Wakimoto realization of Drinfeld current for the elliptic quantum algebra $U_{q,p}(sl_3^)$

W akimoto realization of Drinfeld current for the ellipt ic quan tum algebra U q ,p ( b sl 3 ) No v emb er 5, 20 18 T ak eo K OJIMA Dep artment of Mathematics, Col le ge of Scienc e and T e chnolo gy, Nihon University, Surugadai, Chiyo da-ku, T ok yo 101-0062, JAP AN Abstract W e study a free field r ealizat ion of the ell iptic quan tum alg ebra U q ,p ( c sl 3 ) for arbitrary lev el k . W e giv e th e free field r ealizat ion of elliptic analogue of Drin feld current asso ciated with U q ,p ( c sl 3 ) for arb itrary lev el k . In the limit p → 0 , q → 1 our realizatio n repro du ces W akimoto reali zation for the affine Lie algebra c sl 3 . P A CS n u m b ers : 02.20. Sv, 02.20.Uw, 02.30.Ik 1 In tro ductio n The elliptic quan tum group has been prop o sed in p ap ers [1, 2, 3, 4, 5]. There are t wo types of elliptic quan tum groups, the v ertex t yp e A q ,p ( c sl N ) and the fa ce ty p e B q ,λ ( g ), where g is a Kac-Mo o dy algebra asso ciated with a symmetrizable Cartan matrix. The elliptic quan tum groups hav e the structure of quasi-tr ia ngular quasi-Hopf alg ebras in tro duced b y V.Drinfeld [6]. H.Konno [7] introduced the elliptic quantum algebra U q ,p ( c sl 2 ) as an algebra of the screening currents o f the extended deformed Virasoro algebra in terms of the fusion SOS mo del [8]. M.Jim b o, H.Konno, S.Odak e, J.Shiraishi [9] contin ued to 1 study the elliptic quantum algebra U q ,p ( c sl 2 ). They constructed the elliptic alna lo gue of Drinfeld curren ts a nd iden tified U q ,p ( c sl 2 ) with the tensor pro duct of B q ,λ ( c sl 2 ) and a Heisen b erg algebra H . The elliptic quantum group B q ,λ ( c sl 2 ) is a quasi-Hopf algebra while the elliptic a lgebra U q ,p ( c sl 2 ) is not. The in tert wining relation of the v ertex op erator of B q ,λ ( c sl 2 ) is based on the quasi-Hopf structure of B q ,λ ( c sl 2 ). By the ab ov e isomorphism U q ,p ( c sl 2 ) ≃ B q ,λ ( c sl 2 ) ⊗ H , w e can understand ”in tertwining relation” of the ve rtex op erator for the elliptic algebra U q ,p ( c sl 2 ). Along the ab o v e sc heme the elliptic analo g ue of D rinfeld curren t of U q ,p ( c sl 2 ) is extended t o those of U q ,p ( g ) for non-t wisted affine Lie algebra g [9, 10]. In this pap er w e are inte rested in higher-rank generalization of leve l k free field realization of the elliptic quantum alg ebra. F or the elliptic algebra U q ,p ( c sl 2 ), there exist t wo kind of free field realizations for arbitrary leve l k , the one is parafermion realization [7, 9], the other is W akimoto realization [16]. In this pa p er w e ar e in terested in the higher- rank generalization of W akimoto realization of U q ,p ( c sl 2 ). W e construct lev el k fr ee field realization of Drinfeld current associated with the elliptic a lg ebra U q ,p ( c sl 3 ). This give s the first example of a rbitrary lev el free field realization of the higher-rank elliptic algebra. This free field realization can be a pplied for cons truction of the in tegrals of motion for the elliptic alg ebra U q ,p ( c sl 3 ). F or this purp ose, see references [17, 18 , 19]. The organization of this pap er is as follows. In section 2 w e set the no tation a nd in tro duce b osons. In section 3 we review the leve l k free field realization of the quantum group U q ( c sl 3 ) [15]. In section 4 w e give the lev el k free field realizat io n of the elliptic quan tum algebra U q ,p ( c sl 3 ). In a pp endix w e summarize the normal o r dering of the basic op erators. 2 Boson The purp ose of this section is to se t up the basic nota t io n and to in tro duce the b oson. In this pap er w e fix three para meters q , k , r ∈ C . Let us set r ∗ = r − k . W e assume k 6 = 0 , − 3 and Re( r ) > 0, Re( r ∗ ) > 0. W e assume q is a generic with | q | < 1 , q 6 = 0. Let us set a pair o f parameters p and p ∗ b y p = q 2 r , p ∗ = q 2 r ∗ . 2 W e use the standard sym b o l of q -in teger [ n ] b y [ n ] = q n − q − n q − q − 1 . Let us set the elliptic theta function Θ p ( z ) by Θ p ( z ) = ( z ; p ) ∞ ( p/z ; p ) ∞ ( p ; p ) ∞ , ( z ; p ) ∞ = ∞ Y n =0 (1 − p n z ) . It is conv enien t to w ork with t he additive notatio n. W e use the par a metrization q = e − π √ − 1 /rτ , p = e − 2 π √ − 1 /τ , p ∗ = e − 2 π √ − 1 /τ ∗ , ( r τ = r ∗ τ ∗ ) , z = q 2 u . Let us set Jacobi elliptic theta function [ u ] r , [ u ] r ∗ b y [ u ] r = q u 2 r − u Θ p ( z ) ( p ; p ) 3 ∞ , [ u ] r ∗ = q u 2 r ∗ − u Θ p ∗ ( z ) ( p ∗ ; p ∗ ) 3 ∞ . The function [ u ] r has a zero at u = 0, enjo ys the quasi-p erio dicit y prop ert y [ u + r ] r = − [ u ] r , [ u + r τ ] r = − e − π √ − 1 τ − 2 π √ − 1 u r [ u ] r . Let us set the delta- function δ ( z ) as formal p ow er series. δ ( z ) = X n ∈ Z z n . F ollow ing [15] we in tro duce fr ee b osons a 1 n , a 2 n , b 1 n , b 2 n , b 3 n , c 1 n , c 2 n , c 3 n , ( n ∈ Z 6 =0 ). [ a i n , a j m ] = [( k + 3) n ][ A i,j n ] n δ n + m, 0 , [ p i a , q j a ] = ( k + 3) A i,j , ( i, j = 1 , 2) , (2.1) [ b i n , b j m ] = − [ n ] 2 n δ i,j δ n + m, 0 , [ p i b , q j b ] = − δ i,j , ( i, j = 1 , 2 , 3 ) , (2.2) [ c i n , c j m ] = [ n ] 2 n δ i,j δ n + m, 0 , [ p i c , q j c ] = δ i,j , ( i, j = 1 , 2 , 3 ) . (2.3) Here we hav e used