R-Matrix and Baxter Q-Operators for the Noncompact SL(N,C) Invariant Spin Chain

The problem of constructing the $SL(N,\mathbb{C})$ invariant solutions to the Yang-Baxter equation is considered. The solutions ($\mathcal{R}$-operators) for arbitrarily principal series representations of $SL(N,\mathbb{C})$ are obtained in an explicit form. We construct the commutative family of the operators $\mathcal{Q}_k(u)$ which can be identified with the Baxter operators for the noncompact $SL(N,\mathbb{C})$ spin magnet.


Introduction
The Yang-Baxter equation (YBE) plays an important role in the theory of completely integrable systems. Its solutions, the so-called R-matrices (operators), are basic ingredients of Quantum Inverse Scattering Method (QISM) [1,2]. The problem of constructing solutions to the YBE was thoroughly analyzed in the works of Drinfeld [3,4], Jimbo [5] and many others. In the most studied case of the quantum affine Lie (super)algebras the universal R-matrix was constructed in works of Rosso [6], Kirillov and Reshetikhin [7] and Khoroshkin and Tolstoy [8,9]. The interrelation of YBE with the representation theory was elucidated by Kulish, Reshetikhin and Sklyanin [10,11], who studied solutions of YBE for finite dimensional representations of the GL(N, C) group. The R-matrix on infinite dimensional spaces were not considered till recently.
In the present paper we construct the SL(N, C) invariant R-operator which acts on the tensor product of two principal series representations of the SL(N, C) group. The spin chains with an infinite dimensional Hilbert space, so called noncompact magnets, are interesting in connection with the problem of constructing the Baxter Q-operators [12] and the representation of Separated Variables (SoV) [13,14]. The Baxter Q-operators are known now for a number of models. Beside few exceptions these are the spin chains with a symmetry group of rank one. No regular method of constructing Baxter operators for models with symmetry groups of higher rank exists so far. However, in the studies of the noncompact sl(2) magnets it was noticed that the transfer matrices with generic (infinite dimensional) auxiliary space are factorized into the product of Baxter operators. This property seems to be quite a general feature of the noncompact magnets 1 and can be related to the factorization of R-matrix suggested in [15]. Thus the problem of constructing the Baxter operators is reduced, at least for the noncompact spin chains, to the problem of factorization of the R-matrix. The factorizing operators for the sl(N ) invariant R-matrix acting on the tensor product of two generic lowest weight sl(N ) modules for N = 2, 3 were constructed in [15]. Unfortunately, for a higher N the defining equations for the factorizing operators become too complicated to be solved directly.
To get some insight into a possible structure of solutions for a general N it is instructive to consider the problem of constructing an R-operator for principal series representations of SL(N, C). We remark here that contrary to naive expectations the principal series noncompact magnets appear to be in some respects simpler than their (in)finite dimensional cousins. Different spin chain models with sl(N ) symmetry differ by a functional realization of a quantum space. The less restrictions are imposed on the functions from a quantum space, the simpler becomes the analysis of algebraic properties of a model and the harder its exact solution. For instance, the solution of the noncompact SL(2, C) spin magnet [16,17] presents a quite nontrivial problem already for the spin chain of the length L = 3, while the analysis of this model (constructing the Baxter operators, SoV representation, etc) becomes considerably easier in comparison with its compact analogs. In particularly, the R-operator in this model (SL(2, C) spin magnet) has a rather simple form and, as one can easily verify, admit the factorized representation. All this suggests to consider the problem of constructing the R-operator for the principal series representations of SL(N, C) in first instance. In the paper we give the complete solution to this problem. We will obtain the explicit expression for the R-operator, prove that the latter satisfies Yang-Baxter equation and, as was expected, enjoys the factorization property. It allows us to construct the Baxter Q-operators as traces of a product of the factorizing operators.
