Generalized Drinfeld realization of quantum superalgebras and U_q(hat {frak osp}(1,2))

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

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tham:viXraGENERALIZEDDRINFELDREALIZATIONOFQUANTUMSUPERALGEBRASANDUq(osp (1,2))JINTAIDINGANDBORISFEIGINDedicatedtoourfriendMosheFlatoAbstract.Inthispaper,weextendthegeneralizationofDrin-feldrealizationofquantuma nealgebrastoquantuma nesu-peralgebraswithitsDrinfeldcomultiplicationanditsHopfalgebrastructure,whichdependsonafunctiong(z)satisfyingtherelation:g(z)=g(z 1) 1.Inparticular,wepresenttheDrinfeldrealizationofUq(osp (1,2))anditsSerrerelations.1.Introduction.QuantumgroupsasanoncommutativeandnoncocommutativeHopfalgebraswerediscoveredbyandThestandardde nitionofaquantumgroupisgivenasadeformationofuniversalenvelopingalgebraofasimple(super-)Liealgebrabythebasicgenera-torsandtherelationsbasedonthedatacomingfromthecorrespond-ingCartanmatrix.However,forthecaseofquantuma nealgebras,thereisadi erentaspectofthetheory,namelytheirlooprealizations.The rstapproachwasgivenbyFaddeev,ReshetikhinandTakhtajanandReshetikhinandSemenov-Tian-ShanskywhoobtainedarealizationofthequantumloopalgebraUq(g C[t,t ])viaacanon-icalsolutionoftheYang-Baxterequationdependingonaparameterz∈C.Ontheotherhand,Drinfeldgaveanotherrealizationof

thequantuma nealgebraUq( g)anditsspecialdegenerationcalledtheYangian,whichiswidelyusedinconstructionsofspecialrepresentationofa nequantumInDrinfeldonlygavethereal-izationofthequantuma nealgebrasasanalgebra,andasanalgebrathisrealizationisequivalenttotheapproach

tainGaussdecompositionforthecaseofUq(gl abovethroughcer-

(n)).Certainly,themost

importantaspectofthestructuresofthequantumgroupsisitsHopfalgebrastructure,especiallyitscomultiplication.Drinfeldalsocon-structedanewHopfalgebrastructureforthislooprealization.Thenewcomultiplicationinthisformulation,whichwecalltheDrinfeld

1

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

comultiplication,issimpleandhasveryimportantapplications[DM]

[DI2].

In[DI],weobservethatintheDrinfeldrealizationofquantuma ne (n)),thestructureconstantsarecertainrationalfunc-algebrasUq(sl

tionsgij(z),whosefunctionalpropertyofgij(z)decidescompletelythe 2),itsDrin-Hopfalgebrastructure.Inparticular,forthecaseofUq(sl

feldrealizationisgivencompletelyintermsofafunctiong(z),whichhasthefollowingfunctionproperty:

g(z)=g(z 1) 1.

ly,wecansubstitutegij(z)byotherfunctionsthatsatisfythefunctionalpropertyofgij(z),toderivenewHopfalgebras.

Inthispaper,wewillfurtherextendthegeneralizationoftheDrinfeld 2)toderivequantuma nesuperalgebras.Asanex-realizationofUq(sl

(1,2))ample,wewillalsopresentthequantuma nesuperalgebraUq(osp

intermsofthenewformulation,inparticular,wepresenttheSerrere-lationsintermsofthecurrentoperators.

Thepaperisorganizedasthefollowing:inSection2,werecallthemainresultsin[DI]aboutthegeneralizationofDrinfeldrealization (2));inSection3,wepresentthede nitionofthegeneralizedofUq(sl

Drinfeldrealizationofquantumsuperalgebras;inSection4,wepresent (1,2)).theformulationofUq(osp

2.

n).In[DI],wederiveageneralizationofDrinfeldrealizationofUq(sl

ForthecaseofUq(sl2),we rstpresentthecompletede nition.

