On the R-matrix realization of quantum loop algebras

We consider $\rm R$-matrix realization of the quantum deformations of the loop algebras $\tilde{\mathfrak{g}}$ corresponding to non-exceptional affine Lie algebras of type $\hat{\mathfrak{g}}=A^{(1)}_{N-1}$, $B^{(1)}_n$, $C^{(1)}_n$, $D^{(1)}_n$, $A^{(2)}_{N-1}$. For each $U_q(\tilde{\mathfrak{g}})$ we investigate the commutation relations between Gauss coordinates of the fundamental $\mathbb{L}$-operators using embedding of the smaller algebra into bigger one. The new realization of these algebras in terms of the currents is given. The relations between all off-diagonal Gauss coordinates and certain projections from the ordered products of the currents are presented. These relations are important in applications to the quantum integrable models.

Abstract. We consider R-matrix realization of the quantum deformations of the loop algebrasg corresponding to non-exceptional affine Lie algebras of type g = A N −1 . For each U q (g) we investigate the commutation relations between Gauss coordinates of the fundamental L-operators using embedding of the smaller algebra into bigger one. The new realization of these algebras in terms of the currents is given. The relations between all off-diagonal Gauss coordinates and certain projections from the ordered products of the currents are presented. These relations are important in applications to the quantum integrable models.

