Gauss coordinates vs currents for the Yangian doubles of the classical types

We consider relations between Gauss coordinates of $T$-operators for the Yangian doubles of the classical types corresponding to the algebras $\mathfrak{g}$ of $A$, $B$, $C$ and $D$ series and the current generators of these algebras. These relations are important for the applications in the $\mathfrak{g}$-invariant integrable models and construction of the Bethe vectors in these models.


Introduction
Hierarchical algebraic Bethe ansatz is a powerful tool for investigation of the quantum integrable models associated with g-invariant R-matrices. Since pioneering papers [1,2] this method was mostly developed for the case when g belongs to the type A algebras and their supersymmetric generalizations.
To deal with hierarchical algebraic Bethe ansatz one can identify the monodromy matrix elements of some g-or U q (g)-invariant quantum integrable model with the generating series of the elements of the Yangian doubles or quantum affine algebras [3]. These generating series gathered into T -operators satisfy the same commutation relations defined by some g-or U q (g)invariant R-matrix as monodromy matrix elements of the corresponding quantum integrable model do. These realizations of the Yangian doubles or quantum affine algebras are known as RT T -realization [4]. The same algebras have so called 'current' or 'new' realization [5].
For the type A algebras an explicit relations between generating series in the RT T -formulation and the currents in the 'new' realization were found in [6]. Recent results published in the papers [7][8][9][10] describes similar equivalences for the other type algebras.
A significant development of hierarchical algebraic Bethe ansatz was achieved in the papers [11][12][13] when the main objects of this method, so called Bethe vectors, were constructed in terms of the current generators of the Yangian doubles and quantum affine algebras. This construction uses the projections onto intersections of the different types Borel subalgebras in these infinite dimensional algebras corresponding to their RT T and 'current' realizations.
To prove isomorphism between different formulations of the Yangian doubles and quantum affine algebras it is sufficient to identify the generating series in both pictures corresponding to the simple roots of g as it was done in the papers [6][7][8][9][10]. This identification uses so called Gauss coordinates of T -operators associated with the simple roots of g. To use method of projections in hierarchical algebraic Bethe ansatz one has to express all Gauss coordinates through current generators. This can be realized through the same projection method and allowed in [12] to obtain the explicit formulas for the off-shell Bethe vectors in terms of the monodromy matrix elements for U q (gl N )-invariant integrable models. Analogous results were obtained in [13] for the supersymmetric gl(m|n))-invariant models.
The goal of the present paper is to establish the relations between all Gauss coordinates of T -operators and the currents of the Yangian double for all types classical algebras. It is convenient to use positive and negative integers to index matrix elements of operators from End(C N ) for different algebras. We introduce the sets of integers I A = 1, 2, . . . , N ; I B = −n, −n + 1, . . . , −1, 0, 1, 2, . . . , n; I C = I D = −n + 1, . . . , −1, 0, 1, 2, . . . , n. We will use notation I g to describe all sets of indices simultaneously.

