The central elements of E q,p ( s l 2 ^ ) k with the critical level
aa r X i v : . [ m a t h - ph ] D ec The central elements of E q,p ( c sl ) k with the critical level Wenjing Chang a , Xiang-mao Ding b , Ke Wu a a Capital Normal University, Beijing 100048, People’s Republic of China b Institute of Applied Mathematics, Academy of Mathematics and Systems Science;Chinese Academy of Sciences, P.O.Box 2734, Beijing 100190, People’s Republic of China
Abstract
In this paper we generalize certain results concerning quantum affine algebra U q ( c sl )at the critical level to the corresponding elliptic case E q,p ( c sl ). Using the Wakimotorealization of the algebra E q,p ( c sl ), we construct the central elements of it at the criticallevel. It turns out that the so called Drinfeld conjecture originally proposed for Kac-Moody algebras also holds for the elliptic quantum algebras. For any affine Kac-Moody algebra b g , let e U ( b g ) k be the completion of its universal envelopingalgebra. It was conjectured that e U ( b g ) k has a center Z ( b g ) at the level k = − h ∨ , and further-more, Z ( b g ) possesses a Poisson structure and is isomorphic to the classical W-algebra W ( g L ),and g L is Langlands duality of g . This is the so called Drinfeld conjecture. Please see [1], formore detail statements on the Drinfeld conjecture and its applications. When k + h ∨ = 0, thelevel k is nominated as critical. By applying the Wakimoto realization of b g [2, 3], one can con-struct a homomorphism from Z ( b g ) to a commutative algebra H ( g ) at the critical level. Andthe associated map W ( g L ) → H ( g ) is nothing but the Miura transformation, which has beendefined for any arbitrary g [4]. In particular, for the center Z ( c sl ) of e U ( c sl ) k at the criticallevel k = −
2, it is generated by the Sugawara operator and isomorphic to the algebra W ( sl ),which is also called the classical Virasoro algebra. Later, E.Frenkel and N.Reshetikhin [5]generalized the above results to the quantum affine algebra U q ( c sl ). Using the Wakimotorealization of U q ( c sl ) given [6], they constructed a homomorphism from the center Z q ( c sl )to a Poisson algebra H q ( sl ), which is the q-analogue of the Miura transformation. While,in [7], the authors constructed the ~ -deformed Miura map and the corresponding deformed Email: [email protected] Email: [email protected] Email: [email protected] DY ~ ( sl ) k at the critical level k = −
2. These results showed thatthe Drinfeld conjecture held for the quantum affine algebras and the Yangian double withcenter. And they have many applications, such as, they gave a new interpretation of theBethe ansatz in the Gaudin models of statistical mechanics, which allowed us to relate theBethe ansatz approach to the geometric Langlands correspondence [8, 9]. Recently, in [10],the authors applied the above results to study the Langlands duality for representations ofquantum groups.Elliptic quantum groups were introduced to study the infinite dimensional symmetries ofthe elliptic face models in statistical mechanics [11, 12, 13]. They could be considered as theelliptic deformation of quantum affine algebras. For example, on one hand, the elliptic quan-tum algebra U q,p ( c sl ) degenerates to the quantum affine algebra U q ( c sl ), as the parameter p → U q,p ( c sl ) can be obtained by twisting the Drinfeld realizationof U q ( c sl ) [15]. As a result, it is interesting to ask whether the constructions worked