Introduction
Classification of the solutions to the quantum Yang-Baxter equation for the case of non-exceptional quantum affine Lie algebras was found in the pioneering paper [1].
Let g be one of the Lie algebras sl N , o 2n+1 , sp 2n or o 2n corresponding to the series of the classical Lie algebras A N −1 , B n , C n and D n respectively. Let g be one of nonexceptional affine Lie algebras A N −1 . Byg we denote the loop algebra which is the affine algebra g with zero central charge. To save notations we will use the same names for the different loop algebrasg as for the affine algebras g.
Let q ∈ C be arbitrary complex number not equal to zero or root of unity. In this paper we consider quantum deformation U q (g) [2] of the universal enveloping algebra U(g). One may think about U q (g) as the corresponding quantum affine algebra U q ( g) with zero central charge.
Algebra U q (g) has several descriptions. It can be formulated in terms of the finite number of Chevalley generators or countable set of Cartan-Weyl generators. Latter generators can be gathered into finite number of the generating series and the commutation relations between whole set of the Cartan-Weyl generators can be realized as finite number of the formal series relations between these generating series.
For the applications to the quantum integrable models, the second description of U q (g) is more suitable since generating series of the Cartan-Weyl generators can be identified with Gauss coordinates of the fundamental L-operators, which satisfy the same RLL-type commutation relations as quantum monodromies of the integrable systems do. It opens a possibility to construct off-shell Bethe vectors for these integrable models in terms of Cartan-Weyl generators of the algebra U q (g) [3].
Realization of the algebra U q (g) in terms of Cartan-Weyl generators has in turn two faces. One is given by the quadratic RLL-type commutation relations for the fundamental L-operators defined by the solution of the quantum Yang-Baxter equation [1]. This construction was first proposed in the paper [4]. On the other hand the algebra U q (g) can be realized in terms of so called currents [5]. For the case U q ( gl N ) an isomorphism between these two descriptions was found in [6]. Recent papers [7,8] prove similar isomorphisms for the algebras U q (B (1) n ), U q (C (1) n ) and U q (D (1) n ). Also the case U q (A (2) 2 ) was considered in [9]. In our investigation we extend these results to the case of U q (A (2) N −1 ). Key observation is the fact that R-matrix associated with the algebras U q (B (1) n ), U q (C (1) n ), U q (D (1) n ), U q (A (2) N −1 ) has the same structure for all these algebras. The differences are accumulated in one parameter ξ (see (2.1)).
In [1] one more solution to the quantum Yang-Baxter equation was found. It corresponds to the affine algebra D (2) n . This solution has more complicated structure than R-matrices for above mentioned algebras. We will describe the corresponding quantum loop algebra U q (D (2) n ) in our future publications. The paper is composed as follows. In section 2 quantum R-matrix for the algebra U q (g) is defined together with its properties. Section 3 is devoted to definition of the algebra U q (g) and description of its central elements and automorphism. Gauss coordinates of the fundamental L-operators are introduced in section 4. Here we discuss normal ordering of subalgebras in U q (g) induced by the cyclic ordering of the Cartan-Weyl generators in the quantum affine algebras. Section 5 contains the theorem which describes embedding of the smaller rank algebra U q (g) into the bigger one. This embedding is described on the level of matrix entries of the fundamental L-operators and in terms of the Gauss coordinates. Section 6 describes new realization of the algebra U q (g) in terms of the currents. In section 7 so called composed currents are introduced which belong to certain completion of U q (g) and related to off-diagonal Gauss coordinates of the fundamental L-operators. It was shown in [10,11] that analytical properties of the composed currents and the commutation relations between them are equivalent to the Serre relations between simple root currents. Proofs of auxiliary propositions and lemmas are gathered in four appendices.
We introduce functions of the arbitrary complex numbers u and v, which we call the spectral parameters. Define matrices P(u, v) and Q(u, v) acting in the tensor product where rational functions p ij (u, v) and q ij (u, v) are defined as follows (2.6) and One can check that functions (2.6) have a property Definition 2.1. Quantum trigonometric R-matrix acting in the tensor product of two fundamental vector representations ofg [1] for the algebrag = A and for the algebrasg = B (1) n and A For any X ∈ End(C N ) transposed matrix X t is Let D be a diagonal matrix D = diag(q1, q2, . . . , qN ), whereī for i = 1, . . . , N are given by (2.2). Let P be permutation operator (P 2 = I) in C N ⊗ C N and Q be projector Trigonometricg-invariant R-matrix given by (2.8) and (2.9) possesses following properties.
• Transposition symmetry (2.12) • Twist symmetry • Yang-Baxter equation where subscripts of R-matrices mean the indices of the spaces C N where it acts nontrivially.
3 R-matrix formulation of the algebra U q (g) The algebra U q (g) over C(q) (over C(q 1/2 ) forg = B (1) There are also additional relations for the operators L ± i,j [±m] which are due to existence of the central elements in U q (g) described in section 3.1.
The generators of the algebra U q (g) can be gathered into formal series and combined in the matrices which we call L-operators 1 . The commutation relations in the algebra U q (g) are given by the standard RLL commutation relations in ( where µ, ν = ± and rational functions entering R-matrices (2.8) and (2.9) should be understood as series over v/u for µ = +, ν = − and as series over u/v for µ = −, ν = +. For µ = ν these rational functions can be either series over the ratio v/u or the ratio u/v. The commutation relations in the algebra U q (g) may be written in terms of matrix entries (3.2). Using explicit expression (2.4) and (2.5) one gets The sum in the last line of the commutation relations (3.5) is absent for the algebrã N −1 . It follows from the commutation relations (3.4) or (3.5) that modes L + i,j [m] and L − i,j [−m], m ≥ 0 form Borel subalgebras U ± q (g) ⊂ U q (g). Remark 3.1. One can check that the restrictions to the zero mode generators (3.1) are consistent with the commutation relations (3.5). Indeed, taking the limit u → ∞ in (3.5) with µ = + and using the expansion (3.2), one gets where δ i<j = 1 if i < j and 0 otherwise and Now, if one supposes that i > j and applying (3.1) for the zero mode operators L + i,j [0], the l.h.s. of (3.6) vanishes identically. Due to the coefficients δ l<j and δ i<k in the second line of (3.6) and the combinations q in the third line of this equality, the r.h.s. also vanishes for the same reason.
Analogously, one can check that the restriction that zero mode operators L − i,j [0] vanishes for i < j is consistent with the series expansion (3.2) in u of the L-operator L − (u). In that case, the zero modes occur in the limit u → 0, which changes the exchange relations (3.6) and makes everything consistent again. Finally, one can also verify that the limit v → ∞ in (3.5) for L + k,l (v) and the limit v → 0 for L − k,l (v) leads to the same conclusions.

