Level −1/2 realization of quantum N-toroidal algebras in type C n
aa r X i v : . [ m a t h . QA ] J u l Level − / realization of quantum N-toroidal algebras in type C n Naihuan Jing, Qianbao Wang, and Honglian Zhang ⋆ A BSTRACT . We construct a level − vertex representation of the quantum N-toroidal algebra fortype C n , which is a natural generalization of the usual quantum toroidal algebra. The constructionalso provides a vertex representation of the quantum toroidal algebra for type C n as a by-product.
1. Introduction
Let g be a finite-dimensional complex simple Lie algebra, the N -toroidal Lie algebra g N,tor associated to g is the universal central extension of the multi-loop Lie algebra g ⊗ C [ t ± , · · · , t ± N ] ,which generalizes both the affine Lie algebra and toroidal Lie algebra. The quantum group U q ( g ) was introduced by Drinfeld [
6, 7 ] and Jimbo [ ] independently as a q -deformation of the universalenveloping algebra U ( g ) of the Lie algebra g . The quantum group U q (ˆ g ) associated to the affine Liealgebra ˆ g is also called the quantum affine algebra, whose representation theory is very rich includ-ing vertex representations and finite-dimensional representations and so on. For example, Chari andPressely classified the finite-dimensional representations (c.f. [ ]-[ ]) in terms of Drinfeld polyno-mials and affine Hecke algebras. The vertex representation was first constructed by Frenkel andJing [ ] for simply-laced types. Subsequently vertex representations for quantum twisted affinealgebras were obtained by Jing [ ]. Bosonic realizations of the quantum affine algebras in othertypes were constructed in [ ], [ ],[ ],[ ], [ ], [ ], [ ] and [ ] etc.Quantum toroidal algebras U q ( g tor ) were introduced in [ ] through geometric realization re-lated to Langlands reciprocity for algebraic surfaces. Subsequently, Varagnolo and Vasserot [ ] ob-tained the Schur-Weyl duality between the representation of the quantum toroidal algebra U q ( g tor ) and the elliptic Cherednik algebra in type A . Since there exists another two-parameter deformationof the quantum toroidal algebra U q ( g tor ) in type A , there are special interest in the representationsof U q ( g tor ) see for example [
11, 33, 12, 17, 18, 32, 9, 10, 16, 15 ], [ ]-[ ] and the referencestherein. Unlike the quantum affine case, the quantum toroidal algebra U q ( g tor ) is not a quantumKac-Moody algebra, but rather a quantum affnization. Thus these references were focused in type A or the simply-laced cases. Nevertheless there are still lots of unknowns for the quantum toroidalalgebras in type A, and the knowledge on representation theory in other types is very limited. Inthe present paper, we will construct a level- ( − / vertex representation of the quantum toroidalalgebra for the symplectic type. Actually, our motivation is the recent joint paper [ ], in whichquantum N -toroidal algebras (denoted by U q ( g N,tor ) ) were studied as a natural generalization of Mathematics Subject Classification.
Key words and phrases.
Vertex representation, quantum N-toroidal algebra, vertex operator, toroidal Lie algebra. quantum toroidal algebras U q ( g tor ) . In [ ], the quantum N -toroidal algebras are shown to be quo-tients of extended quantized GIM algebras of N -fold affinization. In this paper, we will give a level − vertex representation of the quantum N -toroidal algebra for type C n , based on the method of[ ].The paper is organized as follows. In section 2, we review the definition of the quantum N -toroidal algebra for type C n . We construct the Fock space and vertex operators in section 3. Thenthe main result of constructing a level- − / vertex representation of the quantum N -toroidal alge-bra in type C n is given. In the last section, we verify the quantum algebra relations to show that theabove construction is a realization in detail.
2. Quantum toroidal algebras U q ( g N,tor ) In this paper, we always assume that g is the finite dimensional simple Lie algebra of type C n .In this section, we review the definition of quantum N-toroidal algebra U q ( g N,tor ) for the symplectictype recently given in [ ]. For this we recall the data of the simple Lie algebra, affine Lie algebraand toroidal Lie algebra of type C n .Let I = { , · · · , n } and I = { , · · · , n } . We denote that A = ( a ij )( i, j ∈ I ) is the Cartanmatrix of g and h is the Cartan subalgebra. Let ε , · · · , ε n denote the usual orthonormal basis ofthe Euclidean space R n . The root system Φ for g is {± ( ε i ± ε j ) , ± ε i | i = j } and a base for Φ is ∆ = { α i | i = 1 , · · · , n } , where α i = ε i − ε i +1 for i = 1 , · · · , n − , α n = 2 ε n . Denote the dominantweights by λ i = ε + · · · + ε i ( i = 1 , · · · , n ) and the weight lattice P = Z ε + · · · + Z ε n . Let ˆ g bethe affine Kac-Moody