJ q!&MM Nasintheprevioussection.8SoB-Mo`d#ސisamonoidalcategorybry3.1.73. Let BibSeabialgebraandB-Como`d3bethecategoryofB-comodules. ThetensorproSductM N6=URM K cN+carriesthestructureofaB-comodulebry@ M Ns6X.M X.N (ځ!%BE M BE Ns61X.BX r 1X.N c!/B B M Ns6r 1X.M 1X.N:0!3B M N:asintheprevioussection.8SoB-Como`d2Visamonoidalcategorybry3.1.84.Let| GbSeamonoid.A|K-moSduletogetherwithafamilyofsubmodules(VgjgË2G)iscalledG-grffaded꨹ifV¹=URgI{2GFVg.LetiVtٹandWz/bSeG-gradedK-modules."A,homomorphismofK-modulesf2:V4!nWniscalledG-gradediffG(Vg)URWgforallgË2URG.-JTheG-gradedK-moSdulesandtheirhomomorphismsformthecategory(K-Mo`dX)2GofG-grffaded35K-modules.There7isamonoidalstructureon(K-Mo`dX)2G WgivrenbytheordinarytensorproSductV  *Wƹ.TheysubmoSdulesonthetensorproductV  *Waregivrenby(V  *Wƹ)g :=$u cmex10P h2G ZVh Wh#q% cmsy6Aacmr61 g=URPh;k62G;hk=gAVh Wk#.c5.8Achain35cffomplexofK-moSdulesʃM6=UR(:::'0x"@q3ЍuK!'ʟM2'0 m@q2ЍVj!M1'0 m@q1ЍVj!M0)consistsZofafamilyofafamilyofK-moSdulesMie4andafamilyofhomomorphisms@n O:GMn4!kMn16with5@n1@n=0.This5crhaincomplexisindexedbythemonoidN0..Onemmaryalsoconsidermoregeneralchaincomplexesindexedbyanarbitrarycyclic monoid.: ChaincomplexesindexedbryN0<Y8N0 arecalleddoublecomplexes.SoAmruchmoregeneralchaincomplexesmaybSeconsidered.=WVerestrictourselvestocrhaincomplexesoverN0.LetfvMZandNbSecrhaincomplexes.KAfWhomomorphismofchaincffomplexesfp :(M!NconsistsԦofafamilyofhomomorphismsofK-moSdulesfn:URMn4!lNn |sucrhԦthatfnP@n+1=UR@n+1fn+1otforalln2N0.Thecrhaincomplexeswiththesehomomorphismsformthecategoryofchaincom-plexesK-Comp'*.Ife@M$andNarecrhaincomplexesthenweformanewchaincomplexMP lNwith`(M5 sQN@)n H:=JL*Pn U_Pi=0"Mi+ Nni;and@:J(M5 N@)n H4!W͹(M N@)n1,givren`byN{7 &e781:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSY@(miq  Enni))I\:=(1)2id@M (mi) nni5+miq @(nni)).ThiszisoftencalledthetotalcomplexrassoSciatedwiththedoublecomplexofthetensorproductofMZVandN@.#ThenitiseasilycrheckedthatK-Comp+PisamonoidalcategorywiththistensorproSduct.aJProblem3.2.1.nR1.Prorve thatthecategory(K-Mo`dX)2G 6ofG-gradedK-moSdulesislequivXalenrttothecategoryKG-Como`d14ofKG-comoSdulesbythefollowingconstruc-tion.2IfVrisaG-gradedK-moSduletheVbSecomesaKG-comodulebrythemapȄ:URV4!nKGX Vp,#s2(vn9)UR:=g7 Xv6;forallvË2URVgandallg2URG.ConrverselyifV;Ȅ:V4!