Central elements in U q (g)
Due to the commutation relations (3.4) algebra U q (g) has central elements which are given by the following n and D (1) n the algebra U q (g) has the central elements where parameter ξ is given by the table (2.1). Equation (3.7) means that products of the matrices are equal and proportional to the unit matrix I. The proportionality coefficients are the central elements.
Proof. To find central elements (3.7) one can transform the commutation relations for the fundamental L-operators (3.4) to the form 2 where standard notations are used. Taking the residue at the point u = vξ 2 in this equation and using (2.21) one gets In what follows we will sometimes skip superscripts of L-operators. If these superscripts is not explicitly mentioned it means that the corresponding relation is valid for both values ±. which proves equality in (3.7). To prove centrality of the elements Z ± (v) we consider the chain of equalities where x(u, v) is defined by (2.22). For these calculations one has to use RLL commutation relations and equalities (2.16) and (2.18) for R-matrices (2.8) and (2.9).
Remark 3.2. Existence of the central element Z(u) for the Yangian Y (gl N ) was mentioned in [15]. In this paper a quantum Liouville formula for the Yangian was considered. Analogous relation in the case of the algebra U q (A and can be proved in the same way as in the Yangian case [16]. In (3.8) k ± ℓ (v) are diagonal Gauss coordinates introduced by (4.1).
We set the central elements Z ± (u) equal to 1 in the algebra U q (g). We denote bỹ U q (A (1) N −1 ) the algebra defined by U q (gl N )-invariant R-matrix (2.8) without any restrictions to these central elements.
The pole structure of R-matrix forg = B (1) n , A N −1 given by (2.20) yields other central elements in the corresponding algebras U q (g). We have following N −1 given by the equalities Again, (3.9) means that product of the matrices are proportional to the unity operator I and the proportionality coefficients are central elements. They are related to Z ± (v) by the relations Proof. Calculating residue at u = vξ in the commutation relation (3.4) one gets which is equivalent to This proves (3.9).
To prove that the elements z ± (u) are central elements in the algebra U q (g) we consider the product z(u)I 1 L (2) (v) and chain of equalities Equality (3.10) can be proved by expressing L ± (v) −1 t from (3.9) and substituting it into (3.7).
For the algebras U q (g) withg = B N −1 we set central elements z ± (v) = 1. Then equalities (3.9) take the form where transposed-inversed L-operatorsL ± (u) are defined aŝ Due to (3.10) the central elements Z ± (v) also equal to 1 when z ± (v) = 1. Then equality (3.7) can be written in the form and describes the relations between order of transposition and taking inverse of the fundamental L-operators in the algebra U q (g). One can check that L-operatorsL ± (u) given by (3.12) satisfies the same commutation relations (3.4). Let us apply to (3.4) the transposition (2.10) in both auxiliary spaces and use (2.12) to get Multiplying from both sides of this equality first byL (1) (u) and then byL (2) Summarizing we conclude that the map moves the algebra U q (g) into the algebra given by the same commutation relation (3.4) but for the transposed-inversed L-operatorsL ± (u). One can also check that central elementsẐ ± (v) andẑ ± (v) defined by (3.7) and (3.9) with L ± (v) replaced byL ± (v) are related to the central elements Z ± (v) and z ± (v) as followŝ

Gauss coordinates
It is known [17] that Gauss coordinates of L-operators introduced below by the equality (4.1) are related to the Cartan-Weyl generators of the corresponding to U q (g) quantum affine algebras U q ( g). The Cartan-Weyl generators satisfy certain ordering properties described in details in [3] and shortly presented in the section 4.1. In this paper we consider Gauss decomposition of the fundamental L-operators of the algebra U q (g) where one assumes that F ± i,i (u) = E ± i,i (u) = 1 for 1 ≤ i ≤ N. Gauss decompositions formula for the matrix entries of L-operators is associated with the products of lower triangular, diagonal and upper triangular matrices Equality (4.2) allows to obtain Gauss decomposition of L-operatorsL ± (u). Indeed, using multiplication rule for and taking inverse of both sides of the equality (4.2)

Normal ordering of the Gauss coordinates
In order to obtain commutation relations for Gauss coordinates from (3.5) one can use the normal ordering of the Cartan-Weyl generators [3].
Let U ± f , U ± e and U ± k be subalgebras of U q (g) formed by the modes of the Gauss coordinates F ± j,i (u), E ± i,j (u) and k ± j (u). The fact that these unions of generators are subalgebras follows from the identification of modes of Gauss coordinates with Cartan-Weyl generators [17]. It is known that Cartan-Weyl generators have two natural circular orderings which imply the normal ordering of the subalgebras formed by the Gauss coordinates. These orderings are If one places subalgebras U ± f , U ± e and U ± k onto circles then ordering (4.5) is counterclockwise in the left circle and the ordering (4.6) is clockwise in the right circle of (4.7). The general theory of the Cartan-Weyl basis allows to prove that in both types of ordering the unions of subalgebras U ± f , U ± e and U ± k along smallest arcs between starting and ending points are subalgebras in U q (g). For example, the union of subalgebras and so on are subalgebras in U q (g).
The notion of the normal ordering yields a powerful practical tool to get relations for the Gauss coordinates of the specific type. In any relation which contains Gauss coordinates of the different types one first has to order all monomials according to (4.5) or (4.6) and then single out all the terms which belong to the one of subalgebras which is composed from the Gauss coordinates of the necessary type. We call this procedure a restriction to subalgebras in U q (g) and will use this method to get relations between Gauss coordinates from RLL-commutation relations (3.5).
were already introduced above as Borel subalgebras in U q (g). To descride so called 'new realization' of these algebras in terms of the currents [5] one has to consider different type Borel subalgebras In [18] certain projections P ± f and P ± e onto intersections of the Borel subalgebras of the different types were introduced. These projections were further investigated in [3] for the ordering (4.5) and was used for the first time in [19] to describe the off-shell Bethe vectors or weight functions in terms of the Cartan-Weyl generators. One can check that the action of the projections P ± f and P ± e onto Borel subalgebras U f and U e introduced in [18] coincides with restrictions onto subalgebras U ± f and U ± e defined for the ordering (4.5).