Lie algebra of type C n with the Cartan subalgebra ˆ h associated to the simpleLie algebra g . Let δ be the primitive imaginary root of ˆ g . Let α = δ − (2 α + · · · + 2 α n − + α n ) ,then the set of simple roots for ˆ g is ˆ∆ = { α , · · · , α n } . We fix the nondegenerate symmetricbilinear form ( ·|· ) on the dual space ˆ h ∗ such that ( ε s | ε t ) = δ st for s, t ∈ I , then ( α i | α j ) = d i a ij , ( δ | α i ) = ( δ, δ ) = 0 for all i, j ∈ I where ( d , d , · · · , d n ) = (1 , , · · · , , and ˆ A = ( a ij )( i, j ∈ I ) is the Cartan matrix of ˆ g .Suppose q is not a root of unity. Let q i = q d i and [ k ] i = q ki − q − ki q i − q − i . Let J = { , · · · , N − } , k =( k , k , · · · , k N − ) ∈ Z N − , e s = (0 , · · · , , , , · · · , the s th standard unit vector of ( N − -dimension lattice Z N − and the ( N − -dimensional zero vector. Now we turn to the definitionof the quantum N -toroidal algebra for type C n denoted by U q ( g N,tor ) introduced in [ ].D EFINITION N -toroidal algebra U q ( g N,tor ) is an associative algebra over F generated by x ± i ( k ) , a ( s ) i ( r ) , K ± i and γ ± s ( i ∈ I , s ∈ J, k ∈ Z N − , r ∈ Z \{ } ) , satisfying therelations as follows: γ ± s are central such that γ ± s γ ∓ s = 1 , K ± i K ∓ i = 1 , (2.1) K ± i and a ( s ) j ( r ) commute each other,(2.2) [ a ( s ) i ( r ) , a ( s ′ ) j ( l ) ] = δ s,s ′ δ r + l, [ r a ij ] i r γ rs − γ − rs q j − q − j , (2.3) K i x ± j ( k ) K − i = q ± a ij i x ± j ( k ) , (2.4) [ x ± i ( ke s ) , x ± i ( le s ′ ) ] = 0 , for s = s ′ and kl = 0 , (2.5) EVEL -1/2 REALIZATION OF QUANTUM N-TOROIDAL ALGEBRAS 3 [ a ( s ) i ( r ) , x ± j ( k ) ] = ± [ ra ij ] i r γ ∓ | r | s x ± j ( re s + k ) , (2.6) [ x ± i (( k + 1) e s ) , x ± j ( le s ) ] q ± aiji + [ x ± j (( l + 1) e s ) , x ± i ( ke s ) ] q ± aiji = 0 , (2.7) [ x + i ( ke s ) , x − j ( le s ) ] = δ ij (cid:0) γ k − l s φ ( s ) i (( k + l )) − γ l − k s ψ ( s ) i (( k + l )) q i − q − i (cid:1) , (2.8)where φ ( s ) i ( r ) and ψ ( s ) i ( − r ) ( r ≥ such that φ ( s ) i (0) = K i and ψ ( s ) i (0) = K − i are defined by: ∞ X r =0 φ ( s ) i ( r ) z − rs = K i exp (cid:16) ( q i − q − i ) ∞ X ℓ =1 a ( s ) i ( ℓ ) z − ℓ (cid:17) , ∞ X r =0 ψ ( s ) i ( − r ) z rs = K − i exp (cid:16) − ( q i − q − i ) ∞ X ℓ =1 a ( s ) i ( − ℓ ) z ℓ (cid:17) ,Sym k , ··· ,k m m =1 − a ij X l =0 ( − l h ml i i x ± i ( k e s ) · · · x ± i ( k l e s ) x ± j ( ℓe s ) (2.9) x ± i ( k l +1 e s ) · · · x ± i ( k m e s ) = 0 , for i = j, X k =0 ( − k h k i i x ± i ( e s m ) · · · x ± i ( e s m k ) x ∓ i ( e s ′ ℓ ) x ± i ( e s m k +1 ) · · · x ± i ( e s m ) = 0 , (2.10) for i ∈ I and m m m ℓ = 0 , s = s ′ ∈ J, where the q -bracket is defined as [ a, b ] u . = ab − uba and Sym m , ··· ,m n denotes the symmetrizationwith respect to the indices ( m , · · · , m n ) .R EMARK N = 2 , the quantum N-toroidal algebras are just the quan-tum toroidal algebras [ ]. Therefore the former are natural generalizations of the usual quantumtoroidal algebra, just like N -toroidal Lie algebras vs. the -toroidal Lie algebras.R EMARK s ∈ J , the subalgebra U ( s ) q of U q ( g N,tor ) generated by the elements x ± i ( ke s ) , a ( s ) i ( r ) , K ± i , γ ± s for i ∈ I isomorphic to the quantum 2-toroidal algebra defined in[ ].R EMARK z = ( z , · · · , z N − ) , denote z k = N − Q s =1 z k s s . We set thegenerating functions of formal variables for i ∈ I and s ∈ J as follows, δ ( z ) = X k ∈ Z z k , x ± i ( z ) = X k ∈ Z N − x ± i ( k ) z − k ,x ± i,s ( z ) = X k ∈ Z x ± i ( ke s ) z − k ,φ ( s ) i ( z ) = X m ∈ Z + φ ( s ) i ( m ) z − m , ψ ( s ) i ( z ) = X n ∈ Z + ψ ( s ) i ( − n ) z n . JING, WANG, AND ZHANG
It is not difficult to see that relations from (2.5) to (2.10) are equivalent to the following rela-tions, respectively, lim z → w [ x ± i,s ( z ) , x ± i,s ′ ( w )] = 0 , for s = s ′ , (2.11) ψ ( s ) i ( z ) x ± j ( w ) ψ ( s ) i ( z ) − = g ij (cid:16) zw s γ ∓ s (cid:17) ± x ± j ( w ) , (2.12) φ ( s ) i ( z ) x ± j ( w ) φ ( s ) i ( z ) − = g ij (cid:16) w s z γ ∓ s (cid:17) ∓ x ± j ( w ) , ( z − q ± a ij i w ) x ± i,s ( z ) x ± j,s ( w ) + ( w − q ± a ij i z ) x ± j,s ( w ) x ± i,s ( z ) = 0 , (2.13) [ x + i,s ( z ) , x − j,s ( w ) ] = δ ij ( q i − q − i ) zw (cid:16) φ ( s ) i ( wγ s ) δ ( wγ s z ) − ψ ( s ) i ( wγ − s ) δ ( wγ − s z ) (cid:17) , (2.14) Sym z , ··· z n n =1 − a ij X k =0 ( − k h nk i i x ± i,s ( z ) · · · x ± i,s ( z k ) x ± j,s ( w ) (2.15) × x ± i,s ( z k +1 ) · · · x ± i,s ( z n ) = 0 , for i = j lim z i → w X k =0 ( − k h k i i x ± i,s ( z ) · · · x ± i,s ( z k ) x ∓ i,s ′ ( w ) x ± i,s ( z k +1 ) · · · x ± i,s ( z ) = 0 , (2.16) for i ∈ I and s = s ′ ∈ J. where g ij ( z ) := P n ∈ Z + c ijn z n is the Taylor series expansion of g ij ( z ) = zq aiji − z − q aiji at z = 0 in C .
3. Vertex representations
