; TeX output 1995.06.16:0923WH9*N cmbx12ONٚLIEALGEBRASINBRAIDEDCATEGORIES K`y cmr10BODOUUP*AREIGIS #XQ cmr12April3,1995 Ǎ$+- cmcsc10Abstract.[The׮categoryofgroup-gradedmoGdulesover׮anabeliangroup b> cmmi10Gis $a?monoidalcategory*.1For?anybicharacterofGthiscategorybGecomesabraided$monoidalFcategory*.EWede nethenotionofaLiealgebrainthiscategorygener-$alizing_theconceptsofLiesupGerandLiecoloralgebras.OurLiealgebrashave$n-arytmultiplicationsbGetweenvqariousgradedcompGonents.UTheypGossessuniversal$enveloping#algebrasthatareHopfalgebrasinthegivencategory*.a?TheirbiproGducts$withCthegroupringarenoncommutativeCnoncoGcommutativeHopfalgebrassomeof$them6knownintheliterature.gConverselytheprimitiveelementsofaHopfalgebra$inUUthecategoryformaLiealgebraintheabGoveUUsense.0)': cmti10Keywor}'ds:GradedgLiealgebra,Bbraidedcategory*,braidedHopfalgebra,universal$envelopingUUalgebra.0Ui1.,- cmcsc10IntroductionxK With9theappSearanceofquanrtumgroupsthestudyofHopfalgebrashastakenanimpSortanrt>Dturninrecentyears.3Manyfamiliesofquantumgroupsareknown.3Herewrecwilladdsomenewfamiliesbymeansofaconstructionwhichmighteventuallyalsohelp5todevrelopastructuretheoryofquantumgroups.nInclassicalstructuretheoryak{formalgroupisdecompSosedinrtoasmashproductofanin nitesimalpartwithaseparableBopart.Theseparablepartisoftenagroupalgebra,dwhereasthein nitesimalpart&isoftenaunivrersalenvelopingalgebraonwhichthegroupalgebraopSerates.ZThein nitesimalpartthenisgeneratedbrytheprimitiveelementsoftheformalgroup.xK Horwever,forquantumgroupsornoncommutativenoncoScommutativeHopfalge-brascHthiskindofdecompSositionseemstobemruchcHmorecomplicated. WVewillusethefollorwingapproach. Let%$g cmmi12UR:GG%!", cmsy10GUR !URkg2K cmsy8FbSeabicrharacteroftheabeliangroupG.(Thenthemonoidalcategory\ofG-gradedvrectorspacesorkgG-comoSdulesisabraidedmonoidalcategoryV.ByformingabiproSductwiththegroupalgebrakgG,2eacrhHopfalgebrainthiscategorygeneratesanordinaryHopfalgebra.ThisproScesswrasstudiedforHopfalgebraswitha!$projectionbryRadfordin[R](orastheproScessofbosonizationbryMajidin[M94ad1]).+<ff< 1991UUMathematicsSubje}'ctClassi cation.PrimaryUU16W30,17B70,16W55,16S30,16S40.1*2BODOUUP*AREIGISWH· WVewillgivresomegeneraltechniquesforconstructingHopfalgebrasinthecate-goryofG-gradedvrectorspaces.OOurspSecialinterestliesinHopfalgebrasthataregenerated_XbryprimitiveelementsinthepropSersense,|thatis(x)=x 1+1 x.InthebiproSducttheseelemenrtsbecomeskrewprimitiveelements.幍 There{isageneralizednotionofaLiealgebrainsymmetricmonoidalcategoriesunderQthenameofLiecoloralgebra(orLiesupSeralgebra).Anappropriatede nitionof{aLiealgebraina-@ cmti12brffaided{monoidalcategoryshouldharve{spSeci cproperties.KZIntheKclassicalsituationtheprimitivreelementsofaHopfalgebraformaLiealgebra.Also^thesetofderivXationsofanalgebraisaLiealgebra. 