Embedding theorem
Each algebrag of the type B (1) n and A N −1 has rank n as rank of the underlying finite dimensional algebra. To stress this fact we will use notation U n q (g) to denote explicitly rank for any of the quantum loop algebras considered in this paper. Following ideas of the paper [7] we consider in this section embedding of smaller algebras U n−1 q (g) ֒→ U n q (g). To note that R-matrix corresponds to the algebra U n q (g) we will use superscript R n (u, v), R n (u, v), Q n (u, v), etc.
In this paper we use Gauss decomposition of the L-operators for the algebra U n q (g) given by (4.1) n and A N −1 we have following Theorem 5.1. The commutation relations for the U n−1 q (g) matrix entries M ± i,j (u) follow from the Yang-Baxter equation (2.14) and the commutation relations (3.4) in U n q (g) and take the form (µ, ν = ±) To prove this theorem we formulate auxiliarly lemmas 5.2 and 5.3. Let L (1,2) (u) be fused L-operator defined as (we again skip superscripts of L-operators to avoid bulky notations) One can calculate its (i, j; 1, 1) matrix element The commutation relations for 1 < i < N and (5.4) implies that L-operators M(u) for the algebra U n−1 q (g) can be presented as There is a commutativity of the matrix entries in U n q (g) According to (5.5) matrix entries M i,j (u) are proportional to the matrix entries L i,j;1,1 (u) up to commuting with them invertible operator L 1,1 (v). It yields that the commutation relations for M i,j (u) should coincide with the commutation relations of L i,j;1,1 (u). To find the commutation relations for the matrix entries L i,j;1,1 (u) we need following where |i, k, j, l = |i ⊗ |k ⊗ |j ⊗ |l and i, k, j, l| = i| ⊗ k| ⊗ j| ⊗ l| are vectors in Proofs of the lemmas 5.2 and 5.3 can be found in appendix A.
To prove theorem 5.1 we consider RLL-commutation relations for L-operators L(u) and L(v) (A.6) and for 1 < i 1 , j 1 , i 2 , j 2 < N take the matrix element of this commutation relation Let us transform last line in (5.8) using lemma 5.3, equality (A.7) and Yang-Baxter equation (2.14). We have At the second step of this calculation we used equality (A.7) taken at u → q 2 u and scaling invariance of R-matrix (2.11). Analogously first line in (5.8) can be transformed to where we used (A.8) at v → q −2 v. Equalities (5.9) and (5.10) allow to rewrite (5.8) in the form which proves the statement of theorem (5.3) due to lemma 5.2 and relation (5.5).
Theorem 5.1 implies that in order to find the commutation relations between Gauss coordinates in the algebra U n q (g) it is sufficient to obtain these commutation relations for the smallest rank nontrivial algebras. We will find such commutation relations in the algebras U q (g) forg of the types B (1) N −1 in appendix C. We can formulate analogous statement for the algebra U q (A N −1 ). Using similar arguments as above we can prove following We are not going to provide a proof of this proposition since it can be performed in a similar way as the proof of theorem 5.1. Practical meaning of this proposition is that in order to obtain the commutation relations between Gauss coordinates for the algebra U q (A (1) N −1 ) it is sufficient to consider the commutation relations for the algebras at small values of N.

