; TeX output 2004.02.03:0624Y[n썑kN cmbx12ONٚSYMBOLICCOMPUTATIONSfaTINٚBRAIDEDMONOIDALCATEGORIESWK`y cmr10BODOUUP*AREIGIS*Ս `$- cmcsc10Abstract.\`There aresomepGowerful notationsandtoolstoperformcomputations $in4withtensors,ltheSweedlernotationforcoalgebras,theEinsteinconvention4to$reduce,thenumbGer,ofsummationsignsincomputationswithtensors,"thePenrose$notationBZthathasbGeenfurtherdevelopedbyJoyalandStreettoagraphiccalculus$in2braidedmonoidalcategories.JIn1977IintroGducedamethodofcomputationthat$loGoks4verymuchlikecomputationwithordinaryelementsortensors,lbutcanbGe$pGerformedginarbitrarymonoidalcategories,5+byusingaY*onedaLemmaliketech-$nique.BInthedualofthecategoryofvectorspacesthisallowstoworkwithordinary$coalgebras asiftheywerealgebras.I willshowhowtoexpandthistechniqueto$braidedmonoidalcategories,anddevelopsomeofthegeneralrulesofcomputation.$As+anapplicationI willderivethewellknownresultthattheantipGode+ofaHopf$algebrainabraidedmonoidalcategoryisanalgebraantihomomorphismwhichis$expressedUUbytheformulas b> cmmi10S(1)=1UUandS(ab)= !", cmsy10hS(b)S(a);!Ǹi.'4?XQ cmr121.C- cmcsc10TheBeginnings:TheSweedler-Heynemannot32ation TVodescribSethecomrultiplicationofa) msbm10K-coalgebraintermsofelementsweintro-duceanotation rstinrtroSducedbySweedlerandHeyneman[H-Sw62'} ]similartothenotation !", cmsy10r(g cmmi12a b)UR=abusedforalgebras.8Insteadof(c)=#u cmex10Pc2cmmi8i c2!K cmsy80RAiOwrewriteO(1)(c)UR=Xc|{Ycmr8(1)$ c(2) \|:nObservrevXthatonlythecompleteexpressionontherighthandsidemakessense,Dnotthe4compSonenrtsc(1)=orc(2)whicrhare,@ cmti12notconsideredasfamiliesofelementsofCܞ.>ThisnotationtalonedoSesnothelpmruchtinthecalculationswrehavetopSerformlateron.SowreintroSduceamoregeneralnotation.ՍDe nition1.1.4(SwreedlerRONotation)LetM3bSeanarbitraryK-moduleandC.beaK-coalgebra.8ThenthereisabijectionbSetrweenallmrultilinearmapsfQ:URCF:::C1K{!MandalllinearmapsGkfG ]k:URCF ::: C1K{!M: yThesemapsareassoSciatedtoeacrhotherbytheformula(2)fG(c1;:::ʜ;cnP)UR=f ];(c1j ::: cnP)orgfQ=URfG ] :Xff< B ': cmti10Date[:UUF*ebruary3,2004. 1991MathematicsSubje}'ctClassi cation.PrimaryUU16A10.c 2004ٚMARCELDEKKER,INC.,270MADISONAVE.,NEWYORK,NY10016 ]-o cmr91*Y[썍2vBODOTP:AREIGISn썹ThisfollorwsfromtheuniversalpropSertyofthetensorproSduct.8FVorcUR2CFwede neO(3)XJfG(c(1) \|;:::ʜ;c(n) Dȹ)UR:=f ];( n1̹(c));|⍹where\2n1a(denotesthe(nt1)-fold\applicationof,8forexample2n1=UR(t 1 :::uH 1):::( 1). In1}particularwreobtainforthebilinearmap :CC3(c;d)7!c d2C C(withassoSciatedidenrtitymap)@(4)X2oc(1)$ c(2)ι=UR(c);andforthemrultilinearmap 22V:URCFCC1K{!C C COVXjITc(1)$ c(2) c(3)ι=UR( 1)(c)=(1 )(c):O Withthisnotationonevreri eseasilyDeXX c(1)$ ::: (c(i) R) ::: c(n)=URXc(1)$ ::: c(n+1)andz=ʍ0͟P>xc(1)$ ::: (c(i) R) ::: c(n)ݭ=URPc(1)$ ::: 1 ::: c(n1)ݭ=URPc(1)$ ::: c(n1)BÍ This[notationanditsapplicationtomrultilinearmapswillalsobSeusedinmoregeneralconrtextslikecomoSdules.