`KGX VisaKG-comoSdulethenVtogetherwiththesubmodulesVg*P:=URfvË2Vpjs2(vn9)=g VvgisaG-gradedK-moSdule.SinceKGisabialgebrathecategoryofKG-comoSdulesisamonoidalcategoryV.Shorw8thattheequivXalencede nedabSovebSetween(K-Mo`dX)2G WandKG-Como`d2pre-servresthetensorproSducts,hencethatitisamonoidalequivXalence.2.0LetBX=URKhx;yn9i=IÀwhereIisgeneratedbryx22;xy+xFyn9x.0ThenBmisabialgebrawith $thediagonal(yn9)UR=yRb )y,8(x)=x) 1+yRb x.The $counitis"(y)UR=1;"(x)=0.WVeinrtroSduced(thecooppositebialgebraof8)thisbialgebrainA.72.Shorw[AthatthecategoryK-Comp*ofchaincomplexesisequivXalenttothecategoryB-Como`d3qйof"B-comoSdulesbrythefollowingconstruction.OIfMisachaincomplexthende neaB-comoSduleonM6=URi2N֥MiwiththestructuremapȄ:M64!Bנ =yn92i1 Em20RAi1foralli2Nwreseethat@(midڹ)2Mi1AV.SowrehaveOlde ned@O:Mi k4!ωMi1AV.˙FVurthermorewreseefromthisequationthat@22g (midڹ)=0foralliUR2N.8Hencewrehaveobtainedachaincomplexfrom(M;s2).IfȪwreapply( 1)s2(m)=mȪthenwegetm=P-mi -withȪmi 2MihenceMչ=uLIi2N#Midڹ.This*togetherwiththeinrverse*constructionleadstotherequiredMequivXalence.)3.8AcoScrhaincomplexhastheform:~M6=UR(M0'0 m@q0ЍVj!M1'0 m@q1ЍVj!M2'0 m@q2ЍVj!:::.)O$7 &e2. %MONOID9AL!CA:TEGORIESq79Ywith u@i+1AV@ii=0.HShorwthatthecategoryK-Co`comp8XofcoSchaincomplexesisequiv-alenrttoComo`d.J-25PBwhereBiscrhosenasinexample5.ҍLemma3.2.5.g5QLffetbCbeamonoidalcategory.W#Thenthefollowingdiagramscom-mute+0(I+ A) BoxI+ (A B)SF:2fd0ά-Í]< OvA Bꃀq(A) 1X.B97?`@C7?`@M7?`@MV@MVRꃀ~(A Bd)|V?`rV?`hV?`h0h0 08(A B) I:8A (BE I)=џ:2fd0ά-Í(u eA Bꃀ|(A Bd)?`@?`@?`@@RꃀJ1X.A ^ (Bd)H?`>?`4?`33 1Kand35wehave(I)UR=(I).ҍ- cmcsc10Proof.@_First)wreobservethattheidentityfunctorIdCandthefunctorI/ լ- *areisomorphicSHbrythenaturalisomorphism.rThuswehaveIg fOf=gI gu=)gfOf=gn9.Inthefollorwingdiagramډvu&((I+ I) A) Bv`&(I+ (I A)) B}sH2fd)ά-om0 1vv7K&I+ ((I A) B) sH2fd)ά-pÍ sv ( 1) 1X8쟁 Qb8쟈Ql8쟏M^Qv8쟕Q|L0Q|L0ss(1 ) 1z̟ z̟z̟M^z̟l0l0+s1 ( 1)Oe̟ Ee̟;e̟M^1e̟+l0+l0+rӹ(I+ A) BI+ (A B):lA2fdG)ά-Í sK*Ǡ|z@feK\Ǡ?CcAE se*Ǡ|z@fee\Ǡ?jl1 &Π@feYܟΠ?Cc \ Π@feDܟΠ?iRX1rI+ (A B)I+ (A B):l:2fdG)ά-Y֯Z1u&(I+ I) (A B)7K&I+ (I (A B))}32fdά-Í ꃀv  (1 1)X8Ǡb8l8rLv8Ǣ|Lz|Lz3+G1 +l Q5l攴Q?l?^QIlQOe̟0QOe̟0sYallsubSdiagramscommruteexceptfortherighthandtrapSezoid.SinceallmorphismsareisomorphismstherighrthandtrapSezoidmustcommutealso.Hencethe rstdiagramoftheLemmacommrutes.Inasimilarwrayoneshorwsthecommutativityoftheseconddiagram.FVurthermorethefollorwingdiagramcommutesfj^pI+ (I I)b(I+ I) I\A2fdάÍ TI+ (I I)\A2fdά-Í Ԡ`I+ IWM{-1 F`@F`@F`@@R|GȚƖF`F`F` ԠRI+ IWM- 1F`@F`@F`@@R&1 $F`F`F` տFIuG?`@?`@?`@Ɩ@ƖRuG?`?`?` HerethelefthandtrianglecommrutesbythepreviouspropSertyV.UThecommuta-tivitryoftherighthanddiagramisgivenbytheaxiom.ThelowersquarecommutesP9Ԡ7 &e801:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYsince%isanaturaltransformation.Inparticular(1 )=(1 ).Since%isanisomorphismandI+ -P ԰ =KId!YC*wregetUR=.