g-invariant R-matrix
Let R(u, v) be g-invariant R-matrix [7,15] where I = N i=1 e i,i is the identity operator acting in the space C N and e i,j ∈ End(C N ) are N × N matrices with the only nonzero entry equals to 1 at the intersection of the i-th row and j-th column. The operators P and Q act in C N ⊗ C N such that (2.3) Operators P and Q satisfy the properties and parameter κ in definition of g-invariant R-matrix (2.1) is equal to (2.5) Value of κ = ∞ in case gl N means that one should drop the term with Q-operator in expression for gl N -invariant R-matrix. In (2.1) nonzero c ∈ C is a parameter and u, v ∈ C are so called spectral parameters. Using properties (2.4) one can check that R-matrix (2.1) satisfies the Yang-Baxter equation and unitarity condition The Yangian double DY (g) is an associative algebra formed by the elements T i,j [ℓ], ℓ ∈ Z and i, j ∈ I g . These elements can be gathered into generating series which become the entries of the T -operators satisfying the commutation relations 2 where µ, ν = ±. Sometimes we will call T -operators monodromy matrices remembering that they become quantum monodromies of some integrable model if T i,j [ℓ] are the operators acting in the Hilbert space of the corresponding physical model. Equation (2.8) yields the commutation relations of the monodromy matrix entries (2.9) 2 We consider the Yangian double without central extension.
with ǫ i defined in (2.2). Let us remind that the second line in (2.9) will be missing in case of the Yangian double DY (gl N ). For any matrix M acting in C N we denote by M t the transposition It is related to the 'usual' transposition (·) t by a conjugation by the matrix satisfy the same RT T commutation relations (2.9). This fact is easy to check for the Yangian double DY (gl N ) [14]. Existence of the inverse T -operators T ± (u) −1 follows from (2.7). The commutation relations for the Yangian doubles DY (o 2n+1 ), DY (sp 2n ) and DY (o 2n ) (2.9), imply the relations [7,8] T where z ± (u) are central elements. The RT T algebras given by the commutation relations (2.9) with central elements z ± (u) was denoted in [7] as DX(g). Yangian doubles DY (o 2n+1 ), DY (sp 2n ) and DY (o 2n ) which we will consider in this paper are isomorphic to the quotients of DX(g) by the ideal generated by the modes of the series z ± (u). This means that we can set z ± (u) = 1 and the equality (2.11) can be written in the form (2.12) It proves that a mapping is an automorphism of the RT T -algebra (2.8). Some property of this automorphism is described in the appendix B. The generating matrices T + (u) and T − (u) form two Borel subalgebras in the algebras DY (gl N ), DY (o 2n+1 ), DY (sp 2n ) and DY (o 2n ). We denote each of these standard Borel subalgebras as A ± , B ± , C ± and D ± , respectively. Any of the Yangian doubles DY (gl N ), DY (o 2n+1 ), DY (sp 2n ) and DY (o 2n ) can be constructed by the quantum double construction starting from one of its subalgebras [3]. We denote these doubles as DA, DB, DC and DD. Quantum double construction in these cases uses the Hopf structure which shows that T + (u) and T − (u) generate also Hopf subalgebras in DY (g).

Gauss coordinates and the currents formulation of DY (g)
As it is well known now, the infinite-dimensional algebras which possess the RT T formulation can be reformulated in terms of 'new' or 'current' realizations [5]. An isomorphism between different realizations of the Yangian doubles and quantum affine algebras was established first for the type A algebras. The main ingredients of this construction were the Gauss coordinates of T -operators. Recently, it was discovered in the papers [7][8][9][10] that the same mechanism allows to establish the corresponding isomorphisms between RT T and 'current' realizations of the Yangian doubles and quantum affine algebras for B, C and D series. In this section we describe these isomorphisms for each of the algebras DY (gl N ), DY (o 2n+1 ), DY (sp 2n ) and DY (o 2n ) separately.