Letg(z)beananalyticfunctionssatisfyingthefollowingpropertythatg(z)=g(z 1) 1andδ(z)bethedistributionwithsupportat1.De nition2.1.Uq(g,fsl2)isanassociativealgebrawithunit1andthegenerators:x±(z), (z),ψ(z),acentralelementcandanonzerocomplexparameterq,wherez∈C . (z)andψ(z)areinvertible.In

2

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

termsofthegeneratingfunctions:thede ningrelationsare

(z) (w)= (w) (z),

ψ(z)ψ(w)=ψ(w)ψ(z),

(z)ψ(w) (z)ψ(w) 1 1=g(z/wq c)

2c±1±)x(w),

z ψ(z)x±(w)ψ(z) 1=g(w/zq

q 1 1q

x±(z)x±(w)=g(z/w)±1x±(w)x±(z).δ(z2c) δ(2c),

Theorem2.1.ThealgebraUq(g,fsl2)hasaHopfalgebrastructure,whicharegivenbythefollowingformulae.

Coproduct

(0) (qc)=qc qc,

(1) (x+(z))=x+(z) 1+ (zq

(3) ( (z))= (zq

(4) (ψ(z))=ψ(zqc22c12),),),c2

2

wherec1=c 1andc2=1 c.

Counitε

ε(qc)=1ε( (z))=ε(ψ(z))=1,

ε(x±(z))=0.

Antipodea

(0)a(qc)=q c,

(1)a(x+(z))= (zq

(3)a( (z))= (z) 1,

(4)a(ψ(z))=ψ(z) 1.

Strictlyspeaking,Uq(g,fsl2)isnotanalgebra.Thisconcept,whichwecallafunctionalalgebra,hasalreadybeenusedbefore[S],etc.

3c2) 1,

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

2)[Dr2]asaHopfalgebraTheDrinfeldrealizationforthecaseofUq(sl

isdi erent,anditanalgebraandHopfalgebrade nedwithcurrentoperatorsintermsofformalpowerseries.

Letg(z)beananalyticfunctionsthatsatisfyingthefollowingprop-ertythatg(z)=g(z 1) 1=G+(z)/G (z),whereG±(z)isananalyticfunctionwithoutpoles exceptat0or∞andG±(z)havenocommonzeropoint.Letδ(z)=n∈Zzn,wherezisaformalvariable.

De nition2.2.ThealgebraUq(g,sl2)isanassociativealgebrawithunit1andthegenerators:a¯(l),¯b(l),x±(l),forl∈Zandacentralelementc.Letzbeaformalvariableand

x±(z)=

(z)=

and

ψ(z)= l∈Z x±(l)z l,a¯(m)z m]exp[ (m)z m=exp[m∈Zm∈Z≤0 a¯(m)z m]m∈Z>0

m∈Z ψ(m)z m=exp[m∈Z≤0 ¯b(m)z m]exp[m∈Z>0 ¯b(m)z m].

Intermsoftheformalvariablesz,w,thede ningrelationsarea(l)a(m)=a(m)a(l),

b(l)b(m)=b(m)b(l),

(z)ψ(w) (z)ψ(w) 1 1=g(z/wq c)

2c±1±)x(w),

z ψ(z)x±(w)ψ(z) 1=g(w/zq

q 1 1q

G (z/w)x±(z)x±(w)=G±(z/w)x±(w)x±(z),δ(z2c) δ(2c),

wherebyg(z)wemeantheLaurentexpansionofg(z)inaregionr1>|z|>r2.

Theorem2.2.ThealgebraUq(g,sl2)hasaHopfalgebrastructure.Theformulasforthecoproduct ,thecounitεandtheantipodeaarethesameasgiveninTheorem2.1.

Here,onehastobecarefulwiththeexpansionofthestructurefunc-tionsg(z)andδ(z),forthereasonthattherelationsbetweenx±(z)andx±(z)aredi erentfromthecaseofthefunctionalalgebraabove.

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In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

Example2.1.Letg¯(z)beaananalyticfunctionsuchthatg¯(z 1)= z 1g¯(z).Letg(z)=q 2¯g(q2z)

g(z/wqc),

(z)x±(w) (z) 1=g(z/wq 1

2c) 1x±(w),

{x+(z),x (w)}=1

wq c)ψ(wq1c

wq) (zq1

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

Accordinglywehavethat,forthetensoralgebra,themultiplicationisde nedforhomogeneouselementsa,b,c,dby

(a b)(c d)=( 1)[b][c](ac bd),

where[a]∈Z2denotesthegradingoftheelementa.

