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

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

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1GENERALIZED DRINFELD REALIZATION OF QUANTUM SUPERALGEBRAS AND U q (?osp (1,2))JINTAI DING AND BORIS FEIGIN Dedicated to our friend Moshe Flato Abstract.In this paper,we extend the generalization of Drin-feld realization of quantum a?ne algebras to quantum a?ne su-peralgebras with its Drinfeld comultiplication and its Hopf algebra structure,which depends on a function g (z )satisfying the relation:g (z )=g (z ?1)?1.In particular,we present the Drinfeld realization of U q (?osp (1,2))and its Serre relations.1.Introduction.Quantum groups as a noncommutative and noncocommutative Hopf algebras were discovered by Drinfeld[Dr1]and Jimbo[J1].The standard de?nition of a quantum group is given as a deformation of universal enveloping algebra of a simple (super-)Lie algebra by the basic genera-tors and the relations based on the data coming from the correspond-ing Cartan matrix.However,for the case of quantum a?ne algebras,there is a di?erent aspect of the theory,namely their loop realizations.The ?rst approach was given by Faddeev,Reshetikhin and Takhtajan [FRT]and Reshetikhin and Semenov-Tian-Shansky [RS],who obtained a realization of the quantum loop algebra U q (g ?C [t,t ?])via a canon-ical solution of the Yang-Baxter equation depending on a parameter z ∈C .On the other hand,Drinfeld [Dr2]gave another realization of

the quantum a?ne algebra U q (?g )and its special degeneration called the Yangian,which is widely used in constructions of special representation of a?ne quantum algebras[FJ].In [Dr2],Drinfeld only gave the real-ization of the quantum a?ne algebras as an algebra,and as an algebra this realization is equivalent to the approach

above [DF]through cer-

tain Gauss decomposition for the case of U q (?gl (n )).Certainly,the most

important aspect of the structures of the quantum groups is its Hopf algebra structure,especially its comultiplication.Drinfeld also con-structed a new Hopf algebra structure for this loop realization.The new comultiplication in this formulation,which we call the Drinfeld

1

comultiplication,is simple and has very important applications[DM] [DI2].

In[DI],we observe that in the Drinfeld realization of quantum a?ne algebras U q(?sl(n)),the structure constants are certain rational func-tions g ij(z),whose functional property of g ij(z)decides completely the Hopf algebra structure.In particular,for the case of U q(?sl2),its Drin-feld realization is given completely in terms of a function g(z),which has the following function property:

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

This leads us to generalize this type of Hopf https://www.doczj.com/doc/2016049653.html,ly,we can substitute g ij(z)by other functions that satisfy the functional property of g ij(z),to derive new Hopf algebras.

In this paper,we will further extend the generalization of the Drinfeld realization of U q(?sl2)to derive quantum a?ne superalgebras.As an ex-ample,we will also present the quantum a?ne superalgebra U q(?osp(1,2)) in terms of the new formulation,in particular,we present the Serre re-lations in terms of the current operators.

The paper is organized as the following:in Section2,we recall the main results in[DI]about the generalization of Drinfeld realization of U q(?sl(2));in Section3,we present the de?nition of the generalized Drinfeld realization of quantum superalgebras;in Section4,we present the formulation of U q(?osp(1,2)).

2.

In[DI],we derive a generalization of Drinfeld realization of U q(?sl n). For the case of U q(sl2),we?rst present the complete de?nition.

Let g(z)be an analytic functions satisfying the following property that g(z)=g(z?1)?1andδ(z)be the distribution with support at1. De?nition2.1.U q(g,f sl2)is an associative algebra with unit1and the generators:x±(z),?(z),ψ(z),a central element c and a nonzero complex parameter q,where z∈C?.?(z)andψ(z)are invertible.In

2

terms of the generating functions:the de?ning relations are ?(z)?(w)=?(w)?(z),

ψ(z)ψ(w)=ψ(w)ψ(z),

g(z/wq?c)

?(z)ψ(w)?(z)?1ψ(w)?1=

c)±1x±(w),

2

ψ(z)x±(w)ψ(z)?1=g(w/zq?1

δ(z2c)?δ(z2c) ,

q?q?1

x±(z)x±(w)=g(z/w)±1x±(w)x±(z).

Theorem2.1.The algebra U q(g,f sl2)has a Hopf algebra structure, which are given by the following formulae.

Coproduct?

(0)?(q c)=q c?q c,

(1)?(x+(z))=x+(z)?1+?(zq c1

2),

(3)?(?(z))=?(zq?c22),

(4)?(ψ(z))=ψ(zq c22),

where c1=c?1and c2=1?c.