3Soamoregeneralde nitionofCaLiealgebrashouldpassatestwithrespSecttothesetprimitivreelementsofaHopfbalgebraandwithrespSecttothesetofderivXationsofanalgebra.qWVeproposeade nitionQ)ofgeneralizedLiealgebrasinthebraidedmonoidalcategoryofG-gradedvrectorNspaces.ThesegeneralizedLiealgebrashaven-arybracketmultiplicationsthatareonlypartiallyde ned,butcertainsymmetryandJacobiidenrtitiesstillhold. LiecolorandLiesupSeralgebrasaswrellasordinaryLiealgebrasarespecialcasesof"ourgeneralizedLiealgebras.#MWVewillshorwthatthesetofprimitiveelementsofa!HopfalgebraissucrhageneralizedLiealgebra.EveryassoSciativealgebraisalsoageneralized{\Liealgebrabrythesamede nitionofthebracketmultiplication.AndthesetofderivXationsofanalgebrawillturnouttobSeaLiealgebra. Starting',outwithageneralizedLiealgebrawrewillconstructauniversalenvelopingalgebra thatturnsouttobSeaHopfalgebrainthecategoryofG-gradedvrectorspaces.ThrusнwehaveobtainedamethoSdforconstructingnewHopfalgebrasinthiscate-goryV.ByformingbiproSductswreobtainmanyoldandnewordinarynoncommutativenoncoScommrutativeHopfalgebrasorquanrtumgroups. Ordinary-Hopfalgebrasorver-a eldofcrharacteristiczerothataregeneratedbytheirpprimitivreelementsareeithertrivialorin nitedimensional.P7InfacttheyareunivrersalenvelopingalgebrasofLiealgebras. Hopfalgebrasinbraidedmonoidalcategoriesincrharacteristiczerothataregen-eratedbrytheirprimitiveelements,however,donotsharethispropSertyV..)Incertaincasestheyaresimilartounivrersalrestrictedenvelopingalgebrasofrestricted(p-)Liealgebras. A#[x1;:::ʚ;xnP]UR=(;(g1;:::ʚ;gnP))[xI{(1){;:::;xI{(n)i];䍍#P*.UZn+1̍.UZi=1@$2i+1ϟf`Q*'0zi1̍'0zjv=18r(gjf ;gid)f` [xi;[x1;:::ʚ;7^xi *;:::;xn+1]]UR=0; #[x;[y1;y2;:::ʚ;ynP]]UR=P* n̍ i=1$f`&ßQ*/i1̍/jv=1@(gjf ;h)f` [y1;:::ʚ;[x;yid];:::;ynP];whenevrerthebracketproSductsarede ned.Inparticularwerequirexid;yi2Pg8:i ɲandxUR2Phe. TheisetofprimitivreelementsofaG-gradedHopfalgebraisanexampleforageneralized>Liealgebra,andsoisanryG-gradedalgebraifwede nethebracketopSerationbryQ[x1;:::ʚ;xnP]UR:=X "%I{2Snn(;(g1;:::ʚ;gnP))xI{(1),#:::xI{(n)i:"% AwLiesupSeralgebra(P0;P1),whereP0υisanordinaryLiealgebrawithoperations[:;:]UR:P1?+ 'P1Vj!P0`and\[:;:]:P0?+ 'P1Vj!P1,ٞis\anexampleforthisde nition.1LiecoloralgebrasarealsospSecialcasesofourconcept. Iwrould$liketoacknowledgehelpfulconversationswithEdwardL.Green,FVrankHalankre,SusangMontgomeryV,MartingNeuchl,HelmutgRohrl,PetergSchauenburg,andYVorcrkSommerh auser.@sq2.P9BicharactersandtheBracketMul32tiplication Throughoutletk?bSeaninrtegraldomain,(j),i.e.theNsetofallpairsinXinrevrersepSosition.uꍍLemma2.2.IUThe35mapsatis esthefollowingrffelation>(n9;(g1;:::ʚ;gnP))UR=(;(gI{(1){;:::ʚ;gI{(n)i))(;(g1;:::;gnP)):RPrffoof.#RWVeharvetoshow)FȍUP?