On the universal weight function for the quantum affine algebra U_q(hat{mathfrak{gl}}_N)

ITEP-TH-55/07 On the univ ersal w eigh t function for the quan tum affine algebra U q ( b gl N ) Andrey Oskin • 1 , Stanisla v P akuliak ⋆ • 2 and Alexey Silan t y ev •◦ 3 ⋆ Institute of The or etic al & Exp erimental Physics, 117259 Mosc o w , R ussia • L ab or atory of The or etic al Physics, JINR, 141980 Dubna, Mosc ow r e g., Russia ◦ D´ ep artmen t de Math´ ematiques, Universit ´ e d’Angers, 2 Bd. L avoisi e r, 49045 A ngers, F r anc e Abstract W e con tinue in ve stigation of the unive rsal w eigh t function for the quan tum affine algebra U q ( b gl N ) started in [K P T] and [KP ]. W e obtain tw o rec ur r ence relations for the univ ersal w eigh t function applying th e metho d of pro jections dev elop ed in [EKP]. On the lev el of the ev aluation represen tation of U q ( b gl N ) we repro du ce b oth recur r ence relations for the off-shell Bethe v ectors calculat ed in [TV2] using combinato rial method s. 1 In tro duction Hierarc hical (nested) Bethe ansatz w as designed in [KR] to construct the eigen v ectors of the commuting in tegrals for quan tum in tegrable models ass o ciated w ith the Lie algebra gl N . It is based on t he inductiv e pro cedure whic h relates gl N and gl N − 1 Bethe v ectors. If the parameters of these, so called off-shell Bethe ve ctors, satisfy Bethe equations, then the correspo nding v ectors are eigen v ectors of the comm uting set of op erators in some quan tum in tegrable model. F urther dev elopment of the off-shell Bethe v ectors theory w as a c hiev ed in [TV1], where they were presen ted as part icular matrix elemen ts o f mono drom y op erators. This construction was used in [TV2] to obtain the explicit formulas for the off-shell Bethe v ectors on the tensor pro duct of ev aluatio n mo dules o f U q ( b gl N ). Tw o differen t recurrence relations for the o ff-shell Bethe v ectors on the ev aluation U q ( b gl N )-mo dules w ere obtained in [TV2]. Iteration of these tw o relatio ns allows to obtain differen t explicit formulas for the off- shell Bethe v ectors (see examples (2 .1 6) and (2.18) b elo w). Existence of tw o t yp es of the recurrence relations in the nested Bethe ansatz is a consequenc e of tw o differen t wa ys of em b edding U q ( b gl N − 1 ) into U q ( b gl N ), when they are realized in terms of L-op erators. The L- op erator of U q ( b gl N − 1 ) can b e placed either into the top- left or in t o the do wn-r ig h t corners of the U q ( b gl N ) L - op erator. 1 E-mail: aoskin@theor.jinr .ru 2 E-mail: pakuliak@theor.jinr .r u 3 E-mail: silant @tonton.univ-angers .fr Due to the applicatio ns in the theory of quan tized Knizhnik-Zamolo dchik ov equa- tions the off-shell Bethe v ectors are called wei g ht functions . W e will use b oth names for these ob jects. W e will call a w eigh t function universal if it is defined in an arbitra r y U q ( b gl N )-mo dule generated b y arbitrary singular v ector. An alternativ e a pproac h for the construction of the off-shell Bethe v ectors for an arbitrary quan tum affine algebra w as dev elop ed in [EKP]. This approac h uses the relation b etw een L-op erator realizatio n of U q ( b gl N ) [RS] a nd the curren t r ealizat io n of the same algebra [D]. An isomorphism betw een these realizatio ns was observ ed in the pap er [DF]. Tw o differen t type of Bo r el subalgebras are r elat ed to these t w o realizations of U q ( b gl N ). It w as conjectured in [KPT] tha t the pro jections on to in tersection o f the differen t t yp e Borel subalgebras for the pro duct of Drinfeld curren ts coincide with the off-shell Bethe v ectors obta ined using the construction of [TV1]. The ba ckground for the conjecture from [KPT] w as an observ a tion that b ot h quan- tities satisfy the same copro duct prop ert y [EKP]. It w as prov ed in [KP] that calculation of the pro jection fo r the pro duct of curren ts gives similar nested recurrence relation for the off-shell Bethe v ectors a s it w as obtained in [TV2] on the lev el of tensor pro duct of the ev alua tion U q ( b gl N )-mo dules. This leads to the conclusion that the pro jection metho d yields t he univ ersal off-shell Bethe vec tors for an arbitrary U q ( b gl N )-mo dule generated b y a singuar w eight v ectors. Only one t yp e of the recurren ce relation which leads to the form ula of the t yp e (2.16) w as considered in the pap er [KP]. Here w e generalize the results of this pap er. W e prov e tha t in order to get b oth t yp es of the recurrence relations one has to use tw o differen t isomorphic current realizations of U q ( b gl N ). Origin of these differen t realizations lies in t w o p ossibilities to in tro duce the Gauss decomp ositions of L-op erators, eac h corresponds to the different em b edding of U q ( b gl N − 1 ) L-op erato r s into U q ( b gl N ) L-op erato r (see (3.1)–( 3 .3) and (3.4)–(3.6) b elo w). The pap er is comp o sed as follows. Section 2 serv es as reminder of the L-op erator realization of U q ( b gl N ) and construction of the off-shell Bethe v ectors in terms of the matrix elemen ts of these L-op erator s [TV1]. In Section 3 t w o differen t Gauss decom- p osition are in tro duced as we ll a s the corresponding curren t realizations of U q ( b gl N ). Here w e intro duce the curren t Bor el subalgebras and describ e the pro jections on to in- tersections of the standar d a nd curren t Borel subalgebras. Section 4 dev o ted to the calculations of the pro jections for the pro duct of the curren ts. The main result here is the Theorem 1 whic h yields the univ ersal w eigh t f unctions as the sum ov er the ordered pro ducts of the pro jections of the simple and comp osed ro ots curren ts. Using the fa ct that the pro jections of the comp osed ro o t currents coincide with the G auss co ordinates of L-op erato rs we giv e in the Section 5 the explicit expressions for the univ ersal w eigh t v ectors in terms of the matrix elemen ts of L- op erators generalizing formulas of the pap er [TV2]. The main result of the pap er is form ulated in the form of the Theorem 3 in the Section 6. The pa p er con tains tw o Appendices. One con tains the reform ulation of the Serre relatio ns in the form of the comm utation relations b et w een comp osed cur- ren ts. Thes e relations are necess ary to pro v e the Prop osition B.1, whic h is the most difficult tec hnical r esult of the pap er. It is f o rm ulated in the second App endix and describes the o r dering of the curren ts and the negativ e pro j ections of the curren ts. 2 2 T araso v-V arc henk o construc tion 2.1 U q ( b gl N ) in L -op erator formalism Let E ij ∈ End( C N ) b e a matrix with the only nonzero en try equal to 1 at the intersec - tion of the i -th ro w and j - th column. Let R( u, v ) ∈ End( C N ⊗ C N ) ⊗ C [[ v /u ]], R( u, v ) = X 1 ≤ i ≤ N E ii ⊗ E ii + u − v q u − q − 1 v X 1 ≤ i b , L − a,b =    0 a < b E − 1 a,a a = b ( q − 1 − q ) E − 1 a,a E b,a a > b and the composed ro ots generators are defined as follows E c,a = E c,b E b,a − q − 1 E b,a E c,b , E a,c = E a,b E b,c − q E b,c E a,b , a < b < c . These form ulas ma y b e pro v ed inductiv ely after substitution of (2.6) in to (2.2). The copro duct of the L-op erators (2.4) implies the copro duct of the Chev a lley generators: ∆ E a,a = E a,a ⊗ E a,a , ∆ E a,a +1 = E a,a +1 ⊗ 1 + E − 1 a,a E a +1 ,a +1 ⊗ E a,a +1 and ∆ E a +1 ,a = 1 ⊗ E a +1 ,a + E a +1 ,a ⊗ E a,a E − 1 a +1 ,a +1 . 