Free field realization of commutative family of elliptic Feigin-Odesskii algebra

F ree field realization of comm utativ e family of elliptic F eigin-Odesskii algebra T ak eo Ko jima Departmen t of Ma thematics, College of Science and T ec h nology , Nihon Univ ers ity Abstract In this review, w e study free field realizat ions of the F eigin-Odesskii algebra. W e construct free field realizatio ns of a pair of infi nitely man y commutati v e op erators, asso ciated with th e elliptic algebra U q ,p ( d sl N ). 1 In tro duction In this review, we study free fi eld realization of elliptic ve rsion of the F eigin-Odesskii a lgebra [1]. F or this purp ose w e in tro duce one parameter ” s ” deformation of the F eigin-Odesskii algebra [1]. This review is based on the pap er [9, 10, 11, 16, 17]. Let the function f l ( z 1 · · · z l | w 1 · · · w l ) b e m er omorp hic and symmetric in eac h of v aribles ( z 1 , · · · , z l ) and ( w 1 , · · · , w l ). Let u s set the symmetric function ( f m ◦ f n )( z 1 , · · · , z m + n | w 1 , · · · , w m + n ), dep ending on t hree conti n uous parameters 0 < x < 1 , 0 < r and 0 < s < 2, by ( f m ◦ f n )( z 1 , · · · , z m + n | w 1 , · · · , w m + n ) = 1 (( m + n )!) 2 X σ ∈ S m + n X τ ∈ S m + n f m ( z σ (1) , · · · , z σ ( m ) | w τ (1) , · · · , w τ ( m ) ) × f n ( z σ ( m +1) , · · · , z σ ( m + n ) | w τ ( m +1) , · · · , w τ ( m + n ) ) × m Y i =1 m + n Y j = m +1 h v τ ( i ) − u σ ( j ) + s 2 i r h u σ ( i ) − v τ ( j ) + s 2 i r [ u σ ( i ) − u σ ( j ) ] r [ u σ ( j ) − u σ ( i ) − 1] r . × m Y i =1 m + n Y j = m +1 h u σ ( j ) − v τ ( i ) + s 2 − 1 i r h v τ ( j ) − u σ ( i ) + s 2 − 1 i r [ v τ ( j ) − v τ ( i ) − 1] r [ v τ ( j ) − v τ ( i ) − 1] r , where the symbol [ u ] r represent s the elliptic theta fun ction defined in (2.1). Here we set z j = x 2 u j , w j = x 2 v j . Th is p r o duct ” ◦ ” on symmetric fun ction giv es the structure of the asso ciativ e 1 algebra. W e call this asso ciativ e algebra ”elliptic F eigin-Odesskii algebra”. Let us set the functional G b y using currents F 1 ( z ) , F 2 ( z ), which is one parameter ” s ” deformation of the elliptic algebra U q ,p ( c sl 2 ). They satisfy the follo wing comm u tation relations. h u 1 − u 2 − s 2 i r h u 1 − u 2 + s 2 − 1 i r F 1 ( z 1 ) F 2 ( z 2 ) = h u 2 − u 1 − s 2 i r h u 2 − u 1 + s 2 − 1 i r F 1 ( z 2 ) F 2 ( z 1 ) , [ u 1 − u 2 ] r [ u 1 − u 2 + 1] r F j ( z 1 ) F j ( z 2 ) = [ u 2 − u 1 ] r [ u 2 − u 1 + 1] r F j ( z 2 ) F j ( z 1 ) . Up on t he specialization s → 2 the cur ren t F 1 ( z ) degenerates to the cuu ren t of the elliptic algebra U q ,p ( c sl 2 ), and the current F 2 ( z ) lo oke s like F 1 ( z ) − 1 . Let u s set the functional G by G ( f m ) = I m Y j =1 dz j 2 π iz j I m Y j =1 dw j 2 π iw j F 1 ( z 1 ) · · · F 1 ( z m ) F 2 ( w 1 ) · · · F 2 ( w m ) × Y 1 ≦ j 0. Let us s et z = x 2 u . The sym b ol [ u ] r stands for the Jacobi theta function, [ u ] r = x u 2 r − u Θ x 2 r ( z ) ( x 2 r ; x 2 r ) 3 ∞ , Θ q ( z ) = ( q ; q ) ∞ ( z ; q ) ∞ ( q /z ; q ) ∞ , (2.1) where w e ha v e used standard notation ( z ; q ) ∞ = Q ∞ j =0 (1 − q j z ). The sym b ol [ a ] stand s for q -inte ger, [ a ] = x a − x − a x − x − 1 . (2.2) 2.1 F eigin-Odesskii a lgebra Let u s set parameters 0 < s < 2 and r > 1. W e introdu ce a p air of F eigin-Odesskii algebra: f ◦ g and f ∗ g . Definition 2.1 L et us se t the symmetric function ( f m ◦ f n )( z 1 , · · · , z m + n | w 1 , · · · , w m + n ) by ( f m ◦ f n )( z 1 , · · · , z m + n | w 1 , · · · , w m + n ) = 1 (( m + n )!) 2 X σ ∈ S m + n X τ ∈ S m + n f m ( z σ (1) , · · · , z σ ( m ) | w τ (1) , · · · , w τ ( m ) ) × f n ( z σ ( m +1) , · · · , z σ ( m + n ) | w τ ( m +1) , · · · , w τ ( m + n ) ) 3 × m Y i =1 m + n Y j = m +1 h v τ ( i ) − u σ ( j ) + s 2 i r h u σ ( i ) − v τ ( j ) + s 2 i r [ u σ ( i ) − u σ ( j ) ] r [ u σ ( j ) − u σ ( i ) − 1] r × m Y i =1 m + n Y j = m +1 h u σ ( j ) − v τ ( i ) + s 2 − 1 i r h v τ ( j ) − u σ ( i ) + s 2 − 1 i r [ v τ ( j ) − v τ ( i ) − 1] r [ v τ ( j ) − v τ ( i ) − 1] r . (2.3) L et us g ive the symmetric function ( f m ∗ f n )( z 1 · · · z m + n | w 1 · · · w m + n ) by ( f m ∗ f n )( z 1 , · · · , z m + n | w 1 , · · · , w m + n ) = 1 (( m + n )!) 