* _Us2.nXSymbolicComput32ationswithtensors LetCݹbSeamonoidalcategoryV.8ForobjectsA;XF2URCde ne``EA(X)UR:=MorOC(XJg;A):WVeconsiderAasa\graded"or\vXariable"setwithcompSonenrtA(X)of\degree"X.ActuallyAisa(represenrtable)functorfromCݹintoSet . Let f:At!5_BbSeamorphisminC5.Thenwreget\mapsofvXariablesets"writtenbryabuseofnotationasfQ:URA(X)n!1B(X)with(5)*fG(a)UR:=fa:ThisCde nesanaturaltransformationandbrytheYVonedaLemmathereisabijectionbSetrweenPthemorphismsfromAtoB{VandthenaturaltransformationsfromthefunctorA꨹tothefunctorB. Inparticulartrwomorphismsf;gË:URAn!1Bareequali 6,8XF2URC5;8a2A(X):fG(a)=gn9(a): LetA;B;C12URC5.6,ThenCܞ(X Yp)isafunctorintrwovXariablesXandYp.6,FVurther-moreD|A(X)B(Yp)isalsoafunctorintrwovXariablesdenotedbyAB.F\ADdnaturaltransformationoffunctorsintrwovXariablesfQ:URABX !_7CFiscalledabimorphism. AspSecialexampleofabimorphismis@ UR:A(X)B(Yp)URn!1A B(X+ Yp)with!X (a;b)UR:=a bwherexam bUR:X^ mY M!`A B..Anelemenrta bUR2Am B(X^ Yp)comingfromtrwomorphismsa,biscalledadeffcomposable35tensor. IffQ:URAdBX !_7CisabimorphismandgË:URC1K{!Disamorphismthengn9fQ:AdB!nD>6isabimorphism. Y[썍6ON!SYMBOLICCOMPUT:ATIONS3n썑 If$fQ:URAXBX !_7C¹isabimorphismandgË:URU6!Aandh:V M!`B*aremorphismsthenfG(gh)UR:UV M!`CFisabimorphism.WLemma#2.1.HFor_effachbimorphismf5:H6A B<I!Cthere_isexactlyonemorphismfG2]k:URA BXV!Csuch35that> ԠABԠ@A Bɋ<:2fd@O line10-ͯ$ gC냀Vf, Q,攴Q,?^Q,Q嵌0Q嵌0sǠ@fe Ǡ?MZfG2]~cffommutes.Prffoof.#RThisBusesaYVonedaLemmatrypSeargument. AFVordetailssee[Pa77J,Lemma1.1].& msam10㍑ OccasionallySifh'=fG2] iisSgivrenthenwewritetheassoSciatedbimorphismash2[c:=h ,sothat(fG2];)2[#=URf2and(h2[<)2]=URh. Givren~abimorphismfL=fG2] anda2A(X);b2B(Yp). Lett=a b2A B(X+ Yp)bSeadecomposabletensor.8ThenfG(a;b)UR=f2];(a b)UR=f2];(t). SimilarremarksasabSorveholdformultimorphismsfY:A1\}:::\}An 6 !CwandassoSciatedymorphismsfG2] Es:/8A1 Ν:::g ΝAn ׈ ! MCܞ. ?UInparticularwrehaveforai 2Aidڹ(Xi);iUR=1;:::ʜ;n꨹andtUR=a1j ::: anQfG(a1;:::ʜ;anP)UR=f ];(t): WVeinrtroSducea rstsymbSolicexpressionforalltUR2A1j ::: AnP(X)byf(6)fffrRfffG(t1;:::ʜ;tnP)UR:=f2];(t):rffffffrRk ObservrethattisnotadecompSosabletensoringeneral.8WVehave,however: FVorOthemrultimorphism 2n1q:`A1o3/:::(/An  "!'A1 :::( An andOtheassoSciatedmorphism 2]#=URiduH:URA1j ::: An -!lA1 ::: An wreget(7)t1j ::: tn=URtforall\tensors"tUR2A1j ::: AnP(X).8InparticularwrehavethenQ(8)fG(t1;:::ʜ;tnP)UR=f ];(t1j ::: tnP):GivrenzfG2]k:URA1O ::: An -!lB1 ::: Bm >andztUR2A1O ::: AnP(X).4|ThenzwremayconsiderfG2];(t)asanelemenrtofB1j ::: BmĹ(X)hence$Ѝ(9)荍UfG2];(t)UR=fG2](t)1j ::: fG2](t)m Z=P=URfG(t1;:::ʜ;tnP)=f(t1;:::ʜ;tnP)1j ::: f(t1;:::ʜ;tnP)m:cbSincefG2]3isalsoanelemenrtinB1k @g:::Kn @gBmĹ(A1 :::Kn AnP)wrecanwritefG2]k=URfOG]1V :::Kn f2G]RAmandget΍(10)ʍQa(fOG]1 ::: f2G]RAmĹ)(t)UR=fG2];(t)=fG2](t)1j ::: fG2](t)m#o\orqɍ6KF(fOG]1 ::: f2G]RAmĹ)(t1j ::: tnP)UR=fG(t1;:::ʜ;tn)1j ::: fG(t1;:::ʜ;tn)m:qIfinadditiongn92]ǹ:URB1j ::: Bm Z s!CFisgivrenthenwegetx:gn9(fG ];(t)1;:::ʜ;fG ](t)mĹ)UR=g ] cmmi10K`y cmr10O line10r