Gauss decomposition for DY (gl N )
There are several different ways to introduce Gauss coordinates of T -operators. In this paper we will use following Gauss decomposition of the monodromy matrix elements One can substitute (3.1) into commutation relations (2.9) and check that Gauss coordinates F ± i+1,i (u), E ± i,i+1 (u), i = 1, . . . , N − 1 and k ± j (u), j = 1, . . . , N are generators of the Yangian double DY (gl N ). Let us introduce the rational function of the spectral parameters u and v The commutation relations between generators of the Yangian double DY (gl N ) can be rewritten in terms of so called currents and can be presented in the form (so called 'new' realization of the Yangian double DY (gl N )) 10) and the Serre relations for the currents E i (u) and F i (u) (see, for example, [13]). In (3.10) the symbol δ(u, v) means the additive δ-function given by the formal series The main result of the paper [14] was an explicit presentation of the isomorphism of the Yangian double DY (gl N ) (2.13) in terms of the Gauss coordinates. Gauss decomposition of the monodromiesT ± (u) has literally the same form as in (3.1) with the Gauss coordinates F ± j,i (u), where (recall that indices i ′ , j ′ and ℓ ′ are defined by (2.3)) ElementsF ± j,i (u) andẼ ± i,j (u) given by (3.15) and (3.16) are matrix entries of the inverse matrices F ± (u) −1 and E ± (u) −1 . Diagonal matrix K ± (u) = diag(k ± 1 , . . . , k ± N (u)), upper triangular matrix F ± (u) and lower triangular matrix E ± (u) define the Gauss decomposition (3.1) of the T -operators: and formulas (3.12)-(3.14) were proved in [14] by reordering Gauss coordinates using the commutation relations between them in the T -operatorsT ± (u) given by (2.10).
In the next three subsections we will introduce Gauss coordinates for the T -operators and the corresponding currents for the Yangian doubles DY (o 2n+1 ), DY (sp 2n ) and DY (o 2n ). We will use the same notations for these quantities in each of these subsections, but it will be always clear from the context what Gauss coordinates, currents and T -operators we are considering.

Gauss coordinates for DY (o 2n+1 )
Let us introduce Gauss coordinates for the monodromy matrix T ± (u) ∈ DY (o 2n+1 ) similarly to (3.1) The commutation relations for the Yangian double DY (o 2n+1 ) (2.9) for the monodromy matrix entries T ± i,j (u) for −n ≤ i, j ≤ 0 and 0 ≤ i, j ≤ n satisfy the gl n+1 -type commutation relations except the relations between T ± i,j (u) and T ± k,l (v) when either i = k = 0 or j = l = 0 or i = k = n = l = 0. Using [14] we can express part of the Gauss coordinates of the transposeinverse monodromy matricesT ± (u) given by (2.10) through Gauss coordinates of Recall that i ′ = −i, j ′ = −j and ℓ ′ = −ℓ in this case. Here the Gauss coordinatesF ± i,j (u) andẼ ± i,j (u) are given by the formulas (3.15) and (3.16). To obtain (3.20) we have used only gl n+1 -type commutation between T -operator entries. It explains why these formulas do not differ from the formulas (3.12)-(3.14) up to certain shift in the spectral parameter.
The equation (2.12) allows to identify the matrix elements . (3.26) Equalities (3.23)-(3.25) and (3.26) allow to chose the set of independent generators of subalgebras B ± . This can be either the set .

(3.28)
In what follows we will use the set (3.27) as the set of generators of the algebra DB. Besides rational function (3.2) we introduce the function In order to find current realization of the algebra DY (o 2n+1 ) and due to (2.12) it is enough to consider the commutation relations for the monodromy matrix elements T i,j (u) following from (2.8) for 0 ≤ i, j ≤ n only. Then the commutation relations in DB can be written in terms of the Cartan currents k ± j (u) for 0 ≤ j ≤ n and generating series (currents) and the Serre relations for the currents E i (u) and F i (u), see e.g. [7]. As in the case of A-type algebras all equalities in (3.31)-(3.37) should be understood as equalities between formal series, so that they correspond to a countable number of relations between modes of the currents. The proof that these relations follow from the RT T commutation relations (2.9) is a straightforward repetition of the arguments invented in [6] and exploited in the present situation in [8].