Counitε

ε(q c)=1ε(?(z))=ε(ψ(z))=1,

ε(x±(z))=0.

Antipode a

(0)a(q c)=q?c,

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

2)?1,

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

(4)a(ψ(z))=ψ(z)?1.

Strictly speaking,U q(g,f sl2)is not an algebra.This concept,which we call a functional algebra,has already been used before[S],etc.

3

The Drinfeld realization for the case of U q(?sl2)[Dr2]as a Hopf algebra is di?erent,and it an algebra and Hopf algebra de?ned with current operators in terms of formal power series.

Let g(z)be an analytic functions that satisfying the following prop-erty that g(z)=g(z?1)?1=G+(z)/G?(z),where G±(z)is an analytic function without poles except at0or∞and G±(z)have no common zero point.Letδ(z)= n∈Z z n,where z is a formal variable.

De?nition2.2.The algebra U q(g,sl2)is an associative algebra with unit1and the generators:ˉa(l),ˉb(l),x±(l),for l∈Z and a central element c.Let z be a formal variable and

x±(z)= l∈Z x±(l)z?l,

?(z)= m∈Z?(m)z?m=exp[ m∈Z≤0ˉa(m)z?m]exp[ m∈Z>0ˉa(m)z?m] and

ψ(z)= m∈Zψ(m)z?m=exp[ m∈Z≤0ˉb(m)z?m]exp[ m∈Z>0ˉb(m)z?m]. In terms of the formal variables z,w,the de?ning relations are a(l)a(m)=a(m)a(l),

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

g(z/wq?c)

?(z)ψ(w)?(z)?1ψ(w)?1=

c)±1x±(w),

2

ψ(z)x±(w)ψ(z)?1=g(w/zq?1

δ(z2c)?δ(z2c) ,

q?q?1

G?(z/w)x±(z)x±(w)=G±(z/w)x±(w)x±(z),

where by g(z)we mean the Laurent expansion of g(z)in a region r1> |z|>r2.

Theorem2.2.The algebra U q(g,sl2)has a Hopf algebra structure. The formulas for the coproduct?,the counitεand the antipode a are the same as given in Theorem2.1.

Here,one has to be careful with the expansion of the structure func-tions g(z)andδ(z),for the reason that the relations between x±(z) and x±(z)are di?erent from the case of the functional algebra above.

4

Example2.1.Letˉg(z)be a an analytic function such thatˉg(z?1)=?z?1ˉg(z).Let g(z)=q?2ˉg(q2z)

g(z/wq c)

,

?(z)x±(w)?(z)?1=g(z/wq?1

2

c)?1x±(w),

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

w

q?c)ψ(wq1

w

q c)?(zq1

Accordingly we have that,for the tensor algebra,the multiplication is de?ned for homogeneous elements a,b,c,d by

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

where[a]∈Z2denotes the grading of the element a.

Similarly we have:

Theorem3.1.The algebra U q(g,f s)has a graded Hopf algebra struc-ture,whose coproduct,counit and antipode are given by the same for-mulae of U q(q,f sl2)in Theorem2.1.

As for the case of U q(g,f sl2)is not a graded algebra but rather a a graded functional algebra.

Let

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

where G±(z)is an analytic function without poles except at0or∞and G±(z)have no common zero point.

De?nition3.2.The algebra U q(g,s)is Z2graded associative algebra with unit1and the generators:ˉa(l),ˉb(l),x±(l),for l∈Z and a central element c,where x±(l)are graded1(mod2)and the rest are graded o(mod2).Let z be a formal variable and

x±(z)= l∈Z x±(l)z?l,

?(z)= m∈Z?(m)z?m=exp[ m∈Z≤0ˉa(m)z?m]exp[ m∈Z>0ˉa(m)z?m] and

ψ(z)= m∈Zψ(m)z?m=exp[ m∈Z≤0ˉb(m)z?m]exp[ m∈Z>0ˉb(m)z?m]. In terms of the formal variables z,w,the de?ning relations are ?(z)?(w)=?(w)?(z),

ψ(z)ψ(w)=ψ(w)ψ(z),

g(z/wq?c)

?(z)ψ(w)?(z)?1ψ(w)?1=

c)±1x±(w),

2

ψ(z)x±(w)ψ(z)?1=g(w/zq?1

δ(z2c)?δ(z2c) ,

q?q?1

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

6

where by g(z)we mean the Laurent expansion of g(z)in a region r1> |z|>r2.

The above relations are basically the same as in that of De?nition2.2 except the relation between x±(z)and x±(w)respectively,which di?ers by a negative sign.The expansion direction of the structure functions g(z)andδ(z)is very important,for the reason that the relations be-tween x±(z)and x±(z)are di?erent from the case of the functional algebra above.

