; TeX output 2001.06.04:1330 7
V Xl*N cmbx12ONٚLIEALGEBRASINTHECA TEGORYٚOF Y*YETTER-DRINFELDٚMODULES K`y
cmr10BODOUUP*AREIGIS*
#Z-
cmcsc10Abstra ct. AEThecategoryofY*etter-DrinfeldmoG dules
!",
cmsy10Y }D^ 0er cmmi7Kb K
overa ZHopfalgebra
b>
cmmi10Kb6(withbijektiveantipG odeovera eldkP)isabraidedZmonoidalcategory*.ĞIfH뚲isaHopfalgebrainthiscategorythenZtheΈprimitiveelementsofHdonotformanordinaryLiealgebraZanymore.<0W*eintroG ducethenotionofa(generalized)LiealgebrainZसY }D^G Kb K
csuchݓthatthesetofprimitiveelementsPc(H )isaLiealgebraZinqthissense.AlsotheY*etter-DrinfeldmoG duleofderivqationsofZanalgebraAinY }D^G Kb K ϲisaLiealgebra.F*urthermoreforeachLie *ZalgebrainY }D^G Kb K
XthereisauniversalenvelopingalgebrawhichturnsZoutUUtobG eaHopfalgebrainY }D^Kb Kв. f ':
cmti10Keywor}'ds:Nbraidedcategory*,߅Yetter-DrinfeldmoG dule,߅Liealge-Zbra,UUuniversalenvelopingalgebra.f1991| MathematicsSubje}'ctClassi cation| 16W30,17B70,16W55,Z16S30,UU16S40 ãx BXQ cmr121. ɚ+-
cmcsc10Introduction BहTheconceptofHopfalgebrasinbraidedcategorieshasturnedout 6toxKbSevreryimportanrtinthecontextofunderstandingthestructure6ofYquanrtumgroupsandnoncommutativenoncoScommutativeHopfal-6gebras.InparticularthewrorkofRadford[7],Ma jid[3],Lusztig[2],6andSommerh auser[8]shorwtheimpSortanceofthedecompositionof6quanrtum[groupsintoaproSductofordinaryHopfalgebrasandofHopf6algebrasinbraidedcategories.BSincejbrytheworkofYVetter[9]Hopfalgebrasinbraidedcategories6thatdarede nedonanunderlying( nite-dimensional)vrectorspacecan6bSeconsideredasHopfalgebrasinsomecategoryofYVetter-Drinfeld6moSdules,(wrewillrestrictourattentiontoHopfalgebras.g cmmi12H]inacategory6ofWYVetter-DrinfeldmoSdules1!",
cmsy10Y D2UV/2 cmmi8KRAK aoorverWaHopfalgebraKwithbijectivre6anrtipSode.BThereO+aretrwoO+structurallyinrterestingandimpSortantconceptsthat6survivreK
inthisgeneralizedsituation,jtheconceptofgroup-likeelements6टԉ ff < B Date [:UUMarch11,1996. u 1991MathematicsSubje}'ctClassi cation. Primary16W30,Z17B70,16W55, 16S30,UU16S40. 营Mo cmr9M1 *7
6M2 BODOTP:AREIGISV 6ह((gn9)UR=gB
xg,2"(g)=1)vandtheconceptofprimitivreelements((x)UR= 6x
1+1
x,"(x)UR=0).BFVorordinaryHopfalgebrasHethesetofprimitivreelementsPƹ(H V)of6HRJformsdaLiealgebra.Thisresult(inasomewhatgeneralizedform)6stillholdsforHopfalgebrasinasymmetricmonoidalcategoryV.X
This6is,horwever,nottrueforbraidedmonoidalcategories.BThereeiharvebSeenvXariousattemptstogeneralizethenotionofLie6algebrastobraidedmonoidalcategories.Themainobstructionforsucrh6ageneralizationistheassumptionthatthecategoryisonlybraidedand6notbYsymmetric.OneofthemostimpSortanrtexamplesofsuchbraided6categories,isgivrenbythecategoryofYVetter-DrinfeldmoSdulesY D2UVKRAK6हorver6aHopfalgebraKwithbijectivreantipSode6whichisalwayspropSerly6braided(exceptforK1=URkg ,thebase eld)[6].BWVeUinrtroSduceaconceptofLiealgebrasinY D2UVKRAK ^mthatgeneralizesthe6conceptswofordinaryLiealgebras,LiesupSeralgebras,Liecoloralgebras,6and(G; )-Liealgebrasasgivrenin[5].BTheyLiealgebrasde nedonYVetter-DrinfeldmoSdulesharvey<@ cmti12pffartial ly6de neffdn-arys bracrketopSerationsforeveryn=e2J
msbm10JNandevreryprimitive6n-thkroSotofunitryV.Theysatisfygeneralizationsofthe(anti-)symmetry6andJacabiidenrtities.BOur'qmainaimistoshorwthattheseLiealgebrashaveuniversalen-6vrelopingTalgebraswhichturnouttobSeHopfalgebrasinY D2UVKRAK.5Con-6vrerselythesetofprimitiveelementsofaHopfalgebrainY D2UVKRAK g.issuch6aAgeneralizedLiealgebra. WVealsogivreanexamplethatgeneralizesthe6conceptoforthogonalorsymplecticLiealgebras. I2. BraidSymmetriza32tion BहWVe2bSeginwithtrwo2simplemoduletheoreticobservXations.SThefol-6lorwingUiswellknown: 7ifA;BT[arealgebrasandM9isanA-B -bimoSdule,6then Hom 5A$w(:PS;:M@) isarighrtB -moduleforevreryA-modulePƹ.WVe6needacomoSduleanalogueofthis.BLet?|AbSeanalgebra,ԱC beacoalgebra,ԱandA
`M@2CbeanA-C ܞ-6dimoSdule,i.e. aEleftA-moduleandarighrtC ܞ-comodulesucrhthat6s2(am)UR=(a
1)(m).U6Prop` ositionLJ2.1.ZLffet֤Pxjbea nitelygeneratedleftA-module.PThen6हHomOZ-AV8(:PS;:M@)7isarightC ܞ-cffomodule7withthecffanonical7comodulestruc-6turffe35suchthat)6टf` u
cmex10<وHomUSA\0(PS;M@)
+g UR
!Hom2yA9Wl(P;M@)
C1 [!Hom.g8A5E (P;M@
C ܞ)f` N6=URHom۟A"(P;s2):)56Prffoof.ZLetpp,|{Y cmr81;:::ʚ;pnDbSeageneratingsetofP>6andletfQ2URHom۟A"(:P;:M@).6Letwmi:=@fG(pidڹ).MThenbrythestructuretheoremoncomoSdulesthemi 7
;MON!LIEALGEBRASINTHECA:TEGORY!OFQYETTER-DRINFELDMODULEScS3V 6हare4conrtainedina nitedimensionalsubScomodule4M0 7wMwhichis 6evren
acomoSduleovera nitedimensionalsubScoalgebraC0 {okC ܞ,i.e.6thediagram>b:4 yGM0:4 G+M0j
C0
l:2 fd O
line10-μ: t 8M nM
C G|32 fd > ά-: UǠ @ fe ƈLǠ? ZǠ @ fe?Ǡ?-6हcommrutes.FVurthermore M1 :==AM0 ɳisaC0-comoSdulecontainedin6M@,
esince MDisadimoSdule,andfȉ:P"P ;!gMDobrviouslyfactorsthrough6M1.8SinceM+andM1aredimoSdulesthediagramE*T <*HomU A[闹(:PS;:M1) ՎHom OA ,(:PS;:M1j
C0) IL:2 fd ά-T G2K cmsy8 >QHomVCA]'(:PS;:M@) Hom ZA >(:PS;:M
C ܞ) \32 fd % ά-T V; 1NHomI{ןAPY(:PS;:M1)
C0o|:2 fd `ά/mP ϡ 船= 4HomM-AT
(:PS;:M@)
C|32 fd f`ά bǠ @ feb,Ǡ? 㰺Ǡ @ fe Ǡ? dzǠ @ feeǠ?f6हcommrutes,Hsopeachfohasauniquelyde nedimage(fG)UR2Hom۟A"(:PS;:M@)
6C ܞ. Norw_itiseasytocheckthatthismapinducesacomoSdulestructure6onHom d1A#B(PS;M@).ѕG
msam10GۍBहTheћsecondobservXationisthefollorwing.