In this section, we construct the Fock space and obtain a level- ( − / vertex representation ofquantum N -toroidal algebra U q ( g N,tor ) for type C n , based on the method in [ ].First of all, let us introduce the quantum Heisenberg algebra U q ( h N,tor ) , which is generated by a ( s ) i ( r ) , b ( s ) i ( r ) for i ∈ I, s ∈ J satisfying the following relations: [ a ( s ) i ( r ) , a ( s ′ ) j ( t )] = δ ss ′ δ r + t, [ ra ij ] i r q − r/ − q r/ q j − q − j , (3.1) [ b ( s ) i ( r ) , b ( s ′ ) j ( t )] = rδ ss ′ δ ij δ r + t, , (3.2) [ a ( s ) i ( r ) , b ( s ′ ) j ( t )] = 0 . (3.3)Let S ( h − N,tor ) be the symmetric algebra generated by a ( s ) i ( − l ) , b ( s ) i ( − l ) with l being a positiveinteger. Then S ( h − N,tor ) is a U q ( h N,tor ) -module by letting a ( s ) i ( − l ) , b ( s ) i ( − l ) act as multiplicationoperators and a ( s ) i ( l ) , b ( s ) i ( l ) operate as differentiation subject the Heisenberg algebra relations. Let ˜ P be the affine weight lattice ˜ P = Z λ + · · · + Z λ n , and set ˜ P ′ ≃ ˜ P , an identical copy of ˜ P corresponding to b ′ i s . We define the Fock space F = S ( h − N,tor ) ⊗ C [ ˜ P ] ⊗ C [ ˜ P ′ ] ⊗ C [ Z J ] , where C [ G ] is the group algebra of the abelian group G . EVEL -1/2 REALIZATION OF QUANTUM N-TOROIDAL ALGEBRAS 5
We equip the lattice Z J with the nondegerate bilinear form ( | ) defined by(3.4) ( s i | s j ) = ( − i = j i = j . where we list the elements of J as { s , s , · · · , s N − } .The action of operators e a i , e b i , e s i , a ( s ) i ( m ) , b ( s ) i ( m ) , s i (0) on F is defined by the followingrelations, e a i .e α ⊗ e β ⊗ e s = e α i + α ⊗ e β ⊗ e s , e b i .e α ⊗ e β ⊗ e s = e α ⊗ e ε i + β ⊗ e s ,z a i (0) . ( e α ⊗ e β ⊗ e s ) = z ( α i | α ) ( e α ⊗ e β ⊗ e s ) , z b i (0) . ( e α ⊗ e β ⊗ e s ) = z (2 ε i | β ) ( e α ⊗ e β ⊗ e s ) ,q s ′ . ( e α ⊗ e β ⊗ e s ) = q ( s ′ | s ) ( e α ⊗ e β ⊗ e s ) where α ∈ ˜ P , β ∈ ˜ P ′ , s, s ′ ∈ Z J . Here we have added ε such that ( ε i | ε j ) = δ ij for i, j ∈ I .The normal order : : is defined as usual, : a ( s ) i ( r ) a ( s ) j ( t ) := ( a ( s ) i ( r ) a ( s ) j ( t ) , if r < t ; a ( s ) j ( t ) a ( s ) i ( r ) , if r ≥ t, : e a j z a i (0) :=: z a i (0) e a j := e a j z a i (0) , and : b ( s ) i ( r ) b ( s ) j ( t ) := ( b ( s ) i ( r ) b ( s ) j ( t ) , if r < t ; b ( s ) j ( t ) b ( s ) i ( r ) , if r ≥ t, : e b j z b i (0) :=: z b i (0) e b j := e b j z b i (0) . : e s q s ′ (0) :=: q s ′ (0) e s := e s q s ′ (0) . P ROPOSITION
We have the relations between the operators a i (0) , b j (0) , s, e a i , e b j , q s ′ asfollows [ a i (0) , e a j ] = ( α i | α j ) e a j , [ b i (0) , e b j ] = 2( ε i | ε j ) e b j , [ s (0) , q s ′ ] = ( s | s ′ ) q s ′ . Now we define the following vertex operators for i ∈ I , X ± i ( z ) = exp ± N − X s =1 ∞ X k s =1 a ( s ) i ( − k s )[ k s /d i ] i q ∓ k s / z k s s exp ∓ N − X s =1 ∞ X k s =1 a ( s ) i ( k s )[ k s /d i ] i q ∓ k s / z − k s s × e ± α i N − Y s =1 z ± a i (0)+1 s ,Y ± i,s ( z ) = exp (cid:16) ± ∞ X k =1 a ( s ) i ( − k )[ − (1 / d i ) k ] i q ± k/ z k (cid:17) exp (cid:16) ∓ ∞ X k =1 a ( s ) i ( k )[ − (1 / d i ) k ] i q ± k/ z − k (cid:17) × e ± a i z ∓ a i (0) ,Z ± i,s ( z ) = exp (cid:16) ± ∞ X k =1 b ( s ) i ( − k ) k z k (cid:17) exp (cid:16) ∓ ∞ X k =1 b ( s ) i ( k ) k z − k (cid:17) e ± b i z ± b i (0) . JING, WANG, AND ZHANG
For simplicity, we introduce the following notations for ǫ = ± or ± , i = 1 , · · · , n − , j = 0 , n and s ∈ J , X + iǫ,s ( z ) = Z + i,s ( q ǫ/ z ) Z − i +1 ,s ( z ) Y + i,s ( z ) e s q ǫs (0) , X − iǫ,s ( z ) = Z − i,s ( z ) Z + i +1 ,s ( q ǫ/ z ) Y − i,s ( z ) e − s q − ǫs (0) ,X + jǫ,s ( z ) =: Z + j,s ( q ǫ z ) Z + j,s ( q − ǫ z ) Y + j,s ( z ) : e s q ǫs (0) , X + j ,s ( z ) =: Z + j,s ( qz ) Z + j,s ( q − z ) Y + j,s ( z ) : e s . Now we give the main result of the paper.T
HEOREM
For i ∈ I and s ∈ J , the Fock space F is a U q ( g N,tor ) -module for type C n oflevel − under the action ψ defined by : γ ± s q ∓ / ,K i q a i (0) i ,x ± i ( z ) X ± i ( z ) ,x ± i,s ( z ) X ± i,s ( z ) ,φ ( s ) i ( z ) Φ ( s ) i ( z ) ,ψ ( s ) i ( z ) Ψ ( s ) i ( z ) , where Φ ( s ) i ( z ) and Ψ ( s ) i ( z ) are defined the same way as φ ( s ) i ( z ) and ψ ( s ) i ( z ) respectively, i =1 , · · · , n − , and j = 0 , n , X ± i,s ( z ) = 1( q / − q − / ) z ( X ± i + ,s ( z ) − X ± i − ,s ( z )) ,X + j,s ( z ) = − q / X + j + ,s ( z ) + q − / X + j − ,s ( z ) − [2] X + j ,s ( z )( q − q − )( q / − q − / ) z ,X − j,s ( z ) =: Z − j,s ( q / z ) Z − j,s ( q − / z ) : Y − j,s ( z ) e − s q − s (0) .
4. Proof of Theorem 3.2
In this section, we proceed to prove Theorem 3.2 in detail. First of all, let us give some relationsthat will be used in the sequel.L
EMMA