Theorem3.2.The algebra U q(g,s)has a Hopf algebra structure.The formulas for the coproduct?,the counitεand the antipode a are the same as given in Theorem2.1.

Example3.1.Letˉg(z)= 1.From[CJWW][Z],we can see that U q(1,s)is basically the same as U q(?gl(1,1)).

4.

For a rational function g(z)that satis?es

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

it is clear that g(z)is determined by its poles and its zeros,which are paired to satisfy the relations above.For the simplest case(except g(z)=1)that g(z)has only one pole and one zero,we have

zp?1

g(z)=

zp2?1

z?p1

zp2?1

z?p1

As in[DM][DK],for the case of quantum a?ne algebras,it is very important to understand the poles and zero of the product of current operators.We will start with the relations between X+(z)with itself.

From the de?nition,we know that

(z?p1w)(z?p2w)X+(z)X+(w)=?(zp1?w)(zp2?w)X+(w)X+(z).

From this,we know that X+(z)X+(w)has two poles,which are

located at(z?p1w)=0and(z?p2w)=0.

This also implies that

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

If we assume that U q(g,s)is related to some quantized a?ne super-algebra,then we can see that the best chance we have is U q(?osp(1,2))

by looking at the number of zeros and poles of X+(z)X+(w).

However,for the case of U q(?osp(1,2)),we know we need an extra

Serre relation.For this,we will follow the idea in[FO].

Let Let

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

(z?p1w)(z?p2w)

Y+(z,w)=

(z1?p1z3)(z1?p2z3)

z1?z2

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

z2?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),

ˉF(z

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

1

Let V(z1,z2,z3)be the algebraic variety of the zeros of F(z1,z2,z3).

Let V(a(z1),a(z2),a(z3)),be the image of the action of a on this variety,

where a∈S3,the permutation group on z1,z2,z3.LetˉV(z1,z2,z3)be

the algebraic variety of the zeros ofˉF(z1,z2,z3).LetˉV(a(z1),a(z2),a(z3)),

be the image of the action of a on this variety,where a∈S3.

Proposition4.2.Y+(z,w)has no poles and is symmetry with respect

to z and w.Y+(z1,z2,z3)has no poles and is symmetry with respect

to z1,z2,z3.

8

Following the idea in[FO],we would like to de?ne the following conditions that may be imposed on our algebra.

Zero Condition I:

Y+(z1,z2,z3)is zero on at least one line that crosses(0,0,0),and this line must lie in a V(a(z1),a(z2),a(z3))for some element a∈S3

Zero Condition II:Y+(z1,z2,z3)is zero on at least one line that crosses(0,0,0),and this line must lie in aˉV(a(z1),a(z2),a(z3))for some element a∈S3

For the line that crosses(0,0,0)where Y+(z1,z2,z3)is zero,we call it the zero line of Y+(z1,z2,z3).Because Y+(z1,z2,z3)is symmetric with respect to the action of S3on z1,z2,z3,if a line is the zero line of Y+(z1,z2,z3),then clearly the orbit of the line under the action of S3 is also an zero line.

Remark1.There is a simple symmetry that we would prefer to choose the function F(z1,z2,z3)to determine the variety V(z1,z2,z3). We have that

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

Let S12be the permutation group acting on z1,z2.Let S22be the permu-tation group acting on z2,z3.Let S12be the permutation group acting on z3,z1.Clearly,[FO]we can choose from a family of varieties determined by the functions f(a1(z1),a1(z2))f(a2(z2),a2(z3))f(a3(z3),a3(z1))for a1∈S12,a2∈S22,a3∈S32.For each such a function

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

we can attach a oriented diagram,whose nods are z1,z2,z3,and the arrows are given by(a1(z1)→a1(z2)),(a2(z2)→a2(z3))(a3(z3)→a3(z1)).For example the diagram of F(z1,z2,z3)is given by

Case1Let q2=p?11,which implies that p?21must be either p1or p2. Clearly,it can not be p1,which implies that the impossible condition p1=1

Therefore,we have that

p?21=p2,

which is what we want.

Case2Let q2=p?12,which implies that p1p2must be either p1or p2.Clearly,it can not be p1because it implies p2=1,it can not be neither be p2,which implies that p1=1.

This completes the proof for

p2=p?21.

Similarly,if we have that q1=p2,we can,then,show

p1=p?22.

However from the algebraic point of view,the two condition are equivalent in the sense that p1and p2are symmetric.

Also we have that

Proposition4.5.If we impose the Zero condition II on the algebra U q(g,s),we have

p1=p22,

or

p2=p21.