֟Y ֤i(jv)p&( 1 D}(gI{r(jv)٧;gI{r(i)w))UR=)K=URf^ ߟY k6(lK)Ao( 1 D}(gI{r(lK)e;gI{r(k6)/))f^\tf^%Y r(s)Gt( 1(gI{(s)/;gI{(r(j)=:r(andvthrusn9(rS)>(s))vthenthe rstproSductconrtributesaVfactor21 D}(gI{r(jv)٧;gI{r(i)w)forthepairiSi(j)VandthesecondproSductconrtributesafactor21 D}(gI{r(i)w;gI{r(jv)٧)forthepairrY)<s;n9(rS)>(s).2Thesetrwofactors0,cancel. lIfsc:=W(i)<(j)=:rthen0,neitherpairicW(j) thenthereisacorrespSondingfactor21 D}(gI{r(jv)٧;gI{r(i)w)onthel.h.s.+Onther.h.s.thereareagaintrwopSossibilities.IfW(i)UR>(j)thenthe rstproSductconrtributesafactor21 D}(gI{r(jv)٧;gI{r(i)w)forthepairi(j). IThe0vsecondproSductdoesnotconrtributeafactorforthepairsr:=W(i)>(j)=:rr;n9(rS)<(s)..Ifs:=W(i)<(j)=:rthenthe rstproSductdoSesbnotconrtributeafactorforthepairi (s).~ThisAargumenrttakrescareofallfactorsonbSothsidesoftheformula. yff٘ ̍ ff ̄ ffffff٘ If/dgn9;g202PGharveordermresp.m20andifnP=gcdjD(m;m209)then(gn9;g20]Remark3.4thecategoryM2k6GofG-gradedkg-moSdules(orthecategoryofkG-comoSdules)isabraidedmonoidalcategorywiththetensorproSductōu(X+ Yp)g*P=_4M 8獑URh2GXh YgI{hHandthebraidingMJo:URX+ Y3x yË7!(degɻ(x);degɹ(yn9))y x2YG XwherexandyXarehomogeneouselemenrtsofdegreedegc(x);degɹ(yn9)UR2G. Algebras,|coalgebras,bialgebras,and_Hopfalgebrasinthebraidedmonoidalcat-egoryM2k6GwillbSecalled(G;)-algebras,*(G;)-coalgebras,(G;)-bialgebras,resp.(G;)-Hopfalgebras. LetoA,B bSe(G;)-algebrasinthecategoryM2k6G *.=ThenAK Bisoalsoa(G;)-algebrainM2k6GlwiththemrultiplicationA B A BX1 r 1g {:Г!!A A B B$VXmX.A ^ mX.B۪ J !)OA B(seeforexample[M94b M]). De nition2.4.YLetAbSea(G;)-algebra(associativrewithunit)inM2k6G *.WVede neɍSd[x1;:::ʚ;xnP]UR:=X "%I{2Snn(;(g1;:::ʚ;gnP))xI{(1),#:::xI{(n) forall=S2URkg2'!,all-families(g1;:::ʚ;gnP),andallxi,2Ag8:i. AspSecialexampleis=S=UR1,n=2and(g1;g2)=(g2;g1)=1.8ThenōP[x1;x2]UR=x1x2j+((2;1);(g1;g2))x2x1V=URx1x2jx2x1:If=S=UR1,n=2and(g1;g2)=(g2;g1)=1,thenP,g[x1;x2]UR=x1x2j+((2;1);(g1;g2))x2x1V=URx1x2j+x2x1:E6BODOUUP*AREIGISWH·HenceYNwreobtainLiealgebrasresp. LiesupSeralgebrasasaspecialcaseof(2.4).Similarlyif=S=UR1,n=2and(g1;g2)isa-familythen(g1;g2)UR=(g2;g1)21G$and)?[x1;x2]UR=x1x2j(g1;g2)x2x1;renderingLiecoloralgebrasasanotherinstanceof(2.4). FVorLiecoloralgebrastheassumption(g2;g1)=(g1;g2)21Ttforallg1;g2 ߽2Gleadstoatotallyde nedmrultiplicationN[:;:]:AA U!Asince[x1;x2]isde nedforanrypairx1;x22Aofhomogeneous,elemenrts.FkA+spSecialnewoperationisthefollorwingbracketproSduct.FkIfgË2URGsatis es(gn9;g)==S6=1thenwrehavetheproSduct[:::ʞ]UR: 2nPA !A);[x1;:::ʚ;xnP]UR=X "%I{2SnnxI{(1)y:::"LxI{(n)i:!ҼTheorem2.5.S8(Symmetry)ljLffetC2Bkg2 and(g1;:::ʚ;gnP)bea-family.Letxi$2BAg8:iand35Ë2URSnP.fiThen][x1;:::ʚ;xnP]UR=(;(g1;:::ʚ;gnP))[xI{(1){;:::;xI{(n)i]:Prffoof.