We hope that the obtained results will also be useful for the construction of the representation of Separated Variables. So far the latter was constructed in an explicit form only for a very limited number of models (see [20,21,22,16,23,24,25,26,27]). It was noticed by Kuznetsov and Sklyanin [28,29] that the kernel of a separating operator and Baxter operator have to be related to each other. So the knowledge of the Baxter operator for the SL(N, C) magnets can shed light on the form of the separating operator.
The paper is organized as follows. In Section 2 we recall the basic facts about the principal series representations of the complex unimodular group. We construct the operators which intertwine the equivalent representations and describe their properties. This material is well known [32,33] but we represent intertwinning operators in the form which is appropriate for our purposes. Section 3 starts with the description of our approach for constructing the R-operator. We derive the defining equations on the R-operator and solve them. Finally, we show that the obtained operator satisfies the Yang-Baxter equation. In Section 4 we study the properties of the factorizing operators in more detail and discuss their relation with the Baxter Q-operators. Finally we construct the explicit realization of the operators in question as integral operators.
2 Principal series representations of the group SL(N, C) The unitary principal series representations of the group SL(N, C) can be constructed as follows [30,31]. Let Z and H be the groups of the lower triangular matrices with unit elements on a diagonal and the upper triangular matrices with unit determinant, respectively, Almost any matrix g ∈ G = SL(N, C) admits the Gauss decomposition g = zh (the group Z is the right coset G/H). The element z 1 ∈ Z satisfying the condition will be denoted by zḡ, so that g −1 z = zḡ · h. Thus we can speak about a local right action of G on Z. Later in (2.4), also h, dependent on g and z as in (2.1), will be used again. Let α be a character of the group H defined by the formula whereh kk ≡ (h kk ) * is the complex conjugate of h kk , whereas in general σ * k =σ k . Since det h = 1 the function α(h) depends only on the differences σ k,k+1 ≡ σ k − σ k+1 and can be rewritten in the form h ii and n k =σ k,k+1 − σ k,k+1 , k = 0, . . . , N − 1, are integer numbers 2 . One can always assume that parameters σ k satisfy the restriction defines a principal series representation of the group SL(N, C) on a suitable space of functions on the group Z, Φ(z) = Φ(z 21 , z 31 , . . . , z N N −1 ) [30,31]. The operator T α (g) is a unitary operator on the Hilbert space L 2 (Z), n k is integer and λ k is real.

Generators and right shifts
We will need the explicit expression for the generators of infinitesimal SL(N, C) transformations. The latter are defined in the standard way , are the generators in the fundamental representation of the SL(N, C) group, The generators E ik are linear differential operators in the variables z mn , (1 ≤ n < m ≤ N ) which satisfy the commutation relation It follows from definition (2.4) that the lowering generators E ki , k > i, are the generators of left shifts, Φ(z) → L(z 0 )Φ(z) = Φ(z −1 0 z). In a similar way we define the generators of right shifts, Since right and left shifts commute one concludes that [E ki , D nm ] = 0 (k > i, n > m). Clearly, the generators D ki satisfy the same commutation relation as E ki , equation (2.6) The explicit expression for the generators of left and right shifts reads wherez ki = (z −1 ) ki and we recall that z ii = 1. Let us notice here that the operator D ki depends on the variables in the k-th and i-th columns of the matrix z, or on the variables in the k-th and i-th rows of the inverse matrix z −1 .
The generators E ki can be expressed in terms of the generators D ki as follows where D nm is nonzero only for n > m. To derive equation (2.9) it is sufficient to notice that for the infinitesimal transformation g = 1l + ǫ · E = 1l + ik ǫ ik E ki the elements of the Gauss decomposition of the matrix g −1 z can be represented in the following form: zḡ = z(1l − ǫ − (z)) and h = 1l − ǫ + (z). The matrices ǫ − (z) and ǫ + (z) are the lower and upper diagonal parts of the matrix z −1 (ǫ·E)z. Namely, the matrix ǫ − (z) ki is nonzero only for k > i, ǫ − (z) ki = (z −1 (ǫ·E)z) ki , while ǫ + (z) ki = (z −1 (ǫ·E)z) ki , for k ≤ i and zero otherwise. Thus one finds that the infinitesimal transformation (2.4) is essentially the right shift generated by z 0 = 1l − ǫ − (z). Finally, taking into account that σ k = N (N − 1)/2 one obtains the representation (2.9) for the symmetry generators.