2.3 Com b inatorial form ulas for off-shell Bethe v ectors Let us remind the construction of the off-shell Bethe v ectors [TV1]. Let L- op erator 2 L ( z ) = P ∞ k =0 P N i,j =1 E ij ⊗ L i,j [ k ] z − k of the Bor el subalgebra U q ( b + ) of U q ( b gl N ) satisfies the Y ang-Baxter comm utatio n relation with a R -matrix R( u, v ). W e use the notation L ( k ) ( z ) ∈  C N  ⊗ M ⊗ U q ( b + ) for L- o p erator acting non trivially o n k - th tensor factor in the pro duct  C N  ⊗ M for 1 ≤ k ≤ M . Consider a series in M v aria bles T ( u 1 , . . . , u M ) = L (1) ( u 1 ) · · · L ( M ) ( u M ) · R ( M ,..., 1) ( u M , . . . , u 1 ) (2.7) with co efficien ts in  End( C N )  ⊗ M ⊗ U q ( b + ), where R ( M ,..., 1) ( u M , . . . , u 1 ) = ← − Y M ≥ j > 1 ← − Y j >i ≥ 1 R ( j i ) ( u j , u i ) . (2.8) In the ordered pro duct of R -mat rices (2.8) the factor R ( j i ) is to the left of t he factor R ( ml ) if j > m , or j = m and i > l . Consider the set of v a riables ¯ t [ ¯ n ] = n t 1 1 , . . . , t 1 n 1 ; t 2 1 , . . . , t 2 n 2 ; . . . . . . ; t N − 2 1 , . . . , t N − 2 n N − 2 ; t N − 1 1 , . . . , t N − 1 n N − 1 o . (2.9) F ollowing [TV1 ], let B ( ¯ t [ ¯ n ] ) = Y 1 ≤ a j , n ≥ 0 of the matrix elemen ts of the L-o p erator L + ( z ) and is an eigenv ector of the diagonal matrix en tries L + i,i ( z ) L + i,j ( z ) v = 0 , i > j , L + i,i ( z ) v = λ i ( z ) v , i = 1 , . . . , N . (2.14) F or any U q ( b gl N )-mo dule V with a singular v ector v denote B V ( ¯ t [ ¯ n ] ) = B ( ¯ t [ ¯ n ] ) v . (2.15) V ector v alued function B V ( ¯ t [ ¯ n ] ) w as called in [TV1, TV2] universa l off-shel l Bethe ve ctor . Let M Λ ( z ) b e a n ev aluatio n mo dule generated b y a singular v ector v suc h that E a,a v = q Λ a v . F rom analysis of the relations o f the hierarchical Bethe ansatz [KR] the authors of the pap er [TV2 ] obtained t w o r ecurrence relations for t he off- shell Bethe v ectors B M Λ ( z ) ( ¯ t [ ¯ n ] ). Iterating these relations many equiv alen t formulas fo r t hese ob jects can b e found in terms of the U q ( gl N ) generators E a,b . Tw o extreme cases were presen ted in [TV2]. First, the off- shell Bethe v ector can be written as B M Λ ( z ) ( ¯ t [ ¯ n ] ) = ( q − q − 1 ) P N − 1 a =1 n a X [[ ¯ s ]] ← − Y N − 1 ≥ b>a ≥ 1 q s b a − 1 ( s b a − 1 − s b a ) [ s b a − s b a − 1 ] q ! ˇ E s b a − s b a − 1 b +1 ,a ! v × Sym ( q ) ¯ t [ ¯ n ]   N − 1 Y b =2 b − 1 Y a =1 s b a Y ℓ =1 q Λ a +1 t a ℓ + ˜ s b a − q − Λ a +1 z t a +1 ℓ + ˜ s b a +1 − t a ℓ + ˜ s b a ℓ + ˜ s b a +1 − 1 Y ℓ ′ =1 q t a +1 ℓ ′ − q − 1 t a ℓ + ˜ s b a t a +1 ℓ ′ − t a ℓ + ˜ s b a     , (2.16) where ˇ E b +1 ,a = E b +1 ,a E b +1 ,b +1 and sum is tak en o v er all p ossible collections of nonneg- ativ e integers [[ ¯ s ]] = { s j i } suc h that 0 = s a 0 ≤ s a 1 ≤ · · · ≤ s a a , n a = N − 1 X b = a s b a , a = 1 , . . . , N − 1 (2.17) 5 and ˜ s b a = s a a + s a +1 a + · · · + s b − 1 a . Second, the same off- shell Bethe vector B M Λ ( z ) ( ¯ t [ ¯ n ] ) has a differen t presen tation B M Λ ( z ) ( ¯ t [ ¯ n ] ) = ( q − q − 1 ) P N − 1 a =1 n a X [[ ¯ m ]] − → Y 1 ≤ b ≤ a ≤ N − 1 q m b a +1 ( m b a − m b a +1 ) [ m b a − m b a +1 ] q ! ˇ E m b a − m b a +1 a +1 ,b ! v × Sym ( q ) ¯ t [ ¯ n ]   N − 1 Y a =2 a − 1 Y b =1 m b a − 1 Y ℓ =0 q Λ a t a m b a − ℓ − q − Λ a z t a − 1 m b a − 1 − ℓ − t a m b a − ℓ n a − 1 Y ℓ ′ = m b a − 1 − ℓ +1 q t a m b a − ℓ − q − 1 t a − 1 ℓ ′ t a m b a − ℓ − t a − 1 ℓ ′     , (2.18) where [[ ¯ m ]] = { m j i } is a collection nonnegativ e integers suc h that m a a ≥ m a a +1 ≥ · · · ≥ m a N − 1 ≥ m a N = 0 , n a = a X b =1 m b a , a = 1 , . . . , N − 1 (2.19) and m b a = m 1 a + m 2 a + · · · + m b a . Ordering pro duct of the non-comm utative en tries in (2.16) is the same as in (2.8 ) and the ordering in (2.18) is in v erse. More general formula for the univ ersal off-shell Bethe v ectors was obtained in [KP] using curren t realization of the quan tum affine algebra U q ( b gl N ) and metho d of pro jec- tions intro duced in [ER] a nd dev elop ed in [EKP]. F o rm ula (2.1 6) w as obtained in the latter pap er after sp ecialization to the ev aluation mo dules. The g oal of the presen t pap er is to describe the recurrence relations for the univ ersal off- shell Bethe v ectors or univ ersal w eigh t functions in terms of the mo des of t he U q ( b gl N ) curren ts. T o do this w e hav e to intro duce t o differen t curren t realizations of the a lgebra U q ( b gl N ). 3 Differen t typ e Borel subalgebras 3.1 Tw o Gauss decomp ositions of L -op erators The relation b et w een L-op erator realization of U q ( b gl N ) and its curren t realization [D ] is kno wn since t he w ork [DF]. The main distinction in these t w o realizations of the same algebra lies in the differen t c hoice of the Borel subalgebras and the corresp onding coalgebraic structures. T o build an isomorphism b etw een tw o realizations one has to consider the Gauss decomp o sition [D F] of the L-op erators a nd iden tify linear com bi- nations of certain Gauss co ordinates with the t otal curren ts of U q ( b gl N ) corresponding to the simple ro ots o f gl N . In order to construct the univ ersal off- shell Bethe v ectors in terms of the modes of the currents one has to consider the o rdered pro duct of the simple ro ots curren ts and calculate the pro jection o f this pro duct onto in tersection of the curren t and standard Borel subalgebras in U q ( b gl N ). Curren t Borel subalgebras will b e introduced in the Subsection 3.3. F or the L-o p erators fixe d by the relations (2.2) and (2.3) w e hav e tw o p ossibilities 6 to introduce the Gauss coor dina t es F ± b,a ( t ), E ± b,a ( t ), b > a and k ± c ( t ): L ± a,b ( t ) = F ± b,a ( t ) k + a ( t ) + X 1 ≤ m b , (3.3) or ˆ F ± b,a ( t ), ˆ E ± b,a ( t ), b > a and ˆ k ± c ( t ): L ± a,b ( t ) = ˆ F ± b,a ( t ) ˆ k + b ( t ) + X b b . (3.6) F ormulas (3 .1)–(3.6) can b e in v erted to expres s the Gauss co ordinates in terms of the matrix elemen ts of L-o p erators as w ell as to express one set of the Gauss co ordinates through another. This is p ossible b ecause of the relations: L ± a,a [0] = k ± a [0] = ˆ k ± a [0] , k + a [0] k − a [0] = 1 (3.7) and the mode decompo sition of the Gauss co ordinates F ± b,a ( t ) = X n ≥ 0 n< 0 F ± b,a [ n ] t − n , E ± a,b ( t ) = X n> 0 n ≤ 0 E ± a,b [ n ] t − n , a < b (3.8) whic h follows from the mo de decomp osition of the L-op erators (2.3). The same rules of the mo de decomp ositions is v alid for another set of G auss co ordinates ˆ F ± b,a ( t ) and ˆ E ± a,b ( t ). Let T b e a ( n + m ) × ( n + m ) matrix with no ncomm utativ e en tries presen ted in the form T =  A B C D  , where A , B , C and D are n × n , n × m , m × n and m × m matrices resp ectiv ely . Supp ose that A and D are in v ertible matrices and in tro duce the follow ing notations [BK]:     A B C D     := A − B D − 1 C ,     A B C D     := ˜ D , where ˜ D is an m × m matrix with the en tries ˜ D ij = D ij − n P k ,l =1 B lj ( A − 1 ) k l C ik . Then 7 the Ga uss co or dina t es can b e expre ssed through the L -op erato r en tries as follo ws k ± a ( u ) =          L ± 11 ( u ) . . . L ± 1 ,a − 1 ( u ) L ± 1 a ( u ) . . . . . . . . . L ± a − 1 , 1 ( u ) . . . L ± a − 1 ,a − 1 ( u ) L ± a − 1 ,a ( u ) L ± a 1 ( u ) . . . L ± a,a − 1 ( u ) L ± aa ( u )          , E ± ab ( u ) = k ± a ( u ) − 1          L ± 11 ( u ) . . . L ± 1 ,a − 1 ( u ) L ± 1 a ( u ) . . . . . . . . . L ± a − 1 , 1 ( u ) . . . L ± a − 1 ,a − 1 ( u ) L ± a − 1 ,a ( u ) L ± b 1 ( u ) . . . L ± b,a − 1 ( u ) L ± ba ( u )          , a < b, F ± ab ( u ) =          L ± 11 ( u ) . . . L ± 1 ,b − 1 ( u ) L ± 1 a ( u ) . . . . . . . . . L ± b − 1 , 1 ( u ) . . . L ± b − 1 ,b − 1 ( u ) L ± b − 1 ,a ( u ) L ± b 1 ( u ) . . . L ± b,b − 1 ( u ) L ± ba ( u )          k ± b ( u ) − 1 , a > b. and ˆ k ± a ( u ) =          L ± aa ( u ) L ± a,a +1 ( u ) . . . L ± aN ( u ) L ± a +1 ,a ( u ) L ± a +1 ,a +1 ( u ) . . . L ± a +1 ,N ( u ) . . . . . . . . . L ± N a ( u ) L ± N ,a +1 ( u ) . . . L ± N N ( u )          , ˆ E ab ( u ) = ˆ k ± b ( u ) − 1          L ± ba ( u ) L ± b,b +1 ( u ) . . . L ± bN ( u ) L ± b +1 ,a ( u ) L ± b +1 ,b +1 ( u ) . . . L ± b +1 ,N ( u ) . . . . . . . . . L ± N a ( u ) L ± N ,b +1 ( u ) . . . L ± N N ( u )          , a < b, ˆ F ± ab ( u ) =          L ± ba ( u ) L ± b,a +1 ( u ) . . . L ± bN ( u ) L ± a +1 ,a ( u ) L ± a +1 ,a +1 ( u ) . . . L ± a +1 ,N ( u ) . . . . . . . . . L ± N a ( u ) L ± N ,a +1 ( u ) . . . L ± N N ( u )          ˆ k ± a ( u ) − 1 , a > b. Gauss decomp osition (3.4)–(3.6) was used 3 in [KP] in order to obtain a recurrence relation for the univers al w eigh t function (4.8) and to pro v e the conjecture of [KPT] that the constructions of the o ff - shell Bethe v ectors using L-op erator approac h [TV1] and metho d of pro jection [EKP] coincide for arbitrary U q ( b gl N )-mo dule g enerated b y arbitrary singular v ectors. Here w e will use another Gauss decomp osition (3.1)–( 3 .3) in or der to get an alternative recurrence relation for the unive rsal w eigh t function (4.7) whic h lead to the form ula (2.18) for the off-shell Bethe v ector. 