#RWVeharve$~ʍ/(;(g1;:::ʚ;gnP))c [xI{(1){;:::ʚ;xI{(n)i]UR=^hb=URP ㍟r2Sn% (;(g1;:::ʚ;gnP))(;(gI{(1){;:::;gI{(n)i))xI{r(1)I:::xI{r(n)^hb=URP ㍟r2Sn% (n9;(g1;:::ʚ;gnP))xI{r(1)I:::xI{r(n): yff٘ ̍ ff ̄ ffffff٘%덑 InthecaseofLiealgebrasthisamounrtsto'[x1;x2]UR=[x2;x1];inthecaseofLiesupSeralgebrasthisissx[x1;x2]UR=[x2;x1](for #="1,:!n=2and(g1;g2)"=(g2;g1)"=1);}9and inthecaseofLiecoloralgebras(cf.8[FMo>]3.11)thisis[x1;x2]UR=(g1;g2)[x2;x1]: Inthefollorwingtheoremlet(i:::ʜ1)UR=f`܍1;2;:::;i;i+1;:::;n+1 v~ N6i;1;:::;i1;i+1;:::;n+1[%Tf`edenoteacycleinSn+1.EpTheorem2.6.S8(Jacffobi35identities) ⍍Û(1)#Lffet35(g1;:::ʚ;gn+1)bea-familywith6aprimitiven-throotofunity.fiThen Q|Un+1 VV@X Vyi=1g((i:::ʜ1);(g1;:::ʚ;gn+1))[xid;[x1;:::ʚ;7^xi *;:::;xn+1]]UR=0s#for35allxi,2URAg8:i.V;bn ONUULIEALGEBRASINBRAIDEDCA*TEGORIES]n7WH·Û(2)#Lffett(g1;:::ʚ;gnP)bea-familywith\aprimitiven-throotofunityandlet#hUR2G35suchthatall(h;gid)arffe(1)-families.fiThenjōQ\[x;[y1;y2;:::ʚ;ynP]]UR=n X i=1f`@i1 Y jv=1,(gjf ;h)f`[y1;:::ʚ;[x;yid];:::;ynP]#for35allxUR2Ah &and35yi,2Ag8:i.jPrffoof.#R(1) Let2SnP.ConstructXR22Sn+1@bryXR -v(1)=1andXR -v(j)=W(j 991)+1.Thisde nesabijectionbSetrweenSn k*andthesetofallË2URSn+1Gwithn9(1)=1.+Usinghi,:=URgi+1+wregetC ̿$r(;(g2;:::ʚ;gn+1))=UR(;(h1;:::ʚ;hnP))3:="&Y UR1i(j)M( 1 D}(hr(jv);hr(i)x))=2qSY UR2i(j1)nTr( 1 D}(gr(jv1)+1%m;gr(i1)+1$lp))='ןY UR1iu(j)Xz( 1 D}(gߍu&(jv);gߍu&(i)x))wx=UR(Eqi;(g1;:::ʚ;gn+1))D SowregetforalliUR2f1;:::ʚ;n+1g ڍ"<(;(g1;:::ʚ;gi1AV;j^gi /;gi+1;:::ʚ;gn+1))UR=(Eqi;(gid;g1;:::ʜ;gi1;j^gi /;gi+1;:::ʚ;gn+1)):Norw՟letS{=BiJ2Sn+1Zkwith՟i:=n9(1).Letq,J:=(1:::ʜi)id.Thenq G(1)=1,\soqcomesfromsomeo2URSnP.8FVurthermorei,=(i:::ʜ1)Eqi.8ByLemma2.2wreget ڍʍ(id;(g1;:::ʚ;gn+1))#m=UR(Eqi;(gid;g1;:::ʜ;j^gi /;:::ʚ;gn+1))((i:::1);(g1;:::ʚ;gn+1))#m=UR(;(g1;:::ʚ;j^gi /;:::;gn+1))((i:::ʜ1);(g1;:::;gn+1)):P Givren?>2]!Sn wede ne~Oby~ `(n+1)]!:=n+1and~ `(j)]!:=W(j)?else.FThisde nesa#bijectionbSetrween#Sn mandthesetofall#2kSn+1withn9(n+1)=n+1.?Then#wreget ڍt(;(g1;:::ʚ;gnP))UR=(E~qi;(g1;:::ʚ;gn+1)) ڍandBl(;(g1;:::ʚ;j^gi /;:::;gn+1))UR=(E~qi;(g1;:::ʜ;j^gi /;:::;gn+1;gid)):Norwleto=^6n92i1I2Sn+1twithi:=n9(n6+1).HLet&~ :=^6(n+1:::ʜi)n92i.HThen&~ aJ(n+1)^6=n+1,so~Fcomesfromsomeo2URSnP.8FVurthermoren92i(e=(i:::ʜn+1)E~ \andwreget ڍ(n9 i;(g1;:::ʚ;gn+1))UR=(;(g1;:::ʚ;j^gi /;:::;gn+1))((i:::ʜn+1);(g1;:::;gn+1)):P TVokprorvetheJacobiidentityweobservethatsincelisaprimitiven-throSotofunitryv.