However we also have that:

If we impose the Zero condition II on the algebra U q(g,s), then U q(g,s)is not a Hopf algebra anymore.

The reason is that the Zero condition II can not be satis?ed by comultiplication,which can be checked by direct calculation.

This is the most important reason that we will choose the Zero con-dition I to be imposed on the algebra U q(g,s),which comes actually from the consideration of Hopf algebra https://www.doczj.com/doc/2016049653.html,ly,if we choose V(z1,z2,z3)or the equivalent ones which has the same diagram presen-tation as Diagram I to de?ne the zero line of Y+(z1,z2,z3),then,the quotient algebra derived from the Zero Condition I is still a Hopf algebra with the same Hopf algebra structure(comultiplication,counit and antipode).

11

From now on,we impose the Zero condition I on the algebra U q(g,s),

and let us?x the notation such that

p1=q2,

p2=q?1.

Similarly,we de?ne

(z?p?11w)(z?p?12w)

Y?(z,w)=

(z1?p?11z3)(z1?p?12z3)

z1?z2

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

z2?z3

We now de?ne the q-Serre relation.

q-Serre relations

Y+(z1,z2,z3)is zero on the line

z1=z2q?1=z3q?2.

Y?(z1,z2,z3)is zero on the line

z1=z2q=z3q2.

The q-Serre relations can also be formulated in more algebraic way.

Proposition4.6.The q-Serre relations are equivalent to the following

two relations:

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

X+(z3)X+(z2)X+(z1))?

(z1?z2q?1)(z3?z1q)

((z1?z3q2)(z1?z3q)(z1q?z3q?1)(z1?z2q2)

X+(z2)X+(z1)X+(z3))=0, (z1?z3)(z3?z1q2)(z1?z2q?1)

(z3?z1q)(z3?z1q?3)(z1?z2q?2)

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

X?(z1)X?(z2)X?(z3)?(z1?z3)(z3?z1q?2)

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

[DI]J.Ding,K.Iohara Generalization and deformation of the quantum a?ne algebras Lett.Math.Phys.,41,1997,181-193q-alg/9608002,RIMS-1091 [DI2]J.Ding,K.Iohara Drinfeld comultiplication and vertex operators,Jour.

Geom.Phys.,23,1-13(1997)

[DK]J.Ding,S.Khoroshkin Weyl group extension of quantized current algebras, to appear in Transformation Groups,QA/9804140(1998)

[DM]J.Ding and T.Miwa Zeros and poles of quantum current operators and the condition of quantum integrability,Publications of RIMS,33,277-284

(1997)

[Dr1]V.G.Drinfeld Hopf algebra and the quantum Yang-Baxter Equation,Dokl.

Akad.Nauk.SSSR,283,1985,1060-1064

[Dr2]V.G.Drinfeld Quantum Groups,ICM Proceedings,New York,Berkeley, 1986,798-820

[Dr3]V.G.Drinfeld New realization of Yangian and quantum a?ne algebra, Soviet Math.Doklady,36,1988,212-216

[Er] B.Enriquez,On correlation functions of Drinfeld currents and shu?e al-gebras,math.QA/9809036.

[FRT]L.D.Faddeev,N.Yu,Reshetikhin,L.A.Takhtajan Quantization of Lie groups and Lie algebras,Yang-Baxter equation in Integrable Systems,(Ad-

vanced Series in Mathematical Physics10)World Scienti?c,1989,299-309. [FO] B.Feigin,V.Odesski Vector bundles on Elliptic curve and Sklyanin alge-bras RIMS-1032,q-alg/9509021

[FJ]I.B.Frenkel,N.Jing Vertex representations of quantum a?ne algebras, https://www.doczj.com/doc/2016049653.html,A85(1988),9373-9377

[GZ]M.Gould,Y.Zhang On Super RS algebra and Drinfeld Realization of Quantum A?ne Superalgebras q-alg/9712011

[J1]M.Jimbo A q-di?erence analogue of U(g)and Yang-Baxter equation,Lett.

Math.Phys.10,1985,63-69

[RS]N.Yu.Reshetikhin,M.A.Semenov-Tian-Shansky Central Extensions of Quantum Current Groups,LMP,19,1990

[S] E.K.Sklyanin On some algebraic structures related to the Yang-Baxter equation Funkts.Anal.Prilozhen,16,No.4,1982,22-34

[Z]Y.Zhang Comments on Drinfeld Realization of Quantum A?ne Superal-gebra U q[gl(m|n)(1)]and its Hopf Algebra Structure q-alg/9703020 Jintai Ding,Department of Mathematical Sciences,University of Cincinnati

Boris Feigin,Landau Institute of Theoretical Physics

14

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