R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain

The problem of constructing the SL(N,C) invariant solutions to the Yang-Baxter equation is considered. The solutions (R-operators) for arbitrarily principal series representations of SL(N,C) are obtained in an explicit form. We construct the commutative family of the operators Qk(u) which can be ide...

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Дата:	2006
Автори:	Derkachov, S.É., Manashov, A.N.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут математики НАН України 2006
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/146069
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain / S.É. Derkachov, A.N. Manashov // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 39 назв. — англ.

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spelling	irk-123456789-1460692019-02-07T01:24:13Z R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain Derkachov, S.É. Manashov, A.N. The problem of constructing the SL(N,C) invariant solutions to the Yang-Baxter equation is considered. The solutions (R-operators) for arbitrarily principal series representations of SL(N,C) are obtained in an explicit form. We construct the commutative family of the operators Qk(u) which can be identified with the Baxter operators for the noncompact SL(N,C) spin magnet. 2006 Article R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain / S.É. Derkachov, A.N. Manashov // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 39 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 82B23; 82B20 http://dspace.nbuv.gov.ua/handle/123456789/146069 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The problem of constructing the SL(N,C) invariant solutions to the Yang-Baxter equation is considered. The solutions (R-operators) for arbitrarily principal series representations of SL(N,C) are obtained in an explicit form. We construct the commutative family of the operators Qk(u) which can be identified with the Baxter operators for the noncompact SL(N,C) spin magnet.
format	Article
author	Derkachov, S.É. Manashov, A.N.
spellingShingle	Derkachov, S.É. Manashov, A.N. R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Derkachov, S.É. Manashov, A.N.
author_sort	Derkachov, S.É.
title	R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain
title_short	R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain
title_full	R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain
title_fullStr	R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain
title_full_unstemmed	R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain
title_sort	r-matrix and baxter q-operators for the noncompact sl(n,c) invariant spin chain
publisher	Інститут математики НАН України
publishDate	2006
url	http://dspace.nbuv.gov.ua/handle/123456789/146069
citation_txt	R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain / S.É. Derkachov, A.N. Manashov // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 39 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT derkachovse rmatrixandbaxterqoperatorsforthenoncompactslncinvariantspinchain AT manashovan rmatrixandbaxterqoperatorsforthenoncompactslncinvariantspinchain
first_indexed	2025-07-10T23:05:30Z
last_indexed	2025-07-10T23:05:30Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 084, 20 pages R-Matrix and Baxter Q-Operators for the Noncompact SL(N, C) Invariant Spin Chain? Sergey É. DERKACHOV † and Alexander N. MANASHOV ‡§ † St.-Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, 191023 St.-Petersburg, Russia E-mail: derkach@euclid.pdmi.ras.ru ‡ Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany E-mail: alexander.manashov@physik.uni-regensburg.de § Department of Theoretical Physics, Sankt-Petersburg University, St.-Petersburg, Russia Received October 30, 2006; Published online December 02, 2006 Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper084/ Abstract. The problem of constructing the SL(N, C) invariant solutions to the Yang– Baxter equation is considered. The solutions (R-operators) for arbitrarily principal series representations of SL(N, C) are obtained in an explicit form. We construct the commutative family of the operators Qk(u) which can be identified with the Baxter operators for the noncompact SL(N, C) spin magnet. Key words: Yang–Baxter equation; Baxter operator 2000 Mathematics Subject Classification: 82B23; 82B20 To the memory of Vadim Kuznetsov 1 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 con- structed 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 ?This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”. The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html mailto:derkach@euclid.pdmi.ras.ru mailto:alexander.manashov@physik.uni-regensburg.de http://www.emis.de/journals/SIGMA/2006/Paper084/ http://www.emis.de/journals/SIGMA/kuznetsov.html 2 S.