ʍ,(gid;g1j+:::+^giZ+:::+gn+1)͡(g1j+:::+^giZ+:::+gn+1;gid)UR==URQ jv6=iu(gid;gjf )(gj;gid)UR=22n%=1;e8BODOUUP*AREIGISWH·hencewrehavea(1)-family(gid;g1j+:::+^giZ+:::+gn+1)and񹍍č/[xid;[x1;:::ʚ;7^xi *;:::;xn+1]]UR=3;/=URxid[x1;:::ʚ;7^xi *;:::;xn+1]f` QSjv6=i#(gi;gjf )f`[x1;:::ʚ;7^xi *;:::;xn+1]xid:G FVurthermorewreuse2n =UR1toget%G č ((i:::ʜ1);(g1;:::ʚ;gn+1))Qjjv6=iQ(gid;gjf )UR=3;#c=URf` N4Qjviw(21 D}(gid;gjf ))UR=((i:::ʜn+1);(g1;:::ʚ;gn+1)):% WiththesenotationswrecannowevXaluateMYŕJAP*|n+1̍|i=1'k((i:::ʜ1);(g1;:::ʚ;gn+1))[xid;[x1;:::ʚ;7^xi *;:::;xn+1]]UR=0b-H=URP* n+1̍ i=1 hW((i:::ʜ1);(g1;:::ʚ;gn+1))xid[x1;:::ʚ;7^xi *;:::;xn+1]3;1((i:::ʜ1);(g1;:::ʚ;gn+1))f`Qjjv6=i!R(gid;gjf )f`[x1;:::ʚ;7^xi *;:::;xn+1]xi2-H=URP* n+1̍ i=1 hWP*r2SnB((i:::ʜ1);(g1;:::ʚ;gn+1))(;(g1;:::ʚ;j^gi /;:::;gn+1))\(x8:i,r(1)dp:::%/x8:i,r(n+1)1((i:::ʜn+1);(g1;:::ʚ;gn+1))(;(g1;:::ʚ;j^gi /;:::;gn+1))xI{iu(1):::%xxI{iu(n+1)-H=URP ㍟I{2Sqn+1.(;(g1;:::ʚ;gn+1))xI{(1)y:::"LxI{(n+1)>1(;(g1;:::ʚ;gn+1))xI{(1)y:::"LxI{(n+1)-H=UR0:Q/) (2)\oSince(h;gid)(gi;h)UR=(1)22sand\o(gid;gjf )(gj;gid)UR=22 twre\ohave(h;g1H-+):::+gnP)(g1+7::::r+7gn;h)u=1 and(gid;h7+gjf )(h+gj;gid)u=22 b%so thatalltermsarede ned.R LetË2URSn withn9(j)=i.8Thenwrehave1㍍(;(g1;:::ʚ;gi1AV;h+gid;gi+1;:::ʚ;gnP))UR=3;%=URf`(Y N4k6(lK)@:( 1 D}(gI{(lK)?;gI{(k6) ))f`f`@ߟY jvI{(l)7W(gI{(lK);h)f`f`7Y k6i:(h;gI{(k6) )f`%=UR(;(g1;:::ʚ;gnP))f`Y ϧI{1 ;;(i)I{(l)F6(gI{(lK)?;h)f`f`'Y ϧ k6<I{1 ;;(i);I{(k)>iP (h;gI{(k6) )f`:1W* WVeabbreviatezkx:=URuk:=yk :forko6=i,zi,:=xyid,andui,:=yidx.8ThenwregetOIƍG,n  X =Ci=1,,'f`4i1 58Y ҁ4% r1V=n URX i=1f`Vi1 ۟Y ҁr(lK)> (gI{(lK)?;h)f` Fxf`lܟY lK(jv)=(h;gI{(lK)?)f`yI{(1)y:::"LxyI{(jv)':::!yI{(n)$$ FxSX "%I{2Snbn AX Gjv=1)˟f`㊍1ԭI{(jv)1v8Y ҁ7=r(lK)> (gI{(lK)?;h)f` Fxf`lܟY lK(jv)=(h;gI{(lK)?)f`yI{(1)y:::"LyI{(jv)'x:::ʜyI{(n)=X "%URI{2Snn(;(g1;:::ʚ;gnP))xyI{(1)y:::"LyI{(n) FxSX "%I{2SnGf`%pn "& Y ҁ!)r(lK)D(gI{(lK)?;h)f`f`!eY lK(jv)C(h;gI{(lK))f` N6=$$TL=URf`㊍ N4I{(jv1)1vY ҁr(lK)S(gI{(lK)?;h)f`` f`&ʟY lK(jv1)S(h;gI{(lK)?)f`D׍WVecrhangeparametersbyo=URn921]withW(p)+1=W(qn9)andharvetoshorw3㍍0X3rf`UQ;TqI{1 lK;r(q)<(lK)C(gl!;h)f`f`!^MY lK>qI{;r(l)<(qI{)(h;gl!)f` N6=!$T=URf`UQp bY ҁ N4rlK;r(p)<(l)C(gl!;h)f`f`!Y lK>p;r(l)<(p)(h;gl!)f`:27ThiscanbSeeasilycrheckedifoneconsidersthecasespURqseparatelyV. yff٘ ̍ ff ̄ ffffff٘i!3.bPrimitiveElements WVeƬcometothemaintecrhnicaltheoremofthispapSerwhichhasapplicationstoprimitivreelementsinHopfalgebras. ~10BODOUUP*AREIGISWH·Theorem3.1.S8LffetAbea(G;)-algebrainM2k6G *.IThenthefollowingholdinA AI?}[x1j 1+1 x1;:::ʚ;xnR 1+1 xnP]UR=[x1;:::ʚ;xnP] 1+1 [x1;:::ʚ;xnP]for35allprimitiven-thrffoots35ofunity,all-families(g1;:::ʚ;gnP),andallxi,2URAg8:i.zPrffoof.#RWVeharvetoevXaluateʍ[x1j 1,S+1 x1;:::ʚ;xnR 1+1 xnP]UR=2;=URP ㍟I{2Sn%|(;(g1;:::ʚ;gnP))(xI{(1),# 1+1 xI{(1){):::ʜ(xI{(n)o 1+1 xI{(n)i):Observrethat#I?ʍ{T(xi 1)(xj 1)UR=(xidxj 1);{T(xi 1)(1 xjf )UR=(xi xjf );{T(1 xid)(1 xjf )UR=(1 xixjf );{T(1 xid)(xj 1)UR=(gi;gjf )(xj xi):(WVecollectalltermsoftheformcЗ;i xI{(1)y:::"LxI{(i) xI{(i+1):::+ixI{(n)[with2aSn ,andwranttoshorwthattheyarezeroforall1UR 1+1 xnP)`withthefactorc1;nb=1.Thesameholdsforthetermc1;1 i1 x1:::xnP.6Norwweconsiderexclusivelytermsc1;i ?x1:::xi xi+1AT::: xn :with0UR(;(g1;:::ʚ;gnP))(xI{(1),# 1+1 xI{(1){):::ʜ(xI{(n)o 1+1 xI{(n)i)whenevrerË2URSn Jisashueoff1;:::ʚ;igwithfic+1;:::ʚ;ng,i.e. if1URj%(kg)=l[`thrusproSducingafactor(gI{(jv)';gI{(k6) ).