WVeconsiderkg -algebras6AandB .
͋Let w:B 4y!cAbSeanalgebrahomomorphism. pin-6ducesanunderlyingfunctorV b:mKA-MoSd ]!0]B -MoSd Jwithrighrtadjoint6HomOZ-BVk(A;-):B -MoSd!q #!5A-MoSdi. yYIfU{ s:BY su!2[Aissurjectivrethen6HomOZ-BVk(A;M@) !Hom.h*B5h(B ;M)Pq=*kM6isBRinjectivre,X=sothatwecaniden-6tifyHom d1B#o(A;M@)UR=fm2MjKer( )m=0g.BLetBn DQbSetheArtinbraidgroupwithgenerators idڹ,iUR=1;:::ʚ;n
16andrelationsi` ʍ ^ id j ӹ=UR jf
iKifji j jUR2; o id i+1AV i ӹ=UR i+1AV id i+1: (1)i`6Letͺ=S2URk4bSeinrvertible.Thenͺkg Bn3 i,7! i2kg Bnv
(forͺthegenerators6 iL۹ofBnP)isanalgebraautomorphismdenotedagainbry=S:URkg Bn -!lkBn.6ThisholdstruesincetherelationsforBn arehomogeneous.B(ObservrethatthisconstructioncanbSeperformedforevrerygroup6algebra ifthegroupisgivrenbygeneratorsandhomogeneousrelations.6Thegivrenconstructionofanautomorphismforevery=S2URU@(kg )de nes6agrouphomomorphismU@(kg )UR n!1Aut((kBnP)UR n!1Aut(kBnP- MoSd!).)BNorwconsiderthecanonicalquotienthomomorphismBn -!lSn Wfrom6thebraidgrouponrtothesymmetricgroup.
z*Itinducesasurjective6homomorphism
n:URkg Bn -!lkSn withkrerneli`hKer{(
)UR=h n9( W 2ڍi 1)'j'; Ë2BnP;i=1;:::ʚ;n 1i: G7
6M4 BODOTP:AREIGISV şBहThepcompSosition h:URkg BnK R! OkBnK;z
R!kSnDde nespafunctorkg BnP- MoSd 6 8/!HOkg SnP- MoSd$bry 6 ʍ;M@( )UR:hE=URHom۟k6B 0; cmmi6n,y(kg SnP;M@)hE=URfm2M@j'2 1 \| 2W 2RAi!'(m)= 22m 8'2BnP;i=1;:::ʚ;n 1g:hE=URfm2M@j 2W 2RAi!'(m)= 22'(m) 8'2BnP;i=1;:::ʚ;n 1g: <(2) 6Thisholdssincethemap
T:kg Bn U og!! kSn hasaskrernelthetwo-6sidedZidealgeneratedasakg -subspacebryf n9( 2W 2RAi * 1)'j ;'E2BnP;i=61;:::ʚ;n 1g. SofQ2 Hom xk6B n/.(kg SnP;M@)withfG(1) =m2M,Li 6 2 1 D}( n9( 2W 2RAi 1)')mUR=0יforall ;';i,ii י 2 1 D}( 2W 2RAi 1)'mUR=0forall';i,6i 2W 2RAi!'mUR= 22'mforall';i,i '2 1 \| 2W 2RAi!'(m)= 22mforall';i.BIfRtheactionofBnonM76isgivrenbyanactionofSnandthecanonical6epimorphismBn
ƅ ! GSnP,:/thentheconstructionoftheM@( )bSecomes6trivial,sinceZM@( )Ƕ=fm2Mj 2W 2RAi!'(m)= 22'(m)='(m)g=0Zif6 22 6=h1"oandM@( 1)=M(1)=M.6Observre"othatthemoSduleM@( )6depSendsonlyon 22,butthattheactionofkg Sn LonM@( )dependson .BM@(1)givresasolutionofthefollowinguniversalproblem. 6Prop` osition 2.2. For35everykg BnP-moffduleMtthesubspace6F$M@(1)UR:=fm2Mj' 1 \| W 2ڍi!'(m)=m 8'2BnP;33i=1;:::ʚ;n 1g6is akg SnP-moffduleandtheinclusionM@(1) L\!2OMLpisakBnP-moffduleho- 6momorphism,suchthatforeverykg SnP-moffduleT4andeverykBnP-moffdule6homomorphismfW:nXT v!M therffeisauniquekg SnP-modulehomomor-6phism35gË:URT ]!M@(1)suchthatthediagrffam=
V mM@(1) L/M Ԟ32 fd B ά-ƍ Q gAT vrǠ @ fe Ǡ?, g 냀 &f d?`H d?`H d?`H d?`H d?`Hd?`H*H*jq퍑6cffommutes. 6De nition2.3.WVe callthefunctor- J( )3:kg BnP- MoSd$
&ܕ!6kSnP- MoSd%2ythe6 -symmetrizationofkg BnP-moSdules.BThede nitiongivres6@ҪM@( )UR=fm2Mj' 1 \| W 2ڍi!'(m)= 2m 8'2BnP;i=1;:::ʚ;n 1g:6हTheactionofSn onM@( )isgivrenby/p 5idڹ(m)UR= 1 D} i(m); (3)6whereizresp.u iarethecanonicalgeneratorsofSn
Qresp.uBnP.Thrus 6M@( )isalsoakg BnP-submoSduleofM.qSincethefunctorM7!M( )6isharighrtadjointhfunctor,itpreservreslimits. Likeforeigenspaceswe /B7
;MON!LIEALGEBRASINTHECA:TEGORY!OFQYETTER-DRINFELDMODULEScS5V 6हharve'thatthesumofthesubspacesM@( )forallwithdi erenrt22 is 6ajdirectsum.OnM@( )wrehavetwodistinctkg SnP-structuresidڹ(m)/=6 2 1 D} idڹ(m)andi,=UR 2 1 idڹ(m),%1sinceԹand de nethesamesubspace6M@( )UR=M( )M.BThe -symmetrizationM@()ofM+canalsobSecalculatedbry6Lemma 2.4.>16M@( )UR=fm2Mj W 1ڍi W 1ڍi+1AT::: W 1ڍjv 1B W 2ڍj! jv 1B:::!