É. Derkachov and A.N. Manashov 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 factori- zed into the product of Baxter operators. This property seems to be quite a general feature of the noncompact magnets1 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 op- erators 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 di- rectly. 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 noncom- pact 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 non- trivial 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 rep- resentation. 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 prop- erty. 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 fol- lows [30, 31]. Let Z and H be the groups of the lower triangular matrices with unit elements 1This property holds also for the integrable models with Uq(ŝln) symmetry, at least for n = 2, 3 [18, 19]. R-Matrix and Baxter Q-Operators 3 on a diagonal and the upper triangular matrices with unit determinant, respectively, z =  1 0 0 . . . 0 z21 1 0 . . . 0 z31 z32 1 . . . 0 ... ... ... . . . ... zN1 zN2 . . . zN,N−1 1  ∈ Z, h =  h11 h12 h13 . . . h1N 0 h22 h23 . . . h2N 0 0 h33 . . . h3N ... ... ... . . . ... 0 0 0 . . . hN,N  ∈ H. 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 z1 ∈ Z satisfying the condition g−1 · z = z1 · h (2.1) will be denoted by zḡ, so that g−1z = 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 α(h) = N∏ k=1 h−σk−k kk h̄−σ̄k−k kk , (2.2) where h̄kk ≡ (hkk)∗ is the complex conjugate of hkk, whereas in general σ∗ k 6= σ̄k. Since det h = 1 the function α(h) depends only on the differences σk,k+1 ≡ σk − σk+1 and can be rewritten in the form α(h) = N−1∏ k=1 ∆1−σk,k+1 k ∆̄1−σ̄k,k+1 k = N−1∏ k=1 ∆nk k \|∆k\|2(1−σ̄k,k+1), where ∆k = k∏ i=1 hii and nk = σ̄k,k+1 − σk,k+1, k = 0, . . . , N − 1, are integer numbers2. One can always assume that parameters σk satisfy the restriction σ1 + σ2 + · · ·+ σN = N(N − 1)/2. (2.3) The map g → Tα(g), where [Tα(g)Φ](z) = α(h−1)Φ(zḡ), (2.4) defines a principal series representation of the group SL(N, C) on a suitable space of functions on the group Z, Φ(z) = Φ(z21, z31, . . . , zNN−1) [30, 31]. The operator Tα(g) is a unitary operator on the Hilbert space L2(Z), 〈Φ1\|Φ2〉 = ∫ ∏ 1≤i<k≤N d2zki (Φ1(z))∗ Φ2(z) , if the character α′(h) = α(h) N∏ k=1 \|hkk\|2k is a unitary one, i.e. \|α′\| = 1. This condition holds if σ∗ k,k+1 + σ̄k,k+1 = 0 for k = 1, . . . , N − 1, i.e. σk,k+1 = −nk 2 + iλk, σ̄k,k+1 = nk 2 + iλk, k = 1, 2, . . . , N − 1, (2.5) nk is integer and λk is real. The unitary principal series representation Tα is irreducible. Two representations Tα and Tα′ are unitary equivalent if and only if the corresponding parameters (σ1, . . . , σN ) and (σ′ 1, . . . , σ ′ N ) are related to each other by a permutation [30, 31]. 2From now in, since each variable a comes along with its antiholomorphic twin ā we will write down only holomorphic variant of equations. 4 S.É. Derkachov and A.N. Manashov 2.1 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[ Tα ( 1l + ∑ ik εikEki ) Φ ] (z) = Φ(z) + ∑ ik εikEkiΦ(z) + O(ε2), where Eik, (1 ≤ i, k ≤ N), are the generators in the fundamental representation of the SL(N, C) group, (Eik)nm = δinδkm − 1 N δikδnm. The generators Eik are linear differential operators in the variables zmn, (1 ≤ n < m ≤ N) which satisfy the commutation relation [Eki, Enm] = δinEkm − δkmEni. (2.6) It follows from definition (2.4) that the lowering generators Eki, k > i, are the generators of left shifts, Φ(z) → L(z0)Φ(z) = Φ(z−1 0 z). In a similar way we define the generators of right shifts, Φ(z) → R(z0)Φ(z) = Φ(zz0), Φ ( z ( 1l + ∑ k>i εikEki )) = ( 1 + ∑ k>i εikDki + O ( ε2 )) Φ(z). (2.7) Since right and left shifts commute one concludes that [Eki, Dnm] = 0 (k > i, n > m). Clearly, the generators Dki satisfy the same commutation relation as Eki, equation (2.6) [Dki, Dnm] = δinDkm − δkmDni. The explicit expression for the generators of left and right shifts reads Eki = − i∑ m=1 zim ∂ ∂zkm , Dki = N∑ m=k zmk ∂ ∂zmi = − i∑ m=1 z̃im ∂ ∂z̃km , (2.8) where z̃ki = (z−1)ki and we recall that zii = 1. Let us notice here that the operator Dki 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 Eki can be expressed in terms of the generators Dki as follows Eki = − ∑ mn zim ( Dnm + δnmσm )( z−1 ) nk , (2.9) where Dnm is nonzero only for n > m. To derive equation (2.9) it is sufficient to notice that for the infinitesimal transformation g = 1l + ε · E = 1l + ∑ ik εikEki the elements of the Gauss decomposition of the matrix g−1z can be represented in the following form: zḡ = z(1l − ε−(z)) and h = 1l− ε+(z). The matrices ε−(z) and ε+(z) are the lower and upper diagonal parts of the matrix z−1(ε·E)z. Namely, the matrix ε−(z)ki is nonzero only for k > i, ε−(z)ki = (z−1(ε·E)z)ki, while ε+(z)ki = (z−1(ε·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 z0 = 1l − ε−(z). Finally, taking into account that ∑ σk = N(N − 1)/2 one obtains the representation (2.9) for the symmetry generators. R-Matrix and Baxter Q-Operators 5 It is convenient to rewrite equation (2.9) in the matrix form E = −z (D + σ) z−1, (2.10) where E = ∑ mn emnEnm, D = ∑ n>m emnDnm and σ = ∑ n ennσn and the matrices enm form the standard basis in Mat(N × N), (enm)ik = δinδmk. Let us recall that z and z−1 are lower triangular matrices while D + σ is an upper triangular matrix D + σ =  σ1 D21 D31 . . . DN1 0 σ2 D32 . . . DN2 ... ... . . . ... ... 