; TeX output 1999.03.03:1921 7 Y3N cmbx127.BialgebrasDe nition1.7.1.vaXQ cmr121.hA}@ cmti12bialgebrffa}չ(g cmmi12B;!", cmsy10r;n9;;)consistsofanalgebra(B;r;n9)andacoalgebra(B;;)sucrhthatthediagramsp`FjBE B#"BE B B BkJ0t> HGyGI{ gׁ Al Aq Av AwT>AwT>UAP0BoBE BNL32fd.ά- YYHBE BYuBӔ{fd.ά-ˍPrH""3K` ׁ AĞ Aɞ AΞ A4>A4>UH]Cԟׁ ԟ ԟ ԟ >> j⍒,FKj⍒fK9 {fd*ά--J6idHIB'35 tׁ A: t A? t AD t AE0>AE0>UH]Cd$RiǠWiǠ\iǠaiǠbTׂbTׂ󭍹commrute,i.e.7Gandarehomomorphismsofalgebrasresp.randUarehomomor-phismsofcoalgebras.2.ɱGivrenpCbialgebrasAandB.Ap!mapf:8A4!B IispCcalledahomomorphismofbialgebrffas꨹ifitisahomomorphismofalgebrasandahomomorphismofcoalgebras.3.8ThecategoryofbialgebrasisdenotedbryK-Bialg"S.۳Problem1.1.cR1.^Let '(B;r;n9)bSeanalgebraand(B;;")bSeacoalgebra.^ThefollorwingareequivXalent:3oa) #(B;r;n9;;")isabialgebra. b) #UR:BX4!_7BE Band":BX4!K꨹arehomomorphismsofK-algebras.Xc) #rUR:BE BX4!_7BandË:K4!1BarehomomorphismsofK-coalgebras.2.#LetBF¹bSea nitedimensionalbialgebraorver eldK.ShorwthatthedualspaceB2 Eisabialgebra.۳OneofthemostimpSortanrtpropertiesofbialgebrasBisthatthetensorproductorverKoftrwoB-moSdulesortrwoB-comoSdulesisagainaB-moSdule.Prop`osition1.7.2.O1.s+LffetKB8Qbeabialgebra.4qLetM/andNbffeleftB-modules. #Then35M )ppmsbm8K cNtisaB-moffdulebythemap4N$BE M N2G 1p6 !@B B M N261 r 1p!vq!"VB M B Nh՚ J6 !@M N:f2. #Lffet B2beabialgebra.LetMandNbffeleftB-comodules.ThenM5 K }Nisa #B-cffomodule35bythemapK$M N2L p6 !@BE M BE N261 r 1p!vq!"VB B M N2Gr 1p6 !@B M N:/^+o cmr932!*7 &ebH7. %BIALGEBRAS 33Yrj3. #K35isaB-moffdulebythemapBE KPUR԰n:=B2 "pX E!K.D捍4. #K35isaB-cffomodule35bythemapKh LJUR !BPX԰ @=BE K.%⍍- cmcsc10Proof.@_WVegivreadiagrammaticproSoffor1.8TheassociativitrylawisgivenbyōvBE B M Nv&BE B B M NYsH2fd#ά-o/W1  1 1vvgfBE B M B N0sH2fd#:ά-om1 1 r 1vvBE M NdpsH2fd0ά-o>͍Y1  9~BE B B B M N^BE B B M B N`A2fd(eά-m=1 1 1 r 1NBE B M Nz!@A2fd[ά-7͍uK1 1  Ԡ9~BE B B B M NԠ^BE B M B B N`:2fd(eά-έyyЀk1 1 r(Bd B;M") 1ԠԠNBE M B Nz!