" i+1AV idڹ(m)= 2m 81ij%n 1g6whichFrffeducesthenumberofconditionstobeimposedonthema2M 6in35orffdertobeinM@( ).s6Prffoof.ZGivreninAppSendix. bGBहOneƞoftheinrterestingkg BnP-structures,forwhichwewillapplythe6previouśconstruction,oSccursonn-foldtensorproductsM@2n :=վMP
6:::En
@MLet`:::G`
>`xk6;n V.WVe6often>suppressthesummationindexandsummationsignandsimply6writez+ ==z1p
l:::]x
lzn J2M@2n4( )althoughM@2n( )doSesnotdecompose6inrtoatensorproSduct.٤O3.^bSymmetricMul32tiplicationandJacobiIdentities BहFVor~therestofthepapSerletC^bethecategoryY D2UVKRAK moSdulePtogetherwithoperations6inY D2UVKRAKQ |߹[:;:]UR:PLn
:::
Pƹ( )=P nJ( ) n!1P 6हfor FallndT2JN Fandallprimitivren-throSotsofunityGiscalledaLie6algebrffaifthefollorwingidentitieshold:H(1)\$forSalln2JN, forSallprimitivren-throSotsofunity , forall\$Ë2URSnP,andforallz52PƟ2nJ( )8S ̲[z ]UR=[n9(z)];H(2)\$forFallnUR2JN,forFallprimitivren-throSotsofunity ,andforall \$z52URPƟ2n+1&( ) `)Zn+1
[ X \:i=1jK[xid;[x1;:::ʚ;7^xi *;:::;xn+1̹]]UR=n+1
ӟX w
i=1Z [:;[:;:]](1:::ʜi)(z )UR=0;
O7
6M10 BODOTP:AREIGISV \$wherewreusethenotation(4), H(3)\$forFallnUR2JN,forFallprimitivren-throSotsofunity ,andforall\$z5=URx
y1j
:::
yn2PƟ2n+1&( 1; )wrehaveV㍑}ٶ[x;[y1;:::ʚ;ynP]]UR=n
X i=1[y1;:::ʚ;[x;yidڹ];:::;ynP] ٯ\$wherewreusethenotation(8).6Corollary4.2.rLffet8AbeanalgebrainY D2UVKRAK.
"rThenAcarriesthe6structurffe35ofaLiealgebraA2L
{
withthesymmetricmultiplications:qp6¹[{ ]UR:A nP( ) !A 3EbyC+[z ]:=X "%I{2S nnr nn9(z ): \6for35al lnUR2JN35andallrffoots35ofunity=S2URkg 2'!.7⍍6Prffoof.ZThis+isarephrasingofthe\anrti"-symmetryidentity(5)andthe6Jacobiidenrtities(6)and(7)inTheorems3.4and3.5.JHGhA5.wTheLieAlgebraofPrimitiveElements BहLetJYAbSeanalgebrainC=URY D2UVKRAK.qThenAc8
AJYisanalgebrawiththe6mrultiplication> AJ?
A
A
AUR1
r
1g 3 5!!A
A
A
AURr
rg < c!qA
A.^LetpUR:A6 8/!HOAI
AbSethemapp(x)UR:=xI
1+1
x.8 Thenp(=UR1
+
1)6isinCݹbutpisnotanalgebramorphism.tLetp2n :w]A2n 98!(AF
A)2n bSe6then-foldtensorproSductofpwithitself.6Lemma45.1.%LffetH~beaHopfalgebrainC 5. ~ThenPƹ(H V)w:=fx26H Vj(x)UR=x
1+1
xg35isaYetter-DrinfeldsubmoffduleofH inC 5.7⍍6Prffoof.ZPƹ(H V)UR=KerBm( p). BKGBहInparticularwrehaves2(x)
2Pƹ(H V)3
K`dandx
2P(H V)forall 6xUR2Pƹ(H V)andallUR2K ܞ.6Lemma 5.2. p2nP(A2n( ))UR(A
A)2nP().6Prffoof.ZByTheorem2.5p:A Ȝ!wA4
Aȹinducesp2n
Wa:A2nP( ) Ȝ!(A4
6A)2nP( ).90G6Theorem)5.3.
Lffet beaprimitiven-throotofunityandletz52URA2nP( ).6Then [p nP(z )]UR=p([z]):6Prffoof.ZIf$z=kP:k xk6;1@
:::ˮ
xk6;nU2kA2nP( )thentheequationofthe6theoremreadsas ʍ]#[`ƯPkTkpx|(xk6;1j
1+1
xk6;1
)
:::
(xk6;nR
1+1
xk6;n V)]o=c[[fPq-kxQQxk6;1j
:::
xk6;n V] "w
1+1
[4P|okxk6;1j
:::
xk6;n V]m{: #(10) 7
;MON!LIEALGEBRASINTHECA:TEGORY!OFQYETTER-DRINFELDMODULESU11V 6हWVewranttoevXaluate vʍ;[?$0PIkkN(xk6;1j
1+1
xk6;1
)
:::
(xk6;nR
1+1
xk6;n V)] S`=URP
㍟I{2S n%|r2nPn9(P
;i
(xk6;1j
1+1
xk6;1
)
:::
(xk6;nR
1+1
xk6;n V)) v6where02}_Sn opSeratesonp2nP(z )2(A
A)2nP( )0asdescribSedinsection 62.BLetx;y(2Aand W (x2
yn9)=P8*iui
vidڹ.6Then(1
x)(y
1)=6(r
r)(P
;i1
uiz
vi
1)UR=P
㍟iHeui
vi,= W (x
yn9)=P
㍟iHg(uiz
1)(1
vidڹ).6Sowrehave( v ʍ (x
1)(y
1)UR=(xy
1); (x
1)(1
yn9)UR=(x
y); (1
x)(1
yn9)UR=(1
xy); (1
x)(y
1)UR= W (x
yn9): $(11)+ vBWVeZexpandaproSduct(x1
1+1
x1):::(xn
1+1
xnP).It6proSducesKaftermrultiplication22nsummands,keachaproSductofnterms.6AtrypicalproSductis(x1_
[1)(1
x2)(x3
1):::ʜ,_someofthefactors6bSeing oftheformxj
#z1,ztheothersoftheform1
xjf
.M?TVoevXaluate6sucrh'aproSductweusetheruleofmultiplicationinAl
A'givenby6rA
A-=UR(r
r)(1
1).BTVo^)explainthefollorwingcalculationweconsiderasanexamplethe6proSduct(x1Vr
n1)(1
x2)(x3
1)(1
x4)(x5