2 X σ ∈ S m + n X τ ∈ S m + n f m ( z σ (1) , · · · , z σ ( m ) | w τ (1) , · · · , w τ ( m ) ) × f n ( z σ ( m +1) , · · · , z σ ( m + n ) | w τ ( m +1) , · · · , w τ ( m + n ) ) × m Y i =1 m + n Y j = m +1 h v τ ( i ) − u σ ( j ) − s 2 i r − 1 h u σ ( i ) − v τ ( j ) − s 2 i r − 1 [ u σ ( i ) − u σ ( j ) ] r − 1 [ u σ ( j ) − u σ ( i ) + 1] r − 1 × m Y i =1 m + n Y j = m +1 h u σ ( j ) − v τ ( i ) − s 2 + 1 i r − 1 h v τ ( j ) − u σ ( i ) − s 2 + 1 i r − 1 [ v τ ( j ) − v τ ( i ) + 1] r − 1 [ v τ ( j ) − v τ ( i ) + 1] r − 1 . (2.4) Her e f l ( z 1 , · · · , z l | w 1 , · · · , w l ) ar e mer omorph ic fu nction symmetric in e ach of varibles ( z 1 , · · · , z l ) and ( w 1 , · · · , w l ) . W e ha ve infinitely mny comm utativ e family of F eigin-Odesskii alg ebra. Let us set th eta functions for three parameters α, ν , ϑ m,α ( z 1 , · · · , z m | w 1 , · · · , w m ) =   m X j =1 ( u j − v j ) − ν + α   r   m X j =1 ( v j − u j ) − α   r . (2.5 ) Prop osition 2.2 ϑ m,α and ϑ n,β c ommute with r esp e ct to the pr o duct (2.3). ϑ m,α ◦ ϑ n,β = ϑ n,β ◦ ϑ m,α . (2.6) Let us set theta functions for parmeters α, ν . ϑ ∗ m,α ( z 1 , · · · , z m | w 1 , · · · , w m ) =   m X j =1 ( v j − u j ) − ν + α   r − 1   m X j =1 ( u j − v j ) − α   r − 1 . (2.7) Prop osition 2.3 ϑ ∗ m,α and ϑ ∗ n,β c ommute with r esp e ct to the pr o duct (2.4). ϑ ∗ m,α ∗ ϑ ∗ n,β = ϑ ∗ n,β ∗ ϑ ∗ m,α . (2.8) Pro of of prop ositions are summ arized in [9]. 4 2.2 F ree field realization Let us set a parameter 0 < s < 2. Let us int ro duce b osons β 1 m , β 2 m , ( m 6 = 0) by [ β i m , β j n ] =    m [( r − 1) m ] [ r m ] [( s − 1) m ] [ sm ] δ m + n, 0 , ( i = j ) − m [( r − 1) m ] [ r m ] [ m ] [ sm ] x sm sgn( i − j ) δ m,n , ( i 6 = j ) (2.9) Let us set P , Q by [ P , iQ ] = 1 . (2.10) W e deal with the b osonic F o ck space F l,k ,( l, k ∈ Z ) generated by β i − m ,( m > 0 , i = 1 , 2) o ver the v acuum v ector | l, k i . β i m | l, k i = 0 ( m > 0 , i = 1 , 2) , (2.11) P | l, k i = r r 2( r − 1) l − r r − 1 2 r k ! | l, k i , (2.12) | l, k i = e  q r 2( r − 1) l − q r − 1 2 r k  iQ | 0 , 0 i . (2.13) Definition 2.4 L et us set the curr ents F j ( z ) , E j ( z ) , ( j = 1 , 2) by F 1 ( z ) = z r − 1 r e i q 2( r − 1) r Q z q 2( r − 1) r P : exp   X m 6 =0 1 m ( β 1 m − β 2 m ) z − m   : , (2.14) F 2 ( z ) = z r − 1 r e − i q 2( r − 1) r Q z − q 2( r − 1) r P : exp   X m 6 =0 1 m ( − x sm β 1 m + x − sm β 2 m ) z − m   : , (2.15) E 1 ( z ) = z r r − 1 e − i q 2 r r − 1 Q z − q 2 r r − 1 P : exp   − X m 6 =0 1 m [ r m ] [( r − 1) m ] ( β 1 m − β 2 m ) z − m   : , (2.16) E 2 ( z ) = z r r − 1 e i q 2 r r − 1 Q z q 2 r r − 1 P : exp   − X m 6 =0 1 m [ r m ] x [( r − 1) m ] x ( − x sm β 1 m + x − sm β 2 m ) z − m   : . (2.17) They satisfy the follo wing comm utation relations. Prop osition 2.5 [ u 1 − u 2 ] r [ u 1 − u 2 − 1] r F j ( z 1 ) F j ( z 2 ) = [ u 2 − u 1 ] r [ u 2 − u 1 − 1] r F j ( z 2 ) F j ( z 1 ) , ( j = 1 , 2) (2.18) [ u 1 − u 2 + s 2 − 1] r [ u 1 − u 2 + s 2 ] r F 1 ( z 1 ) F 2 ( z 2 ) = [ u 2 − u 1 + s 2 − 1] r [ u 2 − u 1 + s 2 ] r F 2 ( z 2 ) F 1 ( z 1 ) , (2.19) [ u 1 − u 2 ] r − 1 [ u 1 − u 2 + 1] r − 1 E j ( z 1 ) E j ( z 2 ) = [ u 2 − u 1 ] r − 1 [ u 2 − u 1 + 1] r − 1 E j ( z 2 ) E j ( z 1 ) , ( j = 1 , 2) (2.20) [ u 1 − u 2 − s 2 + 1] r − 1 [ u 1 − u 2 − s 2 ] r − 1 E 1 ( z 1 ) E 2 ( z 2 ) = [ u 2 − u 1 − s 2 + 1] r − 1 [ u 2 − u 1 − s 2 ] r − 1 E 2 ( z 2 ) E 1 ( z 1 ) . (2.21) 5 [ E i ( z 1 ) , F j ( z 2 )] = δ i,j x − x − 1  δ ( xz 2 /z 1 ) H j ( x r z 2 ) − δ ( xz 1 /z 2 ) H j ( x − r z 2 )  , ( i, j = 1 , 2) . (2.22) Her e we have set H 1 ( z ) = e − 1 √ r ( r − 1) iQ z − 1 √ r ( r − 1) P + 1 r ( r − 1) : exp   − X m 6 =0 1 m [ m ] [( r − 1) m ] ( β 1 m − β 2 m ) z − m   : , (2.23) H 2 ( z ) = e 1 √ r ( r − 1) iQ z 1 √ r ( r − 1) P + 1 r ( r − 1) : exp   X m 6 =0 1 m [ m ] [( r − 1) m ] ( x sm β 1 m − x − sm β 2 m ) z − m   : . (2.24) Definition 2.6 L et us se t the functional G by G ( f m ) = I m Y j =1 dz j 2 π iz j I m Y j =1 dw j 2 π iw j F 1 ( z 1 ) · · · F 1 ( z m ) F 2 ( w 1 ) · · · F 2 ( w m ) (2.25) × Y 1 ≦ j 1 we have [ G ( ϑ m,α ) , G ( ϑ n,β )] = 0 , ( m, n ∈ N ) , (2.29) [ G ∗ ( ϑ ∗ m,α ) , G ∗ ( ϑ ∗ n,β )] = 0 , ( m, n ∈ N ) . (2.30) Theorem 2.9 F or 0 < r < 1 we have [ G ( ϑ m,α ) , G ∗ ( ϑ ∗ n,β )] = 0 , ( m, n ∈ N ) . (2.31) Definition of G ∗ ( ϑ ∗ m,α ) for 0 < r < 1 is give n as the same m anner as (4.44). See detailds in [9]. W e h a ve constructed infin itely many commutativ e op erators G ( ϑ m,α ), G ∗ ( ϑ ∗ m,α ), ( m ∈ N ) acting on the b osonic F o ck sp ace, wh ic h is r egarded as the free fi eld realization of commutat iv e family of F eigin-Odesskii algebra (2.3) and (2.4). 