; TeX output 2001.06.04:13367 V #AN cmbx12SKEW-PRIMITIVEٚELEMENTSOFQUANTUMGROUPSoANDٚBRAIDEDLIEALGEBRASWK`y cmr10BODOUUP*AREIGIS(H1. P cmu10HerrnPrqofessorDr.H.Rqohrlzum70.Geburtstaggewidmet.2- cmcsc10Contents XQ cmr121. Quanrtumgroups,YVetter-Drinfel'dalgebras,andg cmmi12G-gradedalgebras23 2. Skrewprimitiveelementsp05 3. SymmetrizationofB2cmmi8nP-moSdulesZ8 4. LiealgebrasGe11 5. PropSertiesofLiealgebrasRZ13 6. DerivXationsandskrew-symmetricendomorphisms14 7. LiestructuresonCp;cmmi6n 唹-gradedmoSdules `18 ReferencescD2034 InthestudyofLiegroups,ofalgebraicgroupsorofformalgroups,theconceptofLieOalgebrasplarysacentralrole.OTheseLiealgebrasconsistoftheprimitiveelements.It7isdiculttoinrtroSduceasimilarconceptforquantumgroups.ManyimpSortantquanrtumZgroupshavebraidedHopfalgebrasasbuildingbloScks.mAswewillseemostprimitivreYelementsliveinthesebraidedHopfalgebras.In[P1 ދ]and[P2]wreintroSducedthe:conceptofbraidedLiealgebrasforthistrypSeofHopfalgebras.(InthispaperwrewillgivreasurveyofandamotivXationforthisconcepttogetherwithsomeinterestingexamples. By;thewrorkofYVetter[Y]weknowthatthecategoryofYVetter-Drinfel'dmoSdulesiscinasensethemostgeneralcategoryofmoSdulescarryinganaturalbraidingonthetensorpSorwerofeacrhmodule(insteadofasymmetricstructure).Thestudyofalgebraic )structuresinsucrhacategoryisageneralizationofthestudyofgroupgradedalgebraicBstructures.AtWVewilldescribSethebraidstructureinthecategoryofYetter-Drinfel'dmoSdules,theconceptofaHopfalgebrainthiscategoryV,andexplainthereasonwhrywewanttostudysuchbraidedHopfalgebras. ff< B ': cmti10Date[:UUApril15,1997. 1991MathematicsSubje}'ctClassi cation.PrimaryUU16A10. ],o cmr91*7 2vBODOTP:AREIGISV Oneofthebigobstaclesinthistheoryisthefact,1thatthesetofprimitivreele-menrts}Pƹ(HV)ofabraidedHopfalgebraHkdoSesnotformaLiealgebraintheordinaryorc6slighrtlygeneralizedsense.WVewillshow,Zhowever,thatc6thereisstillanalgebraicstructureonPƹ(HV)consistingofpartiallyde nedn-arybracrketopSerations,{satisfy-ingcertaingeneralizationsoftheanrti-symmetryandJacobirelations.WVecallthisstructureάabraidedLiealgebra./ThiswillgeneralizeordinaryLiealgebras,ELiesupSeralgebras,AhandLiecoloralgebras.o{FVurthermorewrewillshowthattheuniversalen-vrelopinggalgebraofabraidedLiealgebraisagainabraidedHopfalgebraleadingusbacrktoquantumgroups. PrimitivreelementsofanordinaryHopfalgebraLareelementsx9 !", cmsy102Lsatisfying(x)"=x 1+1 x. The1setofprimitivreelementsPƹ(L)ofLformsaLiealgebrainducedIbrytheLiealgebrastructure[x;yn9]o:=xy#LyxIonL.gInfactonevreri esthat([x;yn9])UR=[x;yn9] 1+1 [x;yn9]ifx;yË2URPƹ(L). SinceprimitivreelementsarecoScommutativetheycanonlygenerateacoScommu-tativreQHopfsubalgebraofL. Moregeneralelements,{skew-primitiveelementswith(x)f=x g+gn92!K cmsy80" x,WareK4neededtogeneratequanrtumgroupsorgeneral(non-commrutative~noncoScommutative)Hopfalgebras.Buttheskew-primitiveelementsdonotformaLiealgebraanrymore. ManryPquantumgroupsareHopfalgebrasofthespSecialformLUR=kgGpn?HB=URkG HwhereTHB@isabraidedgradedHopfalgebraorverTacommrutativeTgroupG[LWs,Ma#,R ,S ].InthissituationtheprimitivreelementsofHVareskew-primitiveelementsofL.Sothe+7structureofabraidedLiealgebraonthesetofprimitivreelementsinHinducesasimilarstructureonasubsetoftheskrew-primitiveelementsofL. ThecenrtralidealeadingtothestructureofbraidedLiealgebrasistheconceptofsymmetrization.lFVor^,anrymoSdulePinthecategoryofYetter-Drinfel'dmoSdulesthen-th(tensorpSorwer(PƟ2n