The following relations for Y ± i,s ( z ) and Z ± i,s ( z ) holds, (4.1) Y ± i,s ( z ) Y ± j,s ( w ) =: Y ± i,s ( z ) Y ± j,s ( w ) : × , if ( α i | α j ) = 0( z − q ± / w ) , if ( α i | α j ) = − (cid:0) ( z − q ± w )( z − w ) (cid:1) − ( α i | α j ) , if ( α i | α j ) = ± (cid:0) ( z − w )( z − qw )( z − q − w )( z − q ± w ) (cid:1) − , if ( α i | α j ) = 2 . (4.2) Y ± i,s ( z ) Y ∓ j,s ( w ) =: Y ± i,s ( z ) Y ∓ j,s ( w ) : EVEL -1/2 REALIZATION OF QUANTUM N-TOROIDAL ALGEBRAS 7 × , if ( α i | α j ) = 0 , ( z − w ) − , if ( α i | α j ) = − , (cid:0) ( z − q − / w )( z − q / w ) (cid:1) ( α i | α j ) , if ( α i | α j ) = ± , ( z − q − w )( z − q w )( z − q − w )( z − q w ) , if ( α i | α j ) = 2 . (4.3) Z ǫi,s ( z ) Z ǫ ′ j,s ( w ) =: Z ǫi,s ( z ) Z ǫ ′ j,s ( w ) : ( z − w ) ǫǫ ′ δ ij , where ( z − w ) − is the power series in w/z as follows: ( z − w ) − = ∞ X k =0 w k z − k − . P ROOF . Here we only show the relation (4.1) in the case of ( α i | α j ) = − / , other relationscan be verified in a similar manner. According to the definitions of Y ± i,s ( z ) and the normal order,we obtain immediately, Y ± i,s ( z ) Y ± j,s ( w )= : Y ± i,s ( z ) Y ± j,s ( w ) : exp (cid:16) − ∞ X k =1 q ± k/ z − k w k [ − (1 / d i ) k ] i [ − (1 / d j ) k ] j [ a ( s ) i ( k ) , a ( s ) j ( − k )] (cid:17) z =: Y ± i,s ( z ) Y ± j,s ( w ) : exp (cid:16) − ∞ X k =1 (cid:0) q ± / wz (cid:1) k k (cid:17) z =: Y ± i,s ( z ) Y ± j,s ( w ) : ( z − q ± / w ) . (cid:3) L EMMA
From the notations of X ± iǫ,s ( z ) , it holds for i = 1 , · · · , n − and j = 0 , n , X ± iǫ,s ( z ) X ± iǫ ′ ,s ( w ) =: X ± iǫ,s ( z ) X ± iǫ ′ ,s ( w ) : ( q ǫ/ z − q ǫ ′ / w )( z − q ± w ) − , (4.4) X + iǫ,s ( z ) X − iǫ ′ ,s ( w ) =: X + iǫ,s ( z ) X − iǫ ′ ,s ( w ) : ( q ǫ/ z − w ) − ( z − q − ǫ ′ / w ) , (4.5) X − iǫ,s ( z ) X + iǫ ′ ,s ( w ) =: X − iǫ,s ( z ) X + iǫ ′ ,s ( w ) : ( z − q − ǫ ′ / w )( q ǫ/ z − w ) − , (4.6) X + iǫ,s ( z ) X +( i +1) ǫ ′ ,s ( w ) =: X + iǫ,s ( z ) X +( i +1) ǫ ′ ,s ( w ) : ( z − q ǫ ′ / w ) − ( z − q / w ) , (4.7) X − iǫ,s ( z ) X +( i +1) ǫ ′ ,s ( w ) = X +( i +1) ǫ ′ ,s ( w ) X − iǫ,s ( z ) (4.8) =: X − iǫ,s ( z ) X +( i +1) ǫ ′ ,s ( w ) : ( q ǫ/ z − q ǫ ′ / w )( z − w ) − ,X + iǫ,s ( z ) X − ( i +1) ǫ ′ ,s ( w ) = X − ( i +1) ǫ ′ ,s ( w ) X + iǫ,s ( z ) =: X + iǫ,s ( z ) X − ( i +1) ǫ ′ ,s ( w ) : , (4.9) X +( i +1) ǫ,s ( z ) X − iǫ ′ ,s ( w ) =: X +( i +1) ǫ,s ( z ) X − iǫ ′ ,s ( w ) : ( q ǫ/ z − q ǫ ′ / w )( z − w ) − , (4.10) X +( i +1) ǫ,s ( z ) X + iǫ ′ ,s ( w ) =: X +( i +1) ǫ,s ( z ) X + iǫ ′ ,s ( w ) : ( q ǫ/ z − w ) − ( z − q / w ) , (4.11) X + jǫ,s ( z ) X − j,s ( w ) =: X + jǫ,s ( z ) X − j,s ( w ) : (cid:16) q − ǫ/ ( z − q ǫ/ w ) q ǫ/ z − w (cid:17) | ǫ | , (4.12) JING, WANG, AND ZHANG X − j,s ( w ) X + jǫ,s ( z ) =: X − j,s ( w ) X + jǫ,s ( z ) : w − q − ǫ/ zw − q ǫ/ z , (4.13) X + nǫ,s ( z ) X − ( n − ǫ ′ ,s ( w ) = X − ( n − ǫ ′ ,s ( w ) X + nǫ,s ( z ) (4.14) =: X + nǫ,s ( z ) X − ( n − ǫ ′ ,s ( w ) : ( q ǫǫ z − q ǫ ′ / w )( q − ǫǫ z − q ǫ ′ / w )( z − q − / w )( z − q / w ) ,X +0 ǫ,s ( z ) X − ǫ ′ ,s ( w ) = X − ǫ ′ ,s ( w ) X +0 ǫ,s ( z ) (4.15) =: X +0 ǫ,s ( z ) X − ǫ ′ ,s ( w ) : 1( z − q − / w )( z − q / w ) .X − i,s ( z ) X − j,s ( w ) =: X − i,s ( z ) X − j,s ( w ) : z − wz − q − w , (4.16) X ± iǫ,s ( z ) X ± iǫ ′ ,s ′ ( w ) =: X ± iǫ,s ( z ) X ± iǫ ′ ,s ′ ( w ) : , for s = s ′ (4.17) X ± iǫ,s ( z ) X ∓ iǫ ′ ,s ′ ( w ) =: X ± iǫ,s ( z ) X ∓ iǫ ′ ,s ′ ( w ) : , for s = s ′ (4.18) X + jǫ,s ( z ) X + jǫ ′ ,s ′ ( w ) =: X + jǫ,s ( z ) X + jǫ ′ ,s ′ ( w ) : , for s = s ′ (4.19) X − j,s ( z ) X − j,s ′ ( w ) =: X − j,s ( z ) X − j,s ′ ( w ) : q − , for s = s ′ (4.20)P ROOF . To check relation (4.4), without loss of generality, we only prove it in the case of ”+”.Using relations (4.1) and (4.3), one gets directly that, X + iǫ,s ( z ) X + iǫ ′ ,s ( w ) = Z + i,s ( q ǫ/ z ) Z − i +1 ,s ( z ) Y + i,s ( z ) Z + i,s ( q ǫ ′ / w ) Z − i +1 ,s ( w ) Y + i,s ( w )= : X + iǫ,s ( z ) X + iǫ ′ ,s ( w ) : ( q ǫ/ z − q ǫ ′ / w )( z − w )( z − qw ) − ( z − w ) − = : X + iǫ,s ( z ) X + iǫ ′ ,s ( w ) : ( q ǫ/ z − q ǫ ′ / w )( z − qw ) − . Since relations (4.5) and (4.6) can be verified similarly, we only check relation (4.5). By relation(4.2), we obtain immediately that X + iǫ,s ( z ) X − iǫ ′ ,s ( w )= : X + iǫ,s ( z ) X − iǫ ′ ,s ( w ) : ( q ǫ/ z − w ) − ( z − q ǫ ′ / w ) − ( z − q − / w )( z − q / w )= : X + iǫ,s ( z ) X − iǫ ′ ,s ( w ) : ( q ǫ/ z − w ) − ( z − q − ǫ ′ / w ) . To check relation (4.7), we have that, X + iǫ,s ( z ) X +( i +1) ǫ ′ ,s ( w ) = Z + i,s ( q ǫ/ z ) Z − i +1 ,s ( z ) Y + i,s ( z ) Z + i +1 ,s ( q ǫ ′ / w ) Z − i +2 ,s ( w ) Y + i +1 ,s ( w )= : X + iǫ,s ( z ) X +( i +1) ǫ ′ ,s ( w ) : ( z − q ǫ ′ / w ) − ( z − q / w ) . Similarly one can check relations (4.8)-(4.11).Below we verify the relation (4.12) directly, and (4.15) can be similarly done, X + jǫ,s ( z ) X − j,s ( w )= : Z + j,s ( q ǫǫ z ) Z + j,s ( q − ǫǫ z ) Y + j,s ( z ) :: Z − j,s ( q / w ) Z − j,s ( q − / w ) : Y − j,s ( w )= : X + jǫ,s ( z ) X − j,s ( w ) : ( q ǫǫ z − q / w ) − ( q ǫǫ z − q − / w ) − ( q − ǫǫ z − q / w ) − × ( q − ǫǫ z − q − / w ) − ( z − q − / w )( z − q / w )( z − q − / w )( z − q / w ) . EVEL -1/2 REALIZATION OF QUANTUM N-TOROIDAL ALGEBRAS 9