Gauss coordinates for DY (sp 2n )
For the Yangian double DY (sp 2n ) we repeat the same approach that was realized in the previous sections. We use the Gauss decomposition (3.19) where now i, j ∈ I C . Due to (2.9) the monodromy matrix elements T ± i,j (u) for the values of the indices −n + 1 ≤ i, j ≤ 0 and 1 ≤ i, j ≤ n satisfy the commutation relations of the Yangian double DY (gl n ). Repeating calculations of the previous section we can express part of the Gauss coordinates of the inverse-transpose monodromy matrix elementsT ± i,j (u) (2.10) through the Gauss coordinates of T ± (u) ∈ DY (sp 2n ). These relations are given by the formulas (3.20)-(3.22), where i ′ = −i + 1, j ′ = −j + 1 and ℓ ′ = −ℓ+1. Then the identification (2.12) for 1 ≤ i, j ≤ n with κ = n+1 yields the identification of the Gauss coordinates where ℓ = 1, . . . , n. Equalities (3.38) allow to chose the set of independent generators of subalgebras C ± . We choose the set of the Gauss coordinates and this will be a set of independent generators of the Yangian double DC. Note, that the set (3.39) includes Gauss coordinates F ± 1,0 (u) and E ± 0,1 (u).
To describe the current realization of the Yangian double DC we replace function f(u, v) introduced by (3.29) by the function To find current realization of the algebra DY (sp 2n ) it is enough to consider the commutation relations for the monodromy matrix elements T i,j (u) following from (2.9) for 0 ≤ i, j ≤ n only. Then the commutation relations in DC can be written in terms of the Cartan currents k ± ℓ (u) for 1 ≤ ℓ ≤ n and generating series (currents) given by the formula (3.30) as follows [7] and the Serre relations for the currents E i (u) and F i (u), 0 ≤ i ≤ n − 1, see e.g. [7]. In (3.46) the Cartan currents k ± 0 (u) are given by the last relation in (3.38) for ℓ = 1 .

(3.47)
For the proof of these relations we address readers to the paper [8].

Gauss coordinates for DY (o 2n )
For the Yangian double DY (o 2n ) we follow the same approach. We use the Gauss decomposition (3.19) with indices i, j ∈ I D and one more restriction (see proposition 5.11 in [8]) As well as in the previous section the monodromy matrix elements T ± i,j for the values of the indices −n + 1 ≤ i, j ≤ 0 and 1 ≤ i, j ≤ n satisfy the commutation relations of the Yangian double DY (gl n ). Gauss coordinates of the inverse-transpose monodromy matrixT ± (u) (2.10) are related to the Gauss coordinates of T ± (u) ∈ DY (o 2n ) by the equalities (3.20)-(3.22). Equations (2.12) for 1 ≤ i, j ≤ n with κ = n − 1 allows to obtain the identification of the Gauss coordinatesF where ℓ = 1, . . . , n. Note that formulas (3.49) for identification of the Gauss coordinates in the Yangian double DY (o 2n ) differ from the formulas (3.38) for the analogous identification in the Yangian double DY (sp 2n ) by the overall shift by 2c in the right hand sides of (3.49). This is because the method of identification of the Gauss coordinates is the same for these algebras, but κ = n + 1 for DY (sp 2n ) and κ = n − 1 for DY (o 2n ). Due to the identification formulas (3.49) and the commutation relations (2.9), the independent set of generators of the Yangian double DY (o 2n ) are the Gauss coordinates F ± i+1,i (u), F ± 2,0 (u), E ± i,i+1 (u), E ± 0,2 (u), i = 1, . . . , n−1 and k ± j (u), j = 1, . . . , n. Note, that we add the Gauss coordinates F ± 2,0 (u) and E ± 0,2 (u) to this set. This is speciality of the Yangian double DY (o 2n ) (see [7]). The currents for the 'new' realization of the Yangian double DY (o 2n ) are defined by the standard Ding-Frenkel formulas and they satisfy the commutation relations and the Serre relations for the currents E i (u) and F i (u), 0 ≤ i ≤ n − 1 [7]. Cartan currents k ± 0 (u) in (3.56) are given by the last equality in (3.49) for ℓ = 1 . (3.57)