1).5Itiscalculatedwith6thefollorwingbraiddiagram7; ट fd fe B fd fe ̟k fd fe ;ޔ fd fe @ fd fe ԟq fd fe
j fd fe 꽹0 fd fe V fd fe |j fd fe dߡv fd fe r fd fe z, fd fe h=
˄ fd fe T0 fd fe @TS fd fe *Du܄ fd fe fd fe Ο҄ fd fe fٖ fd fe ä fd fe ań fd fe Ɵ9 fd fe gW fd fe F4v\ fd fe #TE fd fe ΄ fd fe 5Ą fd fe @ fd fe bB fd fe _d"ʄ fd fe 3۟= fd fe X fd fe ؠr fd fe H fd fe wo fd fe ߂wn fd fe ߃xN fd fe ߅%> fd fe ߉+݄ fd fe ߎ& fd fe ߔ證 fd fe ߜf膐 fd fe ߥ` fd fe ߰Z: fd fe s fd fe fd fe fd fe ^΄ fd fe )/ fd fe n`
fd fe #=_ fd fe 9" fd fe Pf fd fe i( fd fe d fd fe fd fe w5 fd fe ۠Wބ fd fe 9 fd fe 2 fd fe @ fd fe dx, fd fe 16 fd fe r奺 fd fe 剸 fd fe n0 fd fe /S fd fe \b8n fd fe Ɵ S fd fe v fd fe 뵟닄 fd fe ߂ fd fe ߃C fd fe ߇l fd fe ߍrޖ? fd fe ߕЄ fd fe ߠra fd fe ߭ fd fe < fd fe f fd fe ߏ fd fe ߹6 fd fe rDŽ fd fe - X fd fe Kr5 fd fe k_z fd fe fd fe ಜ fd fe - fd fe fd fe /r/O fd fe ]X fd fe rq fd fe fd fe Փ fd fe .$ fd fe h( fd fe ⥂RF fd fe r{ׄ fd fe %h fd fe ir fd fe 㯂 fd fe " fd fe BK fd fe u= fd fe ߂΄ fd fe 1_ fd fe टwn fd fe tM݄ fd fe $L fd fe fd fe ͤ* fd fe 觙 fd fe ~ fd fe tTw fd fe ꔤ* fd fe tU fd fe iĄ fd fe P3 fd fe 5焢 fd fe [ fd fe 1 fd fe t fd fe 鰤^ fd fe ẗ́ fd fe _< fd fe 3a fd fe 8 fd fe fd fe fd fe ltg fd fe 4ք fd fe thE fd fe > fd fe ~# fd fe =뒄 fd fe fd fe p fd fe ktn߄ fd fe EN fd fe t fd fe , fd fe 1џț fd fe >wn fd fe =ПÄ fd fe ;ɞ fd fe 8 fd fe 3" fd fe ,AS fd fe $hL fd fe Ƅ fd fe Ƅ fd fe ՟L fd fe FX fd fe T# fd fe H fd fe k fd fe ڟ҄ fd fe 6} fd fe &Ӿ fd fe pv fd fe W fd fe =H7x fd fe !W fd fe Cw fd fe 娟 fd fe šۄ fd fe > fd fe 6' fd fe \П fd fe 7, fd fe ֟I" fd fe De$ fd fe F퀬 fd fe ل fd fe dn fd fe 6Љ fd fe ҟ* fd fe ՓQ fd fe टP fd fe Z fd fe | fd fe
Z fd fe 2 fd fe t k fd fe Hr fd fe fd fe < fd fe Urr fd fe 'Mf fd fe 6֟(Ʉ fd fe G@ fd fe Y fd fe lP fd fe A fd fe y fd fe 뮑WH fd fe z6
fd fe F fd fe fd fe fd fe 9 fd fe Y fd fe {x fd fe Z fd fe Z= fd fe fd fe T fd fe 7的 fd fe a fd fe m fd fe DP fd fe 訟|5 fd fe Xb fd fe IIm fd fe टwn fd fe ԟ fd fe dʐ fd fe T! fd fe fd fe TGC fd fe dpԄ fd fe ԟe fd fe , fd fe @ԟ퇄 fd fe Wd fd fe pT@ fd fe 다j: fd fe T˄ fd fe d\ fd fe ԟ fd fe ~ fd fe 7ԟ: fd fe adc fd fe T1 fd fe 커 fd fe TS fd fe d fd fe Tԟ3u fd fe 팤] fd fe ԟ톗 fd fe d( fd fe BTٹ fd fe J fd fe T,ۄ fd fe
dVl fd fe Uԟ fd fe fd fe ԟ fd fe =d fd fe w&A fd fe >P fd fe =V fd fe 9Ɵ. fd fe 3֟X fd fe +/ fd fe ֟{ fd fe Ɵ fd fe VY fd fe Ȅ fd fe V_7 fd fe Ɵ5 fd fe ֟ fd fe ℄ fd fe u֟ fd fe UƟb fd fe 3Veф fd fe <@ fd fe V fd fe Ɵ fd fe ֟ fd fe c fd fe 2֟lk fd fe ƟBڄ fd fe VI fd fe ︄ fd fe XV' fd fe Ɵ fd fe ֟s fd fe It fd fe W֟ fd fe ƟR fd fe V fd fe ~0 fd fe 1Vy fd fe ƟP fd fe &} fd feh fd feB fd fek fd feޔ fd fe fd feq fd feğ
j fd fey}0 fd fenПV fd feb|j fd feU(ߡv fd feF6 fd fe5̟, fd fe$
˄ fd fe0 fd fe S fd fe u܄ fd fe { fd fe ҄ fd fe *ٖ fd fe h fd fe b%ń fd fe C9 fd fe #W fd fe v\ fd fe E fd fe ΄ fd fe Ą fd fe m@ fd fe E&B fd fe ("ʄ fd fe = fd fe ȟX fd fe dr fd fe dH fd fe 3uo fd fe >wn fd fe ? fd fe D݄ fd fe I fd fe PV證 fd fe X*膐 fd fe aq` fd fe l : fd fe x7 fd fe Ɵ fd fe fd fe "΄ fd fe / fd fe 2`
fd fe ֟=_ fd fe fd fe sf fd fe %\( fd fe ?ğd fd fe [ fd fe xɟw5 fd fe dWބ fd fe k9 fd fe fd fe ֟ fd fe <, fd fe E6 fd fe m6奺 fd fe ȟ剸 fd fe Ɵn0 fd fe OS fd fe &8n fd fe F S fd fe v: fd fe y닄 fd fe > fd fe ?C fd fe CFl fd fe I6ޖ? fd fe QЄ fd fe \6a fd fe iF fd fe x< fd fe f fd fe ߏ fd fe F߹6 fd fe 6DŽ fd fe 醟 X fd fe 65 fd fe 'F_z fd fe I fd fe nಜ fd fe - fd fe F fd fe 6/O fd fe X fd fe J6q fd fe }F fd fe Փ fd fe ꆟ$ fd fe $( fd fe aFRF fd fe 6{ׄ fd fe ᆟh fd fe %6 fd fe kF fd fe " fd fe K fd fe Ku= fd fe F΄ fd fe Y_ fd fehwn fd fe8M݄ fd fe$L fd fe fd feh* fd fe~觙 fd feq~ fd feb8Tw fd fePh* fd fe<8U fd fe%Ą fd fe 3 fd fe h焢 fd fe Ӹ[ fd fe 1 fd fe 8 fd fe lh^ fd fe E8̈́ fd fe < fd fe a fd fe h8 fd fe fd fe ] fd fe (8g fd fe hք fd fe 8hE fd fe y> fd fe :# fd fe h뒄 fd fe fd fe op fd fe '8n߄ fd fe hEN fd fe 8 fd fe ?, fd fe 핟ț fd fe fei6 ] fe O fe E fe ӾzQ feff<
lcircle10 z>Q feff $u fe Ď Q feff6Q feff >u fe ]⎎6हTheSsecondandfourthfactorsarepulledorverStotherighrtandthenall6factorsaremrultipliedaccordingto(11).Thuswehave(x1
K1)(1
6x2)(x3MA
=1)(1
x4)(x5MA
1)UR=(r23
=r22)'(x1
x2
x3
x4
x5),zwhere6'UR= 3 4 2asde nedbrythegivenbraiddiagram.BWVe"prorvenowbyinductiononnthatforeveryproSduct(x1
1)(1
6x2)(x3
~1):::UwithifactorsoftheformxjM
1andn ifactorsofthe6form1
xjPthereisanelemenrt'UR2Bn sucrhthatʍRn(x1j
1)(1
x2)(x3
1)::: =UR(r i
r n i))'(x1
:::
xnP): v6हFVurthermore UiftdenotesthenrumbSer Uofpairsoffactorsf1;f2 Yinthe6proSductrG(x1t
ƹ1)(1
x2)(x3
1):::7ʍ;(x1j
1)h(1
x2)(x3j
1)::: (1
xn+1̹)UR=h=URf(r2i
r2n i))'(x1j
:::
xnP)g(1
xn+1̹)h=URf(r2i
r2n i))P 9k˹(uk6;1j
:::
uk6;n V)g(1
xn+1̹)h=UR(P
;kuk6;1j::: uk6;i
uk6;i+1:::uk6;n V)(1
xn+1̹)h=URP
㍟k (uk6;1j
1)::: (uk6;i
1)(1
uk6;i+1A\):::(1
uk6;n V)(1
xn+1̹)h=UR(r2i
r2n i+1 )P 9k˹(uk6;1j
:::
uk6;nR
xn+1̹)h=UR(r2i
r2n i+1 )('
1)(x1j
:::
xn+1̹)>76wheret,`thenrumbSeroffactorsinrevrerseposition,`doesnotcrhange,6neitherdoSesthenrumberofgenerators ig~usedintherepresenrtationof6'
1.8ThesecondpSossibilitryis)7ʍ;(x1j
1)h(1
x2)(x3j
1)::: (xn+1/t
1)h=URf(r2i
r2n i))'(x1j
:::
xnP)g(xn+1/t
1)h=UR(P
;kuk6;1j::: uk6;i
uk6;i+1:::uk6;n V)(xn+1/t
1)h=UR(P
;kuk6;1j::: uk6;i
uk6;i+1:::uk6;n 1ҹ)(1
uk6;n V)(xn+1/t
1)!'㍍ʍEƇ=UR(P
;kuk6;1j::: uk6;i
uk6;i+1:::uk6;n 1ҹ)(vk6;nR
1)(1
vk6;n+1)EƇ=UR(r2i+1
r2n i 1 )(P
;kuk6;1j
:::
uk6;n 1/z
vk6;n V)(1
vk6;n+1ҹ)EƇ=UR(r2i+1
r2n i))(
1)(P
;kuk6;1j
:::
uk6;n 1/z
vk6;nR
vk6;n+1ҹ)EƇ=UR(r2i+1
r2n i))(
1)(12n 1/t
W )('
1)(x1j
:::
xn+1̹):*㍑6हwhereyW'(x18
4:::Q
4xnP)UR=P
㍟k uk6;1
>
:::Q
uk6;n V, W (uk6;nk
xn+1̹)=Pvk6;nk
6vk6;n+1ҹ,MHand9(12n 1e,
` W )('
1)(x1d
:::`
xn
xn+1̹)۞=Piٟkiuk6;1j
`:::
6uk6;n 1
vk6;n-h
vk6;n+1ҹ.GWVe+udeterminethenrumbSer+ut( n9)ofgenerators6 ioSccurringin o=6((
1)(12n 1i
W )('
1).cWVeharvebyinduction6t(')=tn }the-nrumbSeroffactorsin(x1~
1)(1
x2)(x3
1):::in6revrerseQpSosition.Alsowehavetn+1 =URtn+r5(n i)QthenumbSeroffactors6inE(x1ʐ
1)(1
x2)(x3
1):::߸(xn+1X
1)EinrevrersepSosition. Then6t( n9)UR=t((
1)(12n 1m
W )('
1))UR=t(
1)+t(12n 1
W )+t('
1)UR=6(n i 1)+1+tn=URtn+1̹.BIfwresumupweobtain;6(x1BE
A1+1
x1):::"(xn*
1+1
xnP)UR=X iX '8:i"8(r i
Ar n i))'idڹ(x1
:::"
xnP); |t6हforcertain'i,2URBn whicrhariseintheevXaluationgivenabSove.
7
;MON!LIEALGEBRASINTHECA:TEGORY!OFQYETTER-DRINFELDMODULESU13V BहNorw letz2A2nP( ).pWVeexpandtheproSductsinr2np2nz=֟Pk;(xk6;1
6ह1+1
xk6;1
):::Bd(xk6;nd8
1+1
xk6;n V).EacrhoftheseproSductsinthe6sumistreatedinthesamewrayasdescribSedaborve.8Using(3)wreget*ޔȍLr2nPp2n(z ) =URP
㍟k (xk6;1j
1+1
xk6;1
)::: (xk6;nR
1+1
xk6;n V) =URP
㍟k P Xi$0P/k'8:i8(r2i
r2n i))'idڹ(xk6;1j
:::
xk6;n V) =URP
㍟iHeP ֠'8:i'
(r2i
r2n i))'idڹ(z ) =URP
㍟iHeP ֠'8:i'
(r2i
r2n i)) 2 t('8:i,r)i'idڹ(z )*ޔ6wherePi,2URSn farethecanonicalimagesofthe'i2URBn fandt('idڹ)isthe6nrumbSeroffactors jPintherepresenrtationof'idڹ.BThisgivresus,3r[p2nP(z )] >=URP
㍟I{2S n%|r2nPp2nn9(z ) >=URP
㍟P şi%{P0 ؟'8:i;@B 2 t('8:i,r)i'(r2i
r2n i))idn9(z )3; >=URP
㍟P şi%{f`-tP8'8:iC9$ 2 t('8:i,r)i'f` b (r2i
r2n i))n9(z )Ѝ >=URP
㍟iHecidڹ(r2i
r2n i))P 936n9(z ).9g6wherethefactorsci=!P"\'8:iX 2 t('8:i,r) H2!kg .jWVewranttoshorwthattheci6हarezeroforall0UR
B::::O
B:xnP)anditscorrespSonding'i.GThecrhosensummandis6completely6determinedbrygivingthepSositionsinf1;:::ʚ;ngofthen i6हfactorsBoftheform(1
xjf
).8TheB rstofthesefactorshas1 UFfactors6oftheform(xj?