3 Gauss co ordinates ˆ F ± b,a ( t ), ˆ E ± a,b ( t ), a < b and ˆ k ± a ( t ) was denoted in [KP] as ˜ F ± b,a ( t ), ˜ E ± a,b ( t ) and k ± a ( t ) (cf. Sect. 6 in [K P]). 8 3.2 The curr ent realization of U q ( b gl N ) Using argumen ts of the pap er [DF ] w e ma y obtain for the linear combinations of the Gauss co ordinates F i ( t ) = F + i +1 ,i ( t ) − F − i +1 ,i ( t ) , E i ( t ) = E + i,i +1 ( t ) − E − i,i +1 ( t ) (3.9) and ‘diag onal’ Gauss co ordinates k ± i ( t ) the comm uta tion relations of the quantum affine algebra U q ( b gl N ) with the zero cen tral c harge a nd the g radation op erator dropp ed out. In terms of the total curren ts F i ( t ), E i ( t ) and the Cartan curren ts k ± i ( t ) these comm utation relations are ( q z − q − 1 w ) E i ( z ) E i ( w ) = E i ( w ) E i ( z )( q − 1 z − q w ) , ( q − 1 z − q w ) E i ( z ) E i +1 ( w ) = E i +1 ( w ) E i ( z )( z − w ) , k ± i ( z ) E i ( w )  k ± i ( z )  − 1 = z − w q − 1 z − q w E i ( w ) , k ± i +1 ( z ) E i ( w )  k ± i +1 ( z )  − 1 = z − w q z − q − 1 w E i ( w ) , k ± i ( z ) E j ( w )  k ± i ( z )  − 1 = E j ( w ) , if i 6 = j, j + 1 , ( q − 1 z − q w ) F i ( z ) F i ( w ) = F i ( w ) F i ( z )( q z − q − 1 w ) , ( z − w ) F i ( z ) F i +1 ( w ) = F i +1 ( w ) F i ( z )( q − 1 z − q w ) , k ± i ( z ) F i ( w )  k ± i ( z )  − 1 = q − 1 z − q w z − w F i ( w ) , k ± i +1 ( z ) F i ( w )  k ± i +1 ( z )  − 1 = q z − q − 1 w z − w F i ( w ) , k ± i ( z ) F j ( w )  k ± i ( z )  − 1 = F j ( w ) , if i 6 = j, j + 1 , [ E i ( z ) , F j ( w )] = δ i,j δ ( z /w ) ( q − q − 1 )  k − i +1 ( z ) / k − i ( z ) − k + i +1 ( w ) /k + i ( w )  , (3.10) and the Serre relations Sym z 1 ,z 2 ( E i ( z 1 ) E i ( z 2 ) E i ± 1 ( w ) − ( q + q − 1 ) E i ( z 1 ) E i ± 1 ( w ) E i ( z 2 )+ + E i ± 1 ( w ) E i ( z 1 ) E i ( z 2 )) = 0 , Sym z 1 ,z 2 ( F i ( z 1 ) F i ( z 2 ) F i ± 1 ( w ) − ( q + q − 1 ) F i ( z 1 ) F i ± 1 ( w ) F i ( z 2 )+ + F i ± 1 ( w ) F i ( z 1 ) F i ( z 2 )) = 0 . (3.11) In order to obtain the comm utation relations (3.10) w e substitute the decomp osition (3.1)-(3.3) in to comm ut a tion relatio ns (2 .2). If w e substitute in to these commutation relations the decomp osition (3.4)-(3 .6) w e obtain instead of (3.10) sligh tly differen t comm utation relations for the total currents ˆ F i ( t ) = ˆ F + i +1 ,i ( t ) − ˆ F − i +1 ,i ( t ) , ˆ E i ( t ) = ˆ E + i,i +1 ( t ) − ˆ E − i,i +1 ( t ) (3.12) 9 and ˆ k ± i ( t ) used in t he pap ers [K PT, KP]: ( q − 1 z − q w ) ˆ E i ( z ) ˆ E i ( w ) = ˆ E i ( w ) ˆ E i ( z )( q z − q − 1 w ) , ( z − w ) ˆ E i ( z ) ˆ E i +1 ( w ) = ˆ E i +1 ( w ) ˆ E i ( z )( q − 1 z − q w ) , ˆ k ± i ( z ) ˆ E i ( w )  ˆ k ± i ( z )  − 1 = z − w q − 1 z − q w ˆ E i ( w ) , ˆ k ± i +1 ( z ) ˆ E i ( w )  ˆ k ± i +1 ( z )  − 1 = z − w q z − q − 1 w ˆ E i ( w ) , ˆ k ± i ( z ) ˆ E j ( w )  ˆ k ± i ( z )  − 1 = ˆ E j ( w ) , if i 6 = j, j + 1 , ( q z − q − 1 w ) ˆ F i ( z ) ˆ F i ( w ) = ˆ F i ( w ) ˆ F i ( z )( q − 1 z − q w ) , (3.13) ( q − 1 z − q w ) ˆ F i ( z ) ˆ F i +1 ( w ) = ˆ F i +1 ( w ) ˆ F i ( z )( z − w ) , ˆ k ± i ( z ) ˆ F i ( w )  ˆ k ± i ( z )  − 1 = q − 1 z − q w z − w ˆ F i ( w ) , ˆ k ± i +1 ( z ) ˆ F i ( w )  ˆ k ± i +1 ( z )  − 1 = q z − q − 1 w z − w ˆ F i ( w ) , ˆ k ± i ( z ) ˆ F j ( w )  ˆ k ± i ( z )  − 1 = ˆ F j ( w ) , if i 6 = j, j + 1 , [ ˆ E i ( z ) , ˆ F j ( w )] = δ i,j δ ( z /w ) ( q − q − 1 )  ˆ k + i ( z ) / ˆ k + i +1 ( z ) − ˆ k − i ( w ) / ˆ k − i +1 ( w )  and the same Serre relations (3.1 1) with curren ts E i ( z ) and F i ( z ) replaced b y ˆ E i ( z ) and ˆ F i ( z ). F ormulae (3.10) and (3.13) should b e considered a s formal series iden tities describ- ing the infinite set o f the relations b etw een mo des of the curren ts. The sym b ol δ ( z ) en tering these relations is a formal series P n ∈ Z z n . These comm utation relations de- scrib es tw o isomorphic curren t realizations of the algebra U q ( b gl N ). Isomorphism follows from the result of [DF]. F or an y series G ( t ) = P m ∈ Z G [ m ] t − m w e denote G ( t ) (+) = P m> 0 G [ m ] t − m , a nd G ( t ) ( − ) = − P m ≤ 0 G [ m ] t − m . The initial conditions (2.3) imply the relat io ns F ± i +1 ,i ( z ) = z  z − 1 F i ( z )  ( ± ) , E ± i,i +1 ( z ) = E i ( z ) ( ± ) (3.14) and the same for the relatio ns b et w een Ga uss coo rdinates ˆ E ± i,i +1 ( t ), ˆ F ± i +1 ,i ( t ) and the curren ts ˆ E i ( t ), ˆ F i ( t ). 3.3 Borel sub algebras and pro jections to their int ersections W e consider t w o t yp es of Bo r el subalgebras of the algebra U q ( b gl N ). Bo rel subalgebras U q ( b ± ) ⊂ U q ( b gl N ) a re generated by t he mo des of the L-o p erators L ( ± ) ( z ) resp ectiv ely . F or the generators in these subalgebras w e can use either mo des of t he Gauss co or- dinates (3.1 ) –(3.3), E ± i,i +1 ( t ), F ± i +1 ,i ( t ), k ± j ( t ) or the mo des of the Ga uss co or dinates (3.4)–(3.6) ˆ E ± i,i +1 ( t ), ˆ F ± i +1 ,i ( t ), ˆ k ± j ( t ). Another types of Borel subalgebras are related to the curren t realizations of U q ( b gl N ) giv en in the previous subsection. W e consider first the current Borel subalgebras gen- erated by the mo des of the curren ts E i ( t ), F i ( t ), k ± j ( t ). 10 The Borel subalgebra U F ⊂ U q ( b gl N ) is generated by mo des of the curren ts F i [ n ], k + j [ m ], i = 1 , . . . , N − 1, j = 1 , . . . , N , n ∈ Z and m ≥ 0. The Borel subalgebra U E ⊂ U q ( b gl N ) is generated b y mo des of the currents E i [ n ], k − j [ − m ], i = 1 , . . . , N − 1 , j = 1 , . . . , N , n ∈ Z and m ≥ 0. W e will consider also a subalgebra U ′ F ⊂ U F , generated b y the elemen ts F i [ n ], k + j [ m ], i = 1 , . . . , N − 1, j = 1 , . . . , N , n ∈ Z and m > 0, and a subalgebra U ′ E ⊂ U E generated b y the elemen ts E i [ n ], k − j [ − m ], i = 1 , . . . , N − 1 , j = 1 , . . . , N , n ∈ Z and m > 0. F urther, we will be in terested in the inters ections, U − f = U ′ F ∩ U q ( b − ) , U + F = U F ∩ U q ( b + ) (3.15) and will describ e pro p erties of pro j ections to these in tersections. W e call U F and U E the curr ent Bor el sub algebr as . Let U f ⊂ U F b e subalgebra of the curren t Borel subalgebra generated by the mo des of the curren ts F i [ n ], i = 1 , . . . , N − 1, n ∈ Z only . In what follo ws w e will use also the subalgebra U + f ⊂ U f defined by the interse ction U + f = U + F ∩ U f . (3.16) In [D] the curr ent Hopf structure for the algebra U q ( b gl N ) has been defined, ∆ ( D ) ( E i ( z )) = 1 ⊗ E i ( z ) + E i ( z ) ⊗ k − i +1 ( z )  k − i ( z )  − 1 , ∆ ( D ) ( F i ( z )) = F i ( z ) ⊗ 1 + k + i +1 ( z )  k + i ( z )  − 1 ⊗ F i ( z ) , ∆ ( D )  k ± i ( z )  = k ± i ( z ) ⊗ k ± i ( z ) . (3.17) With resp ect to the curren t Hopf structure t he curren t Borel subalgebras are Hopf subalgebras of U q ( b gl N ). W e can c hec k [EKP, KPT] that the in tersections U − f and U + F are subalgebras and coideals with resp ect to the Drinfeld copro duct (3.17) ∆ ( D ) ( U + F ) ⊂ U + F ⊗ U q ( b gl N ) , ∆ ( D ) ( U − f ) ⊂ U q ( b gl N ) ⊗ U − f , and the m ult iplicatio n m in U q ( b gl N ) induces an isomorphism of vec tor spaces m : U − f ⊗ U + F → U F . According to the g eneral theory presen ted in [EKP] w e define the pro jection op erator s P + : U F ⊂ U q ( b gl N ) → U + F and P − : U F ⊂ U q ( b gl N ) → U − f b y the prescriptions P + ( f − f + ) = ε ( f − ) f + , P − ( f − f + ) = f − ε ( f + ) , for any f − ∈ U − f , f + ∈ U + F . (3.18) Denote by U F an extension of the algebra U F formed by infinite sums of monomials whic h are ordered pro ducts a i 1 [ n 1 ] · · · a i k [ n k ] with n 1 ≤ · · · ≤ n k , where a i l [ n l ] is either F i l [ n l ] o r k + i l [ n l ]. It w as prov ed in [EKP] that (1) the action of the pro jections (3.18) can b e extended to the agebra U F ; (2) for an y f ∈ U F with ∆ ( D ) ( f ) = P i f ′ i ⊗ f ′′ i w e hav e f = X i P − ( f ′ i ) · P + ( f ′′ i ) . (3.19) 11 In [KPT, KP], w e used the curren t Bo rel subalgebras ˆ U F , ˆ U E generated b y the mo des of the curren ts ˆ F i ( t ), ˆ E i ( t ), ˆ k ± j ( t ) in the same w a y as it w as done abov e for U F , U E . These curren t Borel subalgebras are Hopf subalgebras of U q ( b gl N ) with a different Drinfeld copro duct ˆ ∆ ( D )  ˆ E i ( z )  = ˆ E i ( z ) ⊗ 1 + ˆ k − i ( z )  ˆ k − i +1 ( z )  − 1 ⊗ ˆ E i ( z ) , ˆ ∆ ( D )  ˆ F i ( z )  = 1 ⊗ ˆ F i ( z ) + ˆ F i ( z ) ⊗ ˆ k + i ( z )  ˆ k + i +1 ( z )  − 1 , ˆ ∆ ( D )  ˆ k ± i ( z )  = ˆ k ± i ( z ) ⊗ ˆ k ± i ( z ) . (3.20) The standard Borel subalgebras U q ( b ± ) are defined by the mo des of the Gauss coo r- dinates ˆ E ± i,i +1 ( t ), ˆ F ± i +1 ,i ( t ), ˆ k ± j ( t ) a nd their in tersections with the curren ts Borel subal- gebras ˆ U F are defined by the same form ulas (3.15). Using copro duct (3.20) o ne may c hec k t hat the coalgebraic properties of these in t ersections are c hanged to ˆ ∆ ( D ) ( ˆ U + F ) ⊂ U q ( b gl N ) ⊗ ˆ U + F , ˆ ∆ ( D ) ( ˆ U − f ) ⊂ ˆ U − f ⊗ U q ( b gl N ) . (3.21) Denote b y ˆ U F an extension of the algebra ˆ U F formed b y infinite sums of monomials whic h are ordered pro ducts a i 1 [ n 1 ] · · · a i k [ n k ] with n 1 ≤ · · · ≤ n k , where a i l [ n l ] is either ˆ F i l [ n l ] or ˆ k + i l [ n l ]. Pro jections to the in tersections (3.21) are defined by the form ulas analogous to (3.18) and can b e extended to the algebra ˆ U F . The prop ert y (3.19) is c hanged to f = P i ˆ P − ( f ′′ i ) · ˆ P + ( f ′ i ), where ˆ ∆ ( D ) ( f ) = P i f ′ i ⊗ f ′′ i . 4 Univ ersal w eig h t functio n and p ro jections W e will use the same notations as in [KP]. Let Π b e the set { 1 , . . . , N − 1 } of indices of t he simple p ositiv e ro ots of gl N . A finite collection I = { i 1 , . . . , i n } with a linear ordering i i ≺ · · · ≺ i n and a map ι : I → Π is called an or der e d Π -multiset . T o eac h Π-or dered m ultiset I = { i 1 , . . . , i n } w e attac h an ordered set of v ar ia bles { t i | i ∈ I } = { t i 1 , . . . , t i n } . Eac h elemen t i k ∈ I and eac h v ariable t i k has its o wn ‘t ype’: ι ( i k ) ∈ Π. Our basic calculations are p erfo rmed on a leve l of formal series a t t ac hed to cer- tain ordered m ultisets. F or the sa v e of space w e oft en write some series as rational homogeneous functions with t he following prescription. Let { t i | i ∈ I } = { t i 1 , . . . , t i n } b e the ordered set of v aria bles attached to an o r dered set I = { i 1 ≺ i 2 ≺ · · · ≺ i n } and g ( t i | i ∈ I ) b e a rational function. Then w e asso ciate to g ( t i | i ∈ I ) a Loran series whic h is the expansion of g ( t i | i ∈ I ) in a region | t i 1 | ≪ | t i 2 | ≪ · · · ≪ | t i n | . If, f o r instance, 1 ≺ 2, then w e asso ciate to a ratio nal function ( t 1 − t 2 ) − 1 a series − P k ≥ 0 t k 1 t − k − 1 2 . On the contrary , fo r the o r dering 2 ≺ 1 w e asso ciate t o the same rational f unction ( t 1 − t 2 ) − 1 a series P k ≥ 0 t k 2 t − k − 1 1 . Let ¯ l and ¯ r b e tw o collections of nonnegative integers satisfying a set of inequalities l a ≤ r a , a = 1 , . . . , N − 1 . (4.1) 12 Denote by [ ¯ l , ¯ r ] a set of segmen ts whic h contain p o sitiv e in tegers { l a + 1 , l a + 2 , . . . , r a − 1 , r a } including r a and excluding l a . The length of eac h segmen t is equal to r a − l a . F or a giv en set [ ¯ l, ¯ r ] of segmen ts w e denote by ¯ t [ ¯ l, ¯ r ] the sets of v a riables ¯ t [ ¯ l, ¯ r ] = { t 1 l 1 +1 , . . . , t 1 r 1 ; t 2 l 2 +1 , . . . , t 2 r 2 ; . . . ; t N − 1 l N − 1 +1 , . . . , t N − 1 r N − 1 } . (4.2) The n umber of the v ariables o f the ty p e a is equal to r a − l a . In this notation, the set of v aria bles (2.9) is ¯ t [ ¯ n ] ≡ ¯ t [ ¯ 0 , ¯ n ] . W e will consider (4.2) as a list of ordered v ariables, corresp onding to t w o ordered multis ets: I = { r N − 1 ≺ · · · ≺ l N − 1 + 1 ≺ · · ≺ r 2 ≺ · · · ≺ l 2 + 1 ≺ r 1 ≺ · · · ≺ l 1 + 1 } (4.3) and ˆ I = { l 1 + 1 ≺ · · · ≺ r 1 ≺ l 2 + 1 ≺ · · · ≺ r 2 ≺ · · ≺ l N − 1 + 1 ≺ · · · ≺ r N − 1 } . (4.4) F or any a = 1 , . . . , N − 1 w e denote the sets of v ariables corresp onding to the segmen ts [ l a , r a ] = { l a + 1 , l a + 2 , . . . , r a } as ¯ t a [ l a ,r a ] = { t a l a +1 , . . . , t a r a } . All the v ar ia bles in ¯ t a [ l a ,r a ] ha v e the t yp e a . F or the segmen ts [ l a , r a ] = [0 , n a ] we use the shorten notations ¯ t [ ¯ 0 , ¯ n ] ≡ ¯ t [ ¯ n ] and ¯ t a [0 ,n a ] ≡ ¯ t a [ n a ] . F or a collection of v ariables ¯ t [ ¯ l, ¯ r ] w e consider t w o types of the ordered pro ducts o f the currents F ( ¯ t [ ¯ l, ¯ r ] ) = − → Y 1 ≤ a ≤ N − 1 − → Y l a <ℓ ≤ r a F a ( t a ℓ ) ! = F 1 ( t 1 l 1 +1 ) · · · F 1 ( t 1 r 1 ) · · · F N − 1 ( t N − 1 r N − 1 ) (4.5) and ˆ F ( ¯ t [ ¯ l, ¯ r ] ) = ← − Y N − 1 ≥ a ≥ 1 ← − Y r a ≥ ℓ>l a ˆ F a ( t a ℓ ) ! = ˆ F N − 1 ( t N − 1 r N − 1 ) · · · ˆ F 1 ( t 1 r 1 ) · · · ˆ F 1 ( t 1 l 1 +1 ) , (4.6) where the series F a ( t ) ≡ F a +1 ,a ( t ) and ˆ F a ( t ) ≡ ˆ F a +1 ,a ( t ) are defined b y (3.9) and ( 3 .12), resp ectiv ely . As particular cases, w e ha v e F ( ¯ t a [ ¯ l a , ¯ r a ] ) = F a ( t a l a +1 ) F a ( t a l a +2 ) · · · F a ( t a r a ) and ˆ F ( ¯ t a [ ¯ l a , ¯ r a ] ) = ˆ F a ( t a r a ) · · · ˆ F a ( t a l a +2 ) ˆ F a ( t a l a +1 ). Sym b ols ← − Q a A a and − → Q a A a mean ordered pro ducts of noncomm utative en tries A a , suc h that A a is on the right (resp., o n the left) from A b for b > a : ← − Y j ≥ a ≥ i A a = A j A j − 1 · · · A i +1 A i , − → Y i ≤ a ≤ j A a = A i A i +1 · · · A j − 1 A j . According to [D K, E, EKP, KP1] the pro duct of the curren ts (4.5) is a formal series o v er the rat ios t b k /t c l with b > c and t a i /t a j with i > j taking v alues in the completions U F . Analogo usly , the pro duct of the curren ts (4.6) is a formal series ov er the ra t ios t b k /t c l with b < c and t a i /t a j with i < j taking v alues in the completion ˆ U F . It means t hat these pro ducts hav e the same ana lytical structure as the rational functions of the v ariables ¯ t [ ¯ l, ¯ r ] defined by the m ultisets I and ˆ I , resp ectiv ely . The 13 domains of analyticit y of the rat io nal functions defined b y the multis ets I and ˆ I are differen t. The pro ducts (4.5) and (4.6) ha v e p oles for some v alues of the ratios t b k /t c l and t a i /t a j . The o p erator v alued co efficien t at these p oles tak e v alues in the completions U F and ˆ U F and