rofPqhasanaturalbraidstructure.WVeconstructsubmodulesPƟ2nJ()PƟ2n ǹforqanrynonzeroYinthebase eldkg,sthatcarrya(symmetric)SnP-structure.ThisisessenrtiallyaneigenspaceconstructionforafamilyofopSerators.TheLie)algebramrultiplicationswillbSede nedontheseSnP-modulesPƟ2nJ()forprimitivren-throSotsofunitry. Inthegroupgradedcase,=Fthereisafairlyexplicitconstructionofthesesymmetriza-tions.0PInthelastsectionofthispapSerwredescribethemintheCpn 唹-gradedcaseforacyclicgroupofprimepSorwerorder. Apart8fromtheexplicitexamplesofbraidedLiealgebraswregavein[P1 ދ]weshowedin<[P2 ދ]thatthesetofderivXations-}h! cmsl12Der@(A)ofanalgebraAinYD2UVKbK formsabraidedLiealgebra.$Thisisbasedontheexistenceofinnerhom-objectsinYD2UVKbK.InTheorem6.3pwrewillshowthatthecategoryYD2UVKbKisaclosedmonoidalcategoryV. WepalsoconstructManotherlargefamilyofbraidedLiealgebrasconsistingofskrew-symmetricendomorphismsofaYVetter-Drinfel'dmoSdulewithabilinearform. ThisgeneralizestheconstructionofLiealgebrasofclassicalgroups. I+2wish+BtothankPreterSchauenburgforvXaluableconversationsespSeciallyonTheo-rems2.2and6.3. &7 SKEW-PRIMITIVE!ELEMENTSOFQUANTUMGR9OUPSANDBRAIDEDLIEALGEBRAS o3Vj@1.'mQuantumgroups,Yetter-Drinfel32'dalgebras,andG-gradedalgebras QuanrtumgroupsarisefromdeformationsofuniversalenvelopingalgebrasofLiealgebras.8Theyoftenharvethefollorwingform. LetCK|=\kgGbSethegroupalgebraofacommrutativeCgroup. CLetH0beaHopfalgebraMinthecategoryofYVetter-Drinfel'dmoSdulesorverMKܞ. )ThenthebiproductK?1HXisaHopfalgebra[R,Maz;,FME@],Pwhicrhingeneralisneithercommutativenory2coScommrutative.~MoregenerallyquantumgroupsareHopfalgebrasoftheformH ?KF?H|{Ycmr8+  whereH+andHaredualtoeacrhother[LWs,S q]. WVe]inrvestigatethequestionwhattheprimitive(Lie)elementsofthesequantumgroupsareandwhethertheycarryaspSeci cstructure(ofaLiealgebra). Let-us rstinrtroSducetheconceptofYVetter-Drinfel'dmodulesorver-aHopfalgebraKwithHbijectivreantipSodeH(see[M =]10.6.10).sA.@ cmti12Yetter-Drinfel'd0moffduleorcrossedmoffduleporveraHopfalgebraK^isavrectorspaceM.whichisarightKܞ-moSduleandarighrtKܞ-comoSdulesuchthat (1)SA#u cmex10Xd[(xc)[0] # (xc)[1] >=URXx[0]c(2)$ S׹(c(1) \|)x[1] xc(3) ˠforTallxVw2M,8andTallcVw2Kܞ.:HereTwreusetheSweedlernotation(c)Vw=P"c(1) c(2)withz:K '!'KW ùKdands2(x)=Px[0] < ùx[1]fwith0:MT nm!mM ùKܞ. WThe$YVetter-Drinfel'dmoSdulesformacategoryYD2UVKbK Źintheobrviousway(morphismsaretheKܞ-moSdulehomomorphismswhicrharealsoK-comoSdulehomomorphisms). The>.mostinrterestingstructureonYD2UVKbK FisgivenbyitstensorproSducts.3rItiswellknorwnٙthatthetensorproSductM: MVN}oftwovectorspaceswhichareKܞ-moSdulesisagainaKܞ-moSdule(viathecomrultiplicationordiagonalofK).iIfMdandNareKܞ-comoSdulesthentheirtensorproductisalsoaKܞ-comodule(viathemrultiplicationofPKܞ).jvIfMhandNareYVetter-Drinfel'dmoSdulesorverPKܞ,ithentheirtensorproductis9aYVetter-Drinfel'dmoSdule,]too.So9withthistensorproSductYD2UVKbK LQisamonoidalcategoryV. AnwmonoidalnortensorproSductstructureonanarbitrarycategoryC!