It is convenient to rewrite equation (2.9) in the matrix form where E = mn e mn E nm , D = n>m e mn D nm and σ = n e nn σ n and the matrices e nm form the standard basis in Mat(N × N ), (e nm ) ik = δ in δ mk . Let us recall that z and z −1 are lower triangular matrices while D + σ is an upper triangular matrix The dependence of the generators E on the representation T α resides in the parameters σ n entering the matrix σ. The same formulae hold for the antiholomorphic generatorsĒ. Closing this subsection we give some identities that will be useful in the further analysis: 12c) The first two formulae follow directly from definition (2.7), while the last one is a consequence of the commutation relations for the generators D ki . We remind also that the operators D ki are defined (nonzero) only for k > i.

Intertwining operators
It is known that two principal series representations T α and T α ′ are equivalent if the parameters which specify the representations, σ = (σ 1 , . . . , σ N ) and σ ′ = (σ ′ 1 , . . . , σ ′ N ), are related to each other by a permutation, σ ′ = P σ (of course, it is assumed that the parameters {σ k }, {σ ′ k } in the antiholomorphic sector are related by the same permutation, σ ′ = P σ) [30]. An arbitrary permutation can be represented as a composition of the elementary permutations, P k , which interchange the k-th and k + 1-st component of the vectors, σ ( σ), The operator U k intertwining the representations, T α and T α ′ which differ by the elementary permutation of spins P k , α ′ = P k α, has the form is the generator of the right shift. Let us note that the operator D k+1,k depends on the variables z nm in the k-th and k + 1-st columns of the matrix z. After the change of variables the operator D k+1,k turns into a derivative with respect to x k+1,k , D k+1,k = ∂ x k+1,k , which means that the construction given in equation (2.13) results in well-defined operator. Moreover, if the powers in equation (2.13) satisfy the condition σ * k,k+1 +σ k,k+1 = 0, the operator U k is a unitary operator on L 2 (Z).
To prove that operator U k is an intertwining operator it is sufficient to check that it intertwines the generators E and E ′ in the representations T α and T α ′ , E ′ U k = U k E. To this end it is convenient to use the representation (2.10) for the generators. Using commutation relations (2.12) one finds and where α = σ k,k+1 . (Let us note that the operator D −1 k+1,k commutes with e k+1,k z −1 and with the operators in the sum in (2.16), so its position can be changed). Starting from U k E and moving the operator U k to the right with the help of equations (2.15) and (2.16) one gets after some simplifications (One has to take into account that α = σ k,k+1 and the matrix σ ′ = σ + σ k,k+1 (e k+1,k+1 − e k,k ) differs from the matrix σ by the transposition σ k ↔ σ k+1 .) Making use of the change of variables (2.14) one can represent the operator U k in the form of an integral operator and Let us note that the matrices z and z ζ differ from each other by the elements in the k-th column only.
The operator U k depends on the difference of σ k,k+1 , U k = U (σ k,k+1 ), which is determined by the representation which the operator acts on. We do not display the dependence on this parameter explicitly, but want to note here that in the product U k+1 U k , the first operator is The intertwining operators U k satisfy the same commutation relations as the operators of elementary permutations P k 19c) The first relation had been already explained. The second one is a trivial consequence of the commutativity of the generators of right shifts, [D k+1,k , D n+1,n ] = 0 for |k − n| > 1. The last relation can be checked by the direct calculation with the help of the integral representation (2.17). Obviously, the operator intertwining the representations T α and T α ′ such that the characters α and α ′ are related to each other by some permutation can be constructed as a certain combination of the operators U k , k = 1, . . . , N − 1. The intertwining operators, U k , play an important role in the construction of an R-operator which we will discuss in the next section.