can b e iden tified with comp osed ro ot curren ts (see [KP1, K P]). In what follows w e will consider pro jections of the pro duct of the curren ts W N ( ¯ t [ ¯ n ] ) = P +  F 1 ( t 1 1 ) · · · F 1 ( t 1 n 1 ) · · · F N − 1 ( t N − 1 1 ) · · · F N − 1 ( t N − 1 n N − 1 )  (4.7) and ˆ W N ( ¯ t [ ¯ n ] ) = ˆ P +  ˆ F N − 1 ( t N − 1 n N − 1 ) · · · ˆ F N − 1 ( t N − 1 1 ) · · · ˆ F 1 ( t 1 n 1 ) · · · ˆ F 1 ( t 1 1 )  . (4 .8) It was prov ed in [KP1] tha t these pro jections can b e analytically con tinue d f r om there domains of definitions. This allows to compare the univ ersal w eigh t functions defined b y these pro jections. It w as conjectured in [KPT] and then prov ed in [KP] t ha t the unive rsal weigh t function can b e iden tified with the pro jection (4.8). A metho d of computation of this pro jection w as propo sed in [KP1] and it was further dev elop ed in [KP]. In this pap er w e will calculate the univ ersal weigh t function giv en by the pro jection (4 .7). F or an y w eigh t singular v ector v (2 .14) let the related w eigh t functions w N V ( ¯ t [ ¯ n ] ) = β ( ¯ t [ ¯ n ] ) W N ( ¯ t [ ¯ n ] ) N − 1 Y a =1 n a Y ℓ =1 k + a ( t a ℓ ) v (4.9) ˆ w N V ( ¯ t [ ¯ n ] ) = β ( ¯ t [ ¯ n ] ) ˆ W N ( ¯ t [ ¯ n ] ) N − 1 Y a =1 n a Y ℓ =1 ˆ k + a +1 ( t a ℓ ) v , (4.10) b e t he functions taking v alues in the U q ( b gl N )-mo dule V generated by a singular vec tor v . In [KPT, KP] they w ere called a mo dified w eigh t function or univ e rs al off-she l l Bethe ve ctor . W e will sho w that analytical contin uations of the univ ersal off-shell Bethe ve ctors defined by the unive rsal w eigh t functions (4.8) and (4.7) coincide: w N V ( ¯ t [ ¯ n ] ) = ˆ w N V ( ¯ t [ ¯ n ] ) . (4.11) This will follo w from the fact that ˆ w N V ( ¯ t [ ¯ n ] ) = B V ( ¯ t [ ¯ n ] ) for arbitrary U q ( b gl N )-mo dule V generated b y a singular v ector v [KP]. In this pap er w e will pro v e that w N V ( ¯ t [ ¯ n ] ) = B V ( ¯ t [ ¯ n ] ) , (4.12) th us yielding the equality (4.11). T o prov e (4.12) we use the same argumen ts as in [KP ]. W e calculate the pro jection (4.7), rewrite the corresp onding univ ersal off-shell Bethe v ector ( 4 .9) in terms of the ordered pro ducts of the matrix elemen ts of L-op erators acting onto singular v ector v and 14 sho w that this implie s a n expres sion (2.18) f o r this vec tor o n the ev aluation mo dules. This g ives a different formula than obtained in [KP] for off-shell Bethe v ectors. Using the result of [EKP] w e may c hec k that the univ ersal o ff - shell Bethe v ector (4 .9 ) defined b y the pro jection satisfies the same com ultiplication pro p erties as the v ector B V ( ¯ t [ ¯ n ] ) (see [KPT]). This means tha t equalit y (4.12) is v alid f or arbitrary tensor pr o duct of the ev aluat io n represen tations of U q ( b gl N ). Then the classical result [CP] implies tha t (4.12) is true for ev ery irreducible finite-dimensional U q ( b gl N )-mo dule V generated b y a singular v ector v . W e conclude this subsection b y describing our strategy of calculation of the pro jec- tions. W e will use the same approac h to calculate (4.8) that was use d in [KP] for t he calculation of (4.7). In that pap er w e separate all factors ˆ F a ( t a i ) with a < N − 1 in (4.8) and apply to this pro duct the ordering pro cedure based on the prop ert y (4.24). Here w e will separate the factors F a ( t a i ) with a > 1 in (4 .7). In b oth cases w e get un- der tot a l pro jection a symmetrization of a sum o f terms x i P − ( y i ) P + ( z i ) with rational co efficien ts. Here x i are expressed via mo des of ˆ F N − 1 ( t ), a nd y i , z i via mo des of ˆ F a ( t ) with a < N − 1 in the case of (4.8). In the case of (4.7), x i are expresse d via mo des of F 1 ( t ), and y i , z i via mo des of F a ( t ) with a > 1 . As w ell as in [KP] w e reorder x i and P − ( y i ) in b o th cases. At this stage comp osed curren ts of differen t types collected in so called strings app ear. 4.1 Symmetrization Consider p erm utation group S n and its action on the formal series of n v ariables defined on the eleme ntary transpositions σ i,i +1 as fo llo ws π ( σ i,i +1 ) G ( u 1 , . . . , u i , u i +1 , . . . , u n ) = q u i − q − 1 u i +1 q − 1 u i − q u i +1 G ( u 1 , . . . , u i +1 , u i , . . . , u n ) . The q -dep ending factor in this formula is chose n in suc h a w a y that the pro ducts F a ( u 1 ) · · · F a ( u n ) and ˆ F a ( u n ) · · · ˆ F a ( u 1 ) are inv ar ia n t under this action. Summing the action o v er all the group of p erm utations we obtain the op erat o r Sym u = P σ ∈ S n π ( σ ) acting as follo ws Sym u G ( u ) = X σ ∈ S n Y ℓ<ℓ ′ σ ( ℓ ) >σ ( ℓ ′ ) q t σ ( ℓ ′ ) − q − 1 t σ ( ℓ ) q − 1 t σ ( ℓ ′ ) − q t σ ( ℓ ) G ( σ u ) . (4.13) The pro duct is tak en ov er all pairs ( ℓ, ℓ ′ ), such that conditions ℓ < ℓ ′ and σ ( ℓ ) > σ ( ℓ ′ ) are satisfied simu ltaneously . The indices of the set of fo rmal v ariables u = ( u 1 , . . . , u n ) are from t w o ordered m ultisets: { 1 ≺ 2 ≺ · · · ≺ n } o r { n ≺ n − 1 ≺ · · · ≺ 1 } . The expansions of the rational functions entering the righ t hand side of (4.13) are defined b y these ordered multis ets. W e call op era t o r Sym u – q -symmetrization . This op eratio n differs from the ones used in the paper [TV2]. The exact relation b et w een them will b e g iv en b elo w. The op erator 1 n ! Sym u is the group a v erage with respect to the action π , whic h implies the form ula Sym u Sym u ( · ) = n !Sym u ( · ) . (4.14) 15 An imp ortan t prop ert y of q -symmetrization is the for mula Sym ( u 1 ,...,u n ) = X σ ∈ S ( s ) n π ( σ ) Sym ( u 1 ,...,u s ) Sym ( u s +1 ,...,u n ) , (4.15) where s ∈ [ n, 0 ] is fixed and the sum is tak en o v er t he subset S ( s ) n = { σ ∈ S n | σ (1) < . . . < σ ( s ) , σ ( s + 1) < . . . < σ ( n ) } . As in Subsection 2.3 we denote by S ¯ l, ¯ r = S l 1 ,r 1 × · · · × S l N − 1 ,r N − 1 a direct pr o duct of the groups S l a ,r a p erm uting in teger n um b ers l a + 1 , . . . , r a . The q -symmetrization o v er whole set of v ariables ¯ t [ ¯ l, ¯ r ] is defined b y the form ula Sym ¯ t [ ¯ l, ¯ r ] G ( ¯ t [ ¯ l, ¯ r ] ) = X σ ∈ S ¯ l, ¯ r Y 1 ≤ a ≤ N − 1 Y ℓ<ℓ ′ σ a ( ℓ ) >σ a ( ℓ ′ ) q t a σ a ( ℓ ′ ) − q − 1 t a σ a ( ℓ ) q − 1 t a σ a ( ℓ ′ ) − q t a σ a ( ℓ ) G ( σ ¯ t [ ¯ l, ¯ r ] ) , (4.16) where the set σ ¯ t [ ¯ l, ¯ r ] is defined in the same w a y as in (2.12). The q -symmetrization (4.16) is related to the q - symmetrization (2.13 ) : Sym ( q ) ¯ t [ ¯ l, ¯ r ] G ( ¯ t [ ¯ l, ¯ r ] ) = N − 1 Y a =1 Y l a <ℓ<ℓ ′ ≤ r a q − 1 t a ℓ − q t a ℓ ′ t a ℓ − t a ℓ ′ Sym ¯ t [ ¯ l, ¯ r ] G ( ¯ t [ ¯ r , ¯ l ] ) (4.17) As in [KP ] w e use the q -symmetrization (4.16) to compute the recurrence relat io n for the univ ersal w eigh t function (4.7) and will restore the q - symmetrization (2.13) when express the univers al off-shell Bethe v ector in terms of mat r ix elemen ts of L-op erators. W e sa y that the series Q ( ¯ t [ ¯ l, ¯ r ] ) is q -symmetric, if it is in v aria n t under the action π of eac h group S l a ,r a with resp ect to the p ermutations of the v ariables t a l a +1 , . . . , t r a for a = 1 , . . . , N − 1. The q -symmetrization of q -symmetric series is eq uiv a len t to the m ultiplication b y the order of the group S ¯ l, ¯ r : Sym ¯ t [ ¯ l, ¯ r ] Q ( ¯ t [ ¯ l, ¯ r ] ) = N − 1 Y a =1 ( r a − l a )! Q ( ¯ t [ ¯ l, ¯ r ] ) . (4.18) The q -symmetrization Q ( ¯ t [ ¯ l, ¯ r ] ) = Sym ¯ t [ ¯ l, ¯ r ] G ( ¯ t [ ¯ l, ¯ r ] ) of any series G ( ¯ t [ ¯ l, ¯ r ] ) is a q -symmetric series, whic h follo ws from (4.14). Let G sym ( u 1 , . . . , u n ) b e symmetric function of n v ariables u k , i.e. G sym ( σ ¯ u ) = G sym ( ¯ u ) for any elemen t σ fro m the symmetric group S n . Then