˹allorwstode nethenotionofanalgebraAwithamrultiplicationrUR:A AURn!1AwhichisassoSciativeandunitary(bryulŹ:km!Z4A). Similarlyonecande necoalgebrasinC5.Thereis,horwever,aH.problemwithde ningabialgebraorHopfalgebraH5inC5. QrInthecompatibilitryconditionbSetweenmultiplicationandcomultiplicationofH U}Xf(hh 09)(1)$ (hh 0)(2)ι=URXh(1)$h 0ڍ(1) h(2)h 0ڍ(2) ˠoneRusesintheformationoftherighrthandsidePh(1) th20뀍(1) h(2)h20뀍(2)ι=UR(r r)(1  Ź1)(Ph(1)A h(2) h20뀍(1) h20뀍(2) \|)x=(r r)(1  1)( )(h h209)baswitcrhorexcrhangefunctiono:URH HB\3!H HinthecategoryC5.O7 4vBODOTP:AREIGISV Thereʁexistssucrhanontrivialmorphism)s:VM CN:,!N M einʁthecategoryYD2UVKbK {ofYVetter-Drinfel'dmoSdules.8Itisgivrenby(2)VnEo:URM N63m n7!Xn[0] # mn[1] >2N M:S㍹ThisisanaturaltransformationwiththeadditionalpropSertryofabraidingwhichwewilldiscusslater.~Sowreknownowhowtode neaHopfalgebraHpֹinYD2UVKbK.~ObservethatitheseHopfalgebrasarenotordinaryHopfalgebrassincetheconditionforthecompatibilitryDbSetweenmultiplicationandcomultiplicationinvolvesthenewswitchmorphism. Givren;aHopfalgebraH)inYD2UVKbK ׹wecande nethebiproSductK{?H)[R]betrweenK)andHV.*,TheunderlyingvrectorspaceisthetensorproSductK-* PH.*,WVedenotetheelemenrtsbyPSci hi,=:URPcidڹ#hi.8The(smashproSduct)mrultiplicationisgivenby(3){#(c#h)(c 09#h 0)UR:=Xcc 0ڍ(1) \|#(hc 0ڍ(2))h 0andthe(smashcoproSduct)comrultiplicationisgivenby(4)YX(c#h)UR=X(c(1) \|#(h(1))[0] x) (c(2) \|(h(1))[1]#h(2)):ԍIfK#isaHopfalgebraandH۹isaHopfalgebrainYD2UVKbK 7thenK {HbSecomesaHopfalgebrawiththismrultiplicationandcomultiplication,calledthebiprffoductK?SHv(see[R,M,Ma]). WVeƨwillbSemainlyinrterestedinthecasewhereK\&=kgGisthegroupringofa[@commrutativegroup.ItiswellknownthatthekgG-comoSdulesarepreciselytheG-gradedvrectorspaces([M =]Example1.6.7).[WVedenotethiscategorybyM2k6G *.[FVromthecomoSdulekstructureonaG-gradedvrectorspaceM6=URh2G9Mh\wecanconstructakgG-moSdulepstructuresucrhthatMҹbecomesaYVetter-Drinfel'dmodule.MThisconstructiondepSendsonabicrharacterUR:GGURn!1kg2 sgivenbyagrouphomomorphismUR:G /2@cmbx8Z'G!nkg2.8ThenitiseasytovrerifythatM+isinYD2UVk6Gyk6G(withthekgG-moSdulestructuredmhgË:=UR(h;gn9)mhforhomogeneouselemenrtsmh2V Mhe,/h2GandgD2V G.:&Soanybicharacterde nesa-functorM2k6Gp.!!6YD2UVk6Gyk6G .qThisfunctorpreservrestensorproSducts.InparticularanryalgebraorcoalgebrainM2k6G¹isalsoanalgebraresp.coalgebrainYD2UVk6Gyk6G .5SinceM2k6GcantbSeconsideredasamonoidalsubcategoryofYD2UVk6Gyk6G3viaandthrushasaswitchmapm.:3}MDj Nta!kN Mwrem.canalsode neHopfalgebrasinM2k6G0Xandtheyarealsobpreservredbythefunctorinducedby. InthissituationturnsouttobSesimplydW(mh ng)UR=(h;gn9)ng mhe:: Althoughonemaryde neYVetter-Drinfel'dcategoriesYD2UVKbK eforarbitraryHopfal-gebrasKfwithbijectivreantipSodehenceinparticularforarbitrarygroupringskgG(where Gisnotcommrutative) theabSorve functorthatinducesYVetter-Drinfel'dmod-ules=fromkgG-comoSduleswithabicrharactercanonlybeconstructedforcommrutativegroupsG.,7 SKEW-PRIMITIVE!ELEMENTSOFQUANTUMGR9OUPSANDBRAIDEDLIEALGEBRAS o5V IfAisanalgebrainM2k6GҹthenkgG#Acarriestheinducedalgebrastructure(5)|(gn9#ahe)(g 0likrederivXations.Theprimitiveelements,chomogeneousofdegreegt2R;G,formavrectorspacePg(HV).8In[P1 ދ](aftertheproSofof3.2)and[P2]Lemma5.1wreproved Lemma 2.1.ThesetofprimitiveelementsofaHopfalgebrffaH,inYD2UVKbK gisaYetter-Drinfel'd35moffdulePƹ(HV). IfK1=URkgGandHB2M2k6GtthenPƹ(HV)=LgI{2G#Pg(H)isalsoinM2k6Gt(viathesame>bicharffacter35).ԍ In+generalandespSeciallyinHopfalgebrasoftheformKw?Hwrehavetoconsidermoregeneralconditionsforprimitivreelements.KAnelementg6=s0inaHopfalgebraL꨹(inVVec0)iscalledagrffoup-like35elementifd?