in theKac-Moody algebra cases and the q -(or ~ )-deformed cases, also work in the elliptic case, andwrite out explicitly the central elements of the completion of elliptic quantum algebra. It isbelieved that the statement will be true. For its complexity, the corresponding results arenot so straightforward as the previously known cases, so they haven’t been given explicitly.This is the goal of this paper.In [15, 16], the authors constructed the free field realizations of U q,p ( c sl ) for any givenlevel k . But these results could not be generalized to the higher rank case U q,p ( d sl N ), whosefree field realization with the level k = 1 was given by [17]. For arbitrary level, we solved thisproblem using a different approach [18, 19]. Here we will apply the realization given in [18]to construct the central elements of the elliptic algebra. We also hope that this constructioncan be generalized to the higher rank case by applying the results of [19].This paper is organized as follows. In section 2, we will recall the definition of the ellipticquantum algebra E q,p ( c sl ) generated by twisting the generating currents of U q ( c sl ). Insection 3, for any given level k , we construct the free fields realization of E q,p ( c sl ). In section4, the central elements of E q,p ( c sl ) at the critical level k = − U q ( c sl ) k = − . Interms of the free field realization,there exists a homomorphism from the center Z q,p ( c sl ) toa commutative algebra H q ( sl ). In some sense, it shows that the Drinfeld conjecture holdsin the elliptic case. Throughout this paper, let q ∈ C with q = 0 , | q | < The elliptic algebra E q,p ( c sl ) k In this section, we first review the Drinfeld realization of the quantum affine algebra U q ( c sl ) k .It is an associative algebra over C with Drinfeld generators E ± n ( n ∈ Z ), H n ( n ∈ Z =0 ),invertible q h and c , which satisfy the following relations [20]: c : central element [ h, H n ] = 0 , q h E ± n q − h = q ± E ± n [ H n , H m ] = [2 n ][ cn ] n δ n + m, , [ H n , E ± m ] = ± [2 n ] n q ∓ c n E ± m + n , [ E + n , E − m ] = 1 q − q − (cid:0) q c ( n − m ) ψ + ,n − m − q − c ( n − m ) ψ − ,n − m (cid:1) , [ E ± n +1 , E ± m ] q ± + [ E ± m +1 , E ± n ] q ± = 0 , where ψ ± ,n are defined by X n ∈ Z ψ ± ,n z − n = q ± h exp (cid:16) ± ( q − q − ) X ± n> H n z − n (cid:17) , and the symbol [ A, B ] x for x ∈ C denotes AB − xBA . In terms of the generating functions K ± ( z ), E ± ( z ) given by K ± ( z ) = q ± h exp (cid:16) ± ( q − q − ) X n> [ n ][2 n ] q ∓ n H ∓ n z ± n (cid:17) ,E ± ( z ) = X n ∈ Z E ± n z − n − , the above defining relations can be recast as K ± ( z ) K ± ( w ) = K ± ( w ) K ± ( z ) , (1) K − ( z ) K + ( w ) = f q ( wz q − k ) f q ( wz q k ) K + ( w ) K − ( z ) , (2) K ± ( z ) E + ( w ) = zq ∓ k − − wqzq ∓ k − w E + ( w ) K ± ( z ) , (3) K ± ( z ) E − ( w ) = zq ± k − wzq ± k − − wq E − ( w ) K ± ( z ) , (4) E + ( z ) E + ( w ) = zq − wz − wq E + ( w ) E + ( z ) , (5) E − ( z ) E − ( w ) = z − wq zq − w E − ( w ) E − ( z ) , (6)[ E + ( z ) , E − ( w )] = 1( q − q − ) zw (cid:16) δ (cid:16) wz q k (cid:17) K − ( q k +2 w ) − K − ( q k w ) − − δ (cid:16) wz q − k (cid:17) K + ( q − k +2 w ) − K + ( q − k w ) − (cid:17) , (7)3n which the symbol f q ( x ) is given by f q ( x ) = ( x ; q ) ∞ ( xq ; q ) ∞ ( xq ; q ) ∞ , with ( a ; b ) ∞ = ∞ Y n =0 (1 − ab n );and δ ( x ) = X m ∈ Z x m . Let the parameters p , p ∗ be p = q r , p ∗ = q r ∗ = pq − c ( r ∗ = r − c ; r, r ∗ ∈ R > )where c is the central element of U q ( c sl ). we introduce some dressing currents V ± ( z ; r, r ∗ )and D ± ( z ; r, r ∗ ) in U q ( c sl ) depending on the parameters r and r ∗ : V + ( z ; r, r ∗ ) = exp (cid:16) − X n> [ n ][ r ∗ n ][2 n ] H − n q ( r ∗ − n z n (cid:17) , (8) V − ( z ; r, r ∗ ) = exp (cid:16) X n> [ n ][ rn ][2 n ] H n q ( r +1) n z − n (cid:17) , (9) D + ( z ; r, r ∗ ) = exp (cid:16) X n> r ∗ n ] H − n q ( r ∗ + c ) n z n (cid:17) , (10) D − ( z ; r, r ∗ ) = exp (cid:16) − X n> rn ] H n q ( r − c ) n z − n (cid:17) (11)then using them to twist the generating currents K ± ( z ) and E ± ( z ) of U q ( c sl ), we have Definition 1
The twisting currents of the algebra U q ( c sl ) are given by k + ( z ; r, r ∗ ) = V + ( z ; r, r ∗ ) K + ( z ) V − ( q k z ; r, r ∗ ) , (12) k − ( z ; r, r ∗ ) = V + ( q k z ; r, r ∗ ) K − ( z ) V − ( z ; r, r ∗ ) , (13) e ( z ; r, r ∗ ) = D + ( z ; r, r ∗ ) E + ( z ) , (14) f ( z ; r, r ∗ ) = E − ( z ) D − ( z ; r, r ∗ ) . (15)For brevity, here we just denote them as k ± ( z ; p ), e ( z ; p ) and f ( z ; p ). Applying the com-mutation relations (1)-(7), we can prove Proposition 1
The currents k ± ( z ; p ) , e ( z ; p ) and f ( z ; p ) satisfy k ± ( z ; p ) k ± ( w ; p ) = k ± ( w ; p ) k ± ( z ; p ) , (16) k − ( z ; p ) k + ( w ; p ) = F q, p ( wz q − k ) F q, p ( zw q k pp ∗ ) F q, p ( wz q k ) F q, p ( zw q − k pp ∗ ) k + ( w ; p ) k − ( z ; p ) , (17) k ± ( z ; p ) e ( w ; p ) = q · Θ p ∗ ( q ∓ k − zw )Θ p ∗ ( q ∓ k zw ) e ( w ; p ) k ± ( z ; p ) , (18)4 ± ( z ; p ) f ( w ; p ) = q − · Θ p ( q ± k zw )Θ p ( q ± k − zw ) f ( w ; p ) k ± ( z ; p ) , (19) e ( z ; p ) e ( w ; p ) = q − · Θ p ∗ ( q zw )Θ p ∗ ( q − zw ) e ( w ; p ) e ( z ; p ) , (20) f ( z ; p ) f ( w ; p ) = q · Θ p ( q − zw )Θ p ( q zw ) f ( w ; p ) f ( z ; p ) , (21)[ e ( z ; p ) , f ( w ; p )] = 1( q − q − ) zw (cid:16) δ ( q − c zw )Ψ + ( q c w ; p ) − δ ( q c zw )Ψ − ( q − c w ; p ) (cid:17) (22) where the currents Ψ ± ( z ; p ) are given by Ψ + ( z ; p ) =: k − ( zq ; p ) − k − ( z ; p ) − :Ψ − ( z ; p ) =: k + ( zq ; p ) − k + ( z ; p ) − : and we use the elliptic theta function Θ t ( z ) defined for t ∈ C : Θ t ( z ) = ( z ; t ) ∞ ( tz − ; t ) ∞ ( t ; t ) ∞ , ( z ; t , · · · , t k ) ∞ = Y n , ··· ,n k ≥ (1 − zt n · · · t n k k ) and F q, p ( x ) = ( x ; q , p, p ∗ ) ∞ ( xq ; q , p, p ∗ ) ∞ ( xq ; q , p, p ∗ ) ∞ . It should be remarked that the factor F q, p ( x ) depends on the two parameters q and p .It is two-parameter generalization of the scalar factor f q ( x ) appeared in [5]. It has closerelationship with the algebraic structure defined by (16)-(22), just like the roles played bythe function f q ( x ) given above. And it is also related to the crossing-symmetry property ofthe elliptic type R-matrix. Furthermore, as p → F q, p ( x ) → f q ( x ) , then the relations (16)-(22) degenerate to (1)-(7). Here we just denote the algebra generatedby the twisting currents k ± ( z ; p ), e ( z ; p ), f ( z ; p ) and the central element c with the definingrelations (16)-(22) as E q,p ( c sl ). E q,p ( c sl ) In this section, we will first recall the Wakimoto realization of U q ( c sl ) for any given level [6].Then by introducing some twisting currents, we construct the free fields realization of the5lgebra E q,p ( c sl ). There are three free bosons[ a n , a m ] = [( k + 2) n ][2 n ] n δ n + m, , [ p a , q a ] = 2( k + 2) , [ b n , b m ] = − [ n ] n δ n + m, , [ p b , q b ] = − , [ c n , c m ] = [ n ] n δ n + m, , [ p c , q c ] = 1which generate a quantum Heisenberg algebra H q,k . Using them, one can introduce somefree boson fields A ± ( z ) = ± (cid:0) ( q − q − ) X n> [ n ][2 n ] a ± n z ∓ n + p a q (cid:1) ,a ( z ; α ) = − X n =0 a n [ n ] q − αn z − n + q a + p a ln z,a ± ( z ) = ± (cid:0) ( q − q − ) X n> a ± n z ∓ n + p a ln q (cid:1) where α ∈ C . For brevity, let a ( z ; 0) = a ( z ). Similarly, we can define other free boson fields b ( z ; α ), b ± ( z ) and c ( z ; α ), c ± ( z ). And the normal ordering : : is defined by moving a n ( n > p a to right and moving a n ( n < q a to left. For example,: exp( a ( z ; α )) := exp (cid:0) − X n< a n [ n ] ( q − α z ) − n (cid:1) e q a z p a exp (cid:0) − X n> a n [ n ] ( q α z ) − n (cid:1) . By [6], we know that there exists a homomorphism h q,k from the algebra U q ( c sl ) k to thealgebra H q,k given by h q,k ( K + ( z )) = A − ( zq − ) − exp (cid:0) − b − ( zq − k − ) (cid:1) ,h q,k ( K − ( z )) = A + ( z ) − exp (cid:0) − b + ( zq k ) (cid:1) ,h q,k ( E + ( z )) =: exp (cid:0) b − ( z ) − ( b + c )( zq − ) (cid:1) : − : exp (cid:0) b + ( z ) − ( b + c )( zq ) (cid:1) : h q,k ( E − ( z )) = A + ( zq k ) A + ( zq k +2 ) : exp (cid:0) b + ( zq k +2 ) + ( b + c )( zq k +1 ) (cid:1) : − A − ( zq − k ) A + ( zq − k − ) : exp (cid:0) b − ( zq − k − ) + ( b + c )( zq − k − ) (cid:1) :which is called the Wakimoto realization of U q ( c sl ) with level k . In particular, in terms ofthe free bosons, the generators H n ( n ∈ Z − { } ) are represented by h q,k ( H n ) = a n q −| n | + b n ( q − k | n | + q − ( k +2) | n | ) , then by (8)-(11), we obtain the free boson realization of the twisting currents V ± ( z ; r, r ∗ )and D ± ( z ; r, r ∗ ). Furthermore, applying them to the relations (12)-(15), we obtain the freebosons realization of the algebra E q,p ( c sl ) with level c = k . More precise, we have another6omomorphism h q,p ; k from the algebra E q,p ( c sl ) to the Heisenberg algebra H q,k . For sim-plicity, we will just use the same notations for the generating currents and the correspondingimages of them under the homomorphism h q,p ; k .It should be remarked that when k = −
2, the homomorphism h q,p ; k provides representa-tions of E q,p ( c sl ) in the Fock representation of the Heisenberg algebra H q,k similarly with thediscussion in [6]. These representations have one parameter the action of p a on the highestweight vector. When k = −
2, the generators a n commute among themselves and generatea commutative algebra H q ( sl ). Therefore representations of E q,p ( c sl ) at the critical level k = − h q,p ; k in a smaller space: the tensor product of the Fock rep-resentation of the subalgebra of H q, − generated by b n , c n , n ∈ Z , and a one-dimensionalrepresentation of H q ( sl ). In next section, we will apply the representation at the criticallevel to construct the central element for the elliptic algebra E q,p ( c sl ). E q,p ( c sl ) In this section we will apply the Wakimoto realization of E q,p ( c sl ) at the critical level toconstruct the center of E q,p ( c sl ). We first recall the Ding-Frenkel correspondence of thealgebra U q ( c sl ) k [21], which was used to construct the isomorphism between the Drinfeldrealization and the RS realization of U q ( c sl ). More precise, the generating functions L ± ( z )of the RS realization can be decomposed as: L ± ( z ) = (cid:18) e ± ( z ) 1 (cid:19) (cid:18) K ± ( z ) 00 K ± ( zq ) − (cid:19) (cid:18) f ± ( z )0 1 (cid:19) in which the notations e ± ( z ) and f ± ( z ) are called the half currents of E + ( z ) and E − ( z ),since they satisfy E + ( z ) = e + ( zq k ) − e − ( zq − k ) ,E − ( z ) = f + ( zq − k ) − f − ( zq k ) . Using D ± ( z ; r, r ∗ ) defined by (10)-(11) to dress the half currents e ± ( z ) and f ± ( z ), we definethe twisting of above half currents as e + ( z ; r, r ∗ ) = D + ( zq − k ; r, r ∗ ) e + ( z ) ,e − ( z ; r, r ∗ ) = D + ( zq k ; r, r ∗ ) e + ( z ) ,f + ( z ; r, r ∗ ) = f + ( z ) D − ( zq k ; r, r ∗ ) ,f − ( z ; r, r ∗ ) = f + ( z ) D − ( zq − k ; r, r ∗ );7espectively, and we will simply denote them as e ± ( z ; p ) and f ± ( z ; p ); then by direct com-putation, we obtain e ( z ; p ) = e + ( zq k ; p ) − e − ( zq − k ; p ) ,f ( z ; p ) = f + ( zq − k ; p ) − f − ( zq k ; p ) . For any k , if we further introduce the generating function l ( z ; p ) = q − : k + ( zq − k ; p ) k − ( zq k ; p ) − : + q : k − ( zq k +2 ; p ) k + ( zq − k +2 ; p ) − :+ k + ( zq − k ; p ) : e ( z ; p ) f ( z ; p ) : k − ( zq k +2 ; p )where : e ( z ; p ) f ( z ; p ) : = e + ( zq k ; p ) f ( z ; p ) − f ( z ; p ) e − ( zq − k ; p ) , then we can prove the following theorem Theorem 1
When k = − , the coefficients of the generating function l ( z ; p ) = q − : k + ( zq ; p ) k − ( zq − ; p ) − : + q : k − ( zq ; p ) k + ( zq ; p ) − :+ k + ( zq ; p ) : e ( z ; p ) f ( z ; p ) : k − ( zq ; p ) are the central elements of E q,p ( c sl ) k . The above identity could be considered as a transfermation from transfer matrix toquantum trace (T-Q transfermation, for short) depending on two parameters. Here we willbriefly present how to prove Thm.1 by applying the Wakimoto realization of the algebra E q,p ( c sl ). In terms of the free bosons fields, we first consider the normally ordered product: e ( z ; p ) f ( z ; p ) :. On one hand, using the Wakimoto realization of D ± ( z ; r, r ∗ ) and E ± ( z ),we have e ( w ; p ) f ( z ; p ) = D + ( w ; r, r ∗ ) E + ( w ) E − ( z ) D − ( z ; r, r ∗ ) , in which E + ( w ) E − ( z ) = qA + ( zq − ) A + ( zq ) e b − ( w ) e ( b + c )( zq − ) − ( b + c )( wq − ) e b + ( z ) + q − A − ( zq − ) A − ( zq ) e ( b − ( z )) e ( b + c )( zq ) − ( b + c )( wq ) e b + ( w ) − w − zwq − zq − A + ( zq − ) A + ( zq ) e ( b + c )( zq − ) − ( b + c )( wq ) e b + ( z )+ b + ( w ) − w − zwq − − zq A − ( zq − ) A − ( zq ) e b − ( z )+ b − ( w ) e ( b + c )( zq ) − ( b + c )( wq − ) . Similarly with E + ( w ) E − ( z ) discussed in [5], e ( w ; p ) f ( z ; p ) makes sense in the region | w | > z | , | w | > q | z | and | w | > q − | z | . On the other hand, we have f ( z ; p ) e ( w ; p ) = E − ( z ) D − ( z ; r, r ∗ ) D + ( w ; r, r ∗ ) E + ( w )= c E − ( z ) D + ( w ; r, r ∗ ) D − ( z ; r, r ∗ ) E + ( w )= c c D + ( w ; r, r ∗ ) E − ( z ) D − ( z ; r, r ∗ ) E + ( w )= c c c D + ( w ; r, r ∗ ) E − ( z ) E + ( w ) D − ( z ; r, r ∗ )= D + ( w ; r, r ∗ ) E − ( z ) E + ( w ) D − ( z ; r, r ∗ ) , in which the coefficients are given by c = (cid:0) pq wz ; p (cid:1) ∞ (cid:0) p ∗ q − wz ; p ∗ (cid:1) ∞ (cid:0) pq − wz ; p (cid:1) ∞ (cid:0) p ∗ wz ; p ∗ (cid:1) ∞ ,c = (cid:0) p ∗ wz ; p ∗ (cid:1) ∞ (cid:0) p ∗ q − wz ; p ∗ (cid:1) ∞ ,c = (cid:0) pq − wz ; p (cid:1) ∞ (cid:0) pq wz ; p (cid:1) ∞ and c c c = 1, then f ( z ; p ) e ( w ; p ) has the same formula as e ( w ; p ) f ( z ; p ), but makes sensein the region | w | < | z | , | w | < q | z | and | w | < q − | z | . Therefore, we can write : e ( z ; p ) f ( z ; p ) :as : e ( z ; p ) f ( z ; p ) := Z C R e ( w ; p ) f ( z ; p ) w − z dw − Z C r f ( z ; p ) e ( w ; p ) w − z dw where the notations C R and C r are circles around the origin of radii R > | w | and r < | w | respectively, and here integrals are the contours on the w plane surrounding the points z, zq , zq − . As a result, we obtain the expression: e ( z ; p ) f ( z ; p ) := qD + ( z ; r, r ∗ ) A + ( zq − ) A + ( zq ) e b − ( z ) e b + ( z ) D − ( z ; r, r ∗ )+ q − D + ( z ; r, r ∗ ) A − ( zq − ) A − ( zq ) e b − ( z ) e b + ( z ) D − ( z ; r, r ∗ ) − q − D + ( z ; r, r ∗ ) A + ( zq − ) A + ( zq ) e b + ( zq − )+ b + ( z ) D − ( z ; r, r ∗ ) − qD + ( z ; r, r ∗ ) A − ( zq − ) A − ( zq ) e b − ( zq )+ b − ( z ) D − ( z ; r, r ∗ ) . Applying the free fields realization of k ± ( z ; p ) and D ± ( z ; r, r ∗ ) given in above section, wehave k + ( zq ; p ) : e ( z ; p ) f ( z ; p ) : k − ( zq ; p )= q − A − ( zq ) A + ( zq ) − + qA − ( zq − ) − A + ( zq − ) − q − W ( q − z ; r, r ∗ ) A − ( zq − ) − A + ( zq − ) W ( q − z ; r, r ∗ ) − e − b − ( z ) e b + ( zq − ) − qW ( qz ; r, r ∗ ) − A − ( zq ) A + ( zq ) − W ( qz ; r, r ∗ ) e b − ( zq ) e − b + ( z ) . W ( z ; r, r ∗ ) = exp (cid:16) − ( q − q − ) X n> [ n ][ r ∗ n ] a − n q rn z n (cid:17) ,W ( z ; r, r ∗ ) = exp (cid:16) ( q − q − ) X n> [2 n ][ r ∗ n ] b − n q ( r +1) n z n (cid:17) . Moreover, q − k + ( zq ; p ) k − ( zq − ; p ) − = q − W ( q − z ; r, r ∗ ) A − ( zq − ) − A + ( zq − ) × W ( q − z ; r, r ∗ ) − e − b − ( z ) e b + ( zq − ) , and qk + ( zq ; p ) − k − ( zq − ; p ) = qW ( qz ; r, r ∗ ) − A − ( zq ) A + ( zq ) − × W ( qz ; r, r ∗ ) e b − ( zq ) e − b + ( z ) ;then under the homomorphism h q,p ; k with k = −
2, the current l ( z ; p ) can be represented as l ( z ; p ) = q − A − ( zq ) A +( zq ) − + qA − ( zq − ) − A + ( zq − ) . It says that in terms of the free fields, the Cartan parts are decoupled with the off-diagonalones. Then we can further show that under the Wakimoto representation h q,p ; k with thecritical level k = − l ( z ; p ) commutes with the twisting currents k ± ( z ; p ), e ( z ; p ) and f ( z ; p ), i.e., the Fourier coefficients of l ( z ; p ) are the central elements of the elliptic algebra E q,p ( c sl ).It should also be noted that although the expression for l ( z ; p ) in terms of free fields isthe same as the one for the trigonometric case U q ( c sl ) k = − [5], their algebraic expressionsare different from each other. In fact, we obtain the elliptic algebra E q,p ( c sl ) by twistingthe quantum affine algebra U q ( c sl ). The twisting currents map a Hopf algebra structure of U q ( c sl ) to a quasi-Hopf algebra structure of E q,p ( c sl ) [14]. As what we have clarified in thispaper, it doesn’t change the center of them in terms of the free fields. In this paper, applying the free field realization of E q,p ( c sl ), we construct the center of itat the critical level. We hope that the construction can be generalized to the higher rankcase. It is also interesting to prove the results without using the free field representationexpressions. We hope to discuss these problems in future.10 Acknowledgments
One of the authors (Ding) is financially supported partly by the Natural Science Foundationsof China through the grands No.10931006 and No.10975180. He would like to thank theInstitute of Mathematical Sciences, the Chinese University of Hong Kong for hospitality,where part of this work was done.
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