one can chec k the follo wing pro p ert y of q -symmetrization: 1 n ! Sym ¯ u  β ( ¯ u ) − 1 G sym ( ¯ u )  = 1 [ n ] q ! Sym ¯ u ( G sym ( ¯ u )) , (4.19) β ( ¯ u ) = Y k l j + s a − 1 ˆ F j +1 ,a ( t j ℓ )   (4.35) for the string asso ciated with the currents ˆ F i ( t ). The string (4.3 5) also dep ends only on the v ariables of the t yp e j : { t j l j +1 , . . . , t j r j } , corresp onding to the segmen t [ r j , l j ]. The set o f nonnegative in tegers ¯ s satisfying the admissibilit y condition (4.34) divides the segmen t [ r j , l j ] in to j subsegmen ts [ l j + s a − 1 , l j + s a ] for a = 1 , . . . , j . This division defines the pro duct o f the comp osed curren ts in the string (4.35). Pro j ection of the string (4.32) will b e presen ted below. Pro jection of the string (4.35) w as calculated in [KP]. F or t w o sets of v ariable ¯ u and ¯ v w e define the series U ( u 1 , . . . , u k ; v 1 , . . . , v k ) = Y 1 ≤ i ≤ k v i /u i 1 − v i /u i Y 1 ≤ mr j − m a − 1 P +  F a,j ( t j ℓ ; t j ℓ +1 , . . . , t j r j )    × Y l j <ℓ<ℓ ′ ≤ r j q − 1 − q t j ℓ /t j ℓ ′ 1 − t j ℓ /t j ℓ ′ Y j +1 ≤ a ≤ N   Y r j − m a − 1 <ℓ<ℓ ′ ≤ r j − m a 1 − t j ℓ /t j ℓ ′ q − q − 1 t j ℓ /t j ℓ ′   , (4.58) 24 ˆ P +  ˆ S j ¯ s ( ¯ t j [ l j ,r j ] )  = − → Y 1 ≤ a ≤ j   − → Y l j + s a − 1 <ℓ ≤ l j + s a ˆ P +  ˆ F j +1 ,a ( t j ℓ ; t j l j +1 , . . . , t j ℓ − 1 )    × Y l j <ℓ<ℓ ′ ≤ r j q − q − 1 t j ℓ ′ /t j ℓ 1 − t j ℓ ′ /t j ℓ Y 1 ≤ a ≤ j   Y l j + s a − 1 <ℓ<ℓ ′ ≤ l j + s a 1 − t j ℓ ′ /t j ℓ q − 1 − q t j ℓ ′ /t j ℓ   . (4.59) Comm utation relations b et w een t o tal and Cartan currents together with the for- m ulas (4 .58), (4.5 9) imply the relations whic h a re necessary for the next section: P +  S j ¯ s ( ¯ t j [ l j ,r j ] )  r j Y ℓ = l j +1 k + j ( t j ℓ ) = Y j +1 ≤ a ≤ N   Y r j − m a − 1 <ℓ<ℓ ′ ≤ r j − m a 1 − t j ℓ /t j ℓ ′ q − q − 1 t j ℓ /t j ℓ ′   × × ← − Y N ≥ a ≥ j +1   ← − Y r j − m a ≥ ℓ>r j − m a − 1 P +  F a,j ( t j ℓ )  k + j ( t j ℓ )   , (4.60) ˆ P +  ˆ S j ¯ s ( ¯ t j [ l j ,r j ] )  r j Y ℓ = l j +1 ˆ k + j +1 ( t j ℓ ) = Y 1 ≤ a ≤ j   Y l j + s a − 1 <ℓ<ℓ ′ ≤ l j + s a 1 − t j ℓ ′ /t j ℓ q − 1 − q t j ℓ ′ /t j ℓ   × × − → Y 1 ≤ a ≤ j   − → Y l j + s a − 1 <ℓ ≤ l j + s a ˆ P +  ˆ F j +1 ,a ( t j ℓ )  ˆ k + j +1 ( t j ℓ )   . (4.61) 5 Univ ersal w eig h t functio ns and L -op e rators In this section w e use factorization formu las (4 .60) and (4.61) to presen t off-shell Bethe v ectors (4.10 ) and ( 4 .9) in terms of L-op erator’s en tries. First w e relate pro jections of the currents to the Gauss co ordinates of L-op erators. It can b e prov ed that for i < j P + ( F j,i ( t )) = ( q − 1 − q ) j − i − 1 F + j,i ( t ) , ˆ P +  ˆ F j,i ( t )  = ( q − q − 1 ) j − i − 1 ˆ F + j,i ( t ) . (5.1) F or j = i + 1 this is Ding-F renk el [DF] relations. F or i < j − 1, form ulas (5.1) fo llo w from the relations b etw een L-op erators ( q − 1 − q ) F + j,i ( t ) = S j − 1  F + j − 1 ,i ( t )  , ( q − q − 1 ) ˆ F + j,i ( t ) = ˆ S i  ˆ F + j,i +1 ( t )  written in terms of Gauss co ordinates (see details in [KPT]). Lemma 5.1 F or any c = 1 , . . . , N de note by I c and ˆ I c the left ide als of U q ( b b + ) , gener- ate d by the mo des of E + i,j ( u ) with i < j ≤ c and by the mo des o f ˆ E + i,j ( u ) with c ≤ i < j . We have inclusions 0 = I 1 ⊂ I 2 ⊂ · · · ⊂ I N and 0 = ˆ I N ⊂ ˆ I N − 1 ⊂ · · · ⊂ ˆ I 1 . 25 (i) Conside r a and b such that a < b , then w e have e q uali ties: L + a,b ( t ) ≡ F + b,a ( t ) k + a ( t ) mo d I c , L + a,a ( t ) ≡ k + a ( t ) mo d I c , a ≤ c , L + a,b ( t ) ≡ ˆ F + b,a ( t ) ˆ k + b ( t ) mo d ˆ I c , L + b,b ( t ) ≡ ˆ k + b ( t ) mo d ˆ I c , b ≥ c . (ii) The le f t id e al I c is gener ate d by mo des of L + j,i ( u ) with i < j ≤ c ; the left ide al ˆ I c is gener ate d by mo des of L + j,i ( u ) with c ≤ i < j , and I N = ˆ I 1 . (iii) F or any a ≥ c the m o des of L + c,a ( t ) normalize the ide al I c : I c · L + c,a ( t ) ⊂ I c . F or any a ≤ c the mo des of L + a,c ( t ) normalize the ide al ˆ I c : ˆ I c · L + a,c ( t ) ⊂ ˆ I c . Pr o of. The it ems (i) and (ii) are pro v ed using the Gauss decomp ositions (3.1)–(3.3) and (3.4)– ( 3 .6). Item (ii i ) follo ws from (ii) and RLL -relations (2.2).  Theorem 2 F or a n y U q ( b gl N ) mo dule V with a weight sin gular ve ctor v we have the fol lowing formulas for mo difie d weight functions w N V ( ¯ t [ ¯ n ] ) = β ( ¯ t [ ¯ n ] ) Sym ¯ t [ ¯ n ] X [[ ¯ m ]] ( q − 1 − q ) N − 1 P b =1 ( n b − m b b ) N − 1 Q a ≥ b ( m b a − m b a +1 )! Y [[ ¯ m ]] ( ¯ t [ ¯ n ] ) × − → Y 1 ≤ b ≤ N − 1 ← − Y N − 1 ≥ a ≥ b n b − m b a +1 Y ℓ = n b − m b a +1  L + b,a +1 ( t b ℓ ) ℓ − 1 Y ℓ ′ = n b − m b a +1 t b ℓ ′ − t b ℓ q − 1 t b ℓ ′ − q t b ℓ  ! n b − m b b Y ℓ =1 L + b,b ( t b ℓ ) ! v (5.2) and ˆ w N V ( ¯ t [ ¯ n ] ) = β ( ¯ t [ ¯ n ] ) Sym ¯ t [ ¯ n ] X [[ ¯ s ]]      ( q − q − 1 ) N − 1 P b =1 ( n b − s b b ) N − 1 Q a ≤ b ( s b a − s b a − 1 )! X [[ ¯ s ]] ( ¯ t [ ¯ n ] ) × ← − Y N − 1 ≥ b ≥ 1 − → Y 1 ≤ a ≤ b s b a Y ℓ = s b a − 1 +1  L + a,b +1 ( t b ℓ ) s b a Y ℓ ′ = ℓ +1 t b ℓ − t b ℓ ′ q − 1 t b ℓ − q t b ℓ ′  ! n b Y ℓ = s b b +1 L + b +1 ,b +1 ( t b ℓ )   v (5.3) (the or dering in the pr o ducts over ℓ is not imp ortant, b e c ause of c om m utativity of the entries of L -op er ators with e qual matrix in dic es ) . 26 Pr o of. L et V b e U q ( b gl N )-mo dule with a w eight singular v ector v . Using the rela- tions (4.60), (4.61), (5 .1 ) and the corolla ry 4.4 w e can write mo dified w eigh t functions w N V ( ¯ t [ ¯ n ] ) and ˆ w N V ( ¯ t [ ¯ n ] ) in the following form: w N V ( ¯ t [ ¯ n ] ) = β ( ¯ t [ ¯ n ] ) Sym ¯ t [ ¯ n ] X [[ ¯ m ]]      ( q − 1 − q ) P N − 1 b =1 ( n b − m b b ) N − 1 Q a ≥ b ( m b a − m b a +1 )! Y [[ ¯ m ]] ( ¯ t [ ¯ n ] ) − → Y 1 ≤ b ≤ N − 1 ← − Y N − 1 ≥ a ≥ b × ← − Y n b − m b a +1 ≥ ℓ>n b − m b a   F + a +1 ,b ( t b ℓ ) k + b ( t b ℓ ) ℓ − 1 Y ℓ ′ = n b − m b a +1 t b ℓ ′ − t b ℓ q − 1 t b ℓ ′ − q t b ℓ     n b − m b b Y ℓ =1 k + b ( t b ℓ )   v (5.4) and ˆ w N V ( ¯ t [ ¯ n ] ) = β ( ¯ t [ ¯ n ] ) Sym ¯ t [ ¯ n ] X [[ ¯ s ]]      ( q − q − 1 ) P N − 1 b =1 ( n b − s b b ) N − 1 Q a ≤ b ( s b a − s b a − 1 )! X [[ ¯ s ]] ( ¯ t [ ¯ n ] ) ← − Y N − 1 ≥ b ≥ 1 − → Y 1 ≤ a ≤ b × − → Y s b a − 1 <ℓ ≤ s b a   ˆ F + b +1 ,a ( t b ℓ ) ˆ k + b +1 ( t b ℓ ) s b a Y ℓ ′ = ℓ +1 t b ℓ − t b ℓ ′ q − 1 t b ℓ − q t b ℓ ′     n b Y ℓ = s b b +1 ˆ k + b +1 ( t b ℓ )   v , (5.5) where series Y [[ ¯ s ]] ( ¯ t [ ¯ n ] ) and X [[ ¯ s ]] ( ¯ t [ ¯ n ] ) a re given by (4.4 9 ) a nd (4 .50) respectiv ely . Then the statemen t of the Theorem follows from (5.4), (5 .5) and Lemma 5.1 in the same w a y as in the pap er [KP].  As in [KP] we can simplify for mulas (5.2) and (5.3) by using the q - symmetrization (2.13) (see [TV2]). Denoting ˜ s b a = n a − s b a = s a a + · · · + s b − 1 a w e form ulate the fo llo wing corollary of the Theorem 2 Corollary 5.2 The o ff -shel l B ethe ve ctors for the quantum a ffi n e algebr a U q ( b gl N ) c an b e written as w N V ( ¯ t [ ¯ n ] ) = Sym ( q ) ¯ t [ ¯ n ] X [[ ¯ m ]] ( q − 1 − q ) N − 1 P a =1 ( n a − m a a ) N − 1 Y a ≤ b 1 [ m a b − m a b +1 ] q ! × × N − 1 Y a =2 a − 1 Y b =1 m b a − 1 Y ℓ =0 t a m b a − ℓ /t a − 1 m b a − 1 − ℓ 1 − t a m b a − ℓ /t a − 1 m b a − 1 − ℓ n a − 1 Y ℓ ′ = m b a − 1 − ℓ +1 q − 1 − q t a m b a − ℓ /t a − 1 ℓ ′ 1 − t a m b a − ℓ /t a − 1 ℓ ′ × − → Y 1 ≤ a ≤ N − 1   ← − Y N − 1 ≥ b ≥ a n a − m a b +1 Y ℓ = n a − m a b +1 L + a,b +1 ( t a ℓ )   n a − m a a Y ℓ =1 L + a,a ( t a ℓ )   v (5.6) 27 and ˆ w N V ( ¯ t [ ¯ n ] ) = Sym ( q ) ¯ t [ ¯ n ] X [[ ¯ s ]] ( q − q − 1 ) N − 1 P b =1 ( n b − s b b ) N − 1 Y a ≤ b 1 [ s b a − s b a − 1 ] q ! × × N − 1 Y b =2 b − 1 Y a =1 s b a Y ℓ =1 1 1 − t a ℓ + ˜ s b a /t a +1 ℓ + ˜ s b a +1 ℓ + ˜ s b a +1 − 1 Y ℓ ′ =1 q − q − 1 t a ℓ + ˜ s b a /t a +1 ℓ ′ 1 − t a ℓ + ˜ s b a /t a +1 ℓ ′ × ← − Y N − 1 ≥ b ≥ 1   − → Y 1 ≤ a ≤ b   s b a Y ℓ = s b a − 1 +1 L + a,b +1 ( t b ℓ )   n b Y ℓ = s b b +1 L + b +1 ,b +1 ( t b ℓ )     v . (5.7) Pr o of. W e prov e this corollary only for (5.6), since (5.7) can b e pro v ed in the same w a y . Consider righ t hand side of formula (5.2). Inside the tota l q -symmetrization there is a symmetric se ries in the sets of v a riables { t b ℓ } f or n b − m b a + 1 ≤ ℓ ≤ n b − m b a +1 for b = 1 , . . . , N − 1 and a = b, . . . , N − 1. It fo llo ws from the comm uta tivit y of matrix elemen t s of L - op erators [L + a,b ( t ) , L + a,b ( t ′ )] = 0 and the explicit form of the series (4.4 9). Pro duct ℓ − 1 Q ℓ ′ = n b − m b a +1 t b ℓ ′ − t b ℓ q − 1 t b ℓ ′ − q t b ℓ is an in v erse to the function β ( t b [ n b − m b a ,n b − m b a +1 ] ) defined in (4.20). Conseque ntly a pplying formulas ( 4 .19), (4.17) a nd using the explicit form of the series (4.49) we obtain (5 .6).  