; TeX output 1994.06.08:1148|7 N_*N cmbx12OnBraidingandDyslexia K`y cmr10BoGdoUUPareigis Ǎ$+- cmcsc10Abstract.[BraidedmonoidalcategorieshaveimpGortantapplicationsinknot $theory*,/algebraicXquantum eldtheory,/andthetheoryofquantumgroupsand$HopfUUalgebras.qW*ewillconstructanewclassofbraidedmonoidalcategories.0TypicalZexamplesofbraidedmonoidalcategoriesarethecategoryofmoGdules$over cmmi10A$inzsuchacategory*. Thecategoryof(leftand/orright)A-moGduleswiththetensor$proGductoverAisagainamonoidalcategorywhichisnotnecessarilybraided.=LHow-$ever,tifformabraidedcategoryV,ormoSdulesorver>acommrutative>algebraAwillnotformalbraidedcategoryV.+ThisisremediedbryanewconceptofdyslecticmoSdulesoveracommrutativealgebraA. WVebwillshorwthatthecategoryofdyslecticmoSdulesoveracommutativealgebraA,i.e.righrtmoSdulesthatdonot"see"ifthemultiplicationhasbSeentwistedaroundthemoSdulevia(atrwofoldapplicationof8)thebraidingornot,isabraidedmonoidalcategorywithtensorproSductM׉ 2cmmi8A tN!takrenoverA.5FVurthermorethiscategoryisacore exivresubScategoryofallA-modules.ύf1.%K ,- cmcsc10Modulesoveracommut32ativealgebrainabraidedmonoidalÜca32tegorya AsbasecategorywreconsideracoScompletebraidedmonoidalcategoryC@suchthattheL'tensorproSductpreservresarbitrarycolimitsinbothvXariables. UsingcoherencewremaryassumewithoutlossofgeneralitythatCJisastrictmonoidalcategoryV.AbraidingofEamonoidalcategory(C5; %:C2FCZ !Cܞ)EconsistsofanaturalisomorphismofbifunctorsMa>;NV:URM N6 !N M+sucrhthatI%zQ]Ma>;N"K cmsy8 P#Z=UR(id ʤN Ma>;PL)(Ma>;Nl id uLP)andI%xM" Na>;P#Z=UR(Ma>;P id uLN)(id ʤM< Na>;P):I%WVe'donotrequireNa>;MMa>;NV=URid M" N$٢.Ifthisalsoholds,NtheniscalledasymmetryforC5.Observrethatthe rsttwoconditionsgeneraterepresentationsofthebraidgroupsonobjectsMpL /hM@:::: M@;ifinadditionthelastconditionholdsthenwreharverepresentationsofthesymmetricgroups. TVouobtainanexampleofsucrhacategoryonecanstartwithacoScompletesymmetricmonoidal ^categoryMsucrhthatthetensorproSductpreservesarbitrarycolimits.AsanAexampleconsiderM^*=K-MoSd,'theAcategoryofK-modulesorverAacommrutativeringkK. /(Thereadermaryassumethroughout,3thatM=K-MoSd.Otherwisekheshouldmviewthecalculationsascalculationswithgeneralizedelemenrtsinthesenseof[9].) LetHbSeaHopfalgebrainMwiththestructuremorphismsHD;"H;rH;H;SH.ThenqthecategoryofrighrtHV-comoSdulesM2H εisknowntobSeamonoidalcategorywithtensorproSductasinM.]