Then we proceed with verification in three subcases: ǫ = ± or . Direct calculation yields that, X + jǫ,s ( z ) X − j,s ( w )= : X + jǫ,s ( z ) X − j,s ( w ) : (cid:16) q − ǫ/ ( z − q ǫ/ w ) q ǫ/ z − w (cid:17) | ǫ | . For relation (4.13) and (4.14), the proofs are similar, here we only check relation (4.14). X +0 ǫ,s ( z ) X − ǫ ′ ,s ( w )= : Z +0 ,s ( q ǫǫ z ) Z +0 ,s ( q − ǫǫ z ) Y +0 ,s ( z ) : Z − ,s ( w ) Z +2 ,s ( q ǫ ′ / w ) Y − ,s ( w )= : X +0 ǫ,s ( z ) X − ǫ ′ ,s ( w ) : 1( z − q − / w )( z − q / w )= X − ǫ ′ ,s ( w ) X +0 ǫ,s ( z ) , which completes the proof of lemma 4.2. (cid:3) With the help of the above two lemmas, we can now prove Theorem 3.2. It means that weneed to check ψ satisfy all defining relations (2.1)-(2.10). It is obvious that relations (2.1)-(2.4)follow from the constructions of the vertex operators. Thus it suffices to show ψ keeps relations(2.5)-(2.10), or equivalently (2.11)-(2.16), which will be explained in more detail as follows.To show relation (2.11), we first look at the case i = 1 , . . . , n − for example. For s = s ′ X ± i,s ( z ) X ± i,s ′ ( w ) = 1( q / − q − / ) zw X ǫ,ǫ ′ ( − ǫǫ ′ X ± iǫ,s ( z ) X ± iǫ ′ ,s ′ ( w )= 1( q / − q − / ) zw X ǫ,ǫ ′ ( − ǫǫ ′ : X ± iǫ,s ( z ) X ± iǫ ′ ,s ′ ( w ) : . Therefore, for s = s ′ . [ X ± i,s ( z ) , X ± i,s ′ ( w )] = 0 . Similarly for s = s ′ and j = 0 , n , by Lemma 4.2 we have (in the following ǫ, ǫ ′ = ± , ) X + js ( z ) X + js ′ ( w ) = P ǫ,ǫ ′ ( − ǫǫ ′ [2 − | ǫ | ] [2 − | ǫ ′ | ] X + jǫ,s ( z ) X + jǫ ′ ,s ′ ( w )( q / − q − / ) ( q − q − ) z w = P ǫ,ǫ ′ ( − ǫǫ ′ [2 − | ǫ | ] [2 − | ǫ ′ | ] : X + jǫ,s ( z ) X + jǫ ′ ,s ′ ( w ) :( q / − q − / ) ( q − q − ) z w , which implies that [ X + j,s ( z ) X + j,s ′ ( w )] = 0 for s = s ′ . The remaining cases are the same.Now we turn to check relation (2.6), it suffices to verify that for i ∈ I, j = 0 , , · · · , n − , Ψ ( s ) i ( z s ) X ± j ( w )Ψ ( s ) i ( z s ) − = g ij (cid:16) z s w s q ∓ (cid:17) ± X ± j ( w ) . Actually, we have that, Ψ ( s ) i ( z s ) X ± j ( w )= q − a i (0) exp (cid:16) − ( q i − q − i ) ∞ X k =1 a ( s ) i ( − k ) z ks (cid:17) exp ± N − X s =1 ∞ X k s =1 a ( s ) j ( − k s )[ k s /d j ] j q ∓ k s / w k s s × exp ∓ N − X s =1 ∞ X k s =1 a ( s ) j ( k s )[ k s /d j ] j q ∓ k s / w − k s s e ± α j N − Y s =1 w ± a j (0)+1 s , = exp (cid:16) ± ( q i − q − i ) ∞ X k s =1 [ a ( s ) i ( − k s ) , a ( s ) j ( k s )][ k s /d j ] j ( q ∓ z s w s ) k s (cid:17) X + j ( w )Ψ ( s ) i ( z s )= (cid:16) g ij ( z s w s q ∓ ) (cid:17) ± X ± j ( w )Ψ ( s ) i ( z s ) . For relation (2.7), we have the proposition as follows.P
ROPOSITION ( z − q ( α i | α j ) w ) X ± i,s ( z ) X ± j,s ( w ) = ( q ( α i | α j ) z − w ) X ± j,s ( w ) X ± i,s ( z ) . P ROOF . Here we only check for the case of ”+”. The proof is divided into several cases asfollows.Case 1. a ij = 0 , it is trivial.Case 2. ( α i | α i +1 ) = − , thanks to relations (4.7) and (4.11), we have immediately, ( z − q − w ) X + iǫ,s ( z ) X +( i +1) ǫ ′ ,s ( w )=: X + iǫ,s ( z ) X +( i +1) ǫ ′ ,s ( w ) : ( z − q ǫ ′ / w ) − ( z − q / w )( z − q − w )=: X +( i +1) ǫ ′ ,s ( w ) X + iǫ,s ( z ) : ( q ǫ ′ / w − z ) − ( w − q / z )( q − z − w )= X +( i +1) ǫ ′ ,s ( w ) X + iǫ,s ( z )( q − z − w ) , which implies the assertion.Case 3. ( α i | α j ) = − , that is ( i, j ) is (1 , , (0 , , ( n − , n ) or ( n, n − . ( z − q − w ) X +0 ǫ,s ( z ) X +1 ǫ ′ ,s ( w )= : Z +0 ,s ( q ǫǫ z ) Z +0 ,s ( q − ǫǫ z ) Y +0 ,s ( z ) : Z +1 ,s ( q ǫ ′ / w ) Z − ,s ( w ) Y +1 ,s ( w )( z − q − w )= : X +0 ǫ,s ( z ) X +1 ǫ ′ ,s ( w ) : ( z − qw )( z − w )( z − q − w ) . On the other hand, it is easy to get that ( q − z − w ) X +1 ǫ ′ ,s ( w ) X +0 ǫ,s ( z )= Z +1 ,s ( q ǫ ′ / w ) Z − ,s ( w ) Y +1 ,s ( w ) : Z +0 ,s ( q ǫǫ z ) Z +0 ,s ( q − ǫǫ z ) Y +0 ,s ( z ) : ( q − z − w )= X +1 ǫ ′ ,s ( w ) X +0 ǫ,s ( z ) : ( w − qz )( w − z )( q − z − w ) . Case 4. ( α i | α j ) = 2 , that is i = j = 0 or n. Using(4.1) and (4.3) , it is clear to see that ( z − q w ) X + jǫ,s ( z ) X + jǫ ′ ,s ( w )= ( z − q w ) : Z + j,s ( q ǫǫ z ) Z + j,s ( q − ǫǫ z ) Y + j,s ( z ) :: Z + j,s ( q ǫ ′ ǫ ′ w ) Z + j,s ( q − ǫ ′ ǫ ′ w ) Y + j,s ( w ) := : X + jǫ,s ( z ) X + jǫ ′ ,s ( w ) : × Q k,l = ± ( q k + ǫǫ z − q l + ǫ ′ ǫ ′ w )( z − w )( z − qw )( z − q − w )( z − q w ) ( z − q w ) . EVEL -1/2 REALIZATION OF QUANTUM N-TOROIDAL ALGEBRAS 11