3 Elliptic algebra U q ,p ( c sl N ) In this section we summarize some of results in [10]. In this section w e fix N = 3 , 4 , · · · . W e set parameters 0 < s < N . 7 3.1 F eigin-Odesskii algebra W e in tro d uce a pair of F eigin-Odesskii algebra. W e set z ( t ) j = x 2 u ( t ) j and understand z ( t + N ) j = z ( t ) j . Definition 3.1 L et us set mer omorph ic fu nction ( f m ◦ f n )( z (1) 1 , · · · , z (1) m + n | · · · | z ( N ) 1 , · · · , z ( N ) m + n ) symmetric in e ach of variables ( z (1) 1 , · · · , z (1) m + n ) , · · · ( z ( N ) 1 , · · · , z ( N ) m + n ) . ( f m ◦ f n )( z (1) 1 , · · · , z (1) m + n | · · · | z ( N ) 1 , · · · , z ( N ) m + n ) = X σ 1 ∈ S m + n X σ 2 ∈ S m + n · · · X σ N ∈ S m + n × f m ( z (1) σ 1 (1) , · · · , z (1) σ 1 ( m ) | · · · | z ( N ) σ N (1) , · · · , z ( N ) σ N ( m ) ) × f n ( z (1) σ 1 ( m +1) , · · · , z (1) σ 1 ( m + n ) | · · · | z ( N ) σ N ( m +1) , · · · , z ( N ) σ N ( m + n ) ) × N Y t =1 m Y i =1 m + n Y j = m +1 h u ( t ) σ t ( i ) − u ( t +1) σ t +1 ( j ) − s N i r h u ( t +1) σ t +1 ( i ) − u ( t ) σ t ( j ) + 1 − s N i r h u ( t ) σ t ( i ) − u ( t ) σ t ( j ) i r h u ( t ) σ t ( j ) − u ( t ) σ t ( i ) − 1 i r . (3.1) Her e mer omorph ic function f l ( z (1) 1 , · · · , z (1) l | · · · | z ( N ) 1 , · · · , z ( N ) l ) is symmetric in e ach of v ariables ( z (1) 1 , · · · , z (1) l ) , · · · , ( z ( N ) 1 , · · · , z ( N ) l ) . L et us set mer omorp hic function ( f m ∗ f n )( z (1) 1 , · · · , z (1) m + n | · · · | z ( N ) 1 , · · · , z ( N ) m + n ) symmetric i n e ach of variables ( z (1) 1 , · · · , z (1) m + n ) , · · · ( z ( N ) 1 , · · · , z ( N ) m + n ) . ( f m ∗ f n )( z (1) 1 , · · · , z (1) m + n | · · · | z ( N ) 1 , · · · , z ( N ) m + n ) = X σ 1 ∈ S m + n X σ 2 ∈ S m + n · · · X σ N ∈ S m + n × f m ( z (1) σ 1 (1) , · · · , z (1) σ 1 ( m ) | · · · | z ( N ) σ N (1) , · · · , z ( N ) σ N ( m ) ) × f n ( z (1) σ 1 ( m +1) , · · · , z (1) σ 1 ( m + n ) | · · · | z ( N ) σ N ( m +1) , · · · , z ( N ) σ N ( m + n ) ) × N Y t =1 m Y i =1 m + n Y j = m +1 h u ( t ) σ t ( i ) − u ( t +1) σ t +1 ( j ) + s N i r − 1 h u ( t +1) σ t +1 ( i ) − u ( t ) σ t ( j ) − 1 + s N i r − 1 h u ( t ) σ t ( i ) − u ( t ) σ t ( j ) i r − 1 h u ( t ) σ t ( j ) − u ( t ) σ t ( i ) + 1 i r − 1 . (3.2) Her e mer omorph ic function f l ( z (1) 1 , · · · , z (1) l | · · · | z ( N ) 1 , · · · , z ( N ) l ) is symmetric in e ach of v ariables ( z (1) 1 , · · · , z (1) l ) , · · · , ( z ( N ) 1 , · · · , z ( N ) l ) . W e hav e a p air of in finitely many commutativ e family of F eigin-Odesskii algebra. Let us set theta function with parameters ν 1 , · · · , ν N and α . ϑ m,α ( u (1) 1 , · · · , u (1) m | · · · | u ( N ) 1 , · · · , u ( N ) m ) = N Y t =1   m X j =1 ( u ( t ) j − u ( t +1) j ) − ν t + α   r . (3.3 ) 8 Prop osition 3.2 ϑ m,ϑ and ϑ n,β c ommute e ach other with r esp e ct to the pr o duct (3.1). ϑ m,α ◦ ϑ n,β = ϑ n,β ◦ ϑ m,α . (3.4) Let us set theta function with parameters ν 1 , · · · , ν N and α . ϑ ∗ m,α ( u (1) 1 , · · · , u (1) m | · · · | u ( N ) 1 , · · · , u ( N ) m ) = N Y t =1   m X j =1 ( u ( t +1) j − u ( t ) j ) − ν t + α   r − 1 . (3.5) Prop osition 3.3 ϑ m,α and ϑ n,β c ommute e ach other with r esp e ct to the pr o duct (3.2). ϑ ∗ m,α ∗ ϑ ∗ n,β = ϑ ∗ n,β ∗ ϑ ∗ m,α . (3.6) Pro of of the ab o v e prop osition is sum marized in [10]. 3.2 F ree field realization Let ǫ j (1 ≦ j ≦ N ) b e an orthonormal basis in R N relativ e to the standard inn er pro du ct ( ǫ i | ǫ j ) = δ i,j . Let us set ¯ ǫ j = ǫ j − ǫ w here ǫ = 1 N P N j =1 ǫ j . W e id entify ǫ j + N = ǫ j . Let the w eighted lattice P = P N j =1 Z ¯ ǫ j . Let us set α j = ¯ ǫ j − ¯ ǫ j +1 ∈ P . Let us intro d uce the b osons β j m ( m ∈ Z 6 =0 ; 1 ≦ j ≦ N ) by [ β i m , β j n ] =    m [( r − 1) m ] [ r m ] [( s − 1) m ] [ sm ] δ m + n, 0 , ( i = j ) − m [( r − 1) m ] [ r m ] [ m ] [ sm ] x sm sgn( i − j ) δ m,n , ( i 6 = j ) (3.7) Let us set the comm u tation relations of P λ , Q µ ( λ, µ ∈ P ) by [ P λ , iQ µ ] = ( λ | µ ) . (3.8) W e deal with th e b osonic F o c k space F l,k ,( l, k ∈ P ) generated by β i − m ,( m > 0 , i = 1 , · · · , N ) o ver the v acuum v ector | l , k i . β i