Similarlywehave:

Theorem3.1.ThealgebraUq(g,fs)hasagradedHopfalgebrastruc-ture,whosecoproduct,counitandantipodearegivenbythesamefor-mulaeofUq(q,fsl2)inTheorem2.1.

AsforthecaseofUq(g,fsl2)isnotagradedalgebrabutratheraagradedfunctionalalgebra.

Let

g(z)=g(z 1) 1=G+(z)/G (z),

whereG±(z)isananalyticfunctionwithoutpolesexceptat0or∞andG±(z)havenocommonzeropoint.

De nition3.2.ThealgebraUq(g,s)isZ2gradedassociativealgebrawithunit1andthegenerators:a¯(l),¯b(l),x±(l),forl∈Zandacentralelementc,wherex±(l)aregraded1(mod2)andtherestaregradedo(mod2).Letzbeaformalvariableand

x(z)=

(z)=

and

ψ(z)= ±l∈Z x±(l)z l,a¯(m)z m (m)z m=exp[m∈Zm∈Z≤0 ]exp[m∈Z>0 a¯(m)z m]

m∈Z ψ(m)z m=exp[m∈Z≤0 ¯b(m)z m]exp[m∈Z>0 ¯b(m)z m].

Intermsoftheformalvariablesz,w,thede ningrelationsare (z) (w)= (w) (z),

ψ(z)ψ(w)=ψ(w)ψ(z),

(z)ψ(w) (z)ψ(w) 1 1=g(z/wq c)

2c±1±)x(w),

z ψ(z)x±(w)ψ(z) 1=g(w/zq

q 1 1q

(G (z/w))x±(z)x±(w)= (G±(z/w))x±(w)x±(z),

6δ(z2c) δ(2c),

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

wherebyg(z)wemeantheLaurentexpansionofg(z)inaregionr1>|z|>r2.

TheaboverelationsarebasicallythesameasinthatofDe nition2.2excepttherelationbetweenx±(z)andx±(w)respectively,whichdi ersbyanegativesign.Theexpansiondirectionofthestructurefunctionsg(z)andδ(z)isveryimportant,forthereasonthattherelationsbe-tweenx±(z)andx±(z)aredi erentfromthecaseofthefunctionalalgebraabove.

Theorem3.2.ThealgebraUq(g,s)hasaHopfalgebrastructure.Theformulasforthecoproduct ,thecounitεandtheantipodeaarethesameasgiveninTheorem2.1.

Example3.1.Letg¯(z)=1.From

Uq(1,s)isbasicallythesameasUq(gl [CJWW][Z],wecanseethat

(1,1)).

4.

Forarationalfunctiong(z)thatsatis es

g(z)=g(z 1) 1,

itisclearthatg(z)isdeterminedbyitspolesanditszeros,whicharepairedtosatisfytherelationsabove.Forthesimplestcase(exceptg(z)=1)thatg(z)hasonlyonepoleandonezero,wehave

g(z)=zp 1

zp2 1

z p1

zp2 1

z p1

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

Asin[DM][DK],forthecaseofquantuma nealgebras,itisveryimportanttounderstandthepolesandzerooftheproductofcurrentoperators.WewillstartwiththerelationsbetweenX+(z)withitself.Fromthede nition,weknowthat

(z p1w)(z p2w)X+(z)X+(w)= (zp1 w)(zp2 w)X+(w)X+(z).Fromthis,weknowthatX+(z)X+(w)hastwopoles,whicharelocatedat(z p1w)=0and(z p2w)=0.

Thisalsoimpliesthat

Proposition4.1.X+(z)X+(w)=0,whenz=w.

IfweassumethatUq(g,s)isrelatedtosomequantizeda nesuper- (1,2))algebra,thenwecanseethatthebestchancewehaveisUq(osp

bylookingatthenumberofzerosandpolesofX+(z)X+(w). (1,2)),weknowweneedanextraHowever,forthecaseofUq(osp

Serrerelation.Forthis,wewillfollowtheideain[FO].

LetLet

f(z1,z2)=(z1 p1z2)(z1 p2z2).