V51)toitsrighrtwith041 8i.,Soitcontributes61 HspairsooffactorsinrevrersepSosition.6Thesecondfactoroftheform6(1 5
xjf
)conrtributes2̹(with0UR2V1i)pairsoffactorsinrevrerse6pSosition,Tand?soon.WVeobtaintUR=1+2+:::+n ipairs?inrevrerse6pSosition.ɘIfp:wreknowtheiwith08n ibO:::;21ip:then6they talsodetermineuniquelythepSositionofthefactorsoftheform6(1s
xjf
).EacrhJpartitionoftL=13+s2+:::+n i;inrtoJ(atmost)Ӎ6n iȹpartseacrhURigivesoneterm 2 t inci,=URP
㍟'8:i 2 t('8:i,r) Pandwe nd=6p(i;n i;t)apartitionsoftinrtoatmostn iapartseachURi. sSoweget ci,=URX t0p(i;n i;t) t C: ⍑6हByatheoremofSylvrester([1]Theorem3.1)wehave Za8 X ayt0qp(i;n i;t)qn9 tUU=ō(1 qn92n)(1 qn92n 1):::ʜ(1 qn92n i+1tW)Qm fe F(1 qn9i)(1 qn9i 1):::ʜ(1 qn9) w6henceci,=UR0for0H+beaHopfalgebrainC 5.Thenthesetofprimitive 6elements35Pƹ(H V)formsaLiealgebrffainC 5.6Prffoof.ZBy>Lemma5.1Pƹ(H V)isaYVetter-DrinfeldsubmoSduleofH.6Let6z2Pƹ(H V)2nP( ). Thendp([z ])=[p2n(z )]=[2n(z )]=([z])dsince6isanalgebrahomomorphism.#Hence[z ]%2Pƹ(H V).SoP(H V)isaLie6subalgebraofH V2L5..CcG6De nition=-5.5.tLetAbSeanalgebrainCcandletend(A)betheinner6endomorphismob jectofAinC 5,i.e.theYVetter-DrinfeldmoSduleend (A)6satisfyingɐC 5(XX
gA;A)PURn9=C(XJg;endG(A))ɐforallXF2URC 5.-ItcanbSeshorwn6that܍ʍ;endN'(A)UR:=ffQ2Hom(A;A)j ʼ9P7f(0)$
f(1)2URHom(A;A)
K ܞ8aUR2A: ʼP K&f(0) \|(a)
f(1)ι=URPfG(a(0))(0)$
fG(a(0))(1)S (a(1))gK06हis theYVetter-DrinfeldmoSdulewiththerequiredunivrersalpropertryV.6endI'(A)opSeratesonAbryacanonicalmapev:p end-(A)
Ap !A6हwithev(f
a)UR=fG(a).BAȷderivationfromAtoAisalinearmap(dϬ:A 7!A)2end(A)6sucrhthatw kd(ab)UR=d(a)b+(1
d)(
1)(d
a
b)6for|alla;b>2A.~]Observre|thatinthesymmetricsituationthismeans6d(ab)UR=d(a)b+ad(b).퍑BItQisclearthatallderivXationsfromAtoAformanob jectDeri.(A)in6CݹandthatthereisanopSerationDer(A)
AUR
!URA.6Corollary 5.6. Der(A)35isaLiealgebrffa.Li6Prffoof.ZLetmdenotethemrultiplicationofA.PAnendomorphismx:6A|; є!|;Ainend(A)isaderivXationi m(x
1+1
x)=xmwhere6(xE
yn9)(a
b)UR=(ev
D
Eev)(1E
0
1)(x
yL
a
b)forelemenrtsaand6b&inAandelemenrtsxandyh_inendA6(A).g\Soxo2end(A)&isaderivXation6i mp(x)UR=xm.BTVoӵshorwthatDer(A)isaLiealgebraitsucestoshowthatitis6closedlunderLiemrultiplicationsinceitisasubSob jectofend(A),which6istanalgebrainthecategoryC 5.Let\bSeaprimitivren-throotofunitryV.6LetrUR:endb(A)
end(A)UR n!1end' A(A)bSethemrultiplicationofend1(A). 7
;MON!LIEALGEBRASINTHECA:TEGORY!OFQYETTER-DRINFELDMODULESU15V BहIfSsx1;x2Ƕ2Derˏ(A)thenmp(x1)p(x2)=x1mp(x2)=x1x2mSsormore 6generallym(r2nPp2n)(x1
:::
xn)W=r2nP(x1
:::
xn)mforallx1
6:::GU
xn2URDer/(A)2nP.8Thruswegetforz52URDer/(A)2nP( )эʍoq
mp([z ]) F=URm[p2nP(z )]=Pm(r2nn9(p2n(z ))) F=URPm(r2nPp2nn9(z ))UR=Pr2nPn9(z )m=[z]m6हhence[z ]UR2Der/(A). eG K>ʹ6.MNTheUniversalEnvelopingAlgebraofaLieAlgebra BहAscin[5]wrecannowconstructtheuniversalenvelopingalgebraofa6Lie>algebraPinCsasU@(Pƹ)UR:=T(P)=Iwhere>T(P)isthetensoralgebra6orverPƹ,*whichlivesagaininC 5,*andwhereIvistheidealgeneratedbythe6relations= [z ]H. P
gr2nPn9(z)= forallz52URPƟ2nJ( ),_forallnandforallprimitivre6n-th_roSotsofunitry . ThenU@(Pƹ)clearlyisauniversalsolutionforthe6follorwinguniversalproblem@- P {>U@(Pƹ) m:2 fd D ά-ƍ ߤ7 냀 f ?`H ?`H ?`H ?`H ?`H ?`H tH tj AǠ @ feE4Ǡ?,gэ6हwhereforeacrhmorphismofLie-algebrasf״thereisauniquemorphism6ofalgebrasgXsucrhthatthediagramcommutes.Kh6Theoremr6.1.kLffet_P%beaLiealgebrainC 5.Thentheuniversalen-6veloping35algebrffaU@(Pƹ)isaHopfalgebrainC 5.6Prffoof.ZItiseasilyseenthatȄ:URP !e(U@(Pƹ)ң
U(Pƹ))2L ȾinM2k6GDgivrenby6s2(x)UR:=& fe R]ڍx A7
>1+1
& fe R]ڍx where& fe R]ڍxιisthecanonicalimageofxUR2PWinU@(Pƹ)6and@wthecounit"a:U@(Pƹ) !Okgivren@wbythezeromorphism0a:P' !k6हde nethestructureofabialgebraonU@(Pƹ)inC 5.BNorw>wewanttode neS:U@(Pƹ) =!U(Pƹ)2op+ybry>theLiehomomor-6phismS<:ePq+ !Z U@(Pƹ)2op+, NS (x)= 9xR.=HereA2op+H\isthealgebraob-ހ6tainedȤfromthealgebraAbrythemultiplicationAZH
A
g UR
!A
A uQrg UR
!A.6Thenforz52URPƟ2nJ( )wrehave&k +;S ([z ])cvA=UR щ fe M
3/[z ]ڟ= P 936r2nPn9(& fe 5]ڍz5)= P 9r2nPn92 1 ʵ(& fe 5]ڍz5):cvA=UR P 938(r2op )2nPn92 1 ʵn9(& fe 5]ڍz5)(bry(3))3j= P 9(r2op )2nPx4n(n 1)4Lʉ fe uF
eO2 C2 1 \|n9(& fe 5]ڍz5)cvA=UR x4n(n 1)4Lʉ fe uF
eO2 C[& fe 5]ڍz5]=( 1)2nP[& fe 5]ڍz]=[щ fe ՟
3/S ןn['(z ) ]%\6where Ë2URBn Glisthebraidmapgivrenbythetwistofallnstrandswith6sourcepf1;:::ʚ;nganddomainfn;:::ʚ;1g,HË=UR( 1):::ʜ( n 2:::"Oh 2 1)( n 1:::"Oh 2 1)6andj6 . h33n(n 1)33Lʉ fe uF
eO2"%Ë=UR2SnP. 7
6M16 BODOTP:AREIGISV BहHence^SisaLiehomomorphismandfactorizesthroughU@(Pƹ).