m | l, k i = 0 ( m > 0 , i = 1 , · · · , N ) , (3.9) P α | l, k i = α      r r ( r − 1) l − r r − 1 r k ! | l, k i , (3.10) | l, k i = e  i q r ( r − 1) Q l − i q r − 1 r Q k  | 0 , 0 i . (3.11) Definition 3.4 We set the scr e ening curr ents F j ( z ) , (1 ≦ j ≦ N ) by F j ( z ) = e i q r − 1 r Q α j ( x ( 2 s N − 1) j z ) q r − 1 r P α j + r − 1 r 9 × : exp   X m 6 =0 1 m B j m z − m   : , (1 ≦ j ≦ N − 1) (3.12) F N ( z ) = e i q r − 1 r Q α N ( x 2 s − N z ) q r − 1 r P ¯ ǫ N + r − 1 2 r z − q r − 1 r P ¯ ǫ 1 + r − 1 2 r × : exp   X m 6 =0 1 m B N m z − m   : , (3.13) We set the scr e ening curr ents E j ( z ) , (1 ≦ j ≦ N ) by E j ( z ) = e − i √ r r − 1 Q α j ( x ( 2 s N − 1) j z ) − √ r r − 1 P α j + r r − 1 × : exp   − X m 6 =0 1 m [ r m ] [( r − 1) m ] B j m z − m   : , (1 ≦ j ≦ N − 1) (3.14) E N ( z ) = e − i √ r r − 1 Q α N ( x 2 s − N z ) − √ r r − 1 P ¯ ǫ N + r 2( r − 1) z √ r r − 1 P ¯ ǫ 1 + r 2( r − 1) × : exp   − X m 6 =0 1 m [ r m ] [( r − 1) m ] B N m z − m   : . (3.15) Her e we have set B j m = ( β j m − β j +1 m ) x − 2 s N j m , (1 ≦ j ≦ N − 1) , (3.16) B N m = ( x − 2 sm β N m − β 1 m ) . (3.17) Prop osition 3.5 The cu rr e nts F j ( z ) , (1 ≦ j ≦ N ; N ≧ 3) satisfy the f ol lowing c ommutation r elations. h u 1 − u 2 − s N i r F j ( z 1 ) F j +1 ( z 2 ) = h u 2 − u 1 + s N − 1 i r F j +1 ( z 2 ) F j ( z 1 ) , (1 ≦ j ≦ N ) , (3.18) [ u 1 − u 2 ] r [ u 1 − u 2 + 1] r F j ( z 1 ) F j ( z 2 ) = [ u 2 − u 1 ] r [ u 2 − u 1 + 1] r F j ( z 2 ) F j ( z 1 ) , (1 ≦ j ≦ N ) , (3.19) F i ( z 1 ) F j ( z 2 ) = F j ( z 2 ) F i ( z 1 ) , ( | i − j | ≧ 2) . (3.20) We r e ad F N +1 ( z ) = F 1 ( z ) . The curr ents E j ( z ) , (1 ≦ j ≦ N ; N ≧ 3) satisfy the fol lowing c ommutation r elations. h u 1 − u 2 + 1 − s N i r − 1 E j ( z 1 ) E j +1 ( z 2 ) = h u 2 − u 1 + s N i r − 1 E j +1 ( z 2 ) E j ( z 1 ) , (1 ≦ j ≦ N ) , (3.21) [ u 1 − u 2 ] r − 1 [ u 1 − u 2 − 1] r − 1 E j ( z 1 ) E j ( z 2 ) = [ u 2 − u 1 ] r − 1 [ u 2 − u 1 − 1] r − 1 E j ( z 2 ) E j ( z 1 ) , (1 ≦ j ≦ N ) , (3.22) E i ( z 1 ) E j ( z 2 ) = E j ( z 2 ) E i ( z 1 ) , ( | i − j | ≧ 2) . (3.23 ) 10 We r e ad E N +1 ( z ) = E 1 ( z ) . Prop osition 3.6 The scr e ening curr ents E j ( z ) , F j ( z ) , (1 ≦ j ≦ N ; N ≧ 3) satisfy the fol lowing r elation. [ E i ( z 1 ) , F j ( z 2 )] = δ i,j x − x − 1 ( δ ( xz 2 /z 1 ) H j ( x r z 2 ) − δ ( xz 1 /z 2 ) H j ( x − r z 2 )) , (1 ≦ i, j ≦ N ) . (3.24) Her e we have set H j ( z ) = x (1 − 2 s N )2 j e − i √ r ( r − 1) Q α j ( x ( 2 s N − 1) j z ) − 1 √ r ( r − 1) P α j + 1 r ( r − 1) × : exp   − X m 6 =0 1 m [ m ] [( r − 1) m ] B j m z − m   : , (1 ≦ j ≦ N − 1) , (3.25) H N ( z ) = x 2( N − 2 s ) e − i √ r ( r − 1) Q α N ( x 2 s − N z ) − 1 √ r ( r − 1) P ¯ ǫ N + 1 2 r ( r − 1) z − 1 √ r ( r − 1) P ¯ ǫ 1 + 1 2 r ( r − 1) × : exp   − X m 6 =0 1 m [ m ] [( r − 1) m ] B N m z − m   : . (3.26) Definition 3.7 L et us se t the functional G by G ( f m ) = I · · · I N Y t =1 m Y j =1 dz ( t ) j 2 π iz ( t ) j F 1 ( z (1) 1 ) · · · F 1 ( z (1) m ) · · · F N ( z ( N ) 1 ) · · · F N ( z ( N ) m ) × N Y t =1 Y 1 ≦ i 1 we have [ G ( ϑ m,α ) , G ( ϑ n,β )] = 0 , ( m, n ∈ N ) , (3.31) [ G ∗ ( ϑ ∗ m,α ) , G ∗ ( ϑ ∗ n,β )] = 0 , ( m, n ∈ N ) . (3.32) Theorem 3.10 F or 0 < r < 1 we have [ G ( ϑ m,α ) , G ∗ ( ϑ ∗ n,β )] = 0 , ( m, n ∈ N ) . (3.33) Definition of G ∗ ( ϑ ∗ m,α ) for 0 < r < 1 is giv en as the same manner as (4.44). See detailds in [10]. W e h a ve constructed infin itely many commutativ e op erators G ( ϑ m,α ), G ∗ ( ϑ ∗ m,α ), ( m ∈ N ) acting on the b osonic F o ck sp ace, wh ic h is r egarded as the free fi eld realization of commutat iv e family of F eigin-Odesskii algebra (3.1) and (3.2). 4 Lev el k generalization of U q ,p ( c sl 2 ) In this section we consider leve l k generaliztion of section 2. Main cont ribution is construction of free field realizatio n for one parameter s deformation of L evel k elliptic alb egra U q ,p ( c sl 2 ). 12 4.1 F eigin-Odesskii algebra Let u s set parameters r , k ∈ R su ch that r > 0 , r − k > 0. It’s n ot difficult to giv e Lev el k generalizat ion of F eigin-Odesskii algebra: f ◦ g and f ∗ g . Definition 4.1 L et us set the symmetric function ( f m ◦ f n )( z 1 , · · · , z m + n | w 1 , · · · , w m + n ) by the same r elation (2.3). L et us set the symmetric function ( f m ∗ f n )( z 1 · · · z m + n | w 1 · · · w m + n ) by mo dific ation of (2.4). ( f m ∗ f n )( z 1 , · · · , z m + n | w 1 , · · · , w m + n ) = 1 (( m + n )!) 