R-operator
Let us recall that an R-operator is a linear operator which acts on the tensor product of two spaces V 1 ⊗ V 2 , depends on the spectral parameter u and satisfies the Yang-Baxter relation As usual, it is implied that the operator R ik (u) acts nontrivially on the tensor product V i ⊗ V k . In the case under consideration each space V i (V k ) is assumed to be a vector space of some representation of the group SL(N, C). We are interested in constructing an R-operator which acts on the space L 2 (Z) ⊗ L 2 (Z) and is invariant with respect to SL(N, C) transformations, The form of the R-operator strongly depends on the spaces V 1 and V 2 which it acts on. The Roperator has the simplest form when both space V 1 and V 2 is the vector space of the fundamental This solution can also be represented in the form Substituting the generators e nm i by the generators E nm in some generic representation of the SL(N, C) group in the above formula one gets the R-operator on the space V 1 ⊗ V f . Such an operator is called Lax operator. Let us recall here that in a generic representation of the SL(N, C) group there are two sets of generators, holomorphic E ik and the antiholomorphicĒ ik , so that we define two Lax operators, the holomorphic one, L(u), and the antiholomorphic one, L(ū), (3. 2) The Yang-Baxter equation (3.1) on the spaces V 1 ⊗ V 2 ⊗ V f takes the form These equations can be considered as defining equations for the operator R 12 (u,ū). Let us notice that the R-operator depends on two spectral parameters u andū, which are not supposed to be related each other. We will show later that the spectral parameters are subject to the restriction, u −ū = n, where n is an integer number 3 . In the next subsection we discuss the approach for solving equations (3.3) suggested in [15].

Factorized ansatz for R-matrix
Let us remark that the Lax operator (3.2) depends on the spectral parameter u and N parameters σ k . Since the character α (equation (2.2)) and, hence, the generators of the SL(N, C) group depend only on the differences σ k,k+1 = σ k − σ k+1 , one concludes that the Lax operators depend on N -independent parameters which can be chosen as u k = u− σ k , k = 1, . . . , N , L(u) = L(u 1 , . . . , u N ). These parameters appear quite naturally when one uses the representation (2.10) for the generators. Indeed, the Lax operator can be represented as and the parameters u k are nothing else as diagonal elements of the matrix (u − σ − D). Let us represent the operator R 12 in the form R 12 = P 12 R 12 . Note, that since V 1 = V 2 = L 2 (Z) the permutation operator P 12 is unambiguously defined on V 1 ⊗ V 2 . We will associate the matrix variables z and w with the spaces V 1 and V 2 , respectively, so that the vectors in V 1 ⊗ V 2 are functions Φ(z, w). The action of the permutation operator P 12 is defined in a conventional way, P 12 Φ(z, w) = Φ(w, z). The Yang-Baxter relation (3.3) can be rewritten as follows 4 It was suggested in [15] to look for the operator R 12 in the factorized form where each operator R (k) interchanges the arguments u k and v k of the Lax operators, Evidently, if such operators can be constructed then equation (3.5) will follow immediately from equations (3.6) and (3.7). We note also that since any permutation inside the string {u} can be carried out with the use of the operators U k (and similarly for the string {v}), it is sufficient to find an operator which interchanges some elements u k and v i of strings {u} and {v}.