(gn9)UR=g g:This@implies"(gn9)UR=1.By@(4)agroup-likreelementgË2URKde nesagroup-likeelementgn9#1UR2L=KF?HV.>Q7 6vBODOTP:AREIGISV Let@gn9;g202URLbSegroup-likreelements.8AnelementxUR2Liscalleda(gn920=h gwithrespSecttotheKܞ-comodulestructureȄ:URHB\3!H KFofHV.8Thenbry(6)w LGع(1#h)UR=1#h gn9#1+1#1 1#hhence1#hUR2P(1;gI{) (L).8SothefollorwingisamonomorphismPg(HV)UR3h7!1#h2P(1;gI{) (L):Theorem 2.2.LffetoKLbeaHopfalgebrawithbijectiveantipodeandH]JbeaHopfalgebrffa35inYD2UVKbK.fiLetLUR=KF?HV.fiFor35everygroup-likeelementgË2URKwehave{P(1;gI{) (L)UR=P(1;gI{)(Kܞ)#11#Pg(HV):oPrffoof.#RLet=URs2(h)whereȄ:H!H KisWthegivrencomoSdulestrukturofHV.ThushishomogeneousofdegreegsothatXBxn9(x)UR21#Pg(HV):B SowrehaveshownP(1;gI{) (L)URP(1;gI{)(Kܞ)#11#Pg(HV).& msam107]Corollary2.3.Lffet6SGbeacommutativegroup,7beabicharacterofG.oLetH#beaHopf35algebrffainM2k6G *.fiLetLUR=kgG?HV.fiFor35everygË2URGwehave}P(1;gI{) (L)UR=kg(g1)#11#Pg(HV):Prffoof.#RTheߊonlythingtocrheckߊisP(1;gI{) (kgG)=k(gm1). WVeߊharve(gm1)=g- g1 1 =(g-1) g+1 (g1).Conrversely ifx =P4 idgin|is inP(1;gq0*)(kgG)thenbrycomparingcoSecientsoneobtainsxUR= 0(g0j1).{ InparticularwrehaveP(1;1)(kgG?HV)UR=1#P1(H). Inmordertostudythe(gn920somekindofJacobiidenrtityV.The>(skew-)commutativityofanordinaryLiealgebra PSҹresultsfromtheactionofS2r(thesymmetricgroup)onP 7Pƹ.&InthecaseofanalgebraAmadeinrtoaLiealgebrathisskew-commutativityresultsfromthefollorwingcompSositionofmapsƍM 9[:;:]UR=rSkSymm3 :URA An!1SkSymmAν(A A)n!1AwhereLSkSymmdenotesthesetofanrti-symmetrictensorsinA ALٹandtheanti-symmetrizationproScessitself.+IngeneraltheLiemrultiplicationmustonlybSede nedonSkSymm(PLn Pƹ)sinceo[x;yn9]UR=[x y]UR=ō1[z ΍2 [x yy x]bwhereFu1۟z@2 (x yy x)UR2SkSymm0_(PLn Pƹ). This{isaspSecialcaseofthefollorwingmoregeneralobservXation._Ifa nitegroupGactsonamoSduleMvthenthemapM 3'm7!P.ԟgI{2G$Ngn9m2G-Inrv(M@)sendsanry>mUR2M5inrtothesetofG-invXariantelementsG-Inv(M@)UR=fm2Mj8gË2G:gn9m=mg:ThisproScessisonlypossiblefor nitegroupsG.IntheaborvecaseS2ĹactsonPr Pbryn9(x y)UR=y x.WVewranttousethisproScesstode neageneralizedLiealgebra.WVewill inotrestrictourselvrestobinarymultiplications,HsincetheJacobiidentityindicatesthat,higherordermrultiplicationsmightbSeofinterest,=toSo.FVurthermoregeneralizedLiemrultiplicationswillonlybSepartiallyde ned,onsubspacesofPLn ::: Pƹ. ThereasonforthefactthattheLiebracrket[x;yn9]oftrwoprimitiveelementsx;y2Pƹ(HV)isnotprimitivreinExample2.4resultsfromthefollowingobservXation.ATheopSeration{oftheswitcrhmap:LfPB  |P,!TP Pinduces{onlyanopSerationofthegroup) msbm10ZratherthanZ=(2)onPT Pƹ.\nObservrethatZi~=B2isthe2-ndbraidgroup,whereasZ=(2)UR=S2isthe2-ndsymmetricgroup. Thehswitcrhmorphism:*P P̣.! P P satis eshthe(quanrtum-)YVang-Baxterequationad( 1)(1 W)( 1)UR=(1 W)( 1)(1 W);henceitinducestheactionofthen-thbraidgroupBn OMwhicrhhastherightadjointHomayk6Bn, (kgSnP;)UR:k6Bn_M!nߟk6Sn M.So?anrybraidmoSduleM#andany=S2URkg2 BԹinducesamoSduleHom8ȟk6Bn+չ(kgSnP;M@)orver9thesymmetricgroup.$;Sincethealgebrahomomorphism!issurjectivrewegetasubmoSduleǍQqM@()UR:=Hom۟k6Bn,y(kgSnP;M)Hom۟k6Bn,y(kgBnP;M)=M:In[P2 ދ]follorwingDe nition2.3weproved2{M@()UR=fm2Mj 1 \| W2ڍi!