2 X σ ∈ S m + n X τ ∈ S m + n f m ( z σ (1) , · · · , z σ ( m ) | w τ (1) , · · · , w τ ( m ) ) × f n ( z σ ( m +1) , · · · , z σ ( m + n ) | w τ ( m +1) , · · · , w τ ( m + n ) ) × m Y i =1 m + n Y j = m +1 h v τ ( i ) − u σ ( j ) − s 2 i r − k h u σ ( i ) − v τ ( j ) − s 2 i r − k [ u σ ( i ) − u σ ( j ) ] r − k [ u σ ( j ) − u σ ( i ) + 1] r − k × m Y i =1 m + n Y j = m +1 h u σ ( j ) − v τ ( i ) − s 2 + 1 i r − k h v τ ( j ) − u σ ( i ) − s 2 + 1 i r − k [ v τ ( j ) − v τ ( i ) + 1] r − k [ v τ ( j ) − v τ ( i ) + 1] r − k . (4.1) Her e f l ( z 1 , · · · , z l | w 1 , · · · , w l ) ar e mer omorph ic fu nction symmetric in e ach of varibles ( z 1 , · · · , z l ) and ( w 1 , · · · , w l ) . W e ha ve infinitely man y comm u tativ e s olutions ϑ m,α and ϑ ∗ m,α with resp ect with pro d u ct f ◦ g and f ∗ g . Th e solutions ϑ m,α ( z 1 , · · · , z m ) for p ro duct ◦ is giv en as the same as (2.5). Let us set the theta function ϑ ∗ m,α with parmeters α, ν . ϑ m ( z 1 , · · · , z m | w 1 , · · · , w m ) =   m X j =1 ( v j − u j ) − ν + α   r − k   m X j =1 ( u j − v j ) − α   r − k . (4.2 ) Prop osition 4.2 ϑ m,alpha and ϑ n,β c ommute with r esp e ct to the pr o duct (4.1). ϑ m,α ∗ ϑ n,β = ϑ n,β ∗ ϑ m,α . (4.3) 4.2 F ree field realization In this section we giv e one parameter deformation of W akimoto r ealizat ion of elliptic algebra U q ,p ( c sl 2 ) [2, 3]. Let us set deformation p arameter 0 < s < 2. Let u s set the b osons α j m , e α j m , ( j = 1 , 2; m ∈ Z 6 =0 ), [ α j m , α j n ] = − 1 m [2 m ][ r m ] [ k m ][( r − k ) m ] δ m + n, 0 , ( j = 1 , 2) , ( 4.4) 13 [ α 1 m , α 2 n ] = 1 m x ( − r + k ) m ([ sm ] − [( s − 2) m ]) [( r − k ) m ] + x k m ([ sm ] + [( s − 2) m ]) [ k m ] ! δ m + n, 0 , (4.5) [ e α j m , e α j n ] = − 1 m [2 m ][( r − k ) m ] [ k m ][ r m ] δ m + n, 0 , ( j = 1 , 2) , (4.6) [ e α 1 m , e α 2 n ] = 1 m  x r m ( − [ sm ] + [( s − 2) m ]) [ r m ] + x k m ([ sm ] + [( s − 2) m ]) [ k m ]  δ m + n, 0 , (4.7) [ α j m , e α j n ] = − 1 m [2 m ] [ k m ] δ m + n, 0 , ( j = 1 , 2) , (4.8) [ α 1 m , e α 2 n ] = 1 m [ sm ] + [( s − 2) m ] [ k m ] δ m + n, 0 , (4.9) [ e α 1 m , α 2 n ] = 1 m [ sm ] + [( s − 2) m ] [ k m ] δ m + n, 0 . (4.10) W e set the b osons β j m , γ j m , ( j = 1 , 2; m ∈ Z 6 =0 ), [ β j m , β j n ] = [2 m ][( k + 2) m ] m δ m + n, 0 , ( j = 1 , 2) , (4 .11) [ β 1 m , β 2 n ] = − [( k + 2) m ]([ sm ] + [( s − 2) m ]) m δ m + n, 0 , (4.12) [ γ j m , γ j n ] = 1 m [2 m ] [ k m ] δ m + n, 0 , ( j = 1 , 2) , (4.13) [ γ 1 m , γ 2 n ] = − 1 m [ sm ] + [( s − 2) m ] [ k m ] δ m + n, 0 . (4.14) W e set the zero-mod e op erators P 0 , Q 0 , h, α and h 0 , h 1 , h 2 , α 0 , α 1 , α 2 , [ P 0 , iQ 0 ] = 1 , [ h, α ] = 2 , (4.15) [ h 0 , α 0 ] = [ h 1 , α 2 ] = [ h 2 , α 1 ] = (2 − s ) , [ h 1 , α 1 ] = [ h 2 , α 2 ] = 0 . (4.16) W e set the F o ck space F K,L , ( K, L ∈ Z ). F K,L = M n,n 0 ,n 1 ,n 2 ∈ Z C [ α j − m , e α j − m , β j − m , γ j − m , ( j = 1 , 2; m ∈ N 6 =0 )] ⊗ | K , L i n,n 0 ,n 1 ,n 2 , (4.17) | K, L i n,n 0 ,n 1 ,n 2 = e  L q r 2( r − k ) − K q r − k 2 r  iQ ⊗ e nα ⊗ e n 0 α 0 ⊗ e n 1 α 1 ⊗ e n 2 α 2 . (4.18) Up on sp ecializatio n s → 2, simp lification o ccures. α 2 m = − α 1 m , e α 1 m = [( r − k ) m ] [ r m ] α 1 m , e α 2 m = − [( r − k ) m ] [ r m ] α 1 m , (4.19) β 2 m = − β 1 m , γ 2 m = − γ 1 m , h 0 = h 1 = h 2 = α 0 = α 1 = α 2 = 0 . (4.20) The b osons α 1 m , β 1 m , γ 1 m are the same b osons which w ere introd u ced to constru ct the elliptic current asso ciated with the elliptic algebra U q ,p ( c sl 2 ) [2, 3, 4 ]. In ord er to construct in finitly many 14 comm utativ e op erators, w e introduce one pr ameter s d eformation of the b osons in [2, 3, 4]. W e in tr o duce the op erators C j ( z ) , C † j ( z ), ( j = 1 , 2) acting on the F o ck sp ace F J,K . C 1 ( z ) = e − q 2 r k ( r − k ) iQ 0 e − q 2 r k ( r − k ) P 0 log z : exp   − X m 6 =0 α 1 m z − m   : , (4.21) C 2 ( z ) = e q 2 r k ( r − k ) iQ 0 e q 2 r k ( r − k ) P 0 log z : exp   − X m 6 =0 α 2 m z − m   : , (4.22) C † 1 ( z ) = e q 2( r − k ) kr iQ 0 e q 2( r − k ) kr P 0 log z : exp   X m 6 =0 e α 1 m z − m   : , (4.23) C † 2 ( z ) = e − q 2( r − k ) kr iQ 0 e − q 2( r − k ) kr P 0 log z : exp   X m 6 =0 e α 2 m z − m   : . (4.24) W e set the op erators e Ψ j,I ( z ) , e Ψ j,I