Cartan matrix   A 11 A 12 A 21 A 22   =   2 − 1 − 1 2   . F or parameters a 1 , a 2 , b 1 , b 2 , b 3 , c 1 , c 2 , c 3 ∈ R , w e set the v acuum v ector | a, b, c i of the F ock space F a 1 a 2 b 1 b 2 b 3 c 1 c 2 c 3 as f o llo wing. a i n | a, b, c i = b j n | a, b, c i = c j n | a, b, c i = 0 , ( i = 1 , 2; j = 1 , 2 , 3) , (2.4) 3 p i a | a, b, c i = a i | a, b, c i , p j b | a, b, c i = b j | a, b, c i , p j c | a, b, c i = c j | a, b, c i , ( i = 1 , 2; j = 1 , 2 , 3; n > 0) . (2.5) The F o c k space F a 1 a 2 b 1 b 2 b 3 c 1 c 2 c 3 is generated b y b osons a 1 − n , a 2 − n , b 1 − n , b 2 − n , b 3 − n , c 1 − n , c 2 − n , c 3 − n for n ∈ N 6 =0 . The dual F o c k space F ∗ a 1 a 2 b 1 b 2 b 3 c 1 c 2 c 3 is defined as the same manner. In this pap er w e construct the elliptic analogue of Drinfeld curren t for U q ,p ( c sl 3 ) b y these b osons a i n , b j n , c j n acting on the F ock space. 3 F ree Field Realization of U q ( c sl 3 ) The purp ose of this section is to give the free field realization of the quantum affine algebra U q ( c sl 3 ). W e giv e a review o f W akimoto realization of U q ( c sl 3 ) [15]. Let us set the b osonic op erators a i ± ( z ) , b i ± ( z ), γ i ( z ) , β i s ( z ) by a i ± ( z ) = ± ( q − q − 1 ) X n> 0 a i ± n z ∓ n ± p i a log q , ( i = 1 , 2) , (3.1) b i ± ( z ) = ± ( q − q − 1 ) X n> 0 b i ± n z ∓ n ± p i b log q , ( i = 1 , 2 , 3) , (3.2) b i ( z ) = − X n 6 =0 b i n [ n ] z − n + q i b + p i b log z , ( i = 1 , 2 , 3) , (3.3) c i ( z ) = − X n 6 =0 c i n [ n ] z − n + q i c + p i c log z , ( i = 1 , 2 , 3) , (3.4) γ i ( z ) = − X n 6 =0 ( b + c ) i n [ n ] z − n + ( q i b + q i c ) + ( p i b + p i c )log( − z ) , ( i = 1 , 2 , 3) , (3.5) β i 1 ( z ) = b i + ( z ) − ( b i + c i )( q z ) , β i 2 ( z ) = b i − ( z ) − ( b i + c i )( q − 1 z ) , ( i = 1 , 2 , 3) , (3.6) β i 1 ( z ) = b i + ( z ) + ( b i + c i )( q z ) , β i 2 ( z ) = b i − ( z ) + ( b i + c i )( q − 1 z ) , ( i = 1 , 2 , 3) . ( 3.7) W e g iv e a free field realiztaion of Drinfeld curren t for U q ( c sl 3 ). Definition 3.1 We defi n e the b osonic op er ators e + 1 ( z ) , e + 2 ( z ) , e − 1 ( z ) , e − 2 ( z ) by e + 1 ( z ) = − 1 ( q − q − 1 ) z ( e + , 1 1 ( z ) − e + , 2 1 ( z )) , (3.8) e + 2 ( z ) = − 1 ( q − q − 1 ) z ( e + , 1 2 ( z ) − e + , 2 2 ( z ) + e + , 3 2 ( z ) − e + , 4 2 ( z )) , (3.9) e − 1 ( z ) = − 1 ( q − q − 1 ) z ( e − , 1 1 ( z ) − e − , 2 1 ( z ) − e − , 3 1 ( z ) + e − , 4 1 ( z )) , (3.10) e − 2 ( z ) = − 1 ( q − q − 1 ) z ( e − , 1 2 ( z ) − e − , 2 2 ( z ) + e − , 3 2 ( z ) − e − , 4 2 ( z )) . (3.11) 4 ψ ± 1 ( z ) = : exp  b 1 ± ( q ± k z ) + b 1 ± ( q ± ( k + 2) z ) + b 2 ± ( q ± ( k + 3) z ) − b 3 ± ( q ± ( k + 2) z ) + a 1 ± ( q ± k +3 2 z )  : , (3.12) ψ ± 2 ( z ) = : exp  − b 1 ± ( q ± ( k + 1) z ) + b 2 ± ( q ± k z ) + b 3 ± ( q ± ( k + 1) z ) + b 3 ± ( q ± ( k + 3) z ) + a 2 ± ( q ± k +3 2 z )  : , (3.13) Her e we have set e + , 1 1 ( z ) = : exp  β 1 1 ( z )  : , (3.14) e + , 2 1 ( z ) = : exp  β 1 2 ( z )  : , (3.15) e + , 1 2 ( z ) = : exp  γ 1 ( z ) + β 2 1 ( z )  : , (3.16) e + , 2 2 ( z ) = : exp  γ 1 ( z ) + β 2 2 ( z )  : , (3.17) e + , 3 2 ( z ) = : exp  β 3 1 ( q z ) + b 2 + ( z ) − b 1 + ( q z )  : , (3.18) e + , 4 2 ( z ) = : exp  β 3 2 ( q z ) + b 2 + ( z ) − b 1 + ( q z )  : , (3.19) e − , 1 1 ( z ) = : exp  β 1 4 ( q − k − 2 z ) + b 2 − ( q − k − 3 z ) − b 3 − ( q − k − 2 z ) + a 1 − ( q − k +3 2 z )  : , (3.20) e − , 2 1 ( z ) = : exp  β 1 3 ( q k +2 z ) + b 2 + ( q k +3 z ) − b 3 + ( q k +2 z ) + a 1 + ( q k +3 2 z )  : , (3.21) e − , 3 1 ( z ) = : exp  γ 2 ( q k +2 z ) + β 3 1 ( q k +2 z ) + b 2 + ( q k +3 z ) − b 3 + ( q k +2 z ) + a 1 + ( q k +3 2 z  : , (3.22) e − , 4 1 ( z ) = : exp  γ 2 ( q k +2 z ) + β 3 2 ( q k +2 z ) + b 2 + ( q k +3 z ) − b 3 + ( q k +2 z ) + a 1 + ( q k +3 2 z  : , (3.23) e − , 1 2 ( z ) = : exp  γ 2 ( q − k − 1 z ) − β 1 3 ( q − k − 1 z ) + 2 b 3 − ( q − k − 1 z ) + a 2 − ( q − k +3 2 z )  : , (3.24) e − , 2 2 ( z ) = : exp  γ 2 ( q − k − 1 z ) − β 1 4 ( q − k − 1 z ) + 2 b 3 − ( q − k − 1 z ) + a 2 − ( q − k +3 2 z )  : , (3.25) e − , 3 2 ( z ) = : exp  β 3 4 ( q − k − 3 z ) + a 2 − ( q − k +3 2 z )  : , (3.26) e − , 4 2 ( z ) = : exp  β 3 3 ( q k +3 z ) + a 2 + ( q k +3 2 z )  : . (3.27) Here the sym b o l : O : represen ts the normal ordering of O . F or example w e hav e : b k b l :=    b i k b i l , k < 0 b i l b i k , k > 0 . : p i b q i b :=: q i b p i b := q i b p i b . Theorem 3.1 [15] The b osonic op er ators e ± i ( z ) , ψ ± i ( z ) , ( i = 1 , 2) satisfy the fol lowing c ommutation r elations. ( z 1 − q A i,j z 2 ) e + i ( z 1 ) e + j ( z 2 ) = ( q A i,j z 1 − z 2 ) e + j ( z 2 ) e + i ( z 1 ) , (3.28) 5 ( z 1 − q − A i,j z 2 ) e − i ( z 1 ) e − j ( z 2 ) = ( q − A i,j z 1 − z 2 ) e − j ( z 2 ) e − i ( z 1 ) , (3.29) [ ψ ± i ( z 1 ) , ψ ± j ( z 2 )] = 0 , (3.30) ( z 1 − q A i,j − k z 2 )( z 1 − q − A i,j + k z 2 ) ψ ± i ( z 1 ) ψ ∓ j ( z 2 ) = ( z 1 − q A i,j + k z 2 )( z 1 − q − A i,j − k z 2 ) ψ ∓ j ( z 2 ) ψ ± i ( z 1 ) , (3.31) ( z 1 − q ± ( A i,j − k 2 ) z 2 ) ψ + i ( z 1 ) e ± j ( z 2 ) = ( q ± A i,j z 1 − q ∓ k 2 z 2 ) e ± j ( z 2 ) ψ + i ( z 1 ) , (3.32) ( z 1 − q ± ( A i,j − k 2 ) z 2 ) e ± i ( z 1 ) ψ − j ( z 2 ) = ( q ± A i,j z 1 − q ∓ k 2 z 2 ) ψ − j ( z 2 ) e ± i ( z 1 ) , (3.33)  e ± i ( z 1 ) e ± i ( z 2 ) e ± j ( z 3 ) − ( q + q − 1 ) e ± i ( z 1 ) e ± j ( z 3 ) e ± j ( z 2 ) + e ± i ( z 3 ) e ± i ( z 1 ) e ± j ( z 2 )  + { z 1 ↔ z 2 } = 0 , for ( i 6 = j ) , (3.34) [ e + i ( z 1 ) , e − j ( z 2 )] = δ i,j ( q − q − 1 ) z 1 z 2  δ  q − k z 1 z 2  ψ + i ( q − k 2 z 1 ) − δ  q k z 1 z 2  ψ − i ( q − k 2 z 2 )  . (3.35) Hence e ± i ( z ) , ψ ± i ( z ) give lev el k f r ee field realization of U q ( c sl 3 ). 4 F ree Field Realization of U q ,p ( c sl 3 ) The purpose of this sec tion is to giv e a free field realization of the elliptic analogue of Drinfeld curren t for U q ,p ( c sl 3 ) with arbitrary lev el k 6 = 0 , − 3. Let us set the b osonic op erators B ∗ i ± ( z ) , B i ± ( z ) , ( i = 1 , 2 , 3), A ∗ i ( z ) , A i ( z ) , ( i = 1 , 2) b y B ∗ i ± ( z ) = exp ± X n> 0 b i − n [ r ∗ n ] z n ! , ( i = 1 , 2 , 3) , (4.1) B i ± ( z ) = exp ± X n> 0 b i n [ r n ] z − n ! , ( i = 1 , 2 , 3) , (4.2) A i ∗ ( z ) = exp X n> 0 a i − n [ r ∗ n ] z n ! , ( i = 1 , 2) , (4.3) A i ( z ) = exp − X n> 0 a i n [ r n ] z − n ! , ( i = 1 , 2) . (4.4) 6 Definition 4.1 L et us set the b osonic op er ators e i ( z ) , f i ( z ) , Ψ ± i ( z ) , ( i = 1 , 2) by e i ( z ) = U ∗ i ( z ) e + i ( z ) , ( i = 1 , 2) , (4.5) f i ( z ) = e − i ( z ) U i ( z ) , ( i = 1 , 2) , (4.6) Ψ + i ( z ) = U ∗ i ( q k 2 z ) ψ + i ( z ) U i ( q − k 2 z ) , ( i = 1 , 2) , (4.7) Ψ + i ( z ) = U ∗ i ( q − k 2 z ) ψ − i ( z ) U i ( q k 2 z ) , ( i = 1 , 2) . (4.8) Her e we have set U ∗ 1 ( z ) = B ∗ 1 + ( q r ∗ z ) B ∗ 1 + ( q r ∗ − 2 z ) B ∗ 2 + ( q r ∗ − 3 z ) B ∗ 3 − ( q r ∗ − 2 z ) A ∗ 1 ( q r ∗ + k − 3 2 z ) , (4.9) U ∗ 2 ( z ) = B ∗ 3 + ( q r ∗ − 3 z ) B ∗ 3 + ( q r ∗ − 1 z ) B ∗ 2 + ( q r ∗ z ) B ∗ 1 − ( q r ∗ − 1 z ) A ∗ 2 ( q r ∗ + k − 3 2 z ) , (4.10) U 1 ( z ) = B 1 − ( q − r ∗ z ) B 1 − ( q − r ∗ +2 z ) B 2 − ( q − r ∗ +3 z ) B 3 + ( q − r ∗ +2 z ) A 1 ( q − r ∗ − k − 3 2 z ) , (4.11) U 2 ( z ) = B 3 − ( q − r ∗ +1 z ) B 3 − ( q − r ∗ +1 z ) B 2 − ( q − r ∗ z ) B 1 + ( q − r ∗ +1 z ) A 2 ( q − r ∗ − k − 3 2 z ) . (4.12) The ab ov e free field realization of the t wistors U ∗ i ( z ) , U i ( z ), ( i = 1 , 2) is the main result of this pap er. Prop osition 4.1 The b osonic op er ators e i ( z ) , f i ( z ) , Ψ ± i ( z ) , ( i = 1 , 2) satisfy the fo l low - ing c ommutation r elations. e i ( z 1 ) e j ( z 2 ) = q − A i,j Θ p ∗ ( q A i,j z 1 /z 2 ) Θ p ∗ ( q − A i,j z 1 /z 2 ) e j ( z 2 ) e i ( z 1 ) , (4.13) f i ( z 1 ) f j ( z 2 ) = q A i,j Θ p ( q − A i,j z 1 /z 2 ) Θ p ( q A i,j z 1 /z 2 ) f j ( z 2 ) f i ( z 1 ) , (4.14) Ψ ± i ( z 1 )Ψ ± j ( z 2 ) = Θ p ( q − A i,j z 1 /z 2 )Θ p ∗ ( q A i,j z 1 /z 