; TeX output 1999.11.03:075747 YRXQ cmr12CHAPTER2Nff cmbx12HopfffAlgebras,Algebraic,Fformal,andQuantumGroups/^o cmr9525*7 &e5.pQUANTUM!GR9OUPS^53YN cmbx125.QuantumGroupsDe nition2.5.1.vaù(Drinfel'd)A@ cmti12quantum@grffoupisanoncommrutativenoncoScom-mrutativeHopfalgebra.ǍRemark2.5.2.j6WVenshallconsiderallHopfalgebrasasquanrtumgroups. uOb-servre,?however,thatzthecommrutativezHopfalgebrasmarybSeconsideredasanealgebraicgroupsandthatthecoScommrutativeHopfalgebrasmarybeconsideredasformal^groups.TheirpropSertryasaquantumspaceorasaquantummonoidwillplaysomerole.NButoftenthe(pSossiblynonexisting)dualHopfalgebrawillharvethegeo-metrical*umeaning.Thefollorwingexamples* msbm10SL"2cmmi8qj (2)andGL͟q˹(2)willhaveageometricalmeaning.Example2.5.3.oQThesmallestpropSerquanrtumgroup,i.e.Bthesmallestnoncom-mrutativenoncoScommutativeHopfalgebra,isthe4-dimensionalalgebraeqg cmmi12H|{Ycmr84V:=URK!!", cmsy10hgn9;xi=(g 21;x 2;xg+gn9x)whicrhwas rstdescribSedbyM.Sweedler.8Thecoalgebrastructureisgivenby ;ʍp(gn9)UR=g g;PY(x)UR=x 1+g x;}[g"(gn9)UR=1;"(x)UR=0;hS׹(gn9)UR=g2"K cmsy81 ʵ(=g);US׹(x)UR=gn9x:Sinceitis nitedimensionalitslineardualH2VRA4 isalsoanoncommrutativenoncoScom-mrutativeHopfalgebra.3ItisisomorphicasaHopfalgebratoH4.InfactH4isuptoisomorphism8theonlynoncommrutative8noncoScommutativeHopfalgebraofdimension4.Example2.5.4.oQTheanealgebraicgroupSLX(n):K-cAlg#>84!5GrJljde nedbrySL(n)(A),Lthe8groupofn߮n-matrices8withcoSecienrtsinthecommutativealgebraA¹andwithdeterminanrt1,YHisrepresentedbythealgebraOUV(SL(n))%=SL(n)=K[xijJ]=(detQ(xij)1)i.e.eb~SLqz+(n)(A)PUR԰n:=K-cAlgo(K[xijJ]=(detQ(xij)1);A):SincebSL(n)(A)bhasagroupstructurebrythemultiplicationofmatrices,1therepresent-ingcommrutativealgebrahasaHopfalgebrastructurewiththediagonal=12hence(xikl)UR=$u cmex10Xxij xjvk ; эthecounit"(xijJ)UR=ij 'andtheanrtipSodeS׹(xij)UR=adjӹ(X)ij 'whereadj(X)istheadjoinrtmatrixofXFչ=UR(xijJ).8WVelearvethevreri cationofthesefactstothereader.fvWVenconsiderSLU(n)URMn=A2n-:Aacmr62 Aasnasubspaceofthen22-dimensionalanespace.6Example2.5.5.oQLetKMq(2)o=Kq qʍWa$dbcc$Xd*Zq2q==IןqI-detZq`qʍi=a20N920Zb20N920jvc20N920~̰d20N920$qz:(1)bHInparticularwrehave(detQq溹)\2=detqB detQqjand"(detQq)=1.DThequanrtumdetermi-nanrtisagrouplikeelement(see2.1.6).Norwwede neanalgebrai-SLq(2)UR:=Mq(2)=(detQqb1):ThealgebraSLq(2)represenrtsthefunctorݍHZSLWq[(2)(A)UR=fqʍXa20\b20 cc20Βd20$͟q0w2Mq(2)(A)jdetQq渟qʍa203CVb20 JIc202Jd20;qG/=1g:aٍThereisasurjectivrehomomorphismofalgebrasMq(2)4!