Exchange operator
In this subsection we will construct operator S which exchanges the arguments u 1 and v N of the Lax operators L 1 and L 2 , It turns out that the operator S has a surprisingly simple form. Taking into account that the character α(h) (equation (2.2)) can be written in the form it is easy to figure out that operator S has to intertwine the representations T α ⊗ T β and where the characters α and α ′ , (β and β ′ ) are related to each other as follows (see equation (2.2)) It turns out that the simplest operator intertwining the representations in question gives a solution to equation (3.8). To construct an operator with the required transformation properties let us consider the matrix w −1 z. Under the transformation z → g −1 z = (zḡ)h(z, g), w → g −1 w = (wḡ)h(w, g) it transforms as follows Taking into account that the matrices z and w are lower triangular matrices while h(z, g), h(w, g) are upper triangular ones, one finds that the matrix element (w −1 z) N 1 transforms in a simple way It suggests to define the operator S as follows Notice, that the difference γ −γ has to be an integer number.
It follows from equations (3.11) and (3.12) that the operator S has necessary transformation properties (3.10). Thus it remains to show that the operator S satisfies equation (3.8). We start with two useful identities for the Lax operator L(u 1 , . . . , u N ) Then it is straightforward to derive where u is a diagonal matrix, u = k u k e kk . Taking into account that γ = u 1 − v N one finds that the first term in the rhs is the Lax operator L(v N , u 2 , . . . , u N ). Quite similarly, one derives for the Lax operator L 2 where v = k v k e kk . The equation (3.8) then follows immediately from equations (3.14), (3.15) and (3.13).

Permutation group and the star-triangle relation
Let us consider the operators we have constructed in more details. As was already noted, a character α can be written in the form (3.9) so that the representation T α is completely determined by the numbers u 1 , . . . , u N (andū 1 , . . . ,ū N which are always implied). Respectively, the tensor product T α ⊗ T β is determined by the numbers u 1 , . . . , u N and v 1 , . . . , v N where the latter refer to the representation T β . Let us join these parameters into the string (notice the inverse order of the parameters {u} and {v}) and accept the notation T v for the tensor product T α ⊗T β . The string v fixes not only the characters α and β but also the spectral parameters, u and v of the Lax operators L 1 and L 2 . For a definiteness we will assume that the representations T α and T β are unitary. We recall that this results in the restriction (2.5) which can be represented as We will assume also that ( Under this condition the representation T v ′ where v ′ is an arbitrary permutation of the string v is unitary. The vector space of the representation T v will be denoted as V v = L 2 (Z × Z).
In Section 2.2 we have constructed the intertwining operators U k , equation (2.13). Since one has now two sets of such operators and also the exchange operator S, it is convenient to introduce the notation The operators U k depend on the "quantum numbers" of the space they act on, namely It will always be implied that the argument of the operator U k is determined by the representation T v it acts on. It can be formulated in the following way: Let us consider the lattice L v in C 2N formed by vectors {v i } obtained from the vector v by all possible permutation of its components. For each point v i we identify the corresponding space V v i . The operators U k map the spaces attached to the lattice points related by elementary permutations to one another. So one can omit the index k of the operator but show the lattice points v i and v i ′ as the argument of the operator U k (v k+1,k ) → U(v i ′ , v i ). So far we consider operators which connect the points related by elementary permutation. But it is obvious that one can construct the operator which maps V v i to V v j where v i and v j are two arbitrary points of the lattice. We will show that such operator depends only on the points v i and v j , and does not depend on the path connecting these points. To this end one has to show that the operators U k satisfy the same commutation relations as the operators of the elementary permutation P k : P 2 k = 1l, [P k , P n ] = 0 for |n − k| > 1 and P k+1 P k P k+1 = P k P k+1 P k . Thus we have to show that Taking into account equations (2.19) which hold for the operators U k one easily figures out that it is sufficient to check only those of equations (3.17) which involve the operator U N . The first equation, U N U N = 1l, is obvious. The second one, U N U k = U k U N , |N − k| > 1, follows immediately from equations (2.13) and (3.12) if one takes into account that