2 ) Θ p ( q A i,j z 1 /z 2 )Θ p ∗ ( q − A i,j z 1 /z 2 ) Ψ ± j ( z 2 )Ψ ± i ( z 1 ) , (4.15) Ψ ± i ( z 1 )Ψ ∓ j ( z 2 ) = Θ p ( pq − A i,j − k z 1 /z 2 )Θ p ∗ ( p ∗ q A i,j + k z 1 /z 2 ) Θ p ( pq A i,j − k z 1 /z 2 )Θ p ∗ ( p ∗ q − A i,j + k z 1 /z 2 ) Ψ ∓ j ( z 2 )Ψ ± i ( z 1 ) , (4.16) Ψ ± i ( z 1 ) e j ( z 2 ) = Θ p ∗ ( q A i,j ± k 2 z 1 /z 2 ) Θ p ∗ ( q − A i,j ± k 2 z 1 /z 2 ) e j ( z 2 )Ψ ± i ( z 1 ) , (4.17) Ψ ± i ( z 1 ) f j ( z 2 ) = Θ p ∗ ( q − A i,j ∓ k 2 z 1 /z 2 ) Θ p ∗ ( q A i,j ∓ k 2 z 1 /z 2 ) e j ( z 2 )Ψ ± i ( z 1 ) , (4.18) [ e i ( z 1 ) , f j ( z 2 )] = δ i,j ( q − q − 1 ) z 1 z 2  δ  q − k z 1 z 2  Ψ + i ( q − k / 2 z 1 ) − δ  q k z 1 z 2  Ψ − i ( q − k / 2 z 2 )  , ( i 6 = j ) . (4.19) 7 W e intro duce the Heisen b erg algebra H generated by the following P i , Q i , ( i = 1 , 2). [ P i , Q j ] = A i,j 2 , ( i, j = 1 , 2) . (4.20) Definition 4.2 L et us define the b osonic op er ators E i ( z ) , F i ( z ) , H ± i ( z ) ∈ U q ( c sl 3 ) ⊗H , ( i = 1 , 2) by E 1 ( z ) = e 1 ( z ) e 2 Q 1 z − P 1 − 1 r ∗ , E 2 ( z ) = e 2 ( z ) e 2 Q 2 z − P 2 − 1 r ∗ , (4.21) F 1 ( z ) = f 1 ( z ) z 2 p 1 b + p 2 b − p 3 b + p 1 a r z P 1 − 1 r , F 2 ( z ) = f 2 ( z ) z 2 p 3 b + p 2 b − p 1 b + p 2 a r z P 2 − 1 r , (4.22) H ± 1 ( z ) = Ψ ± 1 ( z ) e 2 Q 1 ( q ∓ k 2 z ) 2 p 1 b + p 2 b − p 3 b + p 1 a r ( q ± ( r − k 2 ) z ) P 1 − 1 r − P 1 − 1 r ∗ , (4.23) H ± 2 ( z ) = Ψ ± 2 ( z ) e 2 Q 2 ( q ∓ k 2 z ) 2 p 3 b + p 2 b − p 1 b + p 2 a r ( q ± ( r − k 2 ) z ) P 2 − 1 r − P 2 − 1 r ∗ . (4.24) Theorem 4.2 The b osonic op er ators E i ( z ) , F i ( z ) , H ± i ( z ) , ( i = 1 , 2) satisfy the fol lowing c ommutation r elations. E i ( z 1 ) E j ( z 2 ) =  u 1 − u 2 + A i,j 2  r ∗  u 1 − u 2 − A i,j 2  r ∗ E j ( z 2 ) E i ( z 1 ) , (4.25) F i ( z 1 ) F j ( z 2 ) =  u 1 − u 2 − A i,j 2  r  u 1 − u 2 + A i,j 2  r F j ( z 2 ) F i ( z 1 ) , (4.26) H ± i ( z 1 ) H ± j ( z 2 ) =  u 1 − u 2 − A i,j 2  r  u 1 − u 2 + A i,j 2  r ∗  u 1 − u 2 + A i,j 2  r  u 1 − u 2 − A i,j 2  r ∗ H ± j ( z 2 ) H ± i ( z 1 ) , (4.27) H + i ( z 1 ) H − j ( z 2 ) =  u 1 − u 2 − A i,j 2 − k 2  r  u 1 − u 2 + A i,j 2 + k 2  r ∗  u 1 − u 2 + A i,j 2 − k 2  r  u 1 − u 2 − A i,j 2 + k 2  r ∗ H − j ( z 2 ) H + i ( z 1 ) , (4.28) H ± i ( z 1 ) E j ( z 2 ) =  u 1 − u 2 ± k 4 + A i,j 2  r ∗  u 1 − u 2 ± k 4 − A i,j 2  r ∗ E j ( z 2 ) H ± i ( z 1 ) , (4.29) H ± i ( z 1 ) F j ( z 2 ) =  u 1 − u 2 ∓ k 4 − A i,j 2  r  u 1 − u 2 ∓ k 4 + A i,j 2  r F j ( z 2 ) H ± i ( z 1 ) , (4.30) 8 [ E i ( z 1 ) , F j ( z 2 )] = δ i,j ( q − q − 1 ) z 1 z 2  δ  q − k z 1 z 2  H + i ( q − k 2 z 1 ) − δ  q k z 1 z 2  H − i ( q − k 2 z 2 )  . (4.31) No w we ha