Embedding in terms of the Gauss coordinates
Let us introduce 'alternative' to (4.1) Gauss decomposition of the fundamental Loperators where shifts by q −2(i−1) in the arguments of 'alternative' Gauss coordinatesF j,i (u), E i,j (u) andk i (u) are introduced for the further convenience. In terms of these Gauss coordinates matrix entries of L-operators have the form Our goal is to find relations between Gauss coordinates F j,i (u), E i,j (u), k j (u) and F j,i (u),Ē i,j (u),k j (u). This is given by The proof of this proposition is given in appendix B. Note that equality (5.14) can be written in the form which is consequence of the embedding relation (B.6) at each step of the embedding. Moreover, we can excludek ± ℓ (u) from (5.14) and (5.17) to obtain where 1 ≤ ℓ ≤ n + 1 for odd N = 2n + 1 and 1 ≤ ℓ ≤ n for N = 2n even. Proposition 5.5 has an obvious Corollary 5.6. There are relations between Gauss coordinates of the fundamental Loperators in the algebra U q (g) corresponding to the simple roots of the underlying algebra g for ∀N and 1 ≤ i ≤ n − 1 and for N = 2n + 1.
This corollary together with equalities (5.18) defines the algebraically independent sets of the generators in each of the algebras U q (g) of the typeg = B 6 New realization of the algebra U q (g) New realization of the quantum affine algebras U q ( g) was given in [5] in terms of the formal series called currents labeled by the simple roots of the underlying finitedimensional algebra g. Relations between currents and Gauss coordinates for the al-gebraŨ q (A For the algebras U q (B N −1 ) the currents are introduced by the formulas (6.1) for 1 ≤ i ≤ n. For the algebra U q (D (1) n ) first (n − 1) currents are also introduced by (6.1) with 1 ≤ i ≤ n − 1 and the currents F n (u) and E n (u) by the equalities [8] It is obvious from the commutation relations in the algebra U q (g) (3.5) that term Q(u, v) of the R-matrix (2.9) do not contribute into commutation relations of the matrix entries L ± i,j (u) and L ± k,l (v) for 1 ≤ i, j, k, l ≤ n. Commutation relations between these matrix entries and between corresponding Gauss coordinates are defined by U q (gl n )-invariant R-matrix (2.8). These commutation relations can be translated into commutation relations between currents F i (u), E i (u), 1 ≤ i ≤ n − 1 and Gauss coordinates k ± ℓ (u), 1 ≤ ℓ ≤ n according to the standard approach developed in [6]. We formulate all nontrivial commutation relations between these currents without proofs There are also Serre relations for the currents E i (u) and F i (u) [5,6] Sym where [A, B] q means q-commutator The multiplicative delta function in (6.3) is defined by the formal series for any formal series G(u).
Remark 6.1. The equalities (6.3) should be understood in a sense of equalities between formal series. It means that these commutation relations should be understood as infinite set of equalities between modes of the currents which appear after equating the coefficients at all powers u ℓ v ℓ ′ for ℓ, ℓ ′ ∈ Z. The rational functions in the commutation relations (6.3) should be understood as series over powers of v/u in the relations containing the current k + j (u) and over powers of u/v in the relations with the current k − j (u).
The commutation relations of the currents F n (u), E n (u) and diagonal Gauss coordinates will be specific for each of the algebras U q (B N −1 ). According to the theorem 5.1 these commutation relations can be obtained by considering algebras of the small rank presented in the appendix C.
6.1 New realization of the algebra U q (B (1) n ) The full set of the nontrivial commutation relation for the algebra U q (B (1) n ) is given by the relations (6.3) and [8] where modes of the dependent currents k ± n+1 (u) are defined by the relation following from (5.14) and (5.17) for ℓ = n + 1 and ξ = q 1−2n . Serre relations which include currents E n (u) and F n (u) are (3) ).

New realization of the algebra U q (C
n ) The nontrivial commutation relations for the set of the currents in the algebra U q (C (1) n ) are given by the commutation relations (6.3) and the commutation relations involving the currents F n (u) and E n (u) [7] .
Serre relations which includes the currents F n (u) and E n (u) are Using results presented in appendix C.3 one can obtain that the current realization of the algebra U q (D (1) n ) is given by the commutation relations (6.3) and all nontrivial commutation relations which includes the currents F n (u) and E n (u) are [8] 3) the Gauss coordinates k ± n+1 (u) are given by (5.18) for ℓ = n and ξ = q 2−2n . The Serre relations which includes currents F n (u) and E n (u) can be written in the form [8] Sym

New realization of the algebra U q (A
2n ) is given by the relations (6.3) and additional relations which includes currents F n (u) and E n (u) listed below (most of them was found in case U q (A where k ± n+1 (u) are defined by the relation following from (5.18) at ℓ = n + 1 and ξ = −q −1−2n . Serre relations which include currents E n (u) and F n (u) are the same as in the case of U q (B and there are additional Serre relations for the currents E n (u) and F n (u) which can be presented in the form [5,9] Sym u,v,w