I ( z ) , e Ψ † j,I ( z ) , e Ψ † j,I I ( z ), ( j = 1 , 2) acting on the F o ck sp ace F J,K . e Ψ j,I ( z ) = exp − ( x − x − 1 ) X m> 0 x km 2 [ m ] + β j m z − m ! (4.25) × exp − X m> 0 x − km 2 γ j − m z m ! exp − X m> 0 x km 2 [( k + 1) m ] + [ m ] + γ j m z − m ! , ( j = 1 , 2) , e Ψ j,I I ( z ) = exp ( x − x − 1 ) X m> 0 x km 2 [ m ] + β j − m z m ! (4.26) × exp − X m> 0 x km 2 [( k + 1) m ] + [ m ] + γ j − m z m ! exp − X m> 0 x − km 2 γ j m z − m ! , ( j = 1 , 2) , e Ψ † j,I ( z ) = exp ( x − x − 1 ) X m> 0 x − km 2 [ m ] + β j m z − m ! (4.27) × exp X m> 0 x km 2 γ j − m z m ! exp X m> 0 x − km 2 [( k + 1) m ] + [ m ] + γ j m z − m ! , ( j = 1 , 2) , e Ψ † j,I I ( z ) = exp − ( x − x − 1 ) X m> 0 x − km 2 [ m ] + β j − m z m ! (4.28) × exp X m> 0 x − km 2 [( k + 1) m ] + [ m ] + γ j − m z m ! exp X m> 0 x km 2 γ j m z − m ! , ( j = 1 , 2) . W e set the op erators Ψ j,I ( z ) , Ψ j,I I ( z ) , Ψ † j,I ( z ) , Ψ † j,I I ( z ), ( j = 1 , 2) acting on the F o ck sp ace F J,K . Ψ 1 ,I ( z ) = e Ψ 1 ,I ( z ) e α + α 0 + α 1 x h 2 + h 0 + h 1 z − h k , (4.29) Ψ 1 ,I I ( z ) = e Ψ 1 ,I I ( z ) e α + α 0 + α 1 x − h 2 + h 0 − h 1 z − h k , (4.30) Ψ 2 ,I ( z ) = e Ψ 2 ,I ( z ) e − α − α 0 + α 2 x − h 2 + h 0 + h 2 z h k , (4.31) 15 Ψ 2 ,I I ( z ) = e Ψ 2 ,I I ( z ) e − α − α 0 + α 2 x h 2 + h 0 − h 2 z h k , (4.32) Ψ † 1 ,I ( z ) = e Ψ † 1 ,I ( z ) e − α − α 0 + α 1 x h 2 − h 0 − h 1 z h k , (4.33) Ψ † 1 ,I I ( z ) = e Ψ † 1 ,I I ( z ) e − α − α 0 + α 1 x − h 2 − h 0 + h 1 z h k , (4.34) Ψ † 2 ,I ( z ) = e Ψ † 2 ,I ( z ) e α + α 0 + α 2 x − h 2 − h 0 − h 2 z − h k , (4.35) Ψ † 2 ,I I ( z ) = e Ψ † 2 ,I I ( z ) e α + α 0 + α 2 x h 2 − h 0 + h 2 z − h k . (4.36) Definition 4.3 We set the op er ators E j ( z ) , F j ( z ) , ( j = 1 , 2) , which c an b e r e gar de d as one p ar ameter deformation of the level k el liptic curr ents asso ciate d with the el liptic algebr a U q ,p ( c sl 2 ) [3, 4]. E j ( z ) = C j ( z )Ψ j ( z ) , F j ( z ) = C † j ( z )Ψ † j ( z ) , ( j = 1 , 2) , (4.37) wher e we have set Ψ j ( z ) = 1 x − x − 1 (Ψ j,I ( z ) − Ψ j,I I ( z )) , Ψ † j ( z ) = − 1 x − x − 1 (Ψ † j,I ( z ) − Ψ † j,I I ( z )) , ( j = 1 , 2) . (4.38) Prop osition 4.4 The el liptic cu rr ents E j ( z ) , ( j = 1 , 2) satisfy the fol lowing c ommutation r elations. [ u 1 − u 2 ] r − k [ u 1 − u 2 − 1] r − k E j ( z 1 ) E j ( z 2 ) = [ u 2 − u 1 ] r − k [ u 2 − u 1 − 1] r − k E j ( z 2 ) E j ( z 1 ) , ( j = 1 , 2) , (4.39) h u 1 − u 2 + s 2 i r − k h u 1 − u 2 − s 2 + 1 i r − k E 1 ( z 1 ) E 2 ( z 2 ) = h u 2 − u 1 + s 2 i r − k h u 2 − u 1 − s 2 + 1 i r − k E 2 ( z 2 ) E 1 ( z 1 ) . (4.40) The el liptic curr ents F j ( z ) , ( j = 1 , 2) satisfy the fol lowing c ommutation r elations. [ u 1 − u 2 ] r [ u 1 − u 2 + 1] r F j ( z 1 ) F j ( z 2 ) = [ u 2 − u 1 ] r [ u 2 − u 1 + 1] r F j ( z 2 ) F j ( z 1 ) , ( j = 1 , 2) , (4.41) h u 1 − u 2 − s 2 i r h u 1 − u 2 + s 2 − 1 i r F 1 ( z 1 ) F 2 ( z 2 ) = h u 2 − u 1 − s 2 i r h u 2 − u 1 + s 2 − 1 i r F 2 ( z 2 ) F 1 ( z 1 ) . (4.42) The curr ents E j ( z ) and F j ( z ) satisfy [ E j ( z 1 ) , F j ( z 2 )] = x ( − 1) j ( s − 2) x − x − 1  : C j ( z 1 ) C † j ( z 2 )Ψ j,I ( z 1 )Ψ † j,I ( z 2 ) : δ  x k z 2 z 1  (4.43) − : C j ( z 1 ) C † j ( z 2 )Ψ j,I I ( z 1 )Ψ † j,I I ( z 2 ) : δ  x − k z 2 z 1  , ( j = 1 , 2) . Her e we have use d the delta-function δ ( z ) = P n ∈ Z z n . 16 The definition of the functional G ( f m ) is giv en as the same as (2.3). Definition 4.5 L et us se t the functional G ∗ by fol lowings. G ∗ ( f m ) = I m Y j =1 dz j 2 π iz j I m Y j =1 dw j 2 π iw j E 1 ( z 1 ) · · · E 1 ( z m ) E 2 ( w 1 ) · · · E 2 ( w m ) (4.44) × Y 1 ≦ j 0 and r − k > 0 , we have [ G ( ϑ m,α ) , G ( ϑ n,β )] = 0 , ( m, n ∈ N ) , (4.47) [ G ∗ ( ϑ ∗ m,α ) , G ∗ ( ϑ ∗ n,β )] = 0 , ( m, n ∈ N ) . (4.48) W e h a ve constructed infin itely many commutativ e op erators G ( ϑ m,α ), G ∗ ( ϑ ∗ m,α ), ( m ∈ N ) acting on the b osonic F o ck sp ace, wh ic h is r egarded as the free fi eld realization of commutat iv e family of F eigin-Odesskii algebra (2.3) and (4.1). 5 Lev el k generalization of U q ,p ( c sl N ) In this section we rep ort some results for Lev el k generalization of section 3, which are n o w in progress. Main r esult is free field realizati on of Leve l k elliptic algebra U q ,p ( d sl N ). 