:Since 6U@(Pƹ)U@(Pƹ)
U(Pƹ)@isamorphisminCAgeneralizationofthisconceptofLiealgebrasto6the=groupgradedcaseforanoncommrutative=grouprequirestheuseof6YVetter-DrinfeldmoSdulesorverkg G. Weshorwthat(G; )-Liealgebras6are4LiealgebrasonYVetter-DrinfeldmoSdulesinthesenseofthispaper.6WVeusethenotationof[5].BLetyGbSeanabeliangroupwithabicrharacter UR:G
Kpp msbm8KZ4G n!1kg 2'!. &Let6PhbSe7akg G-comodule.-ThenPhisaYVetter-Drinfeldmoduleorver7kg G[4]6withUthemoSdulestructurexgx=
[ (h;gn9)xUforhomogeneouselemenrts6x㋹=xh I|2Mwithks2(x)=xX
h. Thebraidmapis W (xh I
yg)=6yg
xhgË=UR (h;gn9)yg
xhe,$hencelthebraidinggivrenin[5]afterExample62.3. "{7
;MON!LIEALGEBRASINTHECA:TEGORY!OFQYETTER-DRINFELDMODULESU17V BहLetp=S2URkg 2ÑbSegivren.xLet(g1;:::ʚ;gnP)bea -familyV,Hi.e.x (gid;gjf
) (gj;gidڹ)UR= 6 22.8LetQUR:=P
㍟I{2S n%|Pg7 (1)
:::
Pg7 (n)..֍6Lemma 7.1. Q35isarightSnP-moffdulebyEa0(x1j
:::
xnP)Ë=UR( ;(g1;:::ʚ;gn))xI{(1),#
:::
xI{(n)6for35Ë2URSn ۅandx1j
:::
xn2Pg q1 `
:::
Pg n .6Prffoof.ZWVejharvetoshowthecompatibilityofthisopSerationwiththe 6compSositionLJofpermrutations.-+Let ; o2URSnP.WVeuseLemma2.2of[5].6Then,EʍD-(x1
]:::nh
xnP)(n9 W )UR=]=UR(n9 ;(g1;:::ʚ;gnP))xI{ r(1)I
:::
xI{ r(n)]=UR( ;(g1;:::ʚ;gnP))( ;(gI{(1){;:::;gI{(n)iǹ))xI{ r(1)I
:::
xI{ r(n)]=UR(( ;(g1;:::ʚ;gnP))xI{(1),#
:::
xI{(n)iǹ) ]=UR((x1j
:::
xnP)n9) :.ލLGBQpbSecomesaleftSnP-modulebryn9(x1
:::
ʠ
xnP)UR=(2 1 ʵ;(g1;:::ʚ;gnP))xI{ 1
;;(1)s;
6:::FJ
#xI{ 1
;;(n)[."ThrusALĵxf(g q1*;:::;g n 7) qNK`y
3
cmr10N-family"ghPg q1
0
:::
Pg n
QisAalsoaleftSnP-6moSdule.BThispactionisconnectedwiththeactionofBnDonL
xf(g q1*;:::;g n 7) qN-ffamily%0gk=Pg q1*
6:::GU
Pg n
bbry ) 1 D} idڹ(x1j
:::
xnP)UR=i(x1j
:::
xnP) #(12)ύ6forthecanonicalgenerators iOofBn resp.8iofSnP,since)dg
2 1 D} i t(x1j
:::
xnP)UR= t=UR 2 1 D} (gid;gi+1AV)x1j
:::
xi+1
xi
:::
xnR t=UR(n9 1ԍi ʵ;(g1;:::ʚ;gnP))xR|i@I{ 1hi
;;(1)
:::
xR|i@I{ 1hi
;;(n) t=URidڹ(x1j
:::
xnP):6हInparticularwrehave!zT_w W 1ԍiqE W 1ԍi+1AT::: W 1ԍjv 1B 2W 2RAj! jv 1B:::!
" i+1AV idڹ(x1j
:::
xnP)UR=qE=UR 22n9 1ԍi ʵn9 1ԍi+1AT::: n9 1ԍjv 1B2n92RAj.=jv 1B:::!