New realization of the algebra U q (A
2n−1 ) Finally using results presented in appendix C.5 one can describe the new realization of the algebra U q (A 2n−1 ) as collection of the commutation relations (6.3) and additional relations which includes currents F n (u) and E n (u) The currents F n (u) and E n (u) are series with respect to the odd powers of the spectral parameters and the ratio of the Gauss coordinates k ± n+1 (u)k ± n (u) −1 are series with respect to even powers of the spectral parameters. Serre relations which includes the currents F n (u) and E n (u) are the same as in the case of U q (C (1) n ) (6.6).

Currents and the projections
Currents and the Gauss coordinates can be related through projections P ± f and P ± e onto subalgebras U ± f and U ± e acting on the ordered products of the currents. Rigorous definitions of these projections depends on the type of the cycling ordering of the Cartan-Weyl generators (see [3] for detailed exposition of the properties of the projections for the ordering (4.5)).
Denote by U f [20] an extension of the algebra U f = U − f ∪ U + f ∪ U + k formed by linear combinations of series, given as infinite sums of monomials a i 1 [n 1 ] · · · a i k [n k ] with n 1 ≤ · · · ≤ n k , and n 1 + ... + n k fixed, where a i l [n l ] is either F i l [n l ] or k + i l [n l ]. Analogously, denote by U e an extension of the algebra U e = U − e ∪ U + e ∪ U − k formed by linear combinations of series, given as infinite sums of monomials a i 1 [n 1 ] · · · a i k [n k ] with n 1 ≥ · · · ≥ n k , and n 1 + ... + n k fixed, where a i l [n l ] is either E i l [n l ] or k − i l [n l ]. It was proved in [3,10,20] that the ordered products of the simple roots currents F j−1 (u) · · · F i (u) and E i (u) · · · E j−1 (u) in the algebraŨ q (A (1) N −1 ) are well defined and belong to U f and U e respectively. Moreover, the actions of the projections P ± f and P ± e onto these elements are well defined. The action of the projections P ± f onto product of the currents can be defined as follows. In order to calculate projections from such product one has to substitute each current by the difference of the corresponding Gauss coordinates (6.1) and using the commutation relations between them order the product of the currents F i (u) in a way that all negative Gauss coordinates F − j,i (u) will be on the left of all positive Gauss coordinates F + k,l (v). Then application of the projection P + f to this product of the currents is removing all the terms which have at least one negative Gauss coordinate on the left. Analogously, application of the projection P − f is removing all the terms which have at least one positive Gauss coordinate on the right. The action of the projections P ± e onto product of the currents E i (u) is defined analogously, but the ordering of the Gauss coordinates is inverse: all positive Gauss coordinates E + i,j (u) should be placed on the left of all negative coordinates E − l,k (v) using the commutation relations between them and according to the ordering (4.5). For N −1 ) we introduced currents F i (u) and E i (u), 1 ≤ i ≤ n by the formulas (6.1) and (6.2). Using relations (5.19) we define dependent currents F i (u) and For the algebras U q (B (1) n ) and U q (A (2) 2n ) and according to (5.20) we introduce additional dependent currents N −1 ) and We call elements F j,i (u) and E i,j (u) the composed currents. For the algebra U q (A (1) N −1 ) these currents were investigated in [3,10]. It was shown there that the analytical properties of the products of the composed currents considered in the category of the highest weight representations are equivalent to the Serre relations for the simple root currents.
Using result of [3] that action of the projections P ± f and P ± e can be prolonged to the extensions U f and U e respectively we formulate following Proposition 7.1. There are relations between Gauss coordinates of the fundamental L-operators and projections of the composed currents in the algebra U q (g) Proof of this proposition will be given in appendix D simultaneously for all algebras U q (g) and is based on induction over rank n of these algebras. The base of induction is a verification of (7.4) for all algebras U q (g) of small ranks performed in the appendices C.1, C.2, C.3, C.4 and C.5. In particular, formulas (C.12) and (C.13) are base of induction to prove proposition 7.1 in case of the algebra U q (D (1) n ).