D z k+1,k (w −1 z) N 1 = D w i+1,i (w −1 z) N 1 = 0 for k > 1 and i < N − 1. The last identity, equation (3.17c), is nothing else as a slightly camouflaged version of the integral identity known as the star-triangle relation 5 . If we restore the dependence on the spectral parameters it takes the form Let us consider equation (3.18a). We recall that U N +1 (a) = (D z 21 ) a (Dz 21 )ā. After the change of variables (2.14) this operator takes the form U N +1 (a) = ∂ a∂ā , where ∂ ≡ ∂ x and x = x 21 . It turn, the matrix element (w −1 z) N 1 is a linear function of x, where C and D depend on other variables and can be treated as constants. It is easy to see that equation (3.18a) is equivalent to the statement about commutativity of the operators The latter follows from the observation that both operators are some functions of the operators x∂ andx∂, G a = g a (x∂,x∂) and H b = h a (x∂,x∂). This line of reasoning is due to Isaev [36]. The property of commutativity of the operators H b and G a can be expressed as the integral identity which presents a more conventional form of the star-triangle relation By definition, the vectors v i , v j are related by some permutation P (ij), v j = P (ij)v i . The permutation P (ij) can be always represented as the product of the elementary permutations, P (ij) = P im · · · P i 1 . Then we define the operator U(v j , v i ) as Due to equations (3.17) the operator U(v j , v i ) does not depend on the way the decomposition of the permutation P (ij) onto elementary permutations is done. The operator U(v ′ , v) intertwines the representations T v and T v ′ and interchanges the parameters in the product of Lax operators as follows It follows from equation (3.5) that the operator R 12 (u − v) can be identified with U(v ′ , v) for the special v ′ . Namely, one gets As a consequence the R-operator takes the form Note, that since each operator U k depends on the difference v k,k+1 , the operator U(v ′ , v) depends on the differences σ k,k+1 and ρ k,k+1 , which specify the representations T α and T β in the tensor product T α ⊗ T β , and the spectral parameter u − v.
It is quite easy to show that the constructed R-operator satisfies the Yang-Baxter relation (3.1). The latter can be rewritten in the form where we have shown all arguments of R-operators explicitly, that is R 12 (u − v) = R 12 ( u, v), u = (u 1 , . . . , u N ) and so on. Since P 23 P 12 P 23 = P 12 P 23 P 12 one has to check that the product of R-operators in the l.h.s and r.h.s are equal. One can easily find that the operators on the both sides result in the same permutation of the parameters ( u, v, w) → ( w, v, u). Since the operators are constructed from the operators U k which obey the relations (3.17), these operators, as was explained earlier, are equal.

Factorizing operators and Baxter Q-operators
In the previous section we have obtained the expression for the operator R 12 , equation (3.20). For the construction of the Baxter Q-operators it is quite useful to represent the R-operator in the factorized form (3.6). Each operator R (k) interchanges the components u k and v k of the Lax operators, equation (3.7) and can be expressed in terms of the elementary permutation operators U k as follows Taking into account the definition of the operators U k , equation (3.16), it is straightforward to check that the operator R (k) satisfies equation (3.7). Indeed, sequence of the operators in (4.1) results in the following permutation of the parameters u and v in the product of Lax operators where we have displayed the relevant arguments only.
It is easy to show that the operator (4.1) intertwines the representations T α ⊗ T β and T α k,λ ⊗ T β k,−λ , where The operator R (k) is completely determined by the characters α, β and the parameter λ, (λ), which we will refer to as the spectral parameter, i.e. R (k) = R (k) (λ|α, β). Henceforth we accept the shorthand notation, R (k) (λ|α, β) → R (k) (λ), omitting the dependence on the characters α, β. To avoid misunderstanding we stress that the product of operators R (i) (µ)R (k) (λ) reads in explicit form as In a full analogy with R-operator we accept the notation R (k) ab for the operator which acts nontrivially on the tensor product of spaces V a and V b . The expression (3.6) for the R-operator can be written in the form Equations (4.5a) and (4.5b) follow from equations (4.1) and (3.12), while to prove the last three equations it is sufficient to check that the operators on both sides result in the same permutation of the parameters u, v, w, (see equation (3.21)). Namely, the equations (4.5c)-(4.5e) are the deciphered form of the equations 12 . Let us notice that equation (4.5e) indicates that the operators R k (u k − v k ) in the expression for the R-matrix, equation (4.4), can stand in arbitrary order.