In fact, similarly we have that ( q z − w ) X + jǫ ′ ,s ( w ) X + jǫ,s ( z )= : Z + j,s ( q ǫ ′ ǫ ′ w ) Z + j,s ( q − ǫ ′ ǫ ′ w ) Y + j,s ( w ) :: Z + j,s ( q ǫǫ z ) Z + j,s ( q − ǫǫ z ) Y + j,s ( z ) : ( q z − w )= : X + jǫ ′ ,s ( w ) X + jǫ,s ( z ) : × Q k,l = ± ( q k + ǫǫ z − q l + ǫ ′ ǫ ′ w )( w − z )( w − qz )( w − q − z )( w − q z ) ( q z − w ) , which implies relation (2.7).Case 5. ( α i | α i ) = 1 , that is, i = 1 , · · · , n − . By relation (4.4), one has that ( z − qw ) X + iǫ,s ( z ) X + iǫ ′ ,s ( w )= : X + iǫ,s ( z ) X + iǫ ′ ,s ( w ) : ( q ǫ/ z − q ǫ ′ / w )( z − qw ) − ( z − qw )= : X + iǫ ′ ,s ( w ) X + iǫ,s ( z ) : ( q ǫ ′ / w − q ǫ/ z )( w − qz ) − ( qz − w )= ( qz − w ) X + iǫ ′ ,s ( w ) X + iǫ,s ( z ) . Hence we have proved proposition 4.4. (cid:3)
To check relation (2.8), we prove the following result.P
ROPOSITION
One has that [ X + i,s ( z ) , X − j,s ( w ) ] = δ ij ( q − q − ) zw (cid:16) δ ( zw − q )Φ ( s ) i ( wq − ) − δ ( zw − q − )Ψ ( s ) i ( zq − ) (cid:17) P ROOF . We divided the proof into several cases. For the case of ( α i | α j ) = − such as ( i, j ) =(1 , , it follows from relations (4.9) and (4.15), ( q − − q )( q / − q − / ) z w [ X +0 ,s ( z ) , X − ,s ( w )]= (cid:0) q / X +0+ ,s ( z ) + q − / X +0 − ,s ( z ) − ( q / + q − / ) X +00 ,s ( z ) (cid:1)(cid:0) X − ,s ( w ) − X − − ,s ( w ) (cid:1) − (cid:0) X − ,s ( w ) − X − − ,s ( w ) (cid:1)(cid:0) q / X +0+ ,s ( z ) + q − / X +0 − ,s ( z ) − ( q / + q − / ) X +00 ,s ( z ) (cid:1) = 0 . For the case of j = 0 , n such that a ij = 0 , we have that X + jǫ,s ( z ) X − iǫ ′ ,s ( w ) =: Z + j,s ( q ǫǫ z ) Z + j,s ( q − ǫǫ z ) Y + j,s ( z ) : Z − i,s ( w ) Z + i +1 ,s ( q ǫ ′ / w ) Y − i,s ( w )= Z − i,s ( w ) Z + i +1 ,s ( q ǫ ′ / w ) Y − i,s ( w ) : Z + j,s ( q ǫǫ z ) Z + j,s ( q − ǫǫ z ) Y + j,s ( z ) := X − iǫ ′ ,s ( w ) X + jǫ,s ( z ) Similarly, it holds that X + iǫ,s ( z ) X − j,s ( w ) = Z + i,s ( q ǫ z ) Z − i +1 ,s ( z ) Y + i,s ( z ) : Z − j,s ( q w ) Z − j,s ( q − w ) : Y − j,s ( w )=: Z − j,s ( q / w ) Z − j,s ( q − / w ) : Y − j,s ( w ) Z + i,s ( q ǫ/ z ) Z − i +1 ,s ( z ) Y + i,s ( z )= X − j,s ( w ) X + iǫ,s ( z ) . For the case of ( α i | α i ) = 1 , that is, i = 1 , ..., n − . It follows from (4.5)-(4.6) that [ X + i,s ( z ) , X − i,s ( w )]= 1( q − q − ) zw (cid:16) : X + i + ,s ( z ) X − i − ,s ( w ) : z − q ww δ ( q − wz )+ : X − i + ,s ( w ) X + i − ,s ( z ) : z − q − ww δ ( wq − z ) (cid:17) = 1( q − q − ) zw : X + i + ,s ( z ) X − i − ,s ( w ) : δ ( q − wz )+ 1( q − − q ) zw : X − i + ,s ( w ) X + i − ,s ( z ) : δ ( wq − z )= 1( q − q − ) zw : X + i + ,s ( q − w ) X − i − ,s ( w ) : δ ( q − wz )+ 1( q − − q ) zw : X − i + ,s ( w ) X + i − ,s ( q w ) : δ ( wq − z ) , where we have used the property of the δ -function: f ( z , z ) δ ( z z ) = f ( z , z ) δ ( z z ) = f ( z , z ) δ ( z z ) . Actually, we get by direct calculation : X + i + ,s ( q − w ) X − i − ,s ( w ) :=: Z + j,s ( q / w ) Z + j,s ( q − / w ) Y + j,s ( q − / w ) Z − j,s ( q / w ) Z − j,s ( q − / w ) : Y − j,s ( w ) := Φ ( s ) j ( q − / w ) . In a similar manner, it is easy to see that : X − i + ,s ( w ) X + i − ,s ( q w ) := Z − i,s ( w ) Z + i +1 ,s ( q / w ) Y − i,s ( w ) Z + i,s ( w ) Z − i +1 ,s ( q w ) Y + i,s ( q w )= Ψ ( s ) i ( wq / ) . Inserting the above expressions into the left hand side of proposition 4.5, we get that [ X + i,s ( z ) , X − i,s ( w )]= 1( q / − q − / ) zw × (cid:16) Φ ( s ) i ( wq − / ) δ ( q − / wz ) − Ψ ( s ) i ( wq / ) δ ( wq − / z ) (cid:17) . Lastly, we need consider the case of ( α j | α j ) = 2 for j = 0 , n . In this case we use relations(4.12) and (4.13), for ǫ = 0 , we get X + j ,s ( z ) X − j,s ( w ) =: X + j ,s ( z ) X − j,s ( w ) := X − j,s ( w ) X + j ,s ( z ) , so EVEL -1/2 REALIZATION OF QUANTUM N-TOROIDAL ALGEBRAS 13 we have that [ X + j,s ( z ) , X − j,s ( w )]= − q − q − ) z X ǫ = ± q ǫ/ : X + jǫ,s ( z ) X − j,s ( w ) : q − ǫ/ z − ww (cid:16) q ǫ/ z/w − − − q ǫ/ z/w (cid:17) = − q − q − ) z X ǫ = ± : X + jǫ,s ( z ) X − j,s ( w ) : ( q − ǫ/ − q ǫ/ ) δ ( q ǫ/ zw )= 1( q − q − ) zw (cid:16) Φ ( s ) j ( wq − / ) δ ( q / zw ) − Ψ ( s ) j ( wq / ) δ ( zq − / w ) (cid:17) , where we have used q / − q − / ( q − q − ) z : X + jǫ,s ( z ) X − j,s ( w ) := q − / ( q − q − ) z : X + jǫ,s ( q − / w ) X − j,s ( w ) := q − / ( q − q − ) z : Z + j,s ( q / w ) Z + j,s ( q − / w ) Y + j,s ( q − / w ) Z − j,s ( q / w ) Z − j,s ( q − / w ) : Y − j,s ( w ) := 1( q − q − ) zw exp (cid:16) ∞ X k =1 a ( s ) j ( k )( q − q − )( q − / w ) − k (cid:17) q a j (0) = 1( q − q − ) zw Φ ( s ) j ( q − / w ) , and − q − q − ) zw : X + j − ,s ( q / w ) X − j,s ( w ) := − q − q − ) zw : Z + j,s ( q / w ) Z + j,s ( q − / w ) Y + j,s ( q / w ) :: Z − j,s ( q / w ) Z − j,s ( q − / w ) : Y − j,s ( w )= − q − q − ) zw exp (cid:16) ∞ X k =1 a ( s ) j ( − k )( − q − q − )( q / w ) k (cid:17) q − a j (0) = − q − q − ) zw Ψ ( s ) j ( q / w ) . (cid:3) For the Serre relation (2.9), we have the following proposition.P
ROPOSITION