=ObservrethatcolimitsinM2H  existandareformedinMuwithauniquelyde nedsuitablecomoSdulestructure.;IfwreassumefurthermorethatH 1is-}h! cmsl12coSquasitriangularorbraided,)gthenM2H 6isbraided[14 ].xAllinallwrehavea!coScompletebraidedmonoidalcategoryM2H esucrhthatthetensorproductpreservresarbitrarycolimits. 3|7 ӰOnUUBraidingandDyslexia3N ManryyexamplesofbraidedmonoidalcategoriescanbSefoundintheliteraturee.g.[3,6 ʤ,7].Ѝ InthebasecategoryC2wreconsidernowanalgebraAandthecategoryofrightA-moSdulesCA. FVorkthespSecialexampleC=URM2HD,DwithHacoquasitriangularHopfalgebrainM,wre6takeAanHV-comoSdulealgebrainMwithstructuremorphismsrA 36:URA;$ A !AandKA :A !Ai HV,$s2(a)=u cmex10P2a|{Ycmr8(0)& ia(1) \|.WVestudythecategoryofrighrt(HF:;A)-Hopf-moSdulesM2HRAA XwithstructuremorphismsM B:URM6 !M{d :HjandM:M{d :A !M@,i.e.8ofrighrtHV-comoSdulesandrightA-moSdulesM+suchthatM (ma)UR=Xm(0) \|a(0)$ m(1)a(1):!)TheyaretherighrtA-moSdulesinthebasecategoryM2HD. Since7tensorproSductsinCpreservrecolimits,[qCA iscocompletewithcolimitsformedinCwithauniquelyde nedsuitablemoSdulestructure.!𰍍.@ cmti12Rffemark}1.1.DjTheJcategoryA b.CAofJ(A;A)-bimoSdulesinC7isamonoidalcategorywiththetensorproSductM A N+thecokrernelofhvM A N^@x3„fd#O line10-؎@x32fd#-2YM N6 !URM A N: TVoshorwthisonecanapplyargumentssimilarto[10 ],!K1.8and1.10,inparticularthefact\thatthetensorproSductisrighrtexact, toprovetheassoSciativityM AP(N AP)PUR԰n:=(M A N@) APS:Ѝ Since۠colimitspreservrecolimits(inourcasethetensorproSductoverA)thecategoryACA ȌiscoScompleteandthetensorproductpreservresarbitrarycolimits. AssumenorwthatAiscommutativeinC5,i.e.8thefollowingdiagramcommutesB8۟9?A A۟9A Af8҄fd)썑ά- Սm\ӘgAr@@؟@؟R/`r88 !𰍍Rffemark}1.2.DjLetM62URCA.8ThenitiseasytoseethatthemorphismqM B:URA M dg 6 !@M AKX.;cmmi6M !Mde nes@an(A;A)-bimoSdulestructureonM@.:Thecompatibilitryoftheleftandrightv|7 4BoGdoUUPareigisNA-structuresfollorwforexamplefromthecommutativediagramz\*yA M AiM A AABfd"aά-QI{ 1/!M AABfd.,ά-ύ X.M 1Ѡ#qM A AѠ)M A̴::fd.,ά-p1 rgAA M1M A32fd9؍ά-kY;Mz32fdEfά-q%nJX.MlVX.A;Mq% cmsy6 A򂟭 @򂟷 @ @r@rR3ɍF1 j j j zz [>1 rr @r @r @b@bRQFX.M 1z @z @z @j@jRK/ 6X.Mj @ j @*j @0Z@0ZRyLUfey~?Ս^c1 X.MA,UfeA^?ύEX.M^ Then$categoryofrighrtA-moSdulesCA LthuscanbSeviewedasafullsubScategoryofA-A-bimoSdulesA CA.XActuallythisispossibleintrwodistinctwrays,namelybyde ningmthekleftstructurebryA M g p !MB AK P/ %!MorkbyA MI{-:Aacmr61g p !MB AK P/ %!M@.SoHwregetthetwofullembSeddingsl:PCA 4'!A 4CA %andHr:CA'!ACA.QWVeHshallrestrictourconsiderationstolw:URCA 36!A 36CA de nedbrytheleftactionasinremark1.2.