Y+(z,w)=(z p1w)(z p2w)

(z1 p1z3)(z1 p2z3)

z1 z2

z2 z3X+(z1)X+(z2)X+(z3),

F(z1,z2,z3)=(z1 p1z2)(z1 p2z2)(z3 p1z1)(z3 p2z1)(z2 p1z3)(z2 p2z3)=

f(z1,z2)f(z2,z3)f(z3,z1),

¯(z1,z2,z3)=f(z2,z1)f(z2,z3)f(z3,z1).F

LetV(z1,z2,z3)bethealgebraicvarietyofthezerosofF(z1,z2,z3).LetV(a(z1),a(z2),a(z3)),betheimageoftheactionofaonthisvariety,¯(z1,z2,z3)bewherea∈S3,thepermutationgrouponz1,z2,z3.LetV¯(z1,z2,z3).LetV¯(a(z1),a(z2),a(z3)),thealgebraicvarietyofthezerosofF

betheimageoftheactionofaonthisvariety,wherea∈S3.

Proposition4.2.Y+(z,w)hasnopolesandissymmetrywithrespecttozandw.Y+(z1,z2,z3)hasnopolesandissymmetrywithrespecttoz1,z2,z3.

8

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

Followingtheideain[FO],wewouldliketode nethefollowingconditionsthatmaybeimposedonouralgebra.

ZeroConditionI:

Y+(z1,z2,z3)iszeroonatleastonelinethatcrosses(0,0,0),andthislinemustlieinaV(a(z1),a(z2),a(z3))forsomeelementa∈S3ZeroConditionII:Y+(z1,z2,z3)iszero

crosses(0,0,0),andthislinemustlieinaV¯onatleastonelinethat

(a(z1),a(z2),a(z3))forsome

elementa∈S3

Forthelinethatcrosses(0,0,0)whereY+(z1,z2,z3)iszero,wecallitthezerolineofY+(z1,z2,z3).BecauseY+(z1,z2,z3)issymmetricwithrespecttotheactionofS3onz1,z2,z3,ifalineisthezerolineofY+(z1,z2,z3),thenclearlytheorbitofthelineundertheactionofS3isalsoanzeroline.

Remark1.ThereisasimplesymmetrythatwewouldprefertochoosethefunctionF(z1,z2,z3)todeterminethevarietyV(z1,z2,z3).Wehavethat

F(z1,z2,z3)=f(z1,z2)f(z2,z3)f(z3,z1).

LetS12

tation2bethepermutationgroupactingonz1,z2.LetS

groupactingonz2bethepermu-

2,z3.LetS21bethepermutationgroupactingon

z3,z1.Clearly,[FO]wecanchoosefromafamilyofvarietiesdeterminedbythefunctionsf(a1(z1),a1(z2))f(a2(z2),a2(z3))f(a3(z3),a3(z1))fora1∈S21,a2∈S22,a3∈S23.Foreachsuchafunction

f(a1(z1),a1(z2))f(a2(z2),a2(z3))f(a3(z3),a3(z1)),

wecanattachaorienteddiagram,whosenodsarez1,z2,z3,andthearrowsaregivenby(a1(z1)→a1(z2)),(a2(z2)→a2(z3))(a3(z3)→a3(z1)).ForexamplethediagramofF(z1,z2,z3)isgivenby

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

Diagram II It is not di cult to see that the diagram for F (z1, z2, z3 ) is symmetric in the sense that all the points are equivalent, but for the second situation, the top point z1 is di erent from the other two, in the sense that there are two arrows coming to z1, one to z3 and none to z2 . There is only one other such a diagram given by F (z1, z3, z2 ), which however comes from the S3 action on F (z1, z2, z3 ).

7So S S S S S

From the above, we have the following: Proposition 4.3. Under the action of S3 on the the family of varieties determined by the functions f (a1 (z1 ), a1 (z2 ))f (a2 (z2 ), a2 (z3 ))f (a3 (z3 ), a3 (z1 )) 1 2 3 for a1∈ S2, a2∈ S2, a3∈ S2, there are two orbits. One of the orbit consists of the two varieties determined by F (z1, z2, z3 ) and F (z1, z2, z3 ); and the rest forms another orbit. This shows that indeed we have two choices with respect the zero conditions: the Zero condition I and the Zero condition II.