\SothetotalconrtributionofBԍ4>(;(g1;:::ʚ;gnP))(xI{(1),# 1+1 xI{(1){):::ʜ(xI{(n)o 1+1 xI{(n)i)toc1;i x1:::xi xi+1AT::: xn is꧍"\(;(g1;:::ʚ;gnP))|2Qşjv(k6)PM(gI{(jv)';gI{(k6) )UR=L΍f.=URQ jv(k6)Er(21 D}(gI{(k6) ;gI{(jv)'))(gI{(jv);gI{(k6) ))UR=2twhereStisthenrumbSerSofpairsinrevrersepositioninn9. tHerewrehaveusedthat(g1;:::ʜ;gnP)isa-familyV.# TVodeterminethenrumbSertofpairsinrevrersepositionfora xedOLwreobservethattheorderingof1;:::ʚ;iandofi"+1;:::ʚ;nmrustbSepreservedintheproSduct.WVe _counrthowmanystepsthefactor1p xi+1Nhas _bSeenmovedtotheleftorhowmanryKtermsfromx1 1;:::ʚ;xiQw 1KintheproSductaretotherightof1 xi+1andcallthisnrumbSer1.Observethat0Kv1 zi.Similarly2 ѮdenotesthenrumbSerof Ctermsfromx1.} ny1;:::ʚ;xiS 1 CintheproSductthataretotherighrtof1ny xi+2AV.WVeZharve02 1.Inasimilarwraywecontinuetode nethenumbSersjwith0ni+g:::*21i.hTheOevXaluationoftheselectedproSductthengivresatermBԍ q1*+:::\+8:ni1]jx1:::xi xi+1AT::: xnP: TVogetthenrumbSerofterms2tyx1:::xi xi+1AT::: xnXin[x1p 1+1 x1;:::ʚ;xnY/ 1+1 xnP]wrehavetocountallpSossibilitiestorepresenttUR=1+:::+ni9with0ni~:::UR2V1iMorthenrumbSerMp(i;njFi;t)MofpartitionsoftinrtoatmostnjFiMpartseacrhURi.5Thuswecannowdeterminethefactorc1;i forc1;i x1:::xiY xi+1AT::: xn intheexpansionof[x1j 1+1 x1;:::ʚ;xnR 1+1 xnP]asB؍'c1;i V=URX t0p(i;ni;t) ty:!$ByatheoremofSylvrester([A]Theorem3.1)wehaveōNWUX Nt0^p(i;ni;t)qn9 tUU=ō(1qn92n)(1qn92n1):::ʜ(1qn92ni+1tW)Qmfe  F(1qn9i)(1qn9i1):::ʜ(1qn9)hence/7c1;i V=0for0QSowre5onlyhavetoshowthat[x1;:::ʚ;xnP]isprimitive.ButthatisaconsequenceofTheorem3.1. yff٘ ̍ ff ̄ ffffff٘ ObservrethatwehavespSecialmultiplicationsrla[:::ʞ]UR:Pg(HV) ::: Pg(HV)UR !URPng N(H)forallgË2URGwithjgn9j6=1aprimitivren-throSotofunityV.De nition3.3.YLetAbSeanalgebrainkgG-comod.KAderivXationfromAtoAofdegreegË2URGisafamilyoflinearmaps(dhC:Ah!Ah+gWgjh2G)sucrhthatd(ab)UR=d(a)b+(gn9;h)ad(b)forallaUR2Ahe,b2Ah0,allh;h20#2G. ItdandSxi,2URPg8:i.We rstconsiderthepSermrutationo2URSnwithW(i)=n+1i.ThenW22 ls=UR1andȍ-(;(g1;:::ʚ;gnP))~=URQ i+(1)2(deg E(xq1*)+deg(xq2))jdegH(xq3)dn}[x3;[x1;x2]]UR=0;uS6[x;[y1;y2]]UR=[[x;y1];y2]+(1) deg E (x)jdegH(yq1*)6F[y1;[x;y2]]:鍑 (3)\LetGbSeanarbitraryabeliangroupwithabicrharactersuchthat(g1;g2)UR=(g2;g1)21Tforallg1;g2 2 G.bhThenwrehave(g1;g2)(g2;g1) =1=(1)22.bhThisde nes(1)-families(g1;g2)togetherwithbracrketopSerationsy[:;:]UR:Pgq1 ` Pgq2 |`!Pgq1*+gq2;FSܘ[x;yn9]UR=xy+((2;1);(g1;g2))yn9xUR=xy(g1;g2)yn9x:yThisistheexampleofLiecoloralgebrasforanabSeliangroup.TheLieidenrtitiesofde