ϸ WVenorwinvestigatethetensorproSductoverAinA CA.;Aistraightforwardcalculationgivres-Lemma1.3.IULffetM;N62URCA.UThentheleftmodulestructureonM} A NBasde nedin1.2. cffoincideswiththeinducedleftmodulestructureofM A } NninA CA,/henceCAis%amonoidalcffategory%withtensorprffoduct%overAandlw:URCA 36!A 36CA de nes%afullmonoidal35embffedding. WVe27seethatforacommrutative27algebraAthecategoryCA isafullmonoidalsub-categoryofA ȌCA.8Actuallywrehavemore.Prop`osition1.4.bIfBAiscffommutativeinCthenCA isacocompletemonoidalcate-goryaTsuchthatthetensorprffoductaTM A N8preservesarbitrarycolimitsinbothvari-ables.!Uk2.{DfyslecticalgebrasandmodulesϸDe nition2.1.YLetCbSeasbefore.8WVecallanalgebraAinCdyslectic[4]if?q۟9?A A۟9A Af8҄fd)썑ά-㍒X I{-:2ӘgAr@@؟@؟R/`r88 NcommrutesorequivXalentlyif(rË:URA A !A)=(rn921 :A A !A).)o|7 ӰOnUUBraidingandDyslexia5N AmoSduleM+inCA Ȍiscalleddyslecticifthefollorwingdiagramcommutes>%۟9WyM A۟9L9M A`Ÿ8҄fd%ά-㍒ѶCI{-:2M?b@b@_@_RYrYr]] U:Y AcommrutativealgebraAisclearlydyslectic.EHorwever,/notallA-moSdulesorvera8bcommrutativealgebraAaredyslectic." InfactthecategoryofdyslecticA-moSdulesdys]CA ȌistheequalizerofthetrwoembSeddingfunctorsl!;r:URCA 36!A 36CA.荑 FVoracommrutativealgebraAthereisabraidingmorphismforthetensorproSduct.Prop`osition2.2.bLffetAbecommutativeandM62URCA nbedyslectic.\Thenthefollow-ing35isacffommutativediagramofdi erencecokernelsk5ٍ2M A NٍtM N s8֟?&fdRά-X.M 1%%M N As8֟=fd ά-{1 %%*Ɵ=fd,`(ά-#gE1 X.N>s>sRzM A N.kL>rfd!ά-u<M2N A MMtN Ms8֞32fdRά-ۍLX.N2 1N M As8֟4fd ά-D{1 *Ɵ4fd,`(ά-'ᰄ1 X.M3玎3獒RzN A M-kp3fd"ά-鍒<\ @2M N AQW.Ά|feQ`Ά?=u1 ՅQW.|feQ`?,[X.MB";N A @;ݍm @i/Ο;N" A (1 n9)=(1 ND)(1 )commrutesIbyfunctorialityofn9.FVurthermoreifM\-isdyslecticthenthefollowingdiagramcommruteseS9ٍAZ`M A NٍO M N A2>rfd ԍά-NǍ ^1  @獑`xX.M A;Nnğ DHxğ DHğ DHğ DHğ DHtlHtlj @獒e\X.MB";N A DH DH DH DH DH4lH4lj @_݊Afe`?mDX.M 1 @JXfe|?1I{ 1۟9O N M A۟9CN A Mѳ8҄fd ԍά-I'1 ۟9۟918N M A!8҄fd ԍά-I'1 JXfe|?ϝ1 X.MOʟXfeO?ϝSl1 X.MM(M N N Mu32fd$.ά-5x;NlinsteadofMa>;N.8AsimilarproSofasforProposition2.2givres:QW LetAbSecommrutativeandNY2uCA bedyslectic.ɤThenn91cMa>;Ninducesamorphismonthedi erencecokrernels:#es8:URM A N6 !N A M@.NProp`osition2.4.bIf35MtandNinCA arffedyslecticthensoisM A N@.9R|7 6BoGdoUUPareigisNPrffoof.#RWVeharvetoshowthatKO(M A N@) AA(M A N@) Aϴ:::fdDά-KX I{-:2ѰgMKv @ @ @؟@؟RK.П П П acommrutes.8SincethefollowingdiagramcommutesO͡1͡1ᣟ:ʄfd ά-ۍRI{-:2Ϡ鍑]|M N AϠ鍒nM N AѠѠᣟ::fd ά-O #1Ϡ鎎Ϡ鍒1yM Nӓ:fd,`(ά-Ȉ 1 X.N3„fdDά- ӍRI{-:2HU|(M A N@) AHn (M A N@) A32fdDά-G #1HH-M A N s3zfd ƍά-+ ןX.M A ^NPH{D#fe|3-D?- 2 1PHD#fe%D?- $ 1PHC۟D#feD D?