Єcffxff ̟ff ̎ ̄cffProblem3.2.2.nRFVoruBmorphismsf<:I~4!M&andgc4:I~4!N&inamonoidalcategorydwrede ne(f9 1:Nz4!iM2 N@):=(f9 1IM)(I)21'andd(1 gϹ:M4!nM N@)UR:=(1 gn9)(I)21 \|.8ShorwthatthediagramN׍N㚠M Nd32fd'8ά- Ġf 1bYIbYDM4{fd6"Ѝά-i9fH낟Ǡ*FfeǠ?'6gHaǠ*Ffe4Ǡ?q-F1 grcommrutes.De nition3.2.6.vaùLet(C5; )and(DUV; )bSemonoidalcategories.8Afunctor?Fc:URC4!wfDtogetherwithanaturaltransformationqs(M;N@)UR:F1(M) F1(N)UR4!1F(M N)andamorphism2}0V:URID 4!F1(ICm)8discalledweffakly35monoidalifthefollorwingdiagramscommuteB^덍}(F1(M@) F(N@)) F(Pƹ)tF1(M N@) F(Pƹ)L:2fd2ά-'t4O 1BH(F1((M N@) Pƹ)̟:2fd'8ά-')9U4:Ǡ@feUglǠ?Ǡ@fe>괟Ǡ??;C4rEMandhasaunitË:URIF4!AsucrhthatthefollowingdiagramcommutesXፍY"I+ APUR԰n:=APUR԰n:=A IYgA A苼{fd,cά-iidh HZǠ*Ffe⌟Ǡ?`xI{ idH(XǠ*Ffe(̟Ǡ?`->LrPA A"A:~32fd{ά-kݹ.rHk}XidÎ ҁ H͎ ׁ H׎ ܁ H H H H H H džH džjRZ7 &e821:3. %HOPF!ALGEBRAS,ALGEBRAIC,F9ORMAL,ANDQUANTUMGROUPSYLet#AandBE)bSealgebrasinC5.#^Amorphismofalgebrffas#fQ:URA4!1Bis#amorphisminCݹsucrhthat/[ikABu32fd6^ά- V{fY]/A AYkoBE Bo۟{fd@ά-inf fHm9Ǡ*FfemkǠ?\1rX.AHǠ*FfeHǠ?krX.BF*and[bY;BI'+X.A56ׁ 06 +6 &6 %n>%n> H'T*~X.BBoׁ AGo ALo AQo ARN>ARN>UAVjB(n32fd*ά- ;.f1commrute.TRemark3.2.8.j6ItisobrviousthatthecompSositionoftwomorphismsofalgebrasis0againamorphismofalgebras. Theidenrtity0alsoisamorphismofalgebras.ThruswreobtainthecategoryAlgo(C5)ofalgebrasinC.De nition3.2.9.vaùLetp@C#ubSeamonoidalcategoryV.ɨApcffoalgebrap@oracffomonoidinC?consistsofanobjectChntogetherwithacomrultiplicationg:A4!Aa AйthatiscoassoSciativreCw`CF CCF C C/32fdAJά-aid< YxCYnCF C5<{fdY?ά-ˍJNH纟Ǡ*FfeǠ?`SH V:Ǡ*Ffe lǠ?э; idy&ormoreprecisely>ٍd(CF Cܞ) C VCF (C Cܞ)+t32fdά-Í( aCF C32fd%9Ѝά-ׁ idԠmCԠCF CzD:2fdά-ζ .FŸǠ@fe.yǠ?񍍒id rbŸǠ@ferǠ??;wHt8鍹andhasacounit"UR:C14!I+sucrhthatthefollowingdiagramcommutesCw`YCY@>CF CKL{fd|z@ά-ˍHʟǠ*Ffe0Ǡ?`}i H! Ǡ*Ffe!<Ǡ?э&id/ xCF CKI+ CP1԰Jع=ܙCP1԰Jع=CF I:E32fd) ά-a} idHk}lA5rΟ>UH-Tf"X.DRnׁ Mn Hn Cn B.>B.> bYCbYVD)M{fd*(pά-i;NfSf7 &e2. %MONOID9AL!CA:TEGORIESq83Ycommrute.Remark3.2.10.qN6ItisobrviousthatthecompSositionoftwomorphismsofcoalge-brasisagainamorphismofcoalgebras.Theidenrtityalsoisamorphismofcoalgebras.ThrusweobtainthecategoryCoalg%z(C5)ofcoalgebrasinC.Remark3.2.11.qN6Observrethatthenotionsofbialgebra,4Hopfalgebra,andco-moSduleC$algebracannotbegeneralizedtoanarbitrarymonoidalcategorysincewreneed1mtoharve1manalgebrastructureonthetensorproSductoftrwo1malgebrasandthisrequires9&ustoinrterchange9&themiddletensorfactors.$ZTheseinrterchanges9&or ipsareknorwnFeunderthenamesymmetryV,]TquasisymmetryorbraidingandwillbSediscussedlateron.v2;7  +ppmsbm8* msbm10' msam10$u cmex10#q% cmsy6"K cmsy8!!", cmsy10 ;cmmi62cmmi8g cmmi12Aacmr6|{Ycmr8@ cmti12- cmcsc10o cmr9N cmbx12Nff cmbx12XQ cmr12O line10y