For i = j Sym z , ··· z n n =1 − a ij X k =0 ( − k h nk i i X ± i,s ( z ) · · · X ± i,s ( z k ) X ± j,s ( w ) X ± i,s ( z k +1 ) · · · X ± i,s ( z n ) = 0 . P ROOF . First let us see the case when a ij = − for i = 1 , ...n − . Here we only check it for”+”, it is similar for the case ”-”. We list the relations that will be used, X + iǫ ,s ( z ) X + iǫ ,s ( z ) X +( i +1) ǫ,s ( w )=: X + iǫ ,s ( z ) X + iǫ ,s ( z ) X +( i +1) ǫ,s ( w ) : ( q ǫ / z − q ǫ / z )( z − q / w )( z − q / w )( z − qz )( z − q ǫ/ w )( z − q ǫ/ w ) ,X + iǫ ,s ( z ) X +( i +1) ǫ,s ( w ) X + iǫ ,s ( z )=: X + iǫ ,s ( z ) X +( i +1) ǫ,s ( w ) X + iǫ ,s ( z ) : ( z − q / w )( q ǫ / z − q ǫ / z )( q / z − w )( z − qz )( z − q ǫ/ w )( z − q ǫ/ w ) ,X +( i +1) ǫ,s ( w ) X + iǫ ,s ( z ) X + iǫ ,s ( z )=: X +( i +1) ǫ,s ( w ) X + iǫ ,s ( z ) X + iǫ ,s ( z ) : ( w − q / z )( w − q / z )( q ǫ / z − q ǫ / z )( z − qz )( z − q ǫ/ w )( z − q ǫ/ w ) . Thus we get that X + iǫ ,s ( z ) X + iǫ ,s ( z ) X +( i +1) ǫ,s ( w ) − ( q + q − ) X + iǫ ,s ( z ) X +( i +1) ǫ,s ( w ) X + iǫ ,s ( z )+ X +( i +1) ǫ,s ( w ) X + iǫ ,s ( z ) X + iǫ ,s ( z )=: X + iǫ ,s ( z ) X + iǫ ,s ( z ) X +( i +1) ǫ,s ( w ) : q ǫ / z − q ǫ / z ( z − qz )( z − q ǫ/ w )( z − q ǫ/ w ) × (cid:0) ( z − q / w )( z − q / w ) + ( q / + q − / )( z − q / w )( w − q / z )+ ( w − q / z )( w − q / z ) (cid:1) =: X + iǫ ,s ( z ) X + iǫ ,s ( z ) X +( i +1) ǫ,s ( w ) : ( q ǫ / z − q ǫ / z )( q − / − q / ) w ( z − q ǫ/ w )( z − q ǫ/ w ) . We can see that the last part is antisymmetric under ( z , ǫ ) ( z , ǫ ) , which implies the Serrerelation X + i,s ( z ) X + i,s ( z ) X + i +1 ,s ( w ) + ... + ( z ↔ z ) = 0 . For the same reason, it is easy to seethat X + i,s ( z ) X + i,s ( z ) X + i − ,s ( w ) − ( q / + q − / ) X + i,s ( z ) X + i − ,s ( w ) X + i,s ( z ) + · · · = 0 . In the case of a ij = − for i = 0 , j = 1 or i = n, j = n − . We only show the proof in − cases. First, for i = 0 , j = 1 , in this situation we have the following relations, X − ,s ( z ) X − ,s ( z ) X − ǫ,s ( w ) =: X − ,s ( z ) X − ,s ( z ) X − ǫ,s ( w ) : × ( z − z )( z − q − w )( z − q − w )( z − w )( z − w )( z − q − z ) ,X − ,s ( z ) X − ǫ,s ( w ) X − ,s ( z ) =: X − ,s ( z ) X − ǫ,s ( w ) X − ,s ( z ) : × ( z − z )( z − q − w )( z − w )( q − z − w )( z − w )( z − q − z ) , EVEL -1/2 REALIZATION OF QUANTUM N-TOROIDAL ALGEBRAS 15 X − ǫ,s ( w ) X − ,s ( z ) X − ,s ( z ) =: X − ǫ,s ( w ) X − ,s ( z ) X − ,s ( z ) : × ( z − z )( q − z − w )( z − w )( q − z − w )( z − w )( z − q − z ) . Using the above expressions, we get that, X − ,s ( z ) X − ,s ( z ) X − ǫ,s ( w ) − ( q + q − ) X − ,s ( z ) X − ǫ,s ( w ) X − ,s ( z ) + X − ǫ,s ( w ) X − ,s ( z ) X − ,s ( z )=: X − ,s ( z ) X − ,s ( z ) X − ǫ,s ( w ) : z − z z − q − z (cid:16) ( z − q − w )( z − q − w )( z − w )( z − w ) − ( q + q − )( z − q − w )( z − w )( q − z − w )( z − w )+ ( q − z − w )( z − w )( q − z − w )( z − w ) (cid:17) =: X − ,s ( z ) X − ,s ( z ) X − ǫ,s ( w ) : ( q + q − ) w ( z − z )( z − w )( z − w ) . Obviously the antisymmetry with regard to ( z ↔ z ) implies the case.Second, for i = n , j = n − , similarly we have the following relations, X − n,s ( z ) X − n,s ( z ) X − ( n − ǫ,s ( w ) =: X − n,s ( z ) X − n,s ( z ) X − ( n − ǫ,s ( w ) : × ( z − z )( z − q − w )( z − q − w )( z − q − z )( z − q ǫ w )( z − q ǫ w ) ,X − n,s ( z ) X − ( n − ǫ,s ( w ) X − n,s ( z ) =: X − n,s ( z ) X − ( n − ǫ,s ( w ) X − n,s ( z ) : × z − z q ( z − q − z ) (cid:16) z − q − wz − qw (cid:17) ǫ (cid:16) z − qwz − q − w (cid:17) − ǫ ,X − ( n − ǫ,s ( w ) X − n,s ( z ) X − n,s ( z ) =: X − ( n − ǫ,s ( w ) X − n,s ( z ) X − n,s ( z ) : × z − z q ǫ ( z − q − z ) (cid:16) ( w − q − z )( w − q − z )( w − qz )( w − qz ) (cid:17) − ǫ . It is easy to get that X − n,s ( z ) X − n,s ( z ) X − ( n − ,s ( w ) − ( q + q − ) X − n,s ( z ) X − ( n − ,s ( w ) X − n,s ( z )+ X − ( n − ,s ( w ) X − n,s ( z ) X − n,s ( z )=: X − n,s ( z ) X − n,s ( z ) X − ( n − ,s ( w ) : z − z z − q − z × (cid:16) ( z − q − w )( z − q − w )( z − qw )( z − qw ) − (1 + q − ) z − q − wz − qw + q − (cid:17) =: X − n,s ( z ) X − n,s ( z ) X − ( n − ,s ( w ) : z − z ( z − qw )( z − qw ) . X − n,s ( z ) X − n,s ( z ) X − ( n − − ,s ( w ) − ( q + q − ) X − n,s ( z ) X − ( n − − ,s ( w ) X − n,s ( z )+ X − ( n − − ,s ( w ) X − n,s ( z ) X − n,s ( z )=: X − n,s ( z ) X − n,s ( z ) X − ( n − − ,s ( w ) : z − z z − q − z × (cid:16) − (1 + q − ) z − qwz − q − w + q ( w − q − z )( w − q − z )( w − qz )( w − qz ) (cid:17) =: X − n,s ( z ) X − n,s ( z ) X − ( n − − ,s ( w ) : ( z − z ) w ( q − q − )( z − q − w )( z − q − w ) . So we get the conclusion through the antisymmetry of the above two expressions.In the case of a ij = − for i = 1 , j = 0 and i = n − , j = n , here we only give the proof for i = n − , j = n . First we have that, X +( n − ǫ ,s ( z ) X +( n − ǫ ,s ( z ) X +( n − ǫ ,s ( z ) X + nǫ,s ( w )=: X +( n − ǫ ,s ( z ) X +( n − ǫ ,s ( z ) X +( n − ǫ ,s ( z ) X + nǫ,s ( w ) : × Y ≤ k Successively taking limit with respect to z → w, z → w, z → w imply that lim z i → w X k =0 ( − k h k i X − ,s ( z ) · · · X − ,s ( z k ) X +0 ,s ′ ( w ) X − ,s ( z k +1 ) · · · X − ,s ( z ) = 0 . Thus we have completed the proof of Theorem 3.1. ACKNOWLEDGMENT N. Jing would like to thank the support of Simons Foundation grant 523868 and NSFC grant11531004. H. Zhang would like to thank the support of NSFC grant 11871325. References [1] D. Bernard, Vertex operator representations of quantum affine algebras U q ( B (1) n ) , Lett. Math. Phys. (1989)239–245.