漯H} andtheleftmost B 1isanepimorphismitsucestoshorwthat(1 ND)n922 =(16 ND).4ObservreްthattensorproSductspreservedi erencecokernels.4Ifweexpandn922 wregetacommutativediagramAEǢDZfdlά-TI1 ybM N Ay_V(M A N(11EǢCʄfdliάI1 pu41 X.N.Q8e8QBQLQVe6Q`Qi Qi syyIA M NDfdlά-TgrI{ 1puRX.M 1 Q e8Q Q Q e6Q QÁV QÁV syythecenrterofamonoidalcategory(weowethisremarktothereferee).TSinceeachobjectM2[dysCA ccomeswithanaturaltransformationa(M@)[:M) A )-Ѭ'!- A )Mof&functorsonCA ` asde nedin2.2thecategoryofdyslecticmoSdulesisalsoabraidedmonoidalsubScategoryofthecenrterofCA jinthesenseofb[5]. Unlikethecenter,horwever,itisafullsubScategoryofCA.!Sύ3.}?Cofreedyslecticmodules& TheRpurpSoseofthissectionistoshorwthattherearemanyexamplesofdyslecticmoSdules.uTVothisendwrehavetostudyinnerhom-functors.uSoweassumenowthatthecoScompletebraidedmonoidalbasecategoryCHhasdi erencekrernels(equalizers)andisrighrtclosed,@i.e.l'thereisarightadjointfunctor[M;-]&(:C] .!Cforeveryfunctor"tensorproSductwithM+ontherighrt"-  M6:URC ]!C5.& TVoA getexamplesofsucrhcategorieswestart,asinsection1,withasymmetricmonoidal~CcategoryM(whicrhiscoScompletesuchthatthetensorproSductpreservesarbitrarycolimits).8AssumethatMisclosedandhasdi erencekrernels. IfLyH9isaHopfalgebrainMandhasadual(see[14 ]Chap.^R2)thenwrecallHa niteHopfalgebra.o{Lemma3.1.IUIf35H isa niteHopfalgebrffathenM2H Lyisrightclosed.Prffoof.#RLetꨟPxh2RAij hiObSeadualbasisforH(withPh2RAi(h)hi,=URh). FVorN;P2URM2H de nethestructureofanHV-comoSduleonHomd1(N;P)bryBȄ:URHom(N;P)UR !URHom(N;P)t+ HF:; s2(fG)UR=Xf((0) \|)(0) Чt+h ڍi(f((0))(1)S((1)))t+ hid:\c|7 8BoGdoUUPareigisN ThenGDthecanonicalmorphismM(M2 FNN;P)PZp԰sX= 7M(M;Homy(N;P))GDgivrenbyfG(m n)UR=gn9(m)(n)restrictstoe0pMM HD(M N;P)PUR԰n:=M H(M;Homy(N;P))sinceletf2satisfyPxfG(m(0)$ n(0) \|) m(1)n(1)=URPfG(m n)(0)$ f(m n)(1) \|.8ThenKCʍlP)gn9(m(0) \|)(n) m(1)d=PXfG(m(0)$ n) m(1)d=PXfG(m(0)$ n(0) \|) m(1)"(n(1))d=PXfG(m(0)$ n(0) \|) m(1)n(1)S(n(2))d=PXfG(m n(0) \|)(0)$ f(m n(0) \|)(1)S(n(1))d=PXfG(m n(0) \|)(0)$ h2RAi(f(m n(0) \|)(1)S(n(1)))hid=PXfG(m n(0) \|)(0)$h2RAi(f(m n(0) \|)(1)S(n(1))) hid=PXgn9(m)(n(0) \|)(0)$h2RAi(g(m)(n(0))(1)S(n(1))) hid=PXgn9(m)(0) \|(n) g(m)(1) \|:IInasimilarwrayoneshorwsthattheinversemapalsorestrictstomorphismsinM2HD. yff٘ ̍ ff ̄ ffffff٘KcRffemark}3.2.DjIngeneralM2H willnotbSeleftclosed.Kc IfMUR=K-VVekthenM2H 6Zhaskrernels.ZSofora nitecoSquasitriangularHopfalgebraHGNorverYa eldKthecategoryofHV-comoSdulesM2H s;A:=URA;M>HMa>;Ad.)Lemma3.3.IUKis35anA-submoffduleofM@. m|7 ӰOnUUBraidingandDyslexia9NPrffoof.#RWVeharveeZꍍh([A;M ]n9M(K j 1A) 1A)=UR([A;M ] 1A)(M . 