0 0 . . . σN−1 DN,N−1 0 0 . . . 0 σN  (2.11) The dependence of the generators E on the representation Tα resides in the parameters σn entering the matrix σ. The same formulae hold for the antiholomorphic generators Ē. Closing this subsection we give some identities that will be useful in the further analysis: Dkiz = z eki, (2.12a) Dki z −1 = −ekiz −1, (2.12b) [Dki, D] = ∑ n<i Dkneni − ∑ n>k eknDni. (2.12c) The first two formulae follow directly from definition (2.7), while the last one is a consequence of the commutation relations for the generators Dki. We remind also that the operators Dki are defined (nonzero) only for k > i. 2.2 Intertwining operators It is known that two principal series representations Tα and Tα′ are equivalent if the parameters which specify the representations, ~σ = (σ1, . . . , σN ) 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 {σ̄k}, {σ̄′ 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, Pk, which interchange the k-th and k + 1-st component of the vectors, ~σ (~̄σ), Pk, (. . . , σk, σk+1, . . .) = (. . . , σk+1, σk, . . .). The operator Uk intertwining the representations, Tα and Tα′ which differ by the elementary permutation of spins Pk, α′ = Pkα, has the form Uk = D σk,k+1 k+1,k D̄ σ̄k,k+1 k+1,k , UkT α = Tα′ Uk, (2.13) where σk,k+1 = σk−σk+1 (σ̄k,k+1 = σ̄k− σ̄k+1), and Dk+1,k(D̄k+1,k) is the generator of the right shift. Let us note that the operator Dk+1,k depends on the variables znm in the k-th and k+1-st columns of the matrix z. After the change of variables zk+1,k = xk+1,k, zm,k+1 = xm,k+1, zm,k = xm,k + xk+1,kxm,k+1, k < m ≤ N (2.14) the operator Dk+1,k turns into a derivative with respect to xk+1,k, Dk+1,k = ∂xk+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 + σ̄k,k+1 = 0, the operator Uk is a unitary operator on L2(Z). 6 S.É. Derkachov and A.N. Manashov To prove that operator Uk is an intertwining operator it is sufficient to check that it in- tertwines the generators E and E′ in the representations Tα and Tα′ , E′Uk = UkE. To this end it is convenient to use the representation (2.10) for the generators. Using commutation relations (2.12) one finds Ukz = z ( 1 + αD−1 k+1,kek+1,k ) Uk, Ukz −1 = ( 1− αD−1 k+1,kek+1,k ) z−1Uk (2.15) and UkD = ( D + αD−1 k+1,k (∑ n<k Dk+1,nenk − ∑ n>k+1 ek+1,nDn,k )) Uk, (2.16) where α = σk,k+1. (Let us note that the operator D−1 k+1,k commutes with ek+1,kz −1 and with the operators in the sum in (2.16), so its position can be changed). Starting from UkE and moving the operator Uk to the right with the help of equations (2.15) and (2.16) one gets after some simplifications UkE = −z ( D + σ + α(ek+1,k+1 − ek,k)− α(α− σk,k+1)ek+1,kD −1 k+1,k ) z−1Uk = −z(D + σ′)z−1Uk = E′Uk. (One has to take into account that α = σk,k+1 and the matrix σ′ = σ+σk,k+1(ek+1,k+1−ek,k) differs from the matrix σ by the transposition σk ↔ σk+1.) Making use of the change of variables (2.14) one can represent the operator Uk in the form of an integral operator [UkΦ](z) = A(σk,k+1) ∫ d2ζ[zk+1,k − ζ]−1−σk,k+1Φ(zζ), (2.17) where [z]σ = zσ z̄σ̄, A(σ) def= A(σ, σ̄) = 1 π iσ̄−σΓ(1 + σ)/Γ(−σ̄) (2.18) and zζ = z ( 1 + (ζ − zk+1,k)ek+1,k ) . Let us note that the matrices z and zζ differ from each other by the elements in the k-th column only. The operator Uk depends on the difference of σk,k+1, Uk = U(σk,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 Uk+1Uk, the first operator is Uk = Uk(σk,k+1), while the second one is Uk+1 = Uk+1(σk,k+2), since the operator Uk interchanges the parameter σk with σk+1. Similarly, one finds that Uk Uk = Uk(σk+1,k)Uk(σk,k+1) = 1l. The intertwining operators Uk satisfy the same commutation relations as the operators of elementary permutations Pk UkUk = 1l, (2.19a) UkUn = Uk Un, for \|k − n\| > 1 (2.19b) UkUk+1Uk = Uk+1UkUk+1. (2.19c) The first relation had been already explained. The second one is a trivial consequence of the commutativity of the generators of right shifts, [Dk+1,k, Dn+1,n] = 0 for \|k−n\| > 1. The last re- lation can be checked by the direct calculation with the help of the integral representation (2.17). Obviously, the operator intertwining the representations Tα and Tα′ such that the charac- ters α and α′ are related to each other by some permutation can be constructed as a certain combination of the operators Uk, k = 1, . . . , N − 1. The intertwining operators, Uk, play an important role in the construction of an R-operator which we will discuss in the next section. R-Matrix and Baxter Q-Operators 7 3 R-operator Let us recall that an R-operator is a linear operator which acts on the tensor product of two spaces V1 ⊗ V2, depends on the spectral parameter u and satisfies the Yang–Baxter relation R12(u− v)R13(u− w)R23(v − w) = R23(v − w)R13(u− w)R12(u− w). (3.1) As usual, it is implied that the operator Rik(u) acts nontrivially on the tensor product Vi⊗Vk. In the case under consideration each space Vi (Vk) 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 L2(Z)⊗ L2(Z) and is invariant with respect to SL(N, C) transformations, [Tα1(g)⊗ Tα2(g),R12(u)] = 0. The form of the R-operator strongly depends on the spaces V1 and V2 which it acts on. The R- operator has the simplest form when both space V1 and V2 is the vector space of the fundamental (N -dimensional) representation of the SL(N, C) group, V1 = V2 = Vf . Namely, in the case that V1 = V2 = V3 = Vf the solution of the Yang–Baxter equation (3.1) is given by the operator Rik(u) = u + Pik, where Pik is the permutation operator on Vi ⊗ Vk. This solution can also be represented in the form Rik(u) = u + ∑ mn enm i emn k , Substituting the generators enm i by the generators Enm in some generic representation of the SL(N, C) group in the above formula one gets the R-operator on the space V1 ⊗ Vf . 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 Eik and the antiholomorphic Ēik, so that we define two Lax operators, the holomorphic one, L(u), and the antiholomorphic one, L̄(ū), L(u) = u + ∑ mn emn Enm, L̄(ū) = ū + ∑ mn emnĒnm. (3.2) The Yang–Baxter equation (3.1) on the spaces V1 ⊗ V2 ⊗ Vf takes the form R12(u− v, ū− v̄)L1(u)L2(v) = L2(v)L1(u)R12(u− v, ū− v̄), (3.3a) R12(u− v, ū− v̄)L̄1(ū)L̄2(v̄) = L̄2(v̄)L̄1(ū)R12(u− v, ū− v̄). (3.3b) These equations can be considered as defining equations for the operatorR12(u, ū). Let us notice that the R-operator depends on two spectral parameters u and ū, 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 number3. In the next subsection we discuss the approach for solving equations (3.3) suggested in [15]. 3.1 Factorized ansatz for R-matrix Let us remark that the Lax operator (3.2) depends on the spectral parameter u and N parame- ters σk. Since the character α (equation (2.2)) and, hence, the generators of the SL(N, C) group depend only on the differences σk,k+1 = σk − σk+1, one concludes that the Lax operators depend on N -independent parameters which can be chosen as uk = u−σk, k = 1, . . . , N , L(u) = 3The quantization of the spectral parameters for the SL(2, C) spin magnet was observed in [16]. 8 S.É. Derkachov and A.N. Manashov L(u1, . . . , uN ). These parameters appear quite naturally when one uses the representation (2.10) for the generators. Indeed, the Lax operator can be represented as L(u) = z ( u− σ −D ) z−1, (3.4) and the parameters uk are nothing else as diagonal elements of the matrix (u− σ −D). Let us represent the operator R12 in the form R12 = P12R12. Note, that since V1 = V2 = L2(Z) the permutation operator P12 is unambiguously defined on V1⊗V2. We will associate the matrix variables z and w with the spaces V1 and V2, respectively, so that the vectors in V1⊗V2 are functions Φ(z, w). The action of the permutation operator P12 is defined in a conventional way, P12Φ(z, w) = Φ(w, z). The Yang–Baxter relation (3.3) can be rewritten as follows4 R12(u− v)L1(u1, . . . , uN )L2(v1, . . . , vN ) = L1(v1, . . . , vN )L2(u1, . . . , uN )R12(u− v). (3.5) The parameters vk are defined as vk = v−ρk where the parameters {ρ} specify the representation of SL(N, C) group on the space V2. It is seen from equation (3.5) that action of the operator R12(u−v) results in permutation of the arguments {u} and {v} of the Lax operators L1 and L2. Since we have already constructed the operators {Uk} which interchange the components of the string {u} ({v}) it seems reasonable to try to find operators that carry out permutations of the strings {u} and {v}. It was suggested in [15] to look for the operator R12 in the factorized form R12(u− v) = R(1) R(2) · · ·R(N), (3.6) where each operator R(k) interchanges the arguments uk and vk of the Lax operators, R(k)L1(u1, . . . , uk, . . . uN )L2(v1, . . . , vk, . . . , vN ) = L1(u1, . . . , vk, . . . , uN )L2(v1, . . . , uk, . . . , vN )R(k). (3.7) 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 Uk (and similarly for the string {v}), it is sufficient to find an operator which interchanges some elements uk and vi of strings {u} and {v}. 