(m)= 2m 82BnP;i=1;:::ʜ;n1gandcomputedtheactionofSn onM@()as(7)idڹ(m)UR= 1 D}i(m): SowrehavekgSnP-submoSdulesM@()YMZforeveryA2Ykg2.Sincetheyarecon-structedsimilartoeigenspacesfortheeigenrvXalues22 theyformdirectsumsinM@. IfP2 YD2UVKbK cthenPƟ2n )6=P H-:::Z H-PsʹisinYD2UVKbK candBn zTactsonPƟ2nJ.Thesym-metrization/@withrespSecttoe2}kg2 չgivresamodulePƟ2nJ()}2YD2UVKbK X([P2 ދ]/@Theorem2.5). Norw%ThenM2k6Gisthecategoryof3-gradedIsvrectorspaceswiththebraidingW(xiP 2yjf )=s2ijWyjQ< xidڹ.UCTheIshomogeneousD elemenrtsm$2PƟ2nJ()=Lzf(gq1*;:::\;gn7)w2K`y 3 cmr10-family%rgnGPgq1 . d::: dPgnforP2M2k6GharvetoE̍satisfyU%21 \|W22!(m)UR=s22ij [m=22m. ThepSossibilitiesfor=&are1;1;s;;22Tw;22Tw. BycomputingallpSossible-familieswregetforexample!ylQ(PLn Pƹ)(1)UR=(P P0)+(P0j Pƹ);;ݍT(PLn P Pƹ)(s)UR=(P1j P1 P1)(P2 P2 P2);䍒PƟ 6aʹ(s)UR=P 6ڍ1 rP 6ڍ2;~M(PLn P Pƹ)(s 2Tw)UR=0;PƟ 6aʹ(s 2Tw)UR=0: 7 SKEW-PRIMITIVE!ELEMENTSOFQUANTUMGR9OUPSANDBRAIDEDLIEALGEBRAS11VTheparticularcrhoiceofthenumbSeroftensorfactorsinthisexamplewillbecomeclearinthenextsection.8TheactionofthesymmetricgrouponthesesymmetrizationsisZEn9(xi yjf )UR=yj xid;%@1(x1j y1 z1)UR=y1j x1 z1;i.e.*theordinarypSermrutation{andthisholdsforallelementsinS3 ĹandalsoforelemenrtsinP2j P2 P2,321(x1j y1 z1 u1 v1 w1)UR=y1j x1 z1 u1 v1 w1:LM4.Q,Liealgebras LetJbP(bSeaYVetter-Drinfel'dmoduleinYD2UVKbK.tThenPƟ2nJ()isanSnP-module.tWVewillharvetoconsidermorphisms[;]UR:PƟ2nJ()n!1P?inYD2UVKbK.IIfwresuppressthesummationindexandthesummationsignthenwremaywritethebracketopSerationonelementsz5=URx1j ::: xn2PƟ2nJ()as[x1;:::ʜ;xnP]UR:=[z].8FVurthermorewrede nev(8)FxI{(1),# ::: xI{(n):=URn9 1 ʵ(z)Observre[thatthecompSonentsx1;:::ʜ;xn intheseexpressionsareinterchangedandchangeffdL accordingtotheactionofthebraidgroupresp.thesymmetricgrouponPƟ2nJ(),Ĺso=xI{(1)H I:::^: IxI{(n)%isonlyasymrbSolicexpression,nottheusualpSermrutationofthetensorfactorsgivrenbythepSermutationoftheindices. WVeneedanothersubmoSduleofPƟ2n 4whosespecialpropertieswillnotbeinrvestigated.De neSPƟ n+1&(1;)UR:=PLn PƟ nJ()\fz52URPƟ n+1&j82Sn:(1 ) 1 \| W2ڍ1!(1 )(z)UR=zg:Sincethisisakrernel(limit)constructioninYD2UVKbK, PƟ2n+1&(1;)isagainanobjectinYD2UVKbK. FVorUz5=URxy6 y19: ::: yn2PƟ2n+1&(1;)Uwrewritey19: y6::: y6yi1 x yi ::: yn:=i1AT::: 1(z).8Ifthemorphisms[]n:URPƟ2n h !GPand[]2V:PƟ22  Ч!%Paresuitablyde nedthenwrewrite(9)^#[y1;:::ʜ;[x;yidڹ];:::;ynP]UR:=[:;[:;:]2;:]ni1AT::: 1(z): NorwwehavethetoSolstogivethede nitionofabraidedLiealgebra.-"De nition4.1.AYVetter-Drinfel'dmoSdulePntogetherwithoperationsinYD2UVKbKyӹ[:;:]UR:(PLn ::: Pƹ)()=P nJ()n!1PforwallnUR2Nwandallprimitivren-throSotsofunity=S6=UR1iscalledabrffaidedɎLiealgebrffaoraLie35algebrffainYD2UVKbK {ifthefollorwingidentitieshold:-"(1)%("anti"-symmetry )d]foralln$|2N,ford]allprimitivren-throSotsofunity }6=$|1,%forallË2URSnP,andforallz52PƟ2nJ()}[z]UR=[n9(z)]; h7 12vBODOTP:AREIGISV(2)%(1.xJacffobiidentity )6=foralln2N,I"for6=allprimitivren-throSotsofunity6=1,%andforallz5=URx1j ::: xn+12PƟ2n+1&()$1JwonPoi ͣPissgivrenbythecanonicalswitchmapW(xD yn9)UR=y%} Dx.(ThesinducedactionofS23withrespSectto=S=UR1isthen n9(x y)UR=y xbƍbry(7).Thusaxiom1.gives[x;yn9]V=[(x y)]V=[y;x];theusualanrti-symmetryrelation.With,thisactionofS2윹onPƟ22aʹ(1)UR=P7 &qP^one,getstheusualJacobiidenrtityfrombSothbraidedJacobiidenrtities. 2. Let GA`=Z=2Z=f0;1gwiththenonrtrivialbicharacter(i;jӹ)A`=(1)2ijJ. InExample3.2.2.wresawthattheonlynon-trivialsymmetrizationoSccurswithrespectto=1,laprimitivre2-ndroSotofunityV.Sotheonlybracketisde nedon(P_ Pƹ)(1)UR=P7  qP.TheopSerationofB2[onP7  qP=isthebraidactionandn9(xinK yjf )UR=(1)(1)2ijJyjg xidڹ.Sojcwreget[xi;yjf ].