Gauss coordinates and projections
Before we proceed to the relations between Gauss coordinates and the currents we have to describe the projections onto intersections of the different type Borel subalgebras in the Yangian doubles. The mathematically rigorous definition of these projections for the quantum affine algebras was given in the paper [11]. For the Yangian doubles they can be defined analogously (see, for example, [13]). In this paper we will use more practical definition of these projections which will be described at the end of this section. Description of the different types Borel subalgebras is very similar for all types Yangian doubles DY (g), where g = gl N , o 2n+1 , sp 2n , o 2n . The only difference will be in the sets of algebraically independent sets of generators for each algebra. That is why we will use a notation X ± for the standard Borel subalgebras in each Yangian double DY (g). Namely, X ± = A ± for DY (gl N ), X ± = B ± for DY (o 2n+1 ), X ± = C ± for DY (sp 2n ) and X ± = D ± for DY (o 2n ). In all cases subalgebras X ± are generated by the entries of T -operators T ± i,j (u). These subalgebras are Hopf subalgebras with respect to the coproduct (2.14) and the commutation relations are given by the relations (2.9). The whole Yangian double DY (g) may be obtained by the quantum double construction [3] from any of its Borel subalgebra X + or X − using these algebraic and coalgebraic structures.
The 'current' realizations of the Yangian doubles can be also obtained by the quantum double construction, but in this case one has to chose another type Borel subalgebras and another coalgebraic structure.
Let X ± f , X ± e and X ± k are subalgebras in the Yangian double DY (g) generated by the nonnegative and negative modes of the simple root currents which satisfy the relations (3.28). It is clear that standard Borel subalgebras X ± are X ± f ∪ X ± k ∪ X ± e . Different choice of the initial Borel sualgebras in the quantum double construction of DY (g) is to consider X F = X f ∪ X + k as the union of the two subalgebras X f and X + k formed by all modes of the simple root currents F i (u) and modes of the 'positive' Cartan currents k + j (u) We call this type Borel subalgebra as the current Borel subalgebra. The other or dual current subalgebra X E = X e ∪ X − k is formed by all modes of the simple root currents E i (u) and modes of the 'negative' Cartan currents k − j (u) The fact that X F and X E are subalgebras in the Yangian double DY (g) is clearly seen from the commutation relations in terms of the currents given in the sections 3.1-3.4. Current Borel subalgebras X F and X E are Hopf sualgebras with respect to a coproduct ∆ (D) which differs from the coproduct (2.14). It is sufficient to describe this coalgebraic properties only for the generators of the current Borel subalgebras and they are for DY (o 2n ): i = 0, . . . , n − 1, j = 1, . . . , n. It is clear from these coproduct formulas that current Borel subalgebras X F and X E are Hopf subalgebras in DY (g).