3.2 Exchange operator In this subsection we will construct operator S which exchanges the arguments u1 and vN of the Lax operators L1 and L2, S L1(u1, u2, . . . , uN )L2(v1, . . . , vN−1, vN ) = L1(vN , u2, . . . , uN )L2(v1, . . . , vN−1, u1)S. (3.8) 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 α(h) = N∏ k=1 huk−k kk h̄ūk−k kk , (3.9) 4From now on we will write down the equations in the holomorphic sector only and suppress the dependence on “barred” variables, ū, v̄, etc for brevity. R-Matrix and Baxter Q-Operators 9 it is easy to figure out that operator S has to intertwine the representations Tα ⊗ Tβ and Tα′ ⊗ Tβ′ , S ( Tα ⊗ Tβ ) = ( Tα′ ⊗ Tβ′) S, (3.10) where the characters α and α′, (β and β′) are related to each other as follows (see equation (2.2)) α′(h) = hvN−u1 11 h̄v̄N−ū1 11 α(h), β′(h) = hu1−vN NN h̄ū1−v̄N NN β(h). 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−1z. Under the transformation z → g−1z = (zḡ)h(z, g), w → g−1w = (wḡ)h(w, g) it transforms as follows w−1z = h−1(w, g)(wḡ)−1(zḡ)h(z, g). 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−1z)N1 transforms in a simple way ( w−1z ) N1 = h−1 NN (w, g) ( (wḡ)−1(zḡ) ) N1 h11(z, g). (3.11) It suggests to define the operator S as follows [S(γ, γ̄)Φ] (z, w) = (( w−1z ) N1 )γ(( w−1z )∗ N1 )γ̄Φ(z, w), (3.12) where γ = u1− vN , γ̄ = ū1− v̄N . Notice, that the difference γ− γ̄ 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(u1, . . . , uN )∑ m z−1 NmLmk = uN z−1 Nk, ∑ m Lkmzm1 = u1zk1, (3.13) which follow immediately from the representation (3.4) and a triangularity of the matrix D, equation (2.11). Next, using the definitions of the operator of right shifts (2.7), (2.8) one finds(∑ k>i eikD (z) ki )( w−1z ) N1 = ∑ k>1 e1k ( w−1z ) Nk = − ( w−1z ) N1 e11 + ∑ k≥1 e1k ( w−1z ) Nk . Then it is straightforward to derive S(γ, γ̄) (L1)nm S−1(γ, γ̄) = ( z(u− γe11 −D)z−1 ) nm + γ (w−1z)N1 zn1w −1 Nm. (3.14) where u is a diagonal matrix, u = ∑ k ukekk. Taking into account that γ = u1 − vN one finds that the first term in the rhs is the Lax operator L(vN , u2, . . . , uN ). Quite similarly, one derives for the Lax operator L2 S(γ, γ̄) (L2)nm S−1(γ, γ̄) = ( w(v + γeNN −D)w−1 ) nm − γ (w−1z)N1 zn1w −1 Nm = ( L2(v1, . . . , vN−1, u1) ) nm − γ (w−1z)N1 zn1w −1 Nm. (3.15) where v = ∑ k vkekk. The equation (3.8) then follows immediately from equations (3.14), (3.15) and (3.13). 10 S.É. Derkachov and A.N. Manashov 3.3 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 Tα is completely determined by the numbers u1, . . . , uN (and ū1, . . . , ūN which are always implied). Respectively, the tensor product Tα ⊗ Tβ is determined by the numbers u1, . . . , uN and v1, . . . , vN where the latter refer to the representation Tβ. Let us join these parameters into the string v = (v1, . . . , vN , u1, . . . , uN ) ≡ (v1, . . . , vN , vN+1, . . . , v2N ) (notice the inverse order of the parameters {u} and {v}) and accept the notation Tv for the tensor product Tα⊗Tβ. The string v fixes not only the characters α and β but also the spectral parameters, u and v of the Lax operators L1 and L2. For a definiteness we will assume that the representations Tα and Tβ are unitary. We recall that this results in the restriction (2.5) which can be represented as u∗k,k+1 + ūk,k+1 = 0, v∗k,k+1 + v̄k,k+1 = 0, k = 1, . . . , N − 1. We will assume also that (vN−u1)∗+(v̄N−u1) = v∗N,N+1−v̄N,N+1 = 0, so that v∗k,k+1+v̄k,k+1 = 0 for k = 1, . . . , 2N − 1. Under this condition the representation Tv′ where v′ is an arbitrary permutation of the string v is unitary. The vector space of the representation Tv will be denoted as Vv = L2(Z × Z). In Section 2.2 we have constructed the intertwining operators Uk, 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 Uk =  1l⊗ Uk, k = 1, . . . , N − 1, S, k = N, Uk−N ⊗ 1l, k = N + 1, . . . , 2N − 1. (3.16) The operators Uk depend on the “quantum numbers” of the space they act on, namely Uk = Uk(vk+1,k), vk+1,k = vk+1 − vk, and Uk : Vv 7→ Vvk , where vk = Pkv. It will always be implied that the argument of the operator Uk is determined by the represen- tation Tv it acts on. It can be formulated in the following way: Let us consider the lattice Lv in C2N formed by vectors {vi} obtained from the vector v by all possible permutation of its components. For each point vi we identify the corresponding space Vvi . The operators Uk 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 vi and vi′ as the argument of the operator Uk(vk+1,k) → U(vi′ ,vi). 