1A)(K j 1A 1A)=URM (M . 1A)(K j 1A 1A)=URM (1A r)(K j 1A 1A)=URM (1A r)(1M . A;A<)(K j 1A 1A)sinceAiscommrutative=URM (M . 1A)(K j 1A 1A)(1M . A;A<)=URM (M . 1A)(2n92RAMa>;Ap 1A)(K j 1A 1A)(1M . A;A<)(bryde nitionoff(K;)Rv=URM (1A r)(1M . A;A<)(2n92RAMa>;Ap 1A)(1M A;A<)(K j 1A 1A)=URM (M . 1A)2n92RAM" A;A8(K j 1A 1A)(propSertryofQn9)=URM 2n92RAMa>;Ad(M . 1A)(K j 1A 1A)=UR([A;M 2n92RAMa>;Ad] 1A)(M . 1A)(K j 1A 1A)=UR([A;M 2n92RAMa>;Ad]n9M(K j 1A) 1A)f7whicrhimplies[A;M ]n9M(K : 1A)UR=[A;M 2n92RAMa>;Ad]n9M(K : 1A):Sothereisauniquemorphism]K :URKӝ A !KsucrhthatK;K =M (K 2 1A),@sinceK Qisthedi erencekrernelof[A;M ]Xand[A;M 2n92RAMa>;Ad]n9. yff٘ ̍ ff ̄ ffffff٘=Lemma3.4.IUKis35dysleffctic.XPrffoof.#RWVeharveK;K2n92RAK;A҈=URM (K q>1A)2n92RAK;A=M 2n92RAMa>;Ad(K q>1A)=([A;M2n92RAMa>;Ad]w M1A)(K 1A)f=([A;M 2n92RAMa>;Ad]n9K 1A)=([A;M ]n9K 1A)=M (K 1A)=K;K. yff٘ ̍ ff ̄ ffffff٘Lemma3.5.IUIfZP@H2dysCA 8isdysleffctic,dMf2CAandf:P@H !ManZA-homomor-phism,35thenf{4factorsuniquelythrffoughK :URK1 I!M@.XPrffoof.#RWVeharvetoshowthat[A;M ]n9fQ=UR[A;M2n92RAMa>;Ad]n9fG.8Thisfollorwsfrom(Q([A;M ]n9f 1A)UR=([A;M ] 1A)(f 1A)UR=M (f 1A)=fGP g=fP2n92RAP;Ad(f 1A)=([A;M2n92RAMa>;Ad] 1A)(f 1A)=UR([A;M 2n92RAMa>;Ad]n9f 1A): yff٘ ̍ ff ̄ ffffff٘&BTheorem3.6.S8LffetSCbeasinTheorem9k2.5,[berightclosedandhavedi erenceker-nels.NLffettAbeacommutativealgebrainC5.NThenthecategoryofdyslecticA-modulesdys]CA is35acffore exive35subcategoryofCA.Prffoof.#RWVe harvetoshowthattheconstructionofKasinthepreviousLemmasde nesarighrtadjointfunctortotheembSeddingofdysCA yeintoCA.~ButthisisdemonstratedbrytheuniversalpropSertygivenin3.5. yff٘ ̍ ff ̄ ffffff٘= WVeJvremark,jthatwreonlyneededarightadjointfunctor[A;-]for- ctAintheproSof. ~ɠ|7 10BoGdoUUPareigisNBd4.R)Examplesofsuit32ablebraidedbasecategoriesD WVeF closewithsomespSecialexamplesofbraidedmonoidalcategoriesC?(cocompletesucrhdthatthetensorproSductpreservesarbitrarycolimits),>thatmaybSeusedasbasecategories.|One5spSecialexampleisthecategoryofYVetter-Drinfel'dmodulesYDxHHorver@aHopfalgebraHV.:ByjX[14 ]itcanbSeviewredasacategoryofcomodulesorver@acoSquasitriangularHopfalgebra.D FVorYanarbitrarycoScommrutativeYHopfalgebraHFsucrhacategorycanalsobeob-tained)inthefollorwingwayV. }cConsiderthecategoryofrightHV-moSduleswhichisacoScompletesymmetricmonoidalcategorysucrhthatthetensorproductpreservresarbitrarycolimits. ObservrethatHactsonitselfbytheadjointaction썑k h:URH HB3h ko7!XS(k(1) \|)hk(2)2HF::H is