17 5.1 F eigin-Odesskii algebra W e in tro duce a pair of F eigin-Odesskii algebra. Definition 5.1 L et us set mer omorp hic function ( f m ∗ f n )( z (1) 1 , · · · , z (1) m + n | · · · | z ( N ) 1 , · · · , z ( N ) m + n ) symmetric in e ach of variables ( z (1) 1 , · · · , z (1) m + n ) , · · · ( z ( N ) 1 , · · · , z ( N ) m + n ) . ( f m ∗ f n )( z (1) 1 , · · · , z (1) m + n | · · · | z ( N ) 1 , · · · , z ( N ) m + n ) = X σ 1 ∈ S m + n X σ 2 ∈ S m + n · · · X σ N ∈ S m + n × f m ( z (1) σ 1 (1) , · · · , z (1) σ 1 ( m ) | · · · | z ( N ) σ N (1) , · · · , z ( N ) σ N ( m ) ) × f n ( z (1) σ 1 ( m +1) , · · · , z (1) σ 1 ( m + n ) | · · · | z ( N ) σ N ( m +1) , · · · , z ( N ) σ N ( m + n ) ) × N Y t =1 m Y i =1 m + n Y j = m +1 h u ( t ) σ t ( i ) − u ( t +1) σ t +1 ( j ) + s N i r − k h u ( t +1) σ t +1 ( i ) − u ( t ) σ t ( j ) − 1 + s N i r − k h u ( t ) σ t ( i ) − u ( t ) σ t ( j ) i r − k h u ( t ) σ t ( j ) − u ( t ) σ t ( i ) + 1 i r − k . (5.1) Her e mer omorph ic function f l ( z (1) 1 , · · · , z (1) l | · · · | z ( N ) 1 , · · · , z ( N ) l ) is symmetric in e ach of v ariables ( z (1) 1 , · · · , z (1) l ) , · · · , ( z ( N ) 1 , · · · , z ( N ) l ) . The pro du ct ◦ is giv en by the same as (3.1). Let us set theta fun ction with parameters ν 1 , · · · , ν N and α . ϑ ∗ m,α ( u (1) 1 , · · · , u (1) m | · · · | u ( N ) 1 , · · · , u ( N ) m ) = N Y t =1   m X j =1 ( u ( t +1) j − u ( t ) j ) − ν t + α   r − k . (5.2) Prop osition 5.2 ϑ m,α and ϑ n,β c ommute e ach other with r esp e ct to the pr o duct (5.1). ϑ ∗ m,α ∗ ϑ ∗ n,β = ϑ ∗ n,β ∗ ϑ ∗ m,α . (5.3) 5.2 F ree field realization In this section we giv e fr ee field realizatio n of Lev el k elliptic algebra U q ,p ( d sl N ). The author w ou ld like to emp h asize that the free field realizatio n of Level k is completely different from those of L ev el 1. W e in tro duce free b osons a i n , (1 ≦ i ≦ N − 1; n ∈ Z 6 =0 ), b i,j n , (1 ≦ i < j ≦ N ; n ∈ Z 6 =0 ), c i,j n , (1 ≦ i < j ≦ N ; n ∈ Z 6 =0 ), and the zero-mo de op erators a i , (1 ≦ i ≦ N − 1), b i,j , (1 ≦ i < j ≦ N ), c i,j , (1 ≦ i < j ≦ N ). [ a i n , a j m ] = [( k + N ) n ][ A i,j n ] n δ n + m, 0 , [ p i a , q j a ] = ( k + N ) A i,j , ( 5.4) [ b i,j n , b k ,l m ] = − [ n ] 2 n δ i,k δ j,l δ n + m, 0 , [ p i,j b , q k ,l b ] = − δ i,k δ k ,l , (5.5) [ c i,j n , c k ,l m ] = [ n ] 2 n δ i,k δ j,l δ n + m, 0 , [ p i,j c , q k ,l c ] = δ i,k δ j,l . (5.6) 18 Here the matrix ( A i,j ) 1 ≦ i,j ≦ N − 1 represent s the Cartan matrix of classical s l N . F or parameters a i ∈ R , (1 ≦ i ≦ N − 1), b i,j ∈ R , (1 ≦ i < j ≦ N ) c i,j , ∈ R , (1 ≦ i < j ≦ N ), w e set the v acuum v ector | a, b, c i of the F o c k sp ace F a,b,c as follo wing. a i n | a, b, c i = b j,k n | a, b, c i = c j,k n | a, b, c i = 0 , ( n > 0; 1 ≦ i ≦ N − 1; 1 ≦ j < k ≦ N ) , p i a | a, b, c i = a i | a, b, c i , p j,k b | a, b, c i = b j,k | a, b, c i , p j,k c | a, b, c i = c j,k | a, b, c i , (1 ≦ i ≦ N − 1; 1 ≦ j < k ≦ N ) . The F o ck sp ace F a,b,c is generated b y b osons a i − n , b j,k − n , c j,k − n for n ∈ N 6 =0 . The dual F ock space F ∗ a,b,c is defin ed as the same manner. In this pap er w e construct the elliptic analogue of Drinfeld current for U q ,p ( d sl N ) b y these b osons a i n , b j,k n , c j,k n acting on the F o c k space. Let u s set the b osonic op erators a i ± ( z ) , a i ( z ) , (1 ≦ i ≦ N − 1), b i,j ± ( z ) , b i,j ( z ) , c i.j ( z ) , (1 ≦ i < j ≦ N ) b y a i ± ( z ) = ± ( q − q − 1 ) X n> 0 a i ± n z ∓ n ± p i a log q , (5.7) b i,j ± ( z ) = ± ( q − q − 1 ) X n> 0 b i,j ± n z ∓ n ± p i,j b log q , (5.8) a i ( z ) = − X n 6 =0 a i n [( k + N ) n ] q − k + N 2 | n | z − n + 1 k + N ( q i a + p i a log z ) , (5. 9) b i,j ( z ) = − X n 6 =0 b i,j n [ n ] z − n + q i,j b + p i,j b log z , (5.10) c i,j ( z ) = − X n 6 =0 c i,j n [ n ] z − n + q i,j c + p i,j c log z , (5.11) Let us set the auxiliary op erators γ i,j ( z ) , β i,j 1 ( z ) , β i,j 2 ( z ) , β i,j 3 ( z ) , β i,j 4 ( z ), (1 ≦ i < j ≦ N ) b y γ i,j ( z ) = − X n 6 =0 ( b + c ) i,j n [ n ] z − n + ( q i,j b + q i,j c ) + ( p i,j b + p i,j c )log( − z ) , (5.12) β i,j 1 ( z ) = b i,j + ( z ) − ( b i,j + c i,j )( q z ) , β i,j 2 ( z ) = b i,j − ( z ) − ( b i,j + c i,j )( q − 1 z ) , (5.13) β i,j 3 ( z ) = b i,j + ( z ) + ( b i,j + c i,j )( q − 1 z ) , β i,j 4 ( z ) = b i,j − ( z ) + ( b i,j + c i,j )( q z ) . (5.14) W e giv e a free field realization of Drinfeld