v e costructed lev el k free field realization of Drinfeld curren t E i ( z ) , F i ( z ) , H ± i ( z ) for the elliptic algebra U q ,p ( c sl 3 ). This give s t he fir st example of arbitrary-lev el free field realization of higher-r a nk elliptic alg ebra. Ac kno wledge men t The author w ould lik e to thank the organizing committee of the 2 7 -th International Collo quium of the Group Theoretical Metho d in Phys ics held at Y erev an, Armenia 200 8. The author w ould lik e to thank Prof.A.Kluemp er for his kindness a t Armenia. This work is par t ly supp ort ed by the Grant-in Aid fo r Y oung Scientis t B (18 7 40092) from Japan So ciet y for the Promotion of Science. App endix In a pp endix we summarize the no rmal ordering of the ba sic op erato rs. : e γ i ( z 1 ) : B ∗ i + ( z 2 ) = : e γ i ( z 1 ) B ∗ i + ( z 2 ) : ( q r ∗ +1 z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ − 1 z 2 /z 1 ; p ∗ ) ∞ , : e β i 1 ( z 1 ) : B ∗ i + ( z 2 ) = : e β i 1 ( z 1 ) B ∗ i + ( z 2 ) : ( q r ∗ z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ +2 z 2 /z 1 ; p ∗ ) ∞ , : e β i 2 ( z 1 ) : B ∗ i + ( z 2 ) = : e β i 2 ( z 1 ) B ∗ i + ( z 2 ) : ( q r ∗ z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ +2 z 2 /z 1 ; p ∗ ) ∞ , : e β i 3 ( z 1 ) : B ∗ i + ( z 2 ) = : e β i 3 ( z 1 ) B ∗ i + ( z 2 ) : ( q r ∗ z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ − 2 z 2 /z 1 ; p ∗ ) ∞ , : e β i 4 ( z 1 ) : B ∗ i + ( z 2 ) = : e β i 4 ( z 1 ) B ∗ i + ( z 2 ) : ( q r ∗ z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ +2 z 2 /z 1 ; p ∗ ) ∞ , B i − ( z 1 ) : e γ i ( z 2 ) : = : B i − ( z 1 ) e γ i ( z 2 ) : ( q r +1 z 2 /z 1 ; p ) ∞ ( q r − 1 z 2 /z 1 ; p ) ∞ , B i − ( z 1 ) : e β i 1 ( z 2 ) : = : B i − ( z 1 ) e β i 1 ( z 2 ) : ( q r z 2 /z 1 ; p ) ∞ ( q r +2 z 2 /z 1 ; p ) ∞ , B i − ( z 1 ) : e β i 2 ( z 2 ) : = : B i − ( z 1 ) e β i 2 ( z 2 ) : ( q r z 2 /z 1 ; p ) ∞ ( q r +2 z 2 /z 1 ; p ) ∞ , B i − ( z 1 ) : e β i 3 ( z 1 ) : = : B i − ( z 1 ) e β i 3 ( z 1 ) : ( q r z 2 /z 1 ; p ) ∞ ( q r − 2 z 2 /z 1 ; p ) ∞ , 9 B i − ( z 1 ) : e β i 4 ( z 2 ) : = : B i − ( z 1 ) e β i 4 ( z 2 ) : ( q r z 2 /z 1 ; p ) ∞ ( q r − 2 z 2 /z 1 ; p ) ∞ , e b i + ( z 1 ) B ∗ i + ( z 2 ) = : e b i + ( z 1 ) B ∗ i + ( z 2 ) : ( q r ∗ z 2 /z 1 ; p ∗ ) 2 ∞ ( q r ∗ +2 z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ − 2 z 2 /z 1 ; p ∗ ) ∞ , B i − ( z 1 ) e b i − ( z 2 ) = : B i − ( z 1 ) e b i − ( z 2 ) : ( q r z 2 /z 1 ; p ) 2 ∞ ( q r +2 z 2 /z 1 ; p ) ∞ ( q r − 2 z 2 /z 1 ; p ) ∞ , e a i + ( z 1 ) A ∗ i ( z 2 ) = : e a i + ( z 1 ) A ∗ i ( z 2 ) : ( q r ∗ + k +5 z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ − k − 5 z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ + k +1 z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ − k − 1 z 2 /z 1 ; p ∗ ) ∞ , e a 1 + ( z 1 ) A ∗ 2 ( z 2 ) = : e a 1 + ( z 1 ) A ∗ 2 ( z 2 ) : ( q r ∗ + k +2 z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ − k − 2 z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ + k +4 z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ − k − 4 z 2 /z 1 ; p ∗ ) ∞ , e a 2 + ( z 1 ) A ∗ 1 ( z 2 ) = : e a 2 + ( z 1 ) A ∗ 1 ( z 2 ) : ( q r ∗ + k +2 z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ − k − 2 z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ + k +4 z 2 /z 1 ; p ∗ ) ∞ ( q r ∗ − k − 4 z 2 /z 1 ; p ∗ ) ∞ , A i ( z 1 ) e a i − ( z 2 ) = : A i ( z 1 ) e a i − ( z 2 ) : ( q r + k +5 z 2 /z 1 ; p ) ∞ ( q r − k − 5 z 2 /z 1 ; p ) ∞ ( q r + k +1 z 2 /z 1 ; p ) ∞ ( q r − k − 1 z 2 /z 1 ; p ) ∞ , A 1 ( z 1 ) e a 2 − ( z 2 ) = : A 1 ( z 1 ) e a 2 − ( z 2 ) : ( q r + k +2 z 2 /z 1 ; p ) ∞ ( q r − k − 2 z 2 /z 1 ; p ) ∞ ( q r + k +4 z 2 /z 1 ; p ) ∞ ( q r − k − 4 z 2 /z 1 ; p ) ∞ , A 2 ( z 1 ) e a 1 − ( z 2 ) = : A 2 ( z 1 ) e a 1 − ( z 2 ) : ( q r + k +2 z 2 /z 1 ; p ) ∞ ( q r − k − 2 z 2 /z 1 ; p ) ∞ ( q r + k +4 z 2 /z 1 ; p ) ∞ ( q r − k − 4 z 2 /z 1 ; p ) ∞ , B i − ( z 1 ) B ∗ i + ( z 2 ) = : B i − ( z 1 ) B ∗ i + ( z 2 ) : ( q k z 2 /z 1 ; q 2 k , p ∗ ) 2 ∞ ( q k +2 z 2 /z 1 ; q 2 k , p ∗ ) ∞ ( q k − 2 z 2 /z 1 ; q 2 k , p ∗ ) ∞ × ( q k +2 z 2 /z 1 ; q 2 k , p ) ∞ ( q k − 2 z 2 /z 1 ; q 2 k , p ) ∞ ( q k z 2 /z 1 ; q 2 k , p ) 2 ∞ , A i ( z 1 ) A ∗ i ( z 2 ) = : A i ( z 1 ) A ∗ i ( z 2 ) : ( q 2 k + 5 z 2 /z 1 ; q 2 k , p ∗ ) ∞ ( q − 5 z 2 /z 1 ; q 2 k , p ∗ ) ∞ ( q 2 k + 1 z 2 /z 1 ; q 2 k , p ∗ ) ∞ ( q − 1 z 2 /z 1 ; q 2 k , p ∗ ) ∞ × ( q 2 k + 1 z 2 /z 1 ; q 2 k , p ) ∞ ( q − 1 z 2 /z 1 ; q 2 k , p ) ∞ ( q 2 k + 5 z 2 /z 1 ; q 2 k , p ) ∞ ( q − 5 z 2 /z 1 ; q 2 k , p ) ∞ , A 1 ( z 1 ) A ∗ 2 ( z 2 ) = : A 1 ( z 1 ) A ∗ 2 ( z 2 ) : ( q 2 k + 2 z 2 /z 1 ; q 2 k , p ∗ ) ∞ ( q − 2 z 2 /z 1 ; q 2 k , p ∗ ) ∞ ( q 2 k + 4 z 2 /z 1 ; q 2 k , p ∗ ) ∞ ( q − 4 z 2 /z 1 ; q 2 k , p ∗ ) ∞ × ( q 2 k + 4 z 2 /z 1 ; q 2 k , p ) ∞ ( q − 4 z 2 /z 1 ; q 2 k , p ) ∞ ( q 2 k + 2 z 2 /z 1 ; q 2 k , p ) ∞ ( q − 2 z 2 /z 1 ; q 2 k , p ) ∞ , A 2 ( z 1 ) A ∗ 1 ( z 2 ) = : A 2 ( z 1 ) A ∗ 1 ( z 2 ) : ( q 2 k + 2 z 2 /z 1 ; q 2 k , p ∗ ) ∞ ( q − 2 z 2 /z 1 ; q 2 k , p ∗ ) ∞ ( q 2 k + 4 z 2 /z 1 ; q 2 k , p ∗ ) ∞ ( q − 4 z 2 /z 1 ; q 2 k , p ∗ ) ∞ × ( q 2 k + 4 z 2 /z 1 ; q 2 k , p ) ∞ ( q − 4 z 2 /z 1 ; q 2 k , p ) ∞ ( q 2 k + 2 z 2 /z 1 ; q 2 k , p ) ∞ ( q − 2 z 2 /z 1 ; q 2 k , p ) ∞ . 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