We know that a zero line is always in the form z1= q1 z2= q2 z3 . Then we have Proposition 4.4. If we impose the Zero condition I on the algebra Uq (g, s), we have p1= p 2, 2 or p2= p 2 . 1 Proof. The proof is very simple. Because of the action of S3, we know that one of the line must lie in V (z1, z2, z3 ). Let assume this line to be z1= z2 q1= q2 z3 . We know immediately that q1 must be p1 or p2 . Let us rst deal with the case that q1= p1 . We also know that q2 must be either p 1 or p 1 by looking that the relations between z1 and 1 2 z3 .10

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

1 2Case1Letq2=p 1,whichimpliesthatp1mustbeeitherp1orp2.

Clearly,itcannotbep1,whichimpliesthattheimpossibleconditionp1=1

Therefore,wehavethat

2p 1=p2,

whichiswhatwewant.1Case2Letq2=p 2,whichimpliesthatp1p2mustbeeitherp1orp2.Clearly,itcannotbep1becauseitimpliesp2=1,itcannotbeneitherbep2,whichimpliesthatp1=1.

Thiscompletestheprooffor

2p2=p 1.

Similarly,ifwehavethatq1=p2,wecan,then,show

2p1=p 2.

Howeverfromthealgebraicpointofview,thetwoconditionareequivalentinthesensethatp1andp2aresymmetric.

Alsowehavethat

Proposition4.5.IfweimposetheZeroconditionIIonthealgebraUq(g,s),wehave

p1=p22,

or

p2=p21.

Howeverwealsohavethat:

IfweimposetheZeroconditionIIonthealgebraUq(g,s),thenUq(g,s)isnotaHopfalgebraanymore.

ThereasonisthattheZeroconditionIIcannotbesatis edbycomultiplication,whichcanbecheckedbydirectcalculation.

ThisisthemostimportantreasonthatwewillchoosetheZerocon-ditionItobeimposedonthealgebraUq(g,s),ly,ifwechooseV(z1,z2,z3)ortheequivalentoneswhichhasthesamediagrampresen-tationasDiagramItode nethezerolineofY+(z1,z2,z3),then,thequotientalgebraderivedfromtheZeroConditionIisstillaHopfalgebrawiththesameHopfalgebrastructure(comultiplication,counitandantipode).

11

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

Fromnowon,weimposetheZeroconditionIonthealgebraUq(g,s),andletus xthenotationsuchthat

p1=q2,

p2=q 1.

Similarly,wede ne

Y(z,w)= 1 1(z p 1w)(z p2w)

1 1(z1 p 1z3)(z1 p2z3)

z1 z2

z2 z3Wenowde netheq-Serrerelation.X+(z1)X+(z2)X+(z3),

q-Serrerelations

Y+(z1,z2,z3)iszeroontheline

z1=z2q 1=z3q 2.

Y (z1,z2,z3)iszeroontheline

z1=z2q=z3q2.

Theq-Serrerelationscanalsobeformulatedinmorealgebraicway.Proposition4.6.Theq-Serrerelationsareequivalenttothefollowingtworelations:

(z3 z1q 1)(z3 z1q3)(z1 z2q2)

X+(z3)X+(z2)X+(z1)) (z1 z1q)

((z1 z3q2)(z1 z3q)(z1q z3q 1)(z1 z2q2)z2q 1)(z3

(z1 z3)(z3 z1q2)(z1 z2q 1)

(z3 z1q)(z3 z1q 3)(z1 z2q 2)X+(z2)X+(z1)X+(z3))=0,

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

(z2 z1q 2)(z2 z1q)(z3 z1q)(z3 z1q 3)

X (z1)X (z2)X (z3) (z1 z3)(z3 z1

((z1 z3q 2)(z1 z3q 1)(z1q z3q)(z2 z1q 2)(z2 z1q)q 2)

In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g

[DI]

[DI2]

[DK]

[DM]

[Dr1]

[Dr2]

[Dr3]

[Er]

[FRT]

[FO]

[FJ]

[GZ]

[J1]

[RS]

[S]

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gebraUq[gl(m|n)(1)]anditsHopfAlgebraStructureq-alg/9703020

JintaiDing,DepartmentofMathematicalSciences,UniversityofCincinnati

BorisFeigin,LandauInstituteofTheoreticalPhysics

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