curr en t for U q ( d sl N ). Definition 5.1 L et us set the b osonic op er ators E ± ,i ( z ) , (1 ≦ i ≦ N − 1) by E + ,i ( z ) = − 1 ( q − q − 1 ) z i X j =1 E + ,i j ( z ) , (5.15) E − ,i ( z ) = − 1 ( q − q − 1 ) z N − 1 X j =1 E − ,i j ( z ) , (5.16) 19 wher e we have set E + ,i j ( z ) = : e γ j,i ( q j − 1 z ) ( e β j,i +1 1 ( q j − 1 z ) − e β j,i +1 2 ( q j − 1 z ) ) e P j − 1 l =1 ( b l,i +1 + ( q l − 1 z ) − b l,i + ( q l z )) : , (5.17) E − ,i j ( z ) = : e γ j,i +1 ( q − ( k + j ) z ) ( e − β j,i 4 ( q − ( k + j ) z ) − e − β j,i 3 ( q − ( k + j ) z ) ) × e P i l = j +1 ( b l,i +1 − ( q − ( k + l − 1) z ) − b l,i − ( q − ( k + l ) z ))+ a i − ( q − k + N 2 z )+ P N l = i +1 ( b i,l − ( q − ( k + l ) z ) − b i +1 ,l − ( q − ( k + l − 1) z )) : , for 1 ≦ j ≦ i − 1 , (5.18) E − ,i i ( z ) = : e γ i,i +1 ( q − ( k + i ) z )+ a i − ( q − k + N 2 z )+ P N l = i +1 ( b i,l − ( q − ( k + l ) z ) − b i +1 ,l − ( q − ( k + l − 1) z )) : − : e γ i,i +1 ( q k + i z )+ a i + ( q k + N 2 z )+ P N l = i +1 ( b i,l + ( q k + l z ) − b i +1 ,l + ( q k + l − 1 z )) : , (5.19) E − ,i j ( z ) = : e γ i,j +1 ( q k + j z ) ( e β i +1 ,j +1 2 ( q k + j z ) − e β i +1 ,j +1 1 ( q k + j z ) ) e a i + ( q k + N 2 z )+ P N l = j +1 ( b i,l + ( q k + l z ) − b i +1 ,l + ( q k + l − 1 z )) : , for i + 1 ≦ j ≦ N − 1 . (5.20) L et us se t the b osonic op er ator s ψ ± i ( z ) , (1 ≦ i ≦ N − 1) by ψ i ± ( q ± k 2 z ) =: e P i j =1 ( b j,i +1 ± ( q ± ( k + j − 1) z ) − b j,i ± ( q ± ( k + j ) z ))+ a i ± ( q ± k + N 2 z )+ P N j = i +1 ( b i,j ± ( q ± ( k + j ) z ) − b i +1 ,j ± ( q ± ( k + j − 1) z )) : . (5.21) L et us se t h i = i X j =1 ( p j,i +1 b − p j,i b ) + p i a + N X j = i +1 ( p i,j b − p i +1 ,j b ) . (5.22) Let us in tro duce the auxiliary op erators B ∗ i,j ± ( z ) , B i,j ± ( z ), (1 ≦ i < j ≦ N ) b y B ∗ i,j ± ( z ) = exp ± X n> 0 1 [ r ∗ n ] b i,j − n ( q r ∗ − 1 z ) n ! , (5.23) B i,j ± ( z ) = exp ± X n> 0 1 [ r n ] b i,j n ( q − r ∗ +1 z ) − n ! . (5.24) Let us in tro duce the auxiliary op erators A ∗ i ( z ) , A i ( z ), (1 ≦ i ≦ N − 1) by A ∗ i ( z ) = exp X n> 0 1 [ r ∗ n ] a i − n ( q r ∗ z ) n ! , (5.25) A i ( z ) = exp − X n> 0 1 [ r n ] a i n ( q − r ∗ z ) − n ! . (5.26) Definition 5.2 We define the dr essing op er ators U ∗ i ( z ) , U i ( z ) , (1 ≦ i ≦ N − 1) . U ∗ i ( z ) =   i − 1 Y j =1 B ∗ j,i +1 + ( q 2 − j z ) B ∗ j,i − ( q 1 − j z )   (5.27) 20 × B ∗ i,i +1 + ( q 2 − i z ) B ∗ i,i +1 + ( q − i z )   N Y j = i +2 B ∗ i,j + ( q − j +1 z ) B ∗ i +1 ,j − ( q − j +2 z )   A ∗ i ( q k − N 2 z ) , U i ( z ) =   i − 1 Y j =1 B j,i +1 − ( q − 2+ j z ) B j,i + ( q − 1+ j z )   (5.28) × B i,i +1 − ( q − 2+ i z ) B i,i +1 − ( q i z )   N Y j = i +2 B i,j − ( q j − 1 z ) B i +1 ,j + ( q j − 2 z )   A i ( q − k + N 2 z ) . Definition 5.3 We define the e l liptic deformation of Drinfeld curr ent E i ( z ) , F i ( z ) , H ± i ( z ) , (1 ≦ i ≦ N − 1) , by E i ( z ) = U ∗ i ( z ) E + ,i ( z ) e 2 Q i z − P i − 1 r − k , (5.29) F i ( z ) = E − ,i ( z ) U i ( z ) z h i + P i − 1 r , (5.30) H + i ( z ) = U ∗ i ( q k 2 z ) ψ + i ( z ) U i ( q − k 2 z ) e 2 Q i q − h i ( q ( r − k 2 ) z ) h i + P i − 1 r − P i − 1 r ∗ , (5.31) H − i ( z ) = U ∗ i ( q − k 2 z ) ψ − i ( z ) U i ( q k 2 z ) e 2 Q i q h i ( q − ( r − k 2 ) z ) h i + P i − 1 r − P i − 1 r ∗ . (5.32) Theorem 5.3 The b osonic op er ators E i ( z ) , F i ( z ) , H ± i ( z ) , (1 ≦ i, j ≦ N − 1) satisfy the fol lowing c ommutation r elations. [ u 1 − u 2 − A i,j 2 ] r − k E i ( z 1 ) E j ( z 2 ) = [ u 1 − u 2 + A i,j 2 ] r − k E j ( z 2 ) E i ( z 1 ) , (5.33) [ u 1 − u 2 + A i,j 2 ] r F i ( z 1 ) F j ( z 2 ) = [ u 1 − u 2 − A i,j 2 ] r F j ( z 2 ) F i ( z 1 ) , (5.34) [ 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 )  . (5.35) W e hav e constructed the free fi eld realization of the elliptic algebra U q ,p ( d sl N ) for L evel k 6 = 0 , − N . In order to construct free field realiztion of a pair of F eigin-Odesskii algebra (3.1) and (5.1), w e ha ve to solv e the follo wing problem. Problem (1) C onstruct the free field realizati on of the curr en ts E N ( z ) and F N ( z ), whic h satisfy the relations (5.33), (5.34) and (5.35) w hic h are v alid for all 1 ≦ i, j ≦ N . (2) Constru ct one parameter s deformation of the free field realiztion of E j ( z ) , F j ( z ), (1 ≦ j ≦ N ). 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