We remark here that YBE for the R-operator (3.6) is the corollary of the properties (4.5c), (4.5d) and (4.5e) of the operators R (k) . One can expect that these equations will hold for the R matrix for the generic representation of sl(N ) algebra since they express the property of consistency of equations (3.7). At the same time the possibility to represent the operators R (k) as the product of elementary operators U k is a specific feature of the principal series representations. Taking in mind this possibility we give here the alternative proof of YBE. We recall that YBE is equivalent to the following identity for the operators R ik (see discussion after equation (3.21)) respectively. By virtue of equation (4.5d) one finds that they are equal.
So we have shown that if the operators R (k) obey equations (4.5c)-(4.5e) then the Rmatrix (4.4) satisfies YBE (3.1). Moreover, the operators R (k) can be considered as a special case of the R-operator. Namely, let us take the character β(h) to be β(h) = α k,λ (h). Then it follows from equations (4.2) that the operator is a SL(N, C) invariant operator, Let us note that the operators R (k) (λ) depend only on the part of parameters σ i and ρ i which specify the characters α and β. For instance, an inspection of equation (4.1) shows that the operator R (N ) (λ) depends only on the parameters σ k,k+1 = σ k − σ k+1 (the character α), (R (N ) (λ) ≡ R (k) (λ|σ 12 , . . . , σ N −1,N )), while the operator R (1) (λ) depends on ρ k,k+1 (the character β), and similarly for others. It means that the condition β(h) = α k,λ (h) does not reduce the number of the independent parameters the operator R (k) depends on. Further, let us return to equation (4.4) and put u k − v k = λ. Then, provided that β(h) = α k,λ (h) one finds u i − v i = 0 unless i = k. Taking into account equation (4.5a) and It means that the operator R (k) coincides with the R-operator at the "special point". In case λ +λ * = 0, the operator R The operator W k can be easily read off from equation (4.1). It factorizes into the product of two operators, W k = W (2) k act in the spaces V 1 and V 2 , respectively, Here we display explicitly the arguments of the intertwining operators.
Let us consider the homogeneous spin chain of length L and define the operators Q k (λ) by 6 The operators Q k act on the Hilbert space V 1 ⊗ · · · ⊗ V L = L 2 (Z × · · · × Z). It is assumed that the representation T α of the group SL(N, C) is defined at each site. The trace is taken over the auxiliary space V 0 = L 2 (Z) and exists (we give below the explicit realization of Q k (λ) as an integral operator). The operators (4.9) can be identified as the Baxter operators for the noncompact SL(N, C) invariant spin magnet. Since the operators R (k) i0 (λ) coincide with the R i0 -operator for the special choice of auxiliary space one concludes that the Baxter operators Q k (λ) commute with each other for different values of a spectral parameter As follows from equation (4.8) all Baxter operators are unitary equivalent, namely (4.10) This property is a special feature of the principal series SL(N, C) magnets (see also [16] where the SL(2, C) magnet was studied in detail). For conventional (infinite-dimensional) sl(N ) magnets the property (4.10) does not hold (see [34,37,38]). Another property of the Baxter operators which we want to mention is related to the factorization property of the transfer matrix. The latter is defined as where the trace is taken over the auxiliary space and the index β refers to the representation T β which is defined on the space V 0 . We will suppose that the trace exists (this is true at least for N = 2 [16]). Under this assumption one can prove that the transfer matrix factorizes in the product of Q k operators as follows where P is the operator of cyclic permutation PΦ(z 1 , . . . , z L ) = Φ(z L , z 1 , . . . , z L−1 ) and the parameters ρ k determine the character β, equations (2.2), (2.3). The proof repeats that given in [38] and is based on disentangling of the trace equation (4.11) with the help of the relation (4.5c) for R (k) operators. The analysis of analytical properties of the Baxter Q-operators and a derivation of the T-Q relation will be given elsewhere.