6 Connec tion b et w een tw o w e igh t funct i ons Let us calculate the image of the off-shell Bethe vec tor (5.6 ) for the ev aluation ho- momorphism (2.6). Let M Λ b e a U q ( gl N )-mo dule generated b y a v ector v , satisfying the conditions E a,a v = q Λ a v and E a,b v = 0 for a < b . Then v is a singular w eigh t v ector of the U q ( b gl N ) ev aluation mo dule M λ ( z ). The action of the matrix elemen ts of L-op erato r s in this mo dule is giv en b y the form ulas E v z  L + a,b +1 ( t )  = ( q − q − 1 ) E b +1 ,a E b +1 ,b +1 ≡ ( q − q − 1 ) ˇ E b +1 ,a , E v z  L + a,a ( t )  v =  q Λ a − q − Λ a z t  v = λ a ( t ) v . (6.1) Prop osition 6.1 F or any eva luation U q ( b gl N ) m o dule M λ ( z ) with singular w eight v e c- tor v we have w M λ ( z ) ( ¯ t [ ¯ n ] ) = B M λ ( z ) ( ¯ t [ ¯ n ] ) . Pr o of. Substituting (6.1) in to (5.6) and using reordering of the factors N − 1 Y a =2 n a − m a a Y ℓ =1 λ a ( t a ℓ ) = N − 1 Y a =2 a − 1 Y b =1 m b a − 1 Y ℓ =0 λ a ( t a m b a − ℓ ) (6.2) 28 w e obtain w N M λ ( z ) ( ¯ t [ ¯ n ] ) = ( q − q − 1 ) N − 1 P a =1 n a X [[ ¯ m ]]     − → Y 1 ≤ b ≤ N − 1 ← − Y N − 1 ≥ a ≥ b ˇ E m b a − m b a +1 a +1 ,b [ m b a − m b a +1 ] q !   v × Sym ( q ) ¯ t [ ¯ n ] N − 1 Y a =2 a − 1 Y b =1 m b a − 1 Y ℓ =0 q Λ a t a m b a − ℓ − q − Λ a z t a − 1 m b a − 1 − ℓ − t a m b a − ℓ n a − 1 Y ℓ ′ = m b a − 1 − ℓ +1 q t a m b a − ℓ − q − 1 t a − 1 ℓ ′ t a m b a − ℓ − t a − 1 ℓ ′   . (6.3) Reordering generators ˇ E a +1 ,b using Serre relations (2.5) w e literally obtain equa- tion (2.18).  Denote b y J the left ideal of U q ( b gl N ) defined in t he item (ii) of the Lemma 5.1: J = I N = ˆ I 1 . This ideal is g enerated by the mo des of the en tries of the L- op erator L + j,i ( t ) with 1 ≤ i < j ≤ N . W e use the statemen t prov ed in the Theorem 3 of [KP], whic h w e form ulate as the fo llo wing lemma. Lemma 6.2 Conside r U q ( b b + ) as an alge br a over C [[( q − 1)]] , and two el e ments A , B ∈ U q ( b b + ) . If for any singular weight ve c tor v we have that A v = B v , then A ≡ B mo d J. Theorem 3 Universal weight functions ar e subje ct to the fol l o wing r elations: (i) F or e ach irr e ducible finite-dimensional U q ( b gl N ) -mo d ule V with a singular ve ctor v two weigh t functions ar e e qual: w N V ( ¯ t [ ¯ n ] ) = B V ( ¯ t [ ¯ n ] ) . (ii) w N V ( ¯ t [ ¯ n ] ) = ˆ w N V ( ¯ t [ ¯ n ] ) . (iii) Consid er U q ( b b + ) as an algebr a over C [[( q − 1 ) ]] , then β ( ¯ t [ ¯ n ] ) W N ( ¯ t [ ¯ n ] ) N − 1 Y a =1 n a Y ℓ =1 k + a ( t a ℓ ) ≡ B V ( ¯ t [ ¯ n ] ) mo d J. (iv) W N ( ¯ t [ ¯ n ] ) N − 1 Y a =1 n a Y ℓ =1 k + a ( t a ℓ ) ≡ ˆ W N ( ¯ t [ ¯ n ] ) N − 1 Y a =1 n a Y ℓ =1 ˆ k + a +1 ( t a ℓ ) mo d J. Pr o of. ( i ) Since b oth w N V ( ¯ t [ ¯ n ] ) a nd B V ( ¯ t [ ¯ n ] ) hav e the same comultiplication prop erties (cf. [KPT]), it follo ws from the Prop osition 6.1 that for an y set of ev a luation U q ( b gl N ) mo dules M λ i ( z i ) w N 1 g ( z 0 ) ⊗ M λ 1 ( z 1 ) ⊗···⊗ M λ k ( z k ) ( ¯ t [ ¯ n ] ) = B 1 g ( z 0 ) ⊗ M λ 1 ( z 1 ) ⊗···⊗ M λ k ( z k ) ( ¯ t [ ¯ n ] ) 29 for an y tensor pro duct of ev aluat io n mo dules and a one-dimensional mo dule 1 g ( z 0 ) , in whic h ev ery L ii ( z ) a cts by multiplication on g ( z 0 ) (b oth w eigh t functions in consid- eration are trivial for one-dimensional mo dules). Then we can apply classical result of [CP]: ev ery irreducible finite-dimensional U q ( b gl N )-mo dule with a singular v ector v is isomorphic to a sub quotien t of a tensor pro duct o f one-dimensional mo dules and of ev aluation mo dules, generated b y the tensor pro duct o f their w eigh t singular v ectors. The singular v ector corresp onds to the image t ensor pro duct of singular v ectors within this isomorphism. (ii) It follo ws from (i) and the fact pro v ed in [KP] t ha t ˆ w N V ( ¯ t [ ¯ n ] ) = B V ( ¯ t [ ¯ n ] ) . (iii) It follo ws from (i) and Lemma 6.2. (iv) It follo ws from (ii) and Lemma 6.2.  Ac kno wledge ment The authors t hank S.Khoro shkin fo r nu merous useful discussions. This w ork was par- tially done when the second author (S.P .) visited Lab oratoire d’Annecy-Le-Vieux de Ph ysique Th ´ eorique in 200 6 and 2007 . These visits w ere p ossible due to the financial supp ort o f the CNRS-Russia exc hange program on mathematical phys ics. He t ha nks LAPTH for the hospitality and stim ula t ing scien tific atmosphere. His work w as sup- p orted in part b y RFBR gr an t 05-01- 0 1086 and gra nt f o r sup p o r t of scien tific sc ho ols NSh-8065.200 6 .2. This pap er is a part of PhD thesis of A.S. whic h he is prepared in co-direction o f S.P . a nd V.R. in the Bogoliub o v Lab oratory of Theoretical Ph ysics, JINR, Dubna and in LAREMA, D ´ epartemen t de Math´ ematics, Univ ersit ´ e d’Angers. He is grateful to the CNRS-Russia exc hange progra m on mathematical ph ysics a nd p ersonally to J.-M. Maillet for financial and general supp ort of this thesis pro ject. A Analytical prop erties of comp o sed curren ts In this app endix w e refo r mulate the Serre relations in terms of the comp osed curren t s. W e start from the follow ing prop erties of the ‘regular ized’ pro ducts of usual total curren ts: (i) B 1 ( z , w ) = ( q − 1 z − q w ) F i ( z ) F i ( w ) v a nish at z = w : B 1 ( z , z ) = 0; (ii) B 2 ( z 1 , z 2 , z 3 ) = ( z 1 − z 2 )( q z 2 − q − 1 z 3 )( q − 1 z 1 − q z 3 ) F i ( z 1 ) F i +1 ( z 2 ) F i ( z 3 ) v anish at z 1 = z 2 = q − 2 z 3 : B 2 ( z , z , q 2 z ) = 0; (iii) B 3 ( z 1 , z 2 , z 3 ) = ( q z 1 − q − 1 z 2 )( z 2 − z 3 )( q − 1 z 1 − q z 3 ) F i +1 ( z 1 ) F i ( z 2 ) F i +1 ( z 3 ) v anish at z 1 = z 2 = q 2 z 3 : B 3 ( z , z , q − 2 z ) = 0. The prop erty (i) follo ws f rom the comm uta tion relations (3.10) written as B 1 ( z , w ) = − B 1 ( w , z ) . 