@:2fd[ά-0͍uK1  1  BE M N׎BE B M NMP32fd;Ѝά-/]  1 1%BE M B N32fd;kά-m#1 r 1"M NL32fd0aά- s*徟Ǡ|z@fe+Ǡ?ۏ/pr 1 1sT>ՠ@fepՠ?⏍9 1 1 1 1sDՠ@feDՠ?⏍Ip 1 1 1 1s~ՠ@feՠ?⏍0 1 1T>Π@fepΠ?91 r 1 1 1DΠ@feDΠ?Ip1 r(Bd;B M") 1 1~Π@feΠ?01 r 1T>Ǡ@fepǠ?ԏ9r r 1 1DǠ@feDǠ?ԏIpr 1 r 1~Ǡ@feǠ?'0 豍Theunitlarwisthecommutativityofōv"T,xM NP6԰=@K M Nv"t!BE M NˊLsH2fdDvpά-o>͍LI{ 1 1"`K K M N"ZBE B M N߽ A2fd$+@ά-7͍I{  1 1ԩ"SM NP6԰=@K M K Nԩ"ZBE M B NB:2fd `ά-0͍|I{ 1  1shꟙՠ@feՠ?מ⍍NN=hΠ@feΠ?N1 r 1s2Sꟙՠ@fe2ՠ?⏍79 1 12SΠ@fe2Π?791 r 12SǠ@fe2Ǡ?'79 sdҊΠM@feeΠ?Y=M NiX1ۿ`X?`X`X?`X`XɄ?`Xӄ`X݄?`X焜`X񄜟?`X`X?`XXz豍TherCcorrespSondingpropertiesforcomodulesfollorwsfromthedualizeddiagrams.ThemoSduleandcomodulepropertiesofKareeasilycrhecked.\cffxff ̟ff ̎ ̄cff%⍍De nition1.7.3.vaù1.Let2(B;r;n9;;)bSeabialgebra.LetAbealeftB-modulewith8structuremapUR:B l}A4!1A..Let8furthermore(A;rA;A)8bSeanalgebrasucrhthatqOrA O3andAarehomomorphismsofB-moSdules.Then(A;rA;A;)qOiscalledaB-moffdule35algebra.2.Let(B;r;n9;;)bSeabialgebra.LetCobealeftB-modulewithstructuremapUR:B {C14!Cܞ.RLetUfurthermore(C5;C;"C)UbSeacoalgebrasucrhthatC uand"CarehomomorphismsofB-moSdules.8Then(C5;C;"C;)iscalledaB-moffdule35coalgebra.3.%Let9(B;r;n9;;)bSeabialgebra.%LetAbealeftB-comodulewithstructuremapIDi:[A4!CB A.TLetfurthermore(A;rA;A)IDbSeanalgebrasucrhthatrA '(andA yare"homomorphismsofB-comoSdules.Then(A;rA;A;s2)"iscalledaB-cffomodulealgebrffa."7 ̍34Y4.Let2(B;r;n9;;)bSeabialgebra.LetC?bealeftB-comodulewithstructuremapi:7Cd4!BZ Cܞ.Letfurthermore(C5;C;"C)bSeacoalgebrasucrhthatC ('and"C G are'uhomomorphismsofB-comoSdules.FThen(C5;C;"C;s2)'uiscalledaB-cffomodulecffoalgebra.Remark1.7.4.j6Ifh(C5;C;"C)hisaK-coalgebraand(C5;)isaB-moSdule,Xthen(C5;C;"C;)isaB-moSdulecoalgebrai isahomomorphismofK-coalgebras.If(A;rA;A)isaK-algebraand(A;s2)isaB-comoSdule,Ythen(A;rA;A;s2)isaB-comoSdulealgebrai ]ڹisahomomorphismofK-algebras.SimilarstatemenrtformoSdulealgebrasorcomodulecoalgebrasdonothold.&;7  +o cmr9)ppmsbm8( msbm10 K cmsy8!", cmsy102cmmi8g cmmi12|{Ycmr8@ cmti12- cmcsc10N cmbx12XQ cmr12O line10+