Conclusion
In this paper we investigate quantum loop algebras for all classical series (except U q (D (2) n )) associated to the quantum R-matrices found in [1]. Results obtained in this paper can be used for investigation of the space of states of the quantum integrable models with the different symmetries of the high rank. This investigation can be performed in the framework of the approach to integrable models proposed and developed in [3,10,21]. In this method the states of integrable models are expressed through current generators of the quantum loop algebras. To investigate different physical quantities in such models such as scalar products of the states and form-factors of the local operators it is not necessary to have explicit form of the states in terms of the current generators. Usually, it is sufficient to get the action of monodromy matrix entries onto these states. This approach was called zero modes method and was already used in [22,23,24] to investigate the space of states in quantum integrable models related to Yangian doubles and rational g-invariant R-matrices. Using results of the present paper we plan to develop this method for the integrable models associated with U q (g)-invariant R-matrices.

A Proofs of the lemmas 5.2 and 5.3
Recall that we denote by |i and j|, 1 ≤ i, j ≤ N sets of orthonormal vectors in C N with pairing C N ⊗ C N → C: i|j = δ ij . Consider R-matrix (2.9) for v = q 2 u: and Here |i, j = |i ⊗ |j and i, j| = i| ⊗ j| are vectors from (C N ) ⊗2 . Consider commutation relation (3.4) at v = q 2 u Using (3.4) one can obtain the commutation relations between L-operators L(u) and L(v) where Yang-Baxter equation (2.14) Analogously, one can obtain the commutation relations between L-operators L(u) and L(v) To prove lemma 5.2 one has to obtain two equalities for 1 < ℓ < N One can verify equality (A.7) and equation (A.3) were used. Equality (A.8) can be checked analogously.
Multiplying equality (A.5) from the left by the vector 1, i, 1| and from the right by the vector |1, j, 1 for 1 < i, j < N and using (A.7) and (A.8) one obtains which implies the statement of the lemma (5.2).
Since multiplication of the parameter ξ by q 2 means the change of the rank n → n − 1 for all algebras U q (g) (see table (2.1)) one concludes that
Continuing embedding process and repeating these arguments for the Gauss coor-dinatesĒ i,j (u) one proves proposition 5.5.
C Algebra U q (g) for small ranks In this appendix we obtain the commutation relations for the currents F n (u) and E n (u) for each of the algebra U q (g) of the small rank. Here we will introduce different rational functions denoting them by the same notations valid inside of each subsection. Hope that this will not lead to misunderstanding. In order to find commutation relations of the special currents in case of the algebra U q (B (1) n ) we first perform investigation of the simplest nontrivial example of the algebra U q (B (1) 1 ) as it was done in the paper [23]. In this algebra the algebraically independent series of generators are k ± 1 (u), F ± 2,1 (u) and E ± 1,2 (u) and algebraically dependent generating series are k ± ℓ (u), ℓ = 2, 3 and The modes of k ± ℓ (u), ℓ = 2, 3 are defined by the relations The commutation relations between Gauss coordinates for the algebra U(B 1 ) are and subtracting one equality from another one gets Using (C.4) one can calculate the projection P + f (F 2 (u)F 1 (u)) onto subalgebra U + f assuming that Gauss coordinates in the product of the currents F 2 (u)F 1 (u) are ordered according to the order (4.5). We obtain This relation together with analogous formulas for the projections P − f (F 2 (u)F 1 (u)) and P ± e (E 1 (u)E 2 (u)) are base of the induction proof of the proposition 7.1 which explains the relation between Gauss coordinates and projection of the currents for the algebra U q (B (1) n ). Considering similar commutation relations for the algebra U q (B (1) 2 ) and using embedding theorem 5.1 one obtains besides commutation relations (C.1)-(C.3) for the Gauss coordinates k ± ℓ (u), ℓ = 1, 2, 3 with F ± 3,2 (u) and E ± 2,3 (u) also the commutation relations of these Gauss coordinates with F ± 2,1 (u) and E ± 1,2 (u) This information is sufficient to obtain for the algebra U q (B (1) n ) the commutation relations of the currents F n (u), E n (u), Gauss coordinates k ± ℓ (u), 1 ≤ ℓ ≤ n + 1 and the currents F i (u), E i (u), 1 ≤ i ≤ n − 1 given in section 6.1.
Introduce the rational functions relevant to the considered case The rest commutation relations in U q (C 2 ) can be written in the form where diagonal Gauss coordinate k 3 (u) due to (5.14) and (5.17) is equal to These commutation relations allows to restore the full set of the commutation relations in terms of the currents for the algebra U q (C Considering the latter relation at v = q 2 u we obtain and (C.5). Now one can calculate the projection P + f (F 3 (u)F 2 (u)), where dependent current F 3 (u) = −F 1 (q 4 u) is defined by (7.1) for N = 4 and ξ = q −6 . Restoring in (C.5) superscripts of the matrix entries and setting v = q 4 u we obtain Calculating projection P + f (F 3 (u)F 2 (u)) onto U + f according to the ordering (4.5) one gets Analogously, one can prove that These relations together with analogous relations for the currents E i (u) are base of the induction for the proof of the proposition 7.1 in case of the algebra U q (C 2 ) As above we start to consider algebra U q (D n ) for small n. The case n = 1 is not representative and we begin with the case n = 2 to prove that F ± 3,2 (u) = E ± 2,3 (u) = 0. Excluding term L 21 (v)L 24 (u) from the commutation relation (3.5) with set of indices {i, j, k, l} → {2, 3, 2, 2} and {i, j, k, l} → {2, 4, 2, 1} we have This relation after setting v = q −2 u and projecting onto subalgebras U ± f ∪ U ± k in the algebra U q (D (1) 2 ) yields the equality Since Gauss coordinates k ± 2 (u) are invertible it results that Analogously one can prove E ± 2,3 (u) = 0. (C.9) In order to find relations between Gauss coordinates F j,1 (u), j = 2, 3, 4 one can consider the commutation relation (3.5) for the values of the indices {i, j, k, l} → {1, 3, 1, 2} and {i, j, k, l} → {1, 4, 1, 1}. Excluding term L 11 (v)L 14 (u) we have Using explicit expressions for the matrix entries L 1,j (u) through Gauss coordinates (4.1), multiplying both equalities by the product of k 1 (u) −1 k 1 (v) −1 and using the commutation relations one can get from (C.10) Taking in this equality u = q 2 v and u = q −2 v one can find the relations In the same way one can prove that Equalities (5.12), (5.13), (5.15) and (5.16) yields in this case (C.14) Using (3.5) for {i, j, k, l} → {2, 2, 1, 3} we can calculate where we have used (C.8) and (C.14). Using now (C.11) and identities one can find that Analogously one can obtain Using (C.11) and analogous commutation relations for E 1,j (u) one can calculate from the commutation relation (3.5) at {i, j, k, l} → {1, 3, 3, 1} that The embedding theorem 5.1 and commutation relations between Gauss coordinates obtained for the algebra U q (D 2 ) are sufficient to get full set of the commutation relations for the algebra U q (D (1) n ) in terms of the currents presented in the section 6.3.