Define the intersections
According to definition of X F and X E we have Each intersections in (4.3) is a subalgebra in the Yangian double DY (g) and they are all coideals with respect to the Drinfeld coproduct ∆ (D) According to the Cartan-Weyl construction of the Yangian double we may impose a global ordering of the generators in this algebra. There are two different choices for such an ordering. We denote the ordering relation by the symbol ≺ and introduce the cycling ordering between element of the subalgebras X ± f , X ± e and X ± k as follows [11] · · · ≺ X Using this ordering rule we may say that arbitrary elements F ∈ X F and E ∈ X E are ordered if they are presented in the form where F ± ∈ X ± F and E ± ∈ X ± E . According to the general theory one may define projections of any ordered elements from the subalgebras X F and X E onto subalgebras (4.3) using the formulas where the counit morphism ε : DY (g) → C is defined by the rules In the next section we will introduce certain elements in the current Borel subalgebras X F and X E given by the product of the currents at coinciding values of the spectral parameters. Since the currents F i (u) and E i (u) are given by the infinite series (4.1) and (4.2) we have to assign meaning to such products. This can be done by introducing certain completions of the current Borel subalgebras X F and X E . Let X F be the completion of the algebra X F formed by infinite sums of monomials that are ordered products Define analogously X E as the completion of X E that are ordered products for the detailed description of the properties of these completions in case of the quantum affine algebras see [11]). One can prove that (1) the action of the projections (4.5) extends to the algebras X F and X E respectively; (2) for any F ∈ X F with ∆ (D) (F) = F (1) ⊗ F (2) we have (3) for any E ∈ X E with ∆ (D) (E) = E (1) ⊗ E (2) we have Definition of the projections given by the formulas (4.5) is useful to prove their properties, in particular, (4.7) and (4.8). This properties are power tool to calculate the projections of the products of the currents. But sometimes, one can use more practical way to calculate the projection. For example, to calculate the projection of the product of the currents F i (u) one has to replace each current by the combination of the Gauss coordinates (3.3) and then use the commutation relations to move all 'positive' Gauss coordinates F + j,i (u) to the right and all 'negative' Gauss coordinates F − j,i (u) to the left. After such reordering according to the cyclic ordering (4.4), the application of the projection P + f amounts to remove all the terms containing at least one 'negative' Gauss coordinate on the left. Similarly, the application of the projection P − f amounts to remove all the terms containing at least one 'positive' Gauss coordinate on the right. The action of the projections P + e and P − e is defined analogously according to the cyclic ordering (4.4) which signifies that Gauss coordinates E + i,j (u) should be moved to the left and E − i,j (u) to the right. We will use these prescriptions to calculate projections of the composed currents which will be defined in the next section.

Gauss coordinates and projections of composed currents
Due to the relations (3.26), (3.38) and (3.49) we can introduce currents F i (u) and E i (u) for negative values of the index i For the Yangian doubles DY (gl N ), DY (o 2n+1 ) and DY (sp 2n ) we introduce so called composed currents where for X = A: 1 ≤ i < j ≤ N , for X = B: −n ≤ i < j ≤ n and for X = C: −n + 1 ≤ i < j ≤ n.
For the Yangian double DY (o 2n ) the composed currents F j,i (u) ∈ D F and E i,j (u) ∈ D E are given by the formulas In the last lines of (5.3) and (5.4) the indices i and j are from the intervals −n+1 ≤ i ≤ −1, 2 ≤ j ≤ n. These formulas looks rather complicated with respect to the formulas (5.1) and (5.2), but this is because of restriction (3.48) and commutativity of the currents [F 0 (u), F 1 (v)] = 0 and [E 0 (u), E 1 (v)] = 0. The products F i (u) · · · F −2 (u), F 2 (u) · · · F j−1 (u) in the second, fourth and fifth lines of (5.3) disappear for the values of the indices i = −1 and j = 2. The same is valid for the formula (5.4). According to these remarks the composed currents F 2,−1 (u) and E −1,2 (u) are equal to −F 0 (u)F 1 (u) and −E 0 (u)E 1 (u).
Formulas (5.5) and (5.6) were proved in [13] using definitions of the composed currents as residues of the product of the simple root currents for the Yangian double DY (gl(m|n)). Formulas (5.7) and (5.8) can be proved analogously.
In this paper we present another proof of the proposition 5.1 which uses only the commutation relations (2.8) in the Yangian double DY (g) and properties of the projections (4.5). The reader can find this proof in the appendix A.
Formulas which relate the Gauss coordinates of T -operators to the simple root currents are very important in the application to the g-invariant quantum integrable models. In particular, they are important in calculation of the action of monodromy entries T i,j (z) onto off-shell Bethe vectors in these models. As it was shown in [16] this calculation starts from the action of the right-upper entry T 1,N (z) onto off-shell Bethe vector. Due to the Gauss decomposition (3.1) or (3.19) this action is proportional to the action of the right-upper Gauss coordinate F + N,1 (u) for DY (gl N ), F + n,−n (u) for DY (o 2n+1 ) and F + n,−n+1 (u) for DY (sp 2n ), DY (o 2n ). Results of this paper show that these Gauss coordinates are • for DY (sp 2n ) F + n,−n+1 (u) = (−1) n−1 P + f F n−1 (u − cn) · · · F 1 (u − 2c)F 0 (u)F 1 (u) · · · F n−1 (u) , (5.11) (5.12) Equality (5.9) was used in [16] to calculate the action of the monodromy matrix elements T i,j (u) onto off-shell Bethe vectors for gl(m|n)-integrable models. Equalities (5.10)-(5.12) can be used to calculate the actions of monodromy entries onto corresponding off-shell Bethe vectors in o 2n+1 -, sp 2n -and o 2n -invariant integrable models.