Integral representation
In this subsection we give the integral representation for the operators in question. Due to equations (4.8) and (4.10) it suffices to consider the operators Q N and R (N ) only. The latter can be represented as the product of three operators, R (N ) (λ) = X 1 (λ)S(λ)X 1 . The operator S(λ) defined in equation (3.12) is an multiplication operator The operator X 1 is given by a product of the operators (U 1 ⊗ 1l)(U 2 ⊗ 1l) · · · (U N −1 ⊗ 1l), see equations (4.1) and (3.16). Making use of equation (2.17) one gets (4.13) The factor A is defined in equation (2.18), σ kN = σ k − σ N , and we recall that [z] a ≡ z azā . Similarly, for the operator X 1 (λ) one derives withz Thus the operator R (N ) (λ) can be represented in the following form where To obtain the integral representation for the Baxter operator Q N we notice that the operator R . Making use of equations (4.7) and (4.9) one finds that the kernel Q for a spin chain of a length L has the following form where the periodic boundary conditions are implied z L+1 = z 1 . Further, taking into account equations (4.12), (4.13), (4.14) one can represent the operator Q (N ) in the form The operator X k , which acts non-trivially only on the k-th component in the tensor product V 1 ⊗ V 2 ⊗ · · · ⊗ V L , is given by equation (4.13). In turn, for the operator Q(λ) one obtains [Q(λ)Φ](z 1 , . . . , z L ) = q L (λ) A(λ − σ j ).

SL(2, C) magnet
Closing this section we give the explicit expression for the operator R (N ) and the Baxter Qoperator for the SL(2, C) spin magnet. For N = 2 equation (4.15) takes the form where we put z 12 = z, w 12 = w. Integrating over ζ with help of equation ( Similarly, for the operator R With the help of equations (4.4) and (4.3) one can easily find that the integral kernel of the operator R 12 (λ) takes the form The straightforward check shows that up to a prefactor this expression coincides with the kernel for the R-operator obtained in [16].
The expression for the Q operator for N = 2 case can be rewritten in the form Again it can be checked that the expression (4.17) together with equation (4.16) matches (up to a λ-dependent prefactor) the expression for the kernel of the Baxter Q-operator obtained in [16]. Finally, we give one more representation for the Baxter operator Q (N =2) (λ) which is instructive to compare with the expression for the Baxter Q-operator for su(1, 1) spin chain [39].

Summary
In this paper we developed an approach of constructing the solutions of the Yang-Baxter equation for the principal series representations of SL(N, C). We obtained the R-operator as the product of elementary operators U k , k = 1, . . . , 2N − 1. The latter, except for the operator U N , are the intertwining operators for the principal series representations of SL(N, C). The operator U N is a special one. It intertwines the tensor products of SL(N, C) representation and a product of Lax operators, see equations (3.8), (3.10). The operators U k satisfy the same commutation relations as the operators of the elementary permutation, P k . In other words, they define the representation of the permutation group S N . It means that any two operators constructed from the operator U k corresponding to the same permutation are equal. As a result a proof of the Yang-Baxter relation becomes trivial.
We have represented the R-operator in the factorized form and constructed the factorizing operators R (k) . Having in mind application of this approach to spin chains with generic representations of sl(N ) algebra we figured out which properties of the factorizing operators are vital to proving of the Yang-Baxter equation. The operators R k play the fundamental role in constructing the Baxter operators. Namely, using them as building blocks, one can construct the commutative family of the operators Q k which can be identified as the Baxter operators. We obtain the integral representation for the latter and show that for N = 2 our results coincide with the results of [16].