30 The prop erties ( ii) and (iii) can b e obtained from the Serre relations (3.11) and delta- function iden tities [E]. Note that t he an tisymmetry o f B 1 ( z , w ) implies B 2 ( z , w , u ) = − B 2 ( u, w , z ) and B 3 ( z , w , u ) = − B 3 ( u, w , z ). Let us demonstrate ho w o ne can deriv e the comm utation relation b etw een F i ( z ) and the composed curren t F i +2 ,i ( w ) defined as F i +2 ,i ( w ) = u − 1 ( u − w ) F i ( u ) F i +1 ( w )    u = w = ( q − 1 − q ) F i +1 ( w ) F i ( w ). Using B 2 ( z , w , u ) = − B 2 ( u, w , z ) we ha v e 4 F i ( z ) F i +2 ,i ( w ) = w − 1 z − 1 (1 − w /z ) q z − 1 (1 − q 2 w /z ) B 2 ( w , w , z ) . On the other hand q − 1 − q z / w 1 − z /w F i +2 ,i ( w ) F i ( z ) = w − 1 q − 1 w − 1 (1 − q − 2 z /w ) w − 1 (1 − z /w ) B 2 ( w , w , z ) . No w, t he prop ert y (ii) means q z − 1 1 − q 2 w /z B 2 ( w , w , z ) = − q − 1 w − 1 1 − q − 2 z /w B 2 ( w , w , z ) . The equalit y B 2 ( z , z , z ) = 0, which follows from ( i) , means z − 1 1 − w /z B 2 ( w , w , z ) = − w − 1 1 − z /w B 2 ( w , w , z ) . This prov es the comm utatio n relation: F i +1 ,i ( z ) F i +2 ,i ( w ) = q − 1 − q z /w 1 − z /w F i +2 ,i ( w ) F i +1 ,i ( z ) . (A.1) It follo ws f r om this comm utation relatio n, the commu tatio n relation b et w een F i +1 ,i ( z ) and F i,i − 1 ( w ) and the definition F i +2 ,i − 1 ( w ) = ( q − 1 − q ) F i +2 ,i ( w ) F i,i − 1 ( w ) that F i +1 ,i ( z ) F i +2 ,i − 1 ( w ) = F i +2 ,i − 1 ( w ) F i +1 ,i ( z ) . (A.2) Generalizing relatio ns (A.1), (A.2) we obtain: Prop osition A.1 F or an y i > j > k > l we ha v e the fo l lowing c ommutation r elations F j k ( z ) F ik ( w ) = q − 1 − q z / w 1 − z /w F ik ( w ) F j k ( z ) , (A.3) F ik ( z ) F ij ( w ) = q − 1 − q z / w 1 − z /w F ij ( w ) F ik ( z ) , (A.4) F j k ( z ) F il ( w ) = F il ( w ) F j k ( z ) , (A.5) q − 1 − q w /z 1 − w /z F ij ( z ) F ij ( w ) = q − 1 − q z / w 1 − z /w F ij ( w ) F ij ( z ) . (A.6) 4 Rational function 1 1 − x should b e a lwa ys unders to o d a s formal series P n ≥ 0 x n . 31 B Pro of of the Lemma 4 .3 Before prov ing this lemma w e form ulate sev eral preliminary propositions. F or any j = 1 , ..., N − 1 denote b y U j the subalgebra of U f formed b y the modes of F j ( t ) , . . . , F N − 1 ( t ). Let U ε j = U j ∩ K er ε be the correspo nding augmen tation ideal. Let ¯ m j = { m j , m j +1 , . . . , m N − 1 } b e a collection of non-negative in tegers satisfying admissibilit y conditions: m j ≥ m j +1 ≥ · · · ≥ m N − 1 ≥ m N = 0 . Se t ¯ m j +1 = { m j +1 , . . . , m N − 1 } . Prop osition B.1 F or the pr o duct F ( ¯ t j [ m j ] ) and string S j +1 ¯ m j +1 ( ¯ t j +1 [ m j +1 ] ) we hav e a formal series e quality F ( ¯ t j [ m j ] ) · P −  S j +1 ¯ m j +1 ( ¯ t j +1 [ m j +1 ] )  = N − 1 Y i = j 1 ( m i − m i +1 )! × Sym t j [ m j ]  S j ¯ m j ( ¯ t j [ m j ] ) N − 1 Y a = j + 1 Sym ¯ t j +1 [ m j +1 − m a ,m j +1 − m a +1 ] U ( t j m j − m j +1 +1 , . . . , t j m j ; t j +1 1 , . . . , t j +1 m j +1 )  mo d P −  U ε j +1  · U j , (B.1) wher e a symb ol N − 1 Q a = j + 1 Sym ¯ t j +1 [ m j +1 − m a ,m j +1 − m a +1 ] is a c omp osition of the c orr esp onding q- symmetrizations: Sym ¯ t j +1 [0 ,m j +1 − m j +2 ] · · · Sym ¯ t j +1 [ m j +1 − m N − 1 ,m j +1 ] . If admi s s ibility c ondi- tions m j ≥ m j +1 ≥ · · · ≥ m N − 1 ≥ 0 ar e not s a tisfie d then the right hand sid e of (B.1) is zer o mo dulo P −  U ε j +1  · U j . Pr o of. The Prop osition B.1 can be pro v ed along the same lines as it w as done in the App endix C of the pa p er [KP]. T o p erfo rm this pro of the reader should use fo rm ulas for the comm utations of the comp osed curren ts gathered in the App endix A of this pap er.  Lemma B.2 R ationa l series Sym ¯ t j +2 [ m j +2 ] · · · Sym ¯ t N − 2 [ m N − 2 ] Sym ¯ t N − 1 [ m N − 1 ] Y ( ¯ t ¯ m j +1 ) is sym- metric in e ach gr oup of variab l e s { t j +1 m j +1 − m a +1 , . . . , t j +1 m j +1 − m a +1 } fo r a = j + 1 , . . . , N − 1 . Pr o of. This L emma can b e prov ed b y induction. The case of j = N − 3 follow s fro m the fact prov ed in [KP1] that the q - symmetrization of the formal series (4.36) Sym ¯ v U ( ¯ u ; ¯ v ) is the symmetric series with resp ect to the set o f v ar iables ¯ u . F o r j < N − 3 induction follo ws from the same fact, prop ert y of q - symmetrization (4.15) and the form ula U ( u 1 , . . . , u k ; v 1 , . . . , v k ) = Σ( u s +1 , . . . , u k ; v 1 , . . . , v s ) × U ( u 1 , . . . , u s ; v 1 , . . . , v s ) U ( u s +1 , . . . , u k ; v s +1 , . . . , v k ) , where 0 ≤ s ≤ k and Σ( u s +1 , . . . , u k ; v 1 , . . . , v s ) is a symmetric function with resp ect to eac h set of v ariables { u s +1 , . . . , u k } and { v 1 , . . . , v s } .  32 Prop osition B.3 T her e is a formal series e quality F ( ¯ t j [ m j ] ) · Sym ¯ t [ ¯ m j +1 ]  Y ( ¯ t [ ¯ m j +1 ] ) · P −  S j +1 ¯ m j +1 ( ¯ t j +1 [ m j +1 ] )  = = 1 ( m j − m j +1 )! Sym ¯ t [ ¯ m j ]  Y ( ¯ t [ ¯ m j ] ) · S j ¯ m j ( ¯ t j [ m j ] )  mo d P −  U ε j +1  · U j (B.2) is admissibility c onditions m j ≥ m j +1 ≥ · · · ≥ m N − 1 ≥ 0 ar e sa tisfi e d; otherwise the right hand side of the e quality (B.2) is zer o mo dulo P −  U ε j +1  · U j . Pr o of. Lemma B.2 implies the follo wing relation Sym ¯ t j +1 [ m j +1 ]  Sym ¯ t j +2 [ m j +2 ] · · · Sym ¯ t N − 1 [ m N − 1 ] Y ( ¯ t ¯ m j +1 ) × × N − 1 Y a = j + 1 Sym ¯ t j +1 [ m j +1 − m a ,m j +1 − m a +1 ] U  ¯ t j [ m j ,m j − m j +1 ] ; ¯ t j +1 [ m j +1 ]   = (B.3) = N − 1 Y a = j + 1 ( m a − m a +1 )! Sym ¯ t [ ¯ m j +1 ] Y ( ¯ t ¯ m j ) . The q -symmetrization Sym ¯ t [ ¯ m j +1 ] in the left hand side of (B.2) do not affect the formal series dep ending on the v ariables ¯ t j [ m j ] . This means that the left hand side of ( B.2 ) can b e written in the form Sym ¯ t [ ¯ m j +1 ]  Y ( ¯ t [ ¯ m j +1 ] ) · F ( ¯ t j [ m j ] ) · P −  S j +1 ¯ m j +1 ( ¯ t j +1 [ m j +1 ] )   . Then, substituting (B.1) in to this expression, and using ( B.3 ) w e obtain the righ t hand side of (B.2).  Prop osition B.4 T he pr oje ction P −  F ( ¯ t [ ¯ m 2 ] )  c an b e pr esente d as N − 1 Y a =2 1 ( m a − m a +1 )! Sym ¯ t [ ¯ m 2 ]  Y ( ¯ t [ ¯ m 2 ] ) P −  S 2 ¯ m 2 ( ¯ t 2 [ m 2 ] )   mo d P − ( U ε 3 ) · U 2 (B.4) if admissibility c onditions m 2 ≥ m 3 ≥ · · · ≥ m N − 2 ≥ m N − 1 ≥ m N = 0 ar e satisfie d and is zer o mo dulo P − ( U ε 3 ) · U 2 otherwise. Pr o of. The statemen t o f the Prop osition B.4 is a part icular case of the following formal series equality P −  F ( ¯ t [ ¯ m 2 ] )  = N − 1 Y a = j + 1 1 ( m a − m a +1 )! P −  F ( ¯ t 2 [ m 2 ] ) · · · F ( ¯ t j − 1 [ m j − 1 ] ) × (B.5) F ( ¯ t j [ m j ] ) Sym ¯ t [ ¯ m j +1 ]  Y ( ¯ t [ ¯ m j +1 ] ) S j +1 ¯ m j +1 ( ¯ t j +1 [ m j +1 ] )   mo d P −  U ε j +2  · U 2 . W e pro v e (B.5) by induction. Indeed it is correct f or j = N − 1. Supp ose that it is v alid for some j ≤ N − 1. Using the prop erty (3.19) a nd taking in to account (3 .1 8) one can replace t he string S j +1 ¯ m j +1 ( ¯ t j +1 [ m j +1 ] ) in the righ t hand side of (B.5 ) b y its negativ e pro jection 33 P −  S j +1 ¯ m j +1 ( ¯ t j +1 [ m j +1 ] )  . F rom the Prop osition B.3 w e obta in t he righ t hand side of (B.5) for j − 1 , using also the fa cts that P −  F ( ¯ t 2 [ m 2 ] ) · · · F ( ¯ t j − 1 [ m j − 1 ] ) P − ( U ε j +1 ) · U j  ⊂ P − ( U ε j +1 ) · U 2 and P − ( U ε j +2 ) · U 2 ⊂ P − ( U ε j +1 ) · U 2 . If the a dmissibilit y condition is not satisfied for some j ≥ 2 , namely , m j < m j +1 , then according to the Prop osition B.3 the pro duct F ( ¯ t j [ m j ] ) Sym ¯ t [ ¯ m j +1 ]  Y ( ¯ t [ ¯ m j +1 ] ) S j +1 ¯ m j +1 ( ¯ t j +1 [ m j +1 ] )  is zero mo dulo elemen ts from P − ( U ε j +1 ) · U j . Due to the comm utativit y of the mo des of the curren ts F k [ n ], k = 2 , . . . , j − 1 with an y elemen t fr o m U ε j +1 w e conclude t ha t in this case the right hand side of (B.4) is zero mo dulo P − ( U ε j +1 ) · U 2 ⊂ P − ( U ε 3 ) · U 2 .  Pr o of of the L emma 4.3 . F o r the calculation of pro jection P +  F ( ¯ t 1 [ n 1 ] ) P −  F ( ¯ t [ ¯ m 2 ] )   w e substitute P −  F ( ¯ t [ ¯ m 2 ] )  using (B.4). Since any elemen t of U ε 3 comm utes with the curren t F 1 ( t ), elemen ts of P − ( U ε 3 ) · U 2 do no t con tribute: P +  F ( ¯ t 1 [ n 1 ] ) · P − ( U ε 3 ) · U 2  = 0. Then w e ha v e N − 1 Y a =2 1 ( m a − m a +1 )! P +  F ( ¯ t 1 [ n 1 ] ) Sym ¯ t [ ¯ m 2 ]  Y ( ¯ t [ ¯ m 2 ] ) P −  S 2 ¯ m 2 ( ¯ t 2 [ m 2 ] )   , where m N = 0. The latter express ion is non-zero due to the Pro p osition B.3 only if m 2 ≤ n 1 and is equal t o the righ t hand side of (4.42).  References [BK] Brundan, J., Kleshc hev, A. 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