=[n9(xify yj)]=[yj;xidڹ]jcifatleastoneofthedegreesiorjiiszeroandwreget[x1;y1]z=[n9(x1y y1)]=[y1;x1].zInthiscasewregetthenotionofLiesupSeralgebrassincethebraidedJacobiidenrtitiestranslatetotheJacobiidenrtityforLiesupSeralgebras. 3.LetGbSeanarbitrary niteabeliangroupwithbicrharactersuchthat(h;gn9)UR=(gn9;h)21 \|.̱AgainwregetonlyonebracketopSeration[]UR:P P!ePG߹andanti-symmetryandJacobiidenrtitiestranslatetothoseforLiecoloralgebras. Ӡ7 SKEW-PRIMITIVE!ELEMENTSOFQUANTUMGR9OUPSANDBRAIDEDLIEALGEBRAS13V 4.VThelexampleG2۹=Z=3Z=f0;1;2glϹwithbicrharacter(i;jӹ)2=s2ij L&wherelBisaprimitivre3-rdroSotofunityhasthreebracketopSerations ʍp[]UR:PƟ22aʹ(1)=(PLn P0)+(P0j Pƹ)URn!1PS;W[]UR:PƟ23aʹ(s)=(P1j P1 P1)(P2 P2 P2)URn!1PS;ZŹ[]UR:PƟ26aʹ(s)=P26RA1 rP26RA2  Ч!%PS:Herethe1.Jacobiidenrtitymeansforexamplel![x1;[x2;x3;x4]]+[x2;[x1;x3;x4]]+[x3;[x1;x2;x4]]+[x4;[x1;x2;x3]]UR=0;andthe2.Jacobiidenrtity:l[x;[y1;y2;y3]]UR=[[x;y1];y2;y3]+[y1;[x;y2];y3]+[y1;y2;[x;y3]]: FVurtherexplicitexamplesofbraidedLiealgebrascanbSefoundin[P1 ދ].ፍ}5.nProper32tiesofLiealgebras Thede nitionofbraidedLiealgebras,ifalthoughitgeneralizesthenotionoftheknorwnuLiealgebras,>LiesupSeralgebras,andLiecoloralgebras,gainsitsinrterestfromthepropSertiesthattheseLiealgebrasharve./WVecitesomeofthesepropertiesinbrief.õTheorem5.1.([P2 ދ]sCorollary4.2)LffetAbeanalgebrainYD2UVKbK.BThenAcarriesthestructurffe35ofaLiealgebraA2L { withthesymmetricmultiplicationsv^JX[{]UR:A nP()!A3Ede neffd35bykݹ[z]:=7KXI{2Snnr nn9(z):ۍfor35allnUR2N35andallrffoots35ofunity=S6=UR1inkg2.õ This{de nesafunctor{ pw2L :URAYD2UVKbK j !UILYD2UVKbK !from{thecategoryAYD2UVKbKofalgebrasinYD2UVKbK {toLYD2UVKbK. In[P2 ދ]Theorem5.3wreprovedTheorem5.2.For34anyalgebrffaAthemorphismpUR:A3a7!a 1+1 aUR2A Ais35aLiealgebrffahomomorphisminYD2UVKbK. AneasyconsequenceofthistheoremisTheorem:5.3.([P2 ދ]"Corollary5.4)fLffetHSbeaHopfalgebrainYD2UVKbK.Thenthesetof35primitiveelementsPƹ(HV)formsaLiealgebrffainYD2UVKbK. This de nesafunctorPa:ݛHYD2UVKbK n>! eLYD2UVKbK b8from thecategoryHYD2UVKbK b8ofHopfalgebrasQinYD2UVKbKitoLYD2UVKbK. ThisisthemostinrterestingresultwhichsolvesthequestionforthealgebraicstructureoftheprimitivreelementsofaHopfalgebrainYD2UVKbK.InparticularthebraidedLiebracrketslivealsoonthesetofskewprimitiveelemenrtsKF?Haspartiallyde nedopSerations.Theorem5.4.The@functor{ @2Le:HAYD2UVKbK @0!!LYD2UVKbK has@aleftadjointU:LYD2UVKbKfk!AYD2UVKbK,35cffalledtheunivrersalenvelopingalgebra.TheoremV5.5.C([P2 ދ]c9Theorem6.1)xTheuniversalenvelopingalgebrffaU@(Pƹ)ofabrffaided35LiealgebrffaPisaHopfalgebrainYD2UVKbK.i7 14vBODOTP:AREIGISV Thisode nesaleftadjoinrtfunctorU6:URLYD2UVKbK j !UIHYD2UVKbK ktooP:HYD2UVKbK j !UILYD2UVKbK.ٍExample5.6.wTheՖonedimensionalvrectorspacekgxconsideredasaZ=3Z-gradedspace}2withxofdegree1N2G=Z=3Z=f0;1;2g}2isabraidedLiealgebrainYD2UVk6Gyk6Gwithr[x x x]UR=0and[x x x x x x]UR=0.Theunivrersalenvelopingalgebraofkgxisk[x]=(x23),theHopfalgebradiscussedinExample1.1.K􍍑=%6.LDeriva32tionsandskew-symmetricendomorphisms WVeshallgivretwoexampleswhichshowhowtoconstructlargefamiliesofLiealgebrasfromYVetter-Drinfel'dalgebrasandfromYetter-Drinfel'dmoSduleswithabilinearforminasimilarwrayasonedoSesforclassicalLiealgebras. FVorthispurpSosewreneedinnerhom-objectsinYD2UVKbK.LetV;W/bSeYVetter-Drinfel'dmoSdulesinYD2UVKbK.8ThenHomd1(V;Wƹ)isarighrtKܞ-modulebryD(10)(fGh)(vn9)UR=f(vn9Sן 1 S(h(2) \|))h(1):ThisisequivXalenrtto(11)bi(fGh(1) \|)(vn9h(2))UR=f(vn9h(3) \|Sן 1 S(h(2)))h(1)ι=URf(vn9)h;i.e.8theevXaluationHomd1(V;Wƹ) V M!