Conclusion
In this paper we consider a relations between Gauss coordinates of the T -operators for the Yangian doubles of the classical series and the current generators of the same algebras. These relations are important for the investigation of the quantum integrable models associated with g-invariant R-matrices for g = gl N , o 2n+1 , sp 2n and o 2n . Using approach introduced in [11,12] and further developed in [13,16] we may express the off-shell Bethe vectors in terms of the current generators of the corresponding Yangian doubles. These presentations and relations between T -operators matrix entries and the current generators allows to calculate the action of these entries onto off-shell Bethe vectors. These action formulas for the upper triangular entries of T -operators allows to obtain the recurrent relations for the Bethe vectors. The action of the diagonal entries allows to obtain the Bethe equations, while the action of the lower triangular entries gives possibilities to calculate the scalar products of the Bethe vectors and form-factors of the local operators in the corresponding integrable models. Realization of this program for the integrable models associated with algebras of the classical series and their quantum deformations will be presented in our forthcoming publications.

(A.2) disappears and we have
If we multiply the terms in the right hand side of (A.4) by k + j−1 (v) −1 k − j (u) −1 we can check that normal ordering of all these terms with respect to the ordering (4.4) cannot produces a pole of third order at u = v. So multiplying (A.4) after normal ordering by (u − v) 3 and setting u = v we obtain an equality Let us iterate the equality (A.6) once to obtain One important property of the projections described in [11] and following from (4.7) is that for where the sum includes only the terms such that F ′ ℓ ∈ B − f ∪B + f and ε(F ′ ℓ ) = 0. ε is the counit map in the Yangian double DY (o 2n+1 ) defined by (4.6). It means that we can replace the element since all other terms from (A.8) will be cancelled by the first projection in the right hand side of (A.7). This equality will now have the form Further iterations up to F + k+1,k (v) which can be replaced by F k (v) proves equality (5.5). To prove equality (5.6) for the Yangian double DY (gl N ) we take instead of (A.1) the commutation relation for the inverse monodromyT (u) (3.17) Since equality (5.6) is an equality in subalgebra A − f we multiply the relation (A.10) from the left by the product k + i (u)k − i+1 (v) and restrict resulting equality to this subalgebra. Elements from subalgebra A − f can appear only in the left hand side of this equality and Last equality in (A.12) can be rewritten as where we used the relationF ± i+1,i (u) = −F ± i+1,i (u). Iterating (A.13) and using another property of the projections that such that ε(F ′′ ℓ ) = 0 ∀ℓ (A.14) we prove relation (5.6) for the Yangian double DY (gl N ).
Since Gauss coordinates ofT ± i,j (u) satisfy the same RT T relations (2.9) as T ± i,j (u) do and Gauss coordinates for both monodromy matrices are given by the same formula (3.1), the Ding-Frenkel isomorphism yields the same commutation relations (3.4)-(3.10) for the currentŝ F i (u),Ê i (u) andk ± j (u) given by (3.18). We can repeat all calculations as we did to prove proposition 5.1 to obtain F + j,i (u) = P + f F j,i (u) ,Ê + i,j (u) = P + e Ê i,j (u) , 1 ≤ i < j ≤ N , Double application of the map (3.18) yieldŝ