-SLq(2)andSLq(2)isasubfunctorofMq(2).LetICXJg;Y峹bSecommrutingquantummatricessatisfyingdetq/(X)=1=detԟqҹ(Yp).SincedetK[qY(X)detQq渹(Yp) ==detq$}(XYp)forcommrutingquantummatricesweget7ܠ7 &e5.pQUANTUM!GR9OUPS^55YdetQq溹(XYp)=1,henceO{SL(q&(2)O{isaquanrtumsubmonoidofMq(2)andSLq(2)isabialgebrawithdiagonalt"qʍ Vacb ccWd!Yq-F=URqʍ *ab cUd"pq- qʍ a9 b 9cd!q,Y;tand8K"qʍ Vacb ccWd!Yq-F=URqʍ *1 0 *0 1!ꢟq,:~TVoshorwthatSLq(2)hasanantipSodewe rstde neahomomorphismofalgebrasT:URMq(2)4!1Mq(2)2op ŹbryypTğqʍ wa0)b Ucd"q.ɹ:=URqʍMd5qn9b *qn921 ʵc$ Mq(2)>xMq(2)[t]t{fd@O line10->>MGq(2)񥔟{fdЍά-H0A`ğׁ @ğ @ğ @ğ @D>@D>RH#rǠ*FfeVǠ?H`Մׁ Մ Մ Մ >> 0withtUt?7!deteq&aqʍ#a207b20$c206d20?.qH<1V:ThrusGLɭq^(2)(A)isasubsetofMq(2)(A).ObservethatfpMq(2)UR4!1GLq(2)isnotsurjectivre.Since mthequanrtumdeterminantpreservesproSductsandtheproductofinrvertibleelemenrtsAisagaininvertiblewegetGL(q(2)isaquantumsubmonoidofMq(2),hencetUR:GLq(2)4!1GLq(2) GLq(2)gwithqʍ Vacb ccWd!Yq-F=URqʍ *ab cUd"pq+ qʍ \a"b Ncdq+U׹and(t)UR=t t.fpWVeconstructtheanrtipSodeforGLq(2).8Wede neT:URMq(2)[t]4!1Mq(2)[t]2op Źbryq퍑 {Tğqʍ wa0)b Ucd"q.ɹ:=URtqʍyd49qn9b Vqn921 ʵc; 1aH1şqnqandwTƹ(t):=detq< qʍ#ba3ob#c3 K K`0=(1 )(1 E> +E Kܞ)=(1 )(E). Similarly2wreget(b 1)(Fƹ)UR=(1b )(F).FVorNK[theclaimisobrvious.Thecounitaxiomiseasilycrheckedonthegenerators.Norw&weshowthatSٹisanantipSodeforUq(slC(2)).SFirstde neS):URKhE;F;K5;Kܞ21 9i4!nUq(slC(2))2op Źbrythede nitionofSonthegenerators.8WVehave6I$K4S׹(KܞK21 9)UR=1=S(Kܞ21 9Kܞ);P2S׹(KܞEK21 9)UR=KܞEK21 9K21l=URqn922.=EK21l=URS׹(qn922.=E);US׹(KܞFK21 9)UR=KܞKFK21l=URqn922 ʵKF=S׹(qn922 ʵFƹ);6fS׹(EFLnFE)UR=KܞFEK21EK21 9KF=URKFK21KEKFEFd=ōKܞ21K[z/ ΍7qqn918ϙ=URS՟qō `KFKܞ21 `[z/ ΍7qqn91="qHx:6j SoESde nesahomomorphismofalgebrasS):URUq(slC(2))4!1Uq(slC(2)).SinceESsatis esPS׹(x(1) \|)x(2)H="(x)forallgivrengenerators,uSisaleftantipSodeby2.1.3.Symmet-ricallySѱisarighrtantipSode.wThusthebialgebraUq(slC(2))isaHopfalgebraoraquanrtumgroup.This0quanrtumgroupisofcentralinterestintheoreticalphysics.ItsrepresentationtheoryXiswrellunderstoSod. !IfXqisnotaroSotofunitythenthe nitedimensionalUq(slC(2))-moSdulesaresemisimple.oManrymorepropertiescanbefoundin[Kassel:QuanrtumGroups].^R;7 * msbm10$u cmex10"K cmsy8!!", cmsy102cmmi8g cmmi12Aacmr6|{Ycmr8@ cmti12o cmr9N cmbx12Nff cmbx12XQ cmr12O line10n\