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 Vvi to Vvj where vi and vj are two arbitrary points of the lattice. We will show that such operator depends only on the points vi and vj , and does not depend on the path connecting these points. To this end one has to show that the operators Uk satisfy the same commutation relations as the operators of the elementary permutation Pk: P 2 k = 1l, [Pk, Pn] = 0 for \|n−k\| > 1 and Pk+1PkPk+1 = PkPk+1Pk. Thus we have to show that UkUk = 1l, (3.17a) UkUn = UnUk, \|n− k\| > 1 (3.17b) UkUk+1Uk = Uk+1UkUk+1. (3.17c) R-Matrix and Baxter Q-Operators 11 Taking into account equations (2.19) which hold for the operators Uk one easily figures out that it is sufficient to check only those of equations (3.17) which involve the operator UN . The first equation, UNUN = 1l, is obvious. The second one, UNUk = UkUN , \|N − k\| > 1, follows immediately from equations (2.13) and (3.12) if one takes into account that Dz k+1,k(w −1z)N1 = Dw i+1,i(w −1z)N1 = 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 relation5. If we restore the dependence on the spectral parameters it takes the form UN (b)UN+1(a + b)UN (a) = UN+1(a)UN (a + b)UN+1(b), (3.18a) UN (b)UN−1(a + b)UN (a) = UN−1(a)UN (a + b)UN−1(b). (3.18b) Let us consider equation (3.18a). We recall that UN+1(a) = (Dz 21) a(D̄z̄ 21) ā. After the change of variables (2.14) this operator takes the form UN+1(a) = ∂a ∂̄ā, where ∂ ≡ ∂x and x = x21. It turn, the matrix element (w−1z)N1 is a linear function of x, (w−1z)N1 = Cx + D = C(x + x0), x0 = D/C, 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 Ga = (xa ∂a)(x̄ā∂̄ā) and Hb = (∂bxb)(∂̄ b̄x̄b̄). The latter follows from the observation that both operators are some functions of the operators x∂ and x̄∂̄, Ga = ga(x∂, x̄∂̄) and Hb = ha(x∂, x̄∂̄). This line of reasoning is due to Isaev [36]. The property of commutativity of the operators Hb and Ga can be expressed as the integral identity which presents a more conventional form of the star-triangle relation A(a)A(b) ∫ d2ξ 1 [x− ξ]a+1[ξ − ζ]−a−b[ξ − x′]b+1 = A(a + b) [x− ζ]−b[x− x′]1+a+b[x′ − ζ]−a . (3.19) Here the function A(a) is defined in equation (2.17) and [x]a = xax̄ā. Using equation (3.19) and the integral representation (2.17) for the operators Uk it is straightforward to verify equa- tions (3.18). Thus we have proved that the operators Uk (3.16) satisfy the commutation relations (3.17). Now one can construct the operator U(vj ,vi) which maps Vvi 7→ Vvj where vectors vi, vj belong to the lattice Lv. By definition, the vectors vi, vj are related by some permutation P (ij), vj = P (ij)vi. The permutation P (ij) can be always represented as the product of the elementary permutations, P (ij) = Pim · · ·Pi1 . Then we define the operator U(vj ,vi) as U(vj ,vi) = Uim · · ·Ui1 . Due to equations (3.17) the operator U(vj ,vi) 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 Tv and Tv′ U(v′,v)Tv = Tv′U(v′,v) 5The star-triangle relation is, in some sense, a key feature of the Yang–Baxter equation (3.1), see [35] for a nice review. 12 S.É. Derkachov and A.N. Manashov and interchanges the parameters in the product of Lax operators as follows U(v′,v)L1(vN+1, . . . , v2N )L2(v1, . . . , vN ) = L1(v′N+1, . . . , v ′ 2N )L2(v′1, . . . , v ′ N )U(v′,v). It follows from equation (3.5) that the operator R12(u − v) can be identified with U(v′,v) for the special v′. Namely, one gets R12(u− v) = U(v′,v), (3.20) where v = (v1, . . . , vN , u1, . . . , uN ), v′ = (u1, . . . , uN , v1, . . . , vN ). As a consequence the R-operator takes the form R12(u− v) = P12U(v′,v). Note, that since each operator Uk depends on the difference vk,k+1, the operator U(v′,v) depends on the differences σk,k+1 and ρk,k+1, which specify the representations Tα and Tβ in the tensor product Tα ⊗ Tβ, and the spectral parameter u− v. It is quite easy to show that the constructed R-operator satisfies the Yang–Baxter rela- tion (3.1). The latter can be rewritten in the form (P23P12P23) R23(~u,~v)R12(~u, ~w)R23(~v, ~w) = (P12P23P12) R12(~v, ~w)R23(~u, ~w)R12(~u,~v), (3.21) where we have shown all arguments of R-operators explicitly, that is R12(u − v) = R12(~u,~v), ~u = (u1, . . . , uN ) and so on. Since P23P12P23 = P12P23P12 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 Uk which obey the relations (3.17), these operators, as was explained earlier, are equal. 4 Factorizing operators and Baxter Q-operators In the previous section we have obtained the expression for the operator R12, 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 uk and vk of the Lax operators, equation (3.7) and can be expressed in terms of the elementary permutation operators Uk as follows R(k) = (UN+k−1 · · ·UN+1)(Uk · · ·UN−1)UN (UN−1 · · ·Uk) (UN+1 · · ·UN+k−1) (4.1) Taking into account the definition of the operators Uk, 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 (· · · vk, vk+1 · · · vN \|u1 · · ·uk−1, uk · · · ) UN+1···UN+k−1−−−−−−−−−−→ (· · · vk, vk+1 · · · vN \|uk, u1 · · ·uk−1 · · · ) UN−1···Uk−−−−−−→ (· · · vk+1 · · · vN , vk\|uk, u1 · · ·uk−1 · · · ) UN−−→ (· · · vk+1 · · · vN , uk\|vk, u1 · · ·uk−1 · · · ) Uk···UN−1−−−−−−→ (· · ·uk, vk+1 · · · vN \|vk, u1 · · ·uk−1 · · · ) UN+k−1···UN+1−−−−−−−−−−→ (· · ·uk, vk+1 · · · vN \|u1 · · ·uk−1, vk · · · ), where we have displayed the relevant arguments only. R-Matrix and Baxter Q-Operators 13 It is easy to show that the operator (4.1) intertwines the representations Tα ⊗ Tβ and Tαk,λ ⊗ Tβk,−λ , where αk,λ(h) = h−λ kk h̄−λ̄ kk α(h), βk,−λ(h) = hλ kk h̄λ̄ kk β(h), (4.2) with λ = uk − vk, λ̄ = ūk − v̄k. The operator R(k) is completely determined by the charac- ters α, β and the parameter λ, (λ̄), 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 R(i)(µ)R(k)(λ) ≡ R(i)(µ\|αk,λ,βk,−λ)R(k)(λ\|α,β). (4.3) 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 Va and Vb. The expression (3.6) for the R-operator can be written in the form R12(u− v) = P12R12(u− v) = P12R (1) 12 (u1 − v1)R (2) 12 (u2 − v2) · · ·R(N) 12 (uN − vN ). (4.4) We recall that uk = u−σk and vk = v−ρk, where the parameters σ and ρ define the characters α and β, see equations (2.2) and (2.3). The operators R(k) ab (λ) possess a number of remarkable properties R(k) 12 (0) = 1l, (4.5a) R(k) 12 (λ)R(k) 12 (µ) = R(k) 12 (λ + µ), (4.5b) R(k) 12 (λ)R(j) 23 (µ) = R(j) 23 (µ)R(k) 12 (λ), j 6= k, (4.5c) R(k) 12 (λ)R(k) 23 (λ + µ)R(k) 12 (µ) = R(k) 23 (µ)R(k) 12 (λ + µ)R(k) 23 (λ), (4.5d) R(k) 12 (λ− σk + ρk)R (i) 12 (λ− σi + ρi) = R(i) 12 (λ− σi + ρi)R (k) 12 (λ− σk + ρk). (4.5e) 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 R(k) 12 R(j) 23 = R(j) 23 R(k) 12 , R(k) 12 R(k) 23 R(k) 12 = R(k) 23 R(k) 12 R(k) 23 and R(k) 12 R(i) 12 = R(i) 12R(k) 12 . Let us notice that equation (4.5e) indicates that the operators Rk(uk − vk) 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 Uk 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 Rik (see discussion after equation (3.21)) R23(u− v)R12(u− w)R23(v − w) = R12(v − w)R23(u− w)R12(u− v). (4.6) Using the factorized form (4.4) for the operator Rik and using equations (4.5c) and (4.5e) one can bring the l.h.s and r.h.s of equation (4.6) into the form R(1) 23 (u1 − v1)R (1) 12 (u1 − w1)R (1) 23 (v1 − w1) · · ·R(N) 23 (uN − vN ) × R(N) 12 (uN − wN )R(N) 23 (vN − wN ) 14 S.É. Derkachov and A.N. Manashov and R(1) 12 (v1 − w1)R (1) 23 (u1 − w1)R (1) 12 (u1 − v1) · · ·R(N) 12 (vN − wN ) × R(N) 23 (uN − wN )R(N) 12 (uN − vN ), 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 R- matrix (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) = αk,λ(h). Then it follows from equations (4.2) that the operator R(k) 12 (λ) = P12R (k) 12 (λ) (4.7) is a SL(N, C) invariant operator, R(k) 12 (λ) (Tα ⊗ Tαk,λ) (g) = (Tα ⊗ Tαk,λ) (g)R(k) 12 (λ). Let us note that the operators R(k)(λ) depend only on the part of parameters σi and ρi 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 σk,k+1 = σk − σk+1 (the character α), (R(N)(λ) ≡ R(k)(λ\|σ12, . . . , σN−1,N )), while the operator R(1)(λ) depends on ρk,k+1 (the charac- ter β), and similarly for others. It means that the condition β(h) = αk,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 uk − vk = λ. Then, provided that β(h) = αk,λ(h) one finds ui − vi = 0 unless i = k. Taking into account equation (4.5a) and u − v =∑ k(uk − vk)/N = λ/N one derives R(k) 12 (λ) = P12R (k) 12 (λ) = R12 ( λ N ) ∣∣∣ β(h)=αk,λ(h) . It means that the operator R(k) coincides with the R-operator at the “special point”. In case λ+ λ̄∗ = 0, the operator R(k) 12 (λ) is a unitary operator on the Hilbert space V1⊗V2 = L2(Z×Z) which carries the representation Tα ⊗ Tαk,λ ,( R(k) 12 (λ) )† R(k) 12 (λ) = 1l. Moreover, these operators for different k are unitary equivalent, for example R(k) 12 (λ) = W † kR (N) 12 (λ)Wk. (4.8) The operator Wk can be easily read off from equation (4.1). It factorizes into the product of two operators, Wk = W (1) k W̃ (2) k , where the unitary operators W (1) k and W̃ (2) k act in the spaces V1 and V2, respectively, W (1) k = U2N−1(σk−1,N ) · · ·UN+k−1(σk−1,k), W̃ (2) k = UN−1(σk,N + λ) · · ·Uk(σk,k+1 + λ). Here we display explicitly the arguments of the intertwining operators. Let us consider the homogeneous spin chain of length L and define the operators Qk(λ) by6 Qk(λ + σk) = Tr0 { R(k) 10 (λ) · · ·R(k) L0 (λ) } . (4.9) 6We recall our convention is to display only the holomorphic parameter λ, i.e. the operator Qk(λ) ≡ Qk(λ, λ̄), where λ− λ̄ is integer. R-Matrix and Baxter Q-Operators 15 The operators Qk act on the Hilbert space V1 ⊗ · · · ⊗VL = L2(Z × · · · ×Z). It is assumed that the representation Tα of the group SL(N, C) is defined at each site. The trace is taken over the auxiliary space V0 = L2(Z) and exists (we give below the explicit realization of Qk(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 Ri0-operator for the special choice of auxiliary space one concludes that the Baxter operatorsQk(λ) commute with each other for different values of a spectral parameter [Qk(λ),Qi(µ)] = 0. As follows from equation (4.8) all Baxter operators are unitary equivalent, namely Qk(λ + σk) = ( L∏ i=i W (i) k )† QN (λ + σN ) ( L∏ i=i W (i) k ) . (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 factor- ization property of the transfer matrix. The latter is defined as Tβ(λ) = Tr0 {R10(λ) · · ·RL0(λ)} , (4.11) where the trace is taken over the auxiliary space and the index β refers to the representation Tβ which is defined on the space V0. 