`WnisaKܞ-moSdulehomomorphism. WVede neamap0V:URHom(V;Wƹ)URn!1Hom-=(V;WLn Kܞ)bry(12)zJ0(fG)(vn9)UR:=f(v[0] x)[0] # f(v[0])[1]S׹(v[1])that\dualizes"therighrtmoSdulestructureonHomd1(V;Wƹ). Lethom`(V;Wƹ)bSethepullbacrk(inVVec0)inthediagramQIH{H.homy(V;Wƹ)HϗHomI (V;Wƹ) K{fd'0O line10-'c1z$FHom(V;Wƹ)ϗHomI (V;WLn Kܞ)D32fd&ά-Tc0H~Ǡ*FfeǠ?9H Ǡ*Ffe 4Ǡ?hҴj㍍hom1(V;Wƹ)thruscanbSewrittenas(13)ʍ"7hom8(V;Wƹ)UR=ffQ2jHomI(V;Wƹ)j9Pf0j f1V2URHom(V;Wƹ) Kܞ8vË2URV¹:jP{f0(vn9) f1V=URPfG(v[0] x)[0] # f(v[0] x)[1]S׹(v[1])UR=0(f)(vn9)gLemma6.1.hom1(V;Wƹ)35isaKܞ-cffomodule.Prffoof.#RSinceKFisfaithfully atwregetthatP獍HWhomm:8(V;Wƹ) KHHomr(V;Wƹ) KF K{fd=.ά-'1j KU䗹Homn^ (V;Wƹ) KHomr(V;WLn Kܞ) K32fd<ά-T640j KHǠ*Ffe54Ǡ?=]= KH$iǠ*Ffe$Ǡ?P)O4jW{ K7 SKEW-PRIMITIVE!ELEMENTSOFQUANTUMGR9OUPSANDBRAIDEDLIEALGEBRAS15Vagainisapullbacrk.Sowegetauniquelydeterminedhomomorphismx:Fhom(V;Wƹ)!nhom'(V;Wƹ) KFsucrhthatQ96homO(V;Wƹ)Hʍz<^6@h6@r6@|6@~|@~|RH h|(1 )1vWVecollecttheset3%n eufm10gofskew-symmetricendomor-phismsf2iend)(Vp)forwhicrhinprinciplethefollowinghold:PhfG(vn9);wRii=hv;fG(wR)iforŒallvn9;w2Vp.ɠSincethereisaswitcrhbSetweenf andv3˹intheseexpressionsthecorrectconditionis>dhfG(vn9);wRiUR=XUVhvid;fi(wR)i %whereW(f vn9)UR=Pvi fiOresp.Rq(# 1)(f v wR)UR=(1 #)( 1)(f v wR:Thrusgisthedi erencekerneloftwomorphismsinYD2UVKbK.Hencegisan(universal)objectsatisfyingthediagrammaticcondition7;d< lcircle10rAfdfefdfevfdfef„fdfeȶGfdfefwfdfevfdfeɄfdfefdfe1fdfe,v_UfdfeEf8fdfe`fdfe~f䐄fdfevfdfe;fdfe嶟efdfe fdfe6v'fdfebfNfdfe fdfef3 fdfevZfdfe)ބfdfeafdfefdfevfdfeffdfeX>fdfefc$fdfevfdfe*枪\fdfeuAfdfefflfdfevfdfed37fdfe‰Pfdfe)(sfdfeǹ {fdfeǭ)ιfdfeǜ!܄fdfeLJ)tfdfem "fdfeN)Efdfe*nhfdfe)fdfe fdfeƣ)gфfdfel}fdfe1fdfea:fdfeŬ]fdfebyfdfeZfdfeƄfdfeifdfe uT fdfeë/fdfeDRfdfeMufdfejYfdfefdfe|՟Fބfdfefdfe|Q$fdfe@Gfdfeh͟jfdfe捄fdfeBI9fdfe瞌ӄfdfeŞfdfed3fdfe:r/Pfdfeޏ(sfdfe察{fdfeιfdfe/!܄fdfetfdfe1"fdfePEfdfet/nhfdfe꜏fdfeɯfdfegфfdfe2;fdfemfdfe뭽a:fdfe]fdfefdfe R䐄fdfeBfdfeҟ;fdfeefdfeҟfdfehB'fdfefdfeRc$fdfeBfdfesҞ\fdfe)AfdfeҞfflfdfeBfdfe:/37fdfeӓ ]fe1u ]feǏW ]fed ]fe:wnfdfe9[Äfdfe79ɞfdfe3fdfe.fdfe(AASfdfe mhLfdfe&Ƅfdfe yƄfdfe`LfdfeџXfdfeߟ#fdfeuHfdfekfdfee҄fdfe}fdfe􃱟Ӿfdfel$vfdfeS;fdfe8ӟ7xfdfeW„fdfeΟwfdfe3fdfe,ۄfdfe󟡟>fdfe|'fdfeX[fdfe2,fdfe aI"fdfeϟe$fdfeџ퀬fdfeHلfdfe`qnfdfe2 Љfdfe]*fdfeQfdfe/Pfdfefdfe fdfe☟Zfdfe瓟2؄fdfe kfdfeӟrfdfefdfe ǟfdferrfdfe#oMffdfe2a(ɄfdfeB˟fdfeTfdfeg۟fdfe|Afdfe꒏yfdfeWHfdfe6 fdfemFfdfe/fdferfdfe5 fdfeUfdfevxfdfeZfdfe=fdfe㞟 fdfe ߟfdfe3q的fdfe]ofdfefdfeϟPfdfe3|5fdfebfdfeE"Imfdfe/wnfdfe_fdfeʐfdfeߟ!fdfe/fdfeߟGCfdfepԄfdfe_efdfe(/fdfe<_퇄fdfeRfdfekߟ@fdfe/j:fdfeߟ˄fdfe\fdfe_fdfe /~fdfe3_:fdfe\cfdfeߟ1fdfe/„fdfeߟSfdfe fdfeP_3ufdfe/]fdfe_톗fdfe(fdfe=ߟٹfdfe/Jfdfeߟ,ۄfdfeVlfdfeQ_fdfe/fdfe_fdfe8fdfe&Afdfe:Pfdfe8fdfe5Q.fdfe/aXfdfe'/ fdfea{fdfeQfdfeYfdfeȄfdfe_7fdfeQ5fdfea fdfe℄fdfeqafdfeQQbfdfe.eфfdfe <@fdfefdfeQfdfea񿍄fdfe_fdfe.alkfdfeQBڄfdfeIfdfe︄fdfeS'fdfeQ𜖄fdfeasfdfeItfdfeSafdfe QRfdfefdfez0fdfe,yfdfeQPfdfe>&}fdfe_QfeffAQfeffşufe]⎎ӫQfeff1Qfeff:ufe]⎎wgwրg!?V!?V!?V!?V!Ύ=!