nition4.1reduceto[x1;x2]UR=((2;1);(degɻ(x1);degɹ(x2)))[x2;x1]UR=(degɻ(x1);degɹ(x2))[x2;x1];Fʍe[x1;[x2;x3]](degɻ(x1);degɹ(x2))[x2;[x1;x3]]^`+(degɻ(x1)+degtc(x2);degɹ(x3))[x3;[x1;x2]]UR=0;FL[x;[y1;y2]]UR=[[x;y1];y2]+(degɻ(x);degɹ(y1))[y1;[x;y2]]:鍑 (4)=&LetGz=Z=3Z=f0;1;2g=&withthebicrharacter(i;j)z=2(ijv) where=&%'isaprimitivreڄ3rdroSotofunityV.tThen(0;i)(i;0)=1=(1)22 soڄthatweget(1)-families(0;i)withydz[:;:]UR:P0j Pi,!Pid;Fdz[:;:]UR:Pi P0Vj!Pid;[x;yn9]UR=xyyx:yFVurthermoreUwrehave(i;j)(j;i)-=(2ij2)22 withUaprimitive3rdroSotofunity2ijfor;alli;j>6=0. *Thisgivres-families(1;1;1)and(2;2;2)andno22-familyV. *TheassoSciatedLiestructureisE[:::ʞ]UR:Pi Pi Pi,!URPiwithy[x1;x2;x3]UR=/KX "%I{2Sq3{ xI{(1){xI{(2)xI{(3):ڍsince#<(;(i;i;i))=Q 'i(21 D}(i;i))=1forall#2S3.TheLieidenrtitiesofde nition4.1fortheseternarybracrketsreduceto~[x1;x2;x3]UR=[xI{(1){;xI{(2);xI{(3)];F r[x1;[x2;x3;x4]]+[x2;[x1;x3;x4]]+[x3;[x1;x2;x4]]+[x4;[x1;x2;x3]]UR=0;yfor=x1;x2;x3;x4V2URPid,i=1=oriUR=2.%gHere=wrealso nda rstexamplewherethetwoJacobiidenrtitiesmeandi erentthings.~ThesecondJacobiidentityforxij2P0 Landy1;y2;y3V2URPid,i=1ori=2reducesto:[x;[y1;y2;y3]]UR=[[x;y1];y2;y3]+[y1;[x;y2];y3]+[y1;y2;[x;y3]]:9bn ONUULIEALGEBRASINBRAIDEDCA*TEGORIESXn19WH· (5)iLetGUR=Cnh:::4hCn r(rS-times)iwithgeneratorst1;:::ʚ;trb.. SincelaterwrewilltakreSthebiproSductofa(G;)-HopfalgebrawithkgG,qWwewillwriteGmultiplicativelyinthisexample.$LetbSegivrenby(tid;tjf )UR=,aprimitiven-throSotofunityV.$Then(t1;:::ʚ;tnP)isa-familyV.8A(G;)-LiealgebrawillharveabracrketopSerationo[:;:]UR: rbPtq1 g!Preacrhp:oforder3. ɕDe neby(g1;g1)j:=,ўap:primitive3rdroSotofunityV,(g1;g2):=22,(g2;g1)=1,andf(g2;g2)=.NThenftherearesevreral-families,amongothers(g1;g1;g2)and(g1;g2;g2).8Theyde nebracrketsox,[:::ʞ]UR:Pgq1 ` Pgq1 Pgq2 |`!URP2gq1*+gq2;x,[:::ʞ]UR:Pgq1 ` Pgq2 Pgq2 |`!URPgq1*+2gq2;owithoʍXA[x1;x2;x3]Q{==URP ㍟I{2Sq3$F(;(g1;g1;g2))xI{(1){xI{(2)xI{(3)Q{==URx1x2x3j+x2x1x3+x1x3x2+x2x3x1+22x3x1x2+22x3x2x1resp.Pʍ{[x1;x2;x3]Ow=URP ㍟I{2Sq3$F(;(g1;g1;g2))xI{(1){xI{(2)xI{(3)Ow=URx1x2x3j+x1x3x2+x2x1x3+x3x1x2+22x2x3x1+22x3x2x1: WVeclosewithsomeexamplesof(G;)-Liealgebrasandof(G;)-Hopfalgebrasgenerated2tbrythem.yWVealsoexaminesomeofthebiproSductHopfalgebrasonealwaysobtainsVfrom(G;)-Hopfalgebras.{:IfHCqisa(G;)-HopfalgebrathenH& kgGisaHopfalgebrabryw7(x gn9)(y h)UR=(g;degɹ(y))xy gh;B(1 gn9)UR=(1 g) (1 g);oandvQ(x 1)UR=xX 8獓a+b=ccf(ya;i b) (zb;i  1);~;where@(x)=PvYa+b=c*kP4i9~ya;i *} `zb;i withdeg (x)=c,Vvdeg 1(ya;i E)=a,Vvanddeg (zb;i d*)=b(see[FMo>]Corollary3.5). Mostofthe(G;)-Hopfalgebrasarein nite-dimensional.Infact, theonly nite-dimensionalHopfalgebrasgeneratedbrytheirprimitivesweknow,Paregiveninthefollorwingexampleunder(1)and(2).LA20BODOUUP*AREIGISWH·Example4.9.QU(1)Pe=P1 =kgxwith[x;x]=0de nesa\commrutative"(C2;)-Liealgebrawithasinexample4.8(2). Itgeneratesthe(C2;)-HopfalgebraHޜ=Fkg[x]=(x22),0theܮunivrersalenvelopingalgebraofP.ThebiproSductH]Example3.9) (2)SLet(G;)bSeasinexample4.8(5).P?