"i+1AVidڹ(x1j
:::
xnP)RvqE=UR 22(x1j
:::
xnP);"A=6हsothatx1j
:::
xn2URPƟ2nJ( )bryLemma2.4.8ThuswehaveI X͟MEX7f(g q1*;:::;g n 7) qN-family"g ؖ$Pg q1 `
:::
Pg n URPƟ nJ( ): X6हConrverselynletPDx1
?:::J
?xn2URPƟ2n
h=LqƟf(g q1*;:::;g n 7)g>Pg q1
%
:::J
Pg n
`(with6homogeneous6summandsandassumethatoneofthesummandsisnon-6zeroinPg q1
e
*;:::
*;Pg n
Uwhere(g1;:::ʚ;gnP)isnota -familyforexample6bryo (gid;gi+1AV) (gi+1;gidڹ)x6= 22.5Theno( 2W 2RAi K 4 22)(P 9x1
:::4
xnP)ohasa 4!7
6M18 BODOTP:AREIGISV 6हnon-zeroEcompSonenrtinPg q1
:::
Pg n ,\hencePx1
:::
xn cannot 6bSeinPƟ2nJ( ).8Thisprorves؍6Prop` osition 7.2. Lffet35=S2URkg 2 ZVbegiven.fiThen ,PƟ nJ( )UR='|MEXf(g q1*;:::;g n 7)35 qN-family"g\Pg q1 `
:::
Pg n :# BहByƵLemma7.1and(12)thebracrketƵmultiplicationof[5]isaspSecial6caseofthebracrketmultiplicationofthispapSerand(G; )-Liealgebras6areLiealgebrasorverYVetter-DrinfeldmoSdules.6Example7.3.Asn9anewexampleofLiealgebraswregiveonefamilyof6examplesWof(G; )-Liealgebras.)LetG=C3=f0;1;2gWbSethecyclic6groupVCwith3elemenrts.{De nethestructureofarightkg G-moSduleon6aSkg G-comoSduleV_ù(i.e.+onaC3-gradedvrectorspaceV¹=URV0VZRV1V2)6usingF/thebicrharacter UR:C3
KZ0 C3PV.==C3V
.!5kg 2'!,g (1Z
1)UR=ڢaF/primitive63-rdroSotofunitryV,byv*/gË:=UR (degɻ(vn9)
g)vË=UR (degɻ(v);g)v forg2URG6हandhomogeneouselemenrtsvË2URVp.ՈThenV]isaYVetter-DrinfeldmoSdule.BLetpAUR:=endb(V)bSetheinnerendomorphismob jectofV8inkg G-comod%.6By|Corollary4.2AisaLiealgebra.Onevreri eseasily(see[5])that6theonlynon-zerocompSonenrtsA2nP( )forthepartialLiemultiplication6areQNkmA 2( 1)UR=A0j
(A1A2)(A0A1A2)
A0Ā6हand
A 3( s)UR=A1j
A1
A1A2
A2
A2:BहNorwleth:;:iUR:VG
V M!`kQŹbSeabilinearformonVinC 5.8WVede ne76टO%n
eufm10Og<(Vp)i,:=URffQ2Aidj8vn9;w2V ;degɹ(vn9)=j%:hfG(v);wRi= (i;j ӹ)hvn9;fG(wR)ig:6हThis#spaceisthehomogeneouscompSonenrtofOg
/ (Vp)}A#thatbecomes 6aYVetter-DrinfeldmoSdule.BFVorfQ2UROg `(Vp)0andgË2UROg(Vp)idڹ,iUR2C3,vË2Vjf
,w2Vwrehave+ʍjh[f ;gn9](v);wRi H=URh(fGg gn9f)(v);wRi H=URhfGgn9(v);wRi hgn9f(v);wRi H=UR (i;j ӹ)hvn9;gfG(wR)i (i;j ӹ)hv;fGg(wR)i H=UR (i;j ӹ)hvn9;[f ;g](wR)i;+76हhence[f ;gn9]UR2Og `(Vp)idڹ.8Analogouslyoneshorws[gn9;fG]2Og `(Vp)idڹ.BFVorko=UR1;2;3letfkx2UROg `(Vp)iO(iUR=1ori=2).8Then$7ʍ;h[f1;f2;f3](vn9); wRiUR=P
㍟I{2S q3"HhfI{(1){fI{(2)fI{(3)(vn9);wi =UR( 1)P 9I{2S q3#3 (i;i+i+j ӹ) (i;i+j ӹ) (i;j)hvn9;fI{(3){fI{(2)fI{(1)(wR)i =UR hvn9;[f1;f2;f3](wR)i; J 7
;MON!LIEALGEBRASINTHECA:TEGORY!OFQYETTER-DRINFELDMODULESU19V 6हhence7[f1;f2;f3]ت2Og (Vp)0. _Thrus7wehaveaLiealgebraOg
C0(Vp). _DepSend- 6ing"onthecrhoiceofthebilinearformthisisageneralizationofthe6orthogonalorthesymplecticLiealgebra.% LJ8. ֊Appendix 6Prffoof.ZofLemma2.4:BDe neactionsi;j
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1xmj@A}j ` 6 fe}ל `?ѭf '(i)@At? ` 6 fetrܟ `?qy%\'ʍ6हcommrutesAforallfQ:URPƟ22 Ч!%P㎹inC(wherej%=99~' ȹ(i)). Thebraid'(i)
Ccan6bSeH*givrenexplicitlyV,_butweareonlyinterestedinthefollowingspSecial6formsMʍx$l jv(i) =UR jv+1 if Ȗ^j%>URi;> i 1(i)2=UR id i 1AV;x$l jv(i) =UR j if Ȗ^j%<URi 1; i(i)2=UR id i+1 q7
;MON!LIEALGEBRASINTHECA:TEGORY!OFQYETTER-DRINFELDMODULESU21V 6हwhicrhcanbSeeasilyveri ed. BBy(16)wrehaveforallz52URPƟ2n+1&( 1; ))k W 2ڍi! i 1AT::: 1(z )UR= i 1AT::: 1(z ): #(17)*6Lemma 8.1. For35z52URPƟ2n+1&( 1; ),'2Bn ۅandj%:=99~' ȹ(i)wehave01ʍ '(i)
R i 1AT::: 1(z ) =UR jv 1B:::!
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')(z ); ;'(i)
R id:::/v 1(z ) =UR jf:::0 1(1
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' n9)(z );F '(jv) (i)
R id:::/v 1(z ) Gi=UR'(jv) jf:::0 1(1
n9)(z )= k#:::. 1(1
' n9)(z )/腍6sowreonlyhavetoshowtheserelationsforthegenerators'1= jf
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\ i 1AT::: 1(z ) =UR jf
i 1AT::: 1(z ) =UR i 1AT::: jf
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i 1AV)(z );VuX i(i) , i 1AT::: 1(z ) =UR id i+1AV i 1AT::: 1(z ) =UR id:::/v 1 i+1AV(z ) =UR id:::/v 1(1
idڹ)(z ); "7
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jf
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id:::/v 1(z ) =UR id:::/v jf
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jv+1B:::!
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jf
)(z )AforSjj%<URi 1;Q i 1(i)¨ i 1AT::: 1(z ) =UR id i 1AV i i 1 i 2AT::: 1(z ) =UR i 1AV id 2W 2RAi 1 i 2AT::: 1(z ) =UR i 1AV id i 2AT::: 1(z ) =UR i 1AV i 2AT::: 1 idڹ(z ) =UR i 1AT::: 1(1
i 1AV)(z );f i(i) , id:::/v 1(z ) =UR id i+1AV id:::/v 1(z ) =UR i+1AV id i+1AT::: 1(z ) =UR i+1AV id:::/v 1 i+1(z ) =UR i+1AT::: 1(1
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f
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