Ӣ꫍Theoremj56.6.FForTaYetter-Drinfel'dmoffduleV2URYD2UVKbK withbilinearformUR=h:;:i:V d mVX-!k theYetter-Drinfel'dmoffdulegofskew-symmetricendomorphismsisaLiesubffalgebra35ofend,(Vp)2L { 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Afdferfdfer%&Dʄfdfeqfdfeq."fdfep>3fdfep$Vfdfeoyfdfeo7fdfenqxfdfemfdfem71fdfel(fdfel-&Q=Zf8(1)2n+1R`:n!?sAigrWg2nƍ~;Vƍ2Vl@=Zf(1)2n+1RH,n]θAgؤvg2nƍ!-VƍVl/=Zf(1)2n+1}]1=r2n+1n+1]b}qAYVg2n+1ƍmVƍxV꫍In0the rstandthelasttermwreindicatedthemultiplicationr2n :XDg2n !reA0resp.r2n+1:URg2n+1!%HA.8FVurthermorewreindicatedtheuseofn whereappropriate.Md NorwletbSeaprimitiven-throSotofunityandz52URg2nP().%Then4n(n1)48R\)uF eO2=(1)2n1andnP(z)UR=4n(n1)48R\)uF eO2Cn(z)bry(7)andthede nitionofn,׿wheren {UistheimageofnunderthecanonicalmapBn -!lSnP.8Sowreget8-ʍ-H(# 1)Yz([z] v wR)UR=Yz=URP(# 1)(r2nPn9(z) v wR)Yz=URP(# 1)(r2nR 1 1)(n9(z) v wR)Yz=UR(1)2nNPR(1 #)( 1)(r2nPnR 1 1)(n9(z) v wR)Yz=UR(1)PO(1 #)( 1)(r2nR 1 1)(nPn9(z) v wR)Yz=UR(1)(1 #)( 1)([z] v wR);9Cٍhence[z]UR2g.8ThrusgisaLiesubalgebraofend(Vp). !j[B?7.jELiestructuresonCpn -gradedmodules InthissectionwreassumethatGb=Cսpt|=Z=(p2tʹ)isthecyclicgroupwithp2telemenrts6wherepUR6=26isprimeandthatthe eldkhascharacteristic6=UR2andcontainsap2tʹ-thprimitivreroSotofunitys. !WVewanttogetinformationonthenontrivialsymmetrizationsofG-comoSdules. A8bicrharacter9Cܹ:Gy *ppmsbm8Z cGg!7Ekg2 عisuniquelyde nedbrythevXalueof(1;1)as(i;jӹ)f=(1;1)2ij 2kg.WkSinceZ=(p2tʹ) ZZ=(p2t)Pf԰="Z=(p2t)thisamounrtstoahomomor-phism'Z>:Z=(p2tʹ)3gw7!s2g ï2kg2 forsomeelemenrt2kg2.SucrhahomomorphismhasauniqueimagefactorizationZ=(p2tʹ) !YZ=(p2sn<) !kg2 #withgD7! g7!s2g withthesecondthomomorphisminjectivre.Then (hasorderp2ssoitisaprimitivep2sn<-throSotofunitryV.SzWithoutHlossofgeneralitywemayassumes=tH۹andNaprimitivep2tʹ-throSotofunitryV. WVe1)sothatwrecandeterminethedomainofthepSossibleLiemultiplications. TVovdeterminePƟ2nJ()w =Laf(gq1*;:::\;gn7)q-familymg]Pgq1 nN $:::: $Pgn 0(PropSositionv3.1)wrehaveto1 ndthe-families(g1;:::ʜ;gnP)inGi.e. familieswith(gid;gjf )22=22 ٠foralli6=jӹ.SinceԚ(gid;gjf )hasorderp2rfforsomerS, 6thereisonlyarestrictedcrhoicefortheprimitiveroSotofunitryҩandforthearityofthepSossibleLiemultiplicationonPƹ._7 SKEW-PRIMITIVE!ELEMENTSOFQUANTUMGR9OUPSANDBRAIDEDLIEALGEBRAS19V WVeTharven M>1andN2kg2 ؐaprimitivren-throSotofunityV.wWewishtodeterminealld-families(g1;:::ʜ;gnP)ofelemenrtsinGY=Z=(p2tʹ)dsatisfying(gid;gjf )22 =Y22 Xiors22g8:i,rg8:jl=X22 foralli6=jӹ.>Thisamounrtstos2g8:i,rg8:jh="ԛwith"=1.>Inthecase"=+1wre(getH=Gs2g8:i,rg8:j 2Im(').Hence(theorderofisnG=p2t-:0 lyand(=s2b7for(b2Z=(p2tʹ).In]thecase"UR=1]wreget=S=URs2g8:i,rg8:j2Im](')](and=2Imo(')sincepisoSdd).*Hencethexorderofyisnf=2p2t-:0 8 andxN=s2bǹforb2Z=(p2tʹ).VRTVogetherthissarysN="s2bǹhasordernUR=(Fu33333z@2 QFu1۟z@2")p2t-:0 .9for"UR=1. Ift20u¹=0then=1.If=1thenwregetthetrivialcasePƟ21aʹ(1)=Pƹ.If=1the{primitivre2-ndroSotofunityV,ѷthen(g1;g2){isa-familyi itsatis es(g1;g2)UR=1(thevXalue1isnotpSossible)i g1g2VUR0(p2tʹ). AssumeBnorwthatt20S>R0henceb6=0.PThentherearen>2compSonenrtsgiina-family<(g1;:::ʜ;gnP).,ChoSoserepresenrtatives<giD2Nwith02.`TherearetrwocrhoicesforrF2Vf0;1g.Sowegetfamilieswith3,6,27,and54elements.FVorthecrhoicer=UR1,Am=4,"=+1thes29Tw-families(g1;g2;g3)arecompSosedofgi,=UR43+ai`9withMai,2URf0;1;2g.4mThereisforexamplethes29Tw-family(3;12;21).4mSoanrybraidedLiealgebraPninM2k6GҹwithGUR=Z=(27)hasaLieopSeration[]UR:P3j P12 P21 UZn!9P9.ʍReferences[FM]@DavidaNFischmanandSusanMontgomery*,OASchurDoubleCentr}'alizerTheoremforCotrian-