[2] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press Cambridge, 1994.[3] V. Chari, A. Pressley, Quantum affine algebras, Comm. Math. Phys. (1991) 261–283.[4] V. Chari, A. Pressley, Quantum affine algebras and their representations, Representations of groups (Banff, AB,1994), CMS Conf. Proc., 16, Amer. Math. Soc. Providence, RI, (1995) 59–78.[5] V. Chari, A. Pressley, Quantum affine algebras and affine Hecke algebras, Pacific J. Math. (1996) 295–326.[6] V. G. Drinfeld, Quantum groups, Proc. of the ICM, Berkeley, 1986, pp. 798-820. American MathematicalSociety, Providence Soviet Math. Dokl. (1988)212–216.[8] I. B. Frenkel, N. Jing, Vertex representations of quantum affine algebras, Proc. Nat’l. Acad. Sci. USA. (1998)9373–9377.[9] B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Branching rules for quantum toroidal gl ( n ) , Adv. Math. (2016)229-274.[10] B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Representations of quantum toroidal gl n , J. Algebra (2013)78–108.[11] I. B. Frenkel, N. Jing, W. Wang, Quantum vertex representations via finite groups and the McKay correspon-dence, Comm. Math. Phys. (2000) 365–393.[12] Y. Gao, N. Jing, U q ( gl N ) action on gl N -modules and quantum toroidal algebras, J. Algebra (2004) 320–343.[13] Y. Gao, N. Jing, L. Xia, H. Zhang, Quantum N -toroidal algebras and quantized GIM algebras of N -fold affiniza-tion, arXiv:1907.06301.[14] V. Ginzburg, M. Kapranov, E. Vasserot, Langlands reciprocty for algebric surfaces, Math. Res. Lett. (1995)147–160.[15] N. Guay, X. Ma, From quantum loop algebras to Yangians, J. Lond. Math. Soc. (2012) 683–700.[16] S. Gautam, V. Toledano-Laredo, Yangians and quantum loop algebras, Selecta Math. (N.S.) (2013) 271–336.[17] D. Hernandez, Representations of quantum affinizations and fusion product, Tranformation Group (2005)163–200.[18] D. Hernandez, Quantum toroidal algebras and their representations, Selecta Math. (N.S.) (2009) 701–725.[19] M. Jimbo, A q-difference analogue of U ( g ) and the Yang-Baxter equation, Lett. Math. Phys. (1985) 63–69.[20] N. Jing, Twisted vertex representations of quantum affine algebras, Innven. Math. (1990) 663–690.[21] N. Jing, Higher level representations of the quantum affine algebra U q ( ˆ sl (2)) , J. Algebra (1996) 448–468.[22] N. Jing, Level one representations of the quantum affine algebra U q ( G (1)2 ) , Proc. Amer. Math. Soc. (1999),21–27.[23] N. Jing, S.J. Kang, Y. Koyama, Vertex operators between level one irreducible representations of the quantumaffine algebra U q ( D (1) ) , Comm. Math. Phys. (1995) 367–392. [24] N. Jing, Y. Koyama, K.C. Misra, Bosonic realizations of U q ( C (1) n ) , J. Algebra (1998) 155–172.[25] N. Jing, Y. Koyama, K.C. Misra, Level one representations of quantum affine algebras U q ( C (1) n ) , SelectaMath.,(N.S) (1999) 243–255.[26] N. Jing, K. C. Misra, Vertex operators of level one U q ( B (1) n ) -modules, Lett. Math. Phys. (1996) 127–143.[27] N. Jing, K. C. Misra, Vertex operators for twisted quantum affine algebras, Trans. Amer. Math. Soc. (1999)1663–1690.[28] K. Miki, Toroidal and level 0 U ′ q ( b sl n +1 ) actions on U q ( b gl n +1 ) -modules, J. Math. Phys. (1999) 3191–3210.[29] K. Miki, Toroidal braid group action and an automorphism of toroidal algebra U q ( sl n +1 , tor ) ( n ≥ ), Lett.Math. Phys. (1999) 365–378.[30] K. Miki, Representations of quantum toroidal algebra U q ( sl n +1 , tor )( n ≥ , J. Math. Phys. (2000) 7079–7098.[31] K. Miki, Quantum toroidal algebra U q ( sl , tor ) and R matrices, J. Math. Phys. (2001) 2293–2308.[32] Y. Saito, Quantum toroidal algebras and their vertex representations, Publ. RIMS. Kyoto Univ. (1998) 155–177.[33] Y. Saito, K. Takemura, D. Uglov, Toroidal actions on level 1 modules of U q ( sl n ) , Transform. Groups (1998)75–102.[34] M. Varagnolo, E. Vasserot, Schur duality in the toroidal setting, Comm. Math. Phys. (1996) 469–484.D EPARTMENT OF M ATHEMATICS , S HANGHAI U NIVERSITY , S HANGHAI HINA D EPARTMENT OF M ATHEMATICS , N ORTH C AROLINA S TATE U NIVERSITY , R ALEIGH , NC 27695, USA, E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS , S HANGHAI U NIVERSITY , S HANGHAI HINA D EPARTMENT OF M ATHEMATICS , S HANGHAI U NIVERSITY , S HANGHAI HINA E-mail address ::