C.4 Algebra U q (A
2 ) RLL realization of this algebra is given by the R-matrix (2.9) with ξ = −q −1−2n and N = 2n + 1. In the same way as we investigated the algebra U q (B (1) 1 ) we study first the algebra U q (A (2) 2n ) in the simplest case n = 1.
Introduce the functions The commutation relations for the Gauss coordinates in the algebra U q (A 2 ) between F 2,1 (u), k 1 (u) and E 1,2 (u) are the same as for U q (B (1) 1 ) (see (C.1)). The rest relations are and To prove (C.15) one can use the commutation relation Putting v → ∞ in (C.17) one obtains Setting u = −qv in (C.17) one finds . Using this equality one can calculate where dependent current F 2 (v) = − √ qF 1 (−qv) is defined by (7.2) for n = 1 and ξ = −q −3 . Relation (C.19) as well as analogous relation for the projections P ± e (E 1 (u)E 2 (u)) are base of induction to prove proposition 7.1 in case of the algebra U q (A (2) 2n ).

D Proof of proposition 7.1
We start with the proof of the first equality in (7.4). Assume that it is valid in the algebra U n−1 q (g). It means that in order to prove first equality in (7.4) one has to prove it for the Gauss coordinates F + N,i (u), 1 ≤ i ≤ N − 1 and F + j,1 (u), 2 ≤ j ≤ N using induction assumption that it is valid for all F + j,i (u), 2 ≤ i < j ≤ N − 1. To do this we consider RLL commutation relations (3.4)  To prove the second equality in (7.4) we have to repeat all arguments as above for the transpose-inverse L-operatorsL ± (u) using Gauss decomposition (4.4). Taking the corresponding matrix elements in (D.1) one can find that and By the property of projection P − f [3] P − f F j (u) · · · F i (u) = F j (u) · · · F i (u) + . . . , where . . . stands for the terms which are annihilated by projection P − f and induction assumption one can prove the second equality in the first line of (7.4). The second line in (7.4) can be proved similarly.