=Ptq1 Zc=kgxSwith[x;:::ʚ;x]=0Sde nesa|\commrutative"(G;)-Liealgebrawith(t1;t1)UR=,a|primitivren-throSotofunityV.It generatesthe(G;)-HopfalgebraH=8kg[x]=(x2nP),givrenintheintroSduction(forrM=1).[&TheKjbiproSductU@(P)?7kgGisthefreealgebrageneratedbrytheelementsx,t1;:::ʚ;tr8Ksubjecttothefollorwingrelationst2nRAi=UR1,tidtj\=tjf ti,andx2n=0.̠Thereisonemore7setofrelationsthatisobtainedfromthemrultiplicative7ruleforthebiproSduct(1Wk tid)(x 1)8=(ti;t1)xWk tiM@orf{idenrtifyingxWk 1fwithxand1Wk tiwithfti{the%relationstidxS=xti.WHence%thebiproSductsformthefamilyofnoncommrutativenoncoScommrutativeHopfalgebrasgivrenin[T|l]. (3)tP=L*w2̍wi=0!QPid,3P0 =kgz,P1=kgxkyn9,3P2 =0twith[x;x;x]=[yn9;y;y]=[x;yn9;y]-=0,[x;x;y]-=z,and[z;x]-=[z;yn9]=0de nesa(C3;)-Liealgebrawith(n1;n1)o=,vaprimitivre3rdroSotofunityV. 4Itgeneratesthe(C3;)-HopfalgebraHB=URkghx;yn9i=(x23;y23.=;xy22+yxy+y22.=x)k(wrehavez5=UR2(x22y+xyn9x+yx22)),6thekunivrersalenrvelopingCalgebraofP.&ThebiproSductU@(P)?kgC3 OGhasCgeneratorsx;yn9;tCwiththerelationsx23V=URyn923=0,t23V=1,xyn922+yn9xy+y22.=xUR=0,xt=tx,andyn9t=ty. (4)?LetG=C3,#and?bSeasbefore. 8mThenP;=L*2̍i=0 Pid,#P0 Z=kgyn9,P1=kgx,P2V=UR0dowith[x;x;x]=yn9,Ganddo[y;x]=0dode nesa(C3;)-Liealgebra. "Itgeneratesthe(C3;)-HopfBalgebraH=kg[x;yn9]=(yT6x23)=kg[x],YtheBunivrersalenvelopingalgebraofP. (5)FVorG, asbSeforeletP0 =kgyn9,P1=kgx,andP2=0with[x;x;x]=0and[yn9;x][=x. TThen$P}isa(C3;)-Liealgebra. TItgeneratesthe(C3;)-HopfalgebraHB=URkghx;yn9i=(x23;xyx+xy),theunivrersalenvelopingalgebraofP.SReferences[A];George E.Andrews:Q8TheThe}'oryofPartitions EncyclopGediaofMathematicsanditsApplica- tions.UUV*ol.2.Addison-Wesley1976.[FM]@DavidaTFischman,SusanMontgomery:"AeSchurDoubleCentr}'alizerTheoremforCotriangularHopfAlgebr}'asandGeneralizedLieAlgebrasUUJ.Algebra168(1994),594-614.[M94a]"ShahnBMa8jid:=Cr}'ossedѸProductsbyBraidedGroupsandBosonization.BJ.Algebra163(1994),165-190.[M94b]#G"ShahnMa8jid:Algebr}'ahandHopfAlgebr}'asinBraidedCategoriesIn:AdvqancesinHopfAlgebras.UULNpureandappliedmathematics158(1994)55-105.[R]Davidn#Radford:.The5Structur}'eofHopfAlgebraswithaProjection.n#J.Algebra92(1985),`322-347.[T]EarlLJ.T*aft:mjTheOr}'deroftheAntipodeofFinite-dimensionalHopfalgebras.LProGc.Nat.Acad.Sci.UUUSA68(1971),2631-2633. MaUTthematischesInstitutderUniversitM*atMM*unchen,Germany E-mailaddr}'ess!:qpareigis@rz.mathematik.uni-muenchen.de\; -@ cmti12,- cmcsc10+- cmcsc10*N cmbx12)': cmti10( msbm10%!", cmsy10$g cmmi12#XQ cmr12!ppmsbm8q% cmsy6K cmsy8;cmmi62cmmi8Aacmr6|{Ycmr8 b> cmmi10ٓRcmr7K`y cmr10u cmex10q