yarighrtHV-moSduleHopfalgebrabytheadjointactionascanbSeeasilychecked.Thrus*HisaHopfalgebrainthecategoryMH !nandthecategoryMߍH HofHV-comoSdulesinMH isamonoidalcategoryV.8AK-moSduleM+isinMߍH Hi a)itisarighrtHV-moSduleUR:M HB !M+, b)itisarighrtHV-comoSduleȄ:URM6 !M H,and c)s2(mh)UR=(m)hor(mh)UR=Pm(0) \|h(1) m(1)h(2)=URPm(0) \|h(1) S(h(2) \|)m(1)h(3):InviewofthecoScommrutativityofHthelastconditionisequivXalenrtto썍nX|s(mh1)0j h2(mh1)1V=URXm0h1 m1h25whicrh*istheYVetter-Drinfel'dcondition[13 ].aThusMߍH H&:=URYDxlHlH!`~isabraidedmonoidalcategoryo[2,15],thebraidingmorphismbSeingde nedbryY-Ë:URM N63m n7!Xn(0)$ mn(1)2N M:ObrviouslylMߍH H=isnotsymmetricsincePm(0) \|n(1)_ 5n(0)m(1)n(2).=2m nldoSesnotholdingeneral.D AspSecialcaseforthisstructureisobtainedbrychoSosingHB=URKG,agroupalgebrawith֗a nitegroupG. Inthissituationthenameofdyslecticalgebrawras rstinrtroSducedbyHaran[4]. Norw+wecangiveanexampleofamoSdulewhichisnotdyslectic.+aLetGUR=C2WC2bSe&;theKleinfourgroup.Letcrhar(K)nw6=2.Then&;theHopfalgebraH[:=nwKG=K[s;t]=(s22R1;t22R1)BNiscoSquasitriangularwithr:URH@! HB !KBNde nedbryrS(s s)UR=rS(t' s)UR=r(t' t)UR=1,r(s' t)UR=1.Observrethatr(st' t)UR=r(s' t)r(t t)UR=1.LetNAz,=K1fKxwiths2(1)z,=1f 1,(x)z,=xf t.yThenNAbSecomesacommrutativealgebra'inM2H @Xbryx22|/=+1.$LetM=KyBKzwiths2(yn9)=yB sands2(z)=z\ st.ThenM`bSecomesanA-moduleinM2H 92bryyn9x=zandzx=y.رInparticularwregetn9(yLC x)UR=x yn9,6?22.=(yLC x)UR=yLC x %andn9(zf x)UR=x z,6?n922.=(zf x)=zf x.ThemaximaldyslecticsubmoSduleK&ofM`lturnsouttobezero,H(sinceM (( yy+ Oz) x)UR= >|7 ӰOnUUBraidingandDyslexia11N z6+ Oy]andM n922.=(( y/+ Oz) x)]= z6 Oyn9.GrInparticularM0gisnotdyslectic.Itis-easytocrheck-thatthediagraminProp.2.2withN(=DAhasanon-commrutativeuppSerG lefthandsquaresoitinducesnobraidingforM2HRAAD. NIftherewrasamap# ߫esinducedjbryn9,1then#@es O(y A :x)=x\ Ay=x\ Ay=0.J&SojthereisnoinducedbraidingonM2HRAAD..i WVeclosewithanotherexampleofasuitablemonoidalcategory.)KLetGbSeagroupandXbSea(righrt)G-set. BGitselfcanbeconsideredasaG-setbrythe(right)adjoinrtI5action.TTheFVreyd-YetterI5categoryCrWe(G)ofcrossedG-setsconsistsofpairs(XJg;j:j)pOwithaG-setXaandaG-morphismj:jUR:XF .!GpOasobjectsandG-morphismsf:aXR !Y$jsucrhthatj:jYPf=aj:jX.By[3]Thm.4.2.2thisisabraidedmonoidalcategory@9with(XJg;j:j)6 (Y;j:j)=(X6Y;j:j),sucrh@9thatj(x;yn9)j=jxjjyj,and@9thebraiding?X&;Y(x;yn9)=(y;xjyj).JThe?unitobjectIofthiscategoryistheonepSoinrtsetbSeingmappedinrtotheunitofG. AnalgebrainthiscategoryisasetAwithmapsj:j4:A Y!G,'AWG4!4A,AAUR !URAandf1g !Asucrhthat7uʍa(gn9g20 cmmi10K`y cmr10O line10u cmex10.