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 Qk operators as follows Tβ(λ) = Q1(λ + ρ1)P−1Q2(λ + ρ2)P−1 · · · P−1QN (λ + ρN ), where P is the operator of cyclic permutation PΦ(z1, . . . , zL) = Φ(zL, z1, . . . , zL−1) and the parameters ρk 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. 4.1 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 QN and R(N) only. The latter can be represented as the product of three operators, R(N)(λ) = X̃1(λ)S(λ)X1. The operator S(λ) defined in equation (3.12) is an multiplication operator [S(λ)Φ](z, w) = [w, z]λ Φ(z, w), (4.12) where [w, z]λ ≡ (w−1z)λ N1((w −1z)∗N1) λ̄. The operator X1 is given by a product of the operators (U1 ⊗ 1l)(U2 ⊗ 1l) · · · (UN−1 ⊗ 1l), see equations (4.1) and (3.16). Making use of equation (2.17) one gets [X1Φ](z, w) = N−1∏ k=1 A(σkN ) ∫ d2ζk [zk+1,k − ζk]−1−σkN Φ(zζ , w). (4.13) 16 S.É. Derkachov and A.N. Manashov The factor A is defined in equation (2.18), σkN = σk − σN , zζ = z ( 1 + (ζ1 − z21)e21 ) · · · ( 1 + (ζN−1 − zNN−1)eNN−1 ) = z ( 1 + N−1∑ k=1 (ζk − zk+1,k)ek+1,k ) . and we recall that [z]a ≡ zaz̄ā. Similarly, for the operator X̃1(λ) one derives [X̃1(λ)Φ](z, w) = N−1∏ k=1 A(σNk + λ) ∫ d2ζk [zk+1,k − ζk]−(1+λ+σNk)Φ(z̃ζ , w), (4.14) with z̃ζ = z (1 + (ζN−1 − zNN−1)eNN−1) · · · (1 + (ζ1 − z21)e21) . Thus the operator R(N)(λ) can be represented in the following form [R(N) 12 (λ)Φ](z, w) = N−1∏ k=1 A(σkN )A(σNk + λ) ∫ d2ζk ∫ d2ζ ′k × [zk+1,k − ζk]−(1+λ+σNk)[ζk − ζ ′k] −(1+σkN )[w, z̃ζ ]λΦ(zζζ′ , w), (4.15) where zζζ′ = z̃ζ ( 1 + (ζ ′1 − ζ1)e21 ) · · · ( 1 + (ζ ′N−1 − ζN−1)eNN−1 ) . To obtain the integral representation for the Baxter operator QN we notice that the operator R(N) 12 (λ) leaves the second argument of the function Φ(z, w) intact. Therefore the integral kernel of the operator R(N) 12 (λ), [R(N) 12 (λ)Φ](z, w) = ∫ Dz′Dw′Rλ(z, w\|z′, w′)Φ(z′, w′), where Dz′ = ∏ k>i d2z′ki, Dw′ = ∏ k>i d2w′ ki, has the following form Rλ(z, w\|z′, w′) = Rλ(z, w\|z′)δ(w − w′), with δ(w − w′) = ∏ k>i δ2(wki − w′ ki). Making use of equations (4.7) and (4.9) one finds that the kernel Q (N) λ of the operator QN [QN (λ + σN )Φ](z1, . . . , zL) = ∫ L∏ k=1 Dz′k Q (N) λ (z1, . . . , zL\|z′1, . . . , z′L) Φ(z′1, . . . , z ′ L) . for a spin chain of a length L has the following form Q (N) λ (z1, . . . , zL\|z′1, . . . , z′L) = L∏ k=1 Rλ(zk, zk+1\|z′k+1), where the periodic boundary conditions are implied zL+1 = z1. Further, taking into account equations (4.12), (4.13), (4.14) one can represent the operator Q(N) in the form Q(N)(λ + σN ) = Q(λ + σN )X1X2 · · · XL. (4.16) R-Matrix and Baxter Q-Operators 17 The operator Xk, which acts non-trivially only on the k-th component in the tensor product V1 ⊗ V2 ⊗ · · · ⊗ VL, is given by equation (4.13). In turn, for the operator Q(λ) one obtains [Q(λ)Φ](z1, . . . , zL) = qL(λ) L∏ k=1 N−1∏ j=1 ∫ d2ζk,j [(zk)j+1,j − ζk,j ]−(1+λ−σj)  × [zk+1, (z̃k)ζk ]λ−σN Φ((z̃L)ζL , (z̃1)ζ1 , . . . , (z̃L−1)ζL−1 ), where q(λ) = N−1∏ j=1 A(λ− σj). 4.2 SL(2, C) magnet Closing this section we give the explicit expression for the operator R(N) and the Baxter Q- operator for the SL(2, C) spin magnet. For N = 2 equation (4.15) takes the form[ R(N=2) 12 (λ)Φ ] (z, w) = A(σ12)A(λ− σ12) ∫ d2ζ ∫ d2ζ ′ × [z − ζ]−(1+λ−σ12)[ζ − ζ ′]−(1+σ12)[ζ − w]λΦ(ζ ′, w), where we put z12 = z, w12 = w. Integrating over ζ with help of equation (3.19) one derives[ R(N=2) 12 (λ)Φ ] (z, w) = A(λ)[z − w]σ12 ∫ d2ζ ′[z − ζ ′]−1−λ[ζ ′ − w]λ−σ12 Φ(ζ ′, w) = A(λ) ∫ d2α[α]−1−λ [1− α]λ−σ12Φ(z(1− α) + wα, w). Similarly, for the operator R(1) 12 one finds[ R(1) 12 (λ)Φ ] (z, w) = A(λ)[z − w]ρ12 ∫ d2ζ ′ [w − ζ ′]−1−λ[z − ζ ′]λ−ρ12Φ(z, ζ ′) = (−1)λ−λ̄A(λ) ∫ d2α[α]−1−λ[1− α]λ−ρ12Φ(z, w(1− α) + zα). With the help of equations (4.4) and (4.3) one can easily find that the integral kernel of the operator R12(λ) [R12(λ)Φ] (z, w) = ∫ d2z′d2w′ Rλ(z, w\|z′, w′)Φ(z′, w′) takes the form Rλ(z, w\|z′, w′) = (−1)λ−λ̄−σ1+σ̄1+ρ1−ρ̄1A(λ− σ1 + ρ1)A(λ− σ2 + ρ2) [z − w]−λ+σ2−ρ1 [z′ − w′]−λ+σ1−ρ2 [z − w′]1+λ−σ2+ρ2 [z′ − w]1+λ−σ1+ρ1 . 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 [Q(λ)Φ](z1, . . . , zL) = AL(λ− σ1) L∏ k=1 ∫ d2ζk[zk − ζk+1]−(1+λ−σ1)[ζk − zk]λ−σ2 × Φ(ζ1, ζ2, . . . , ζL). (4.17) 18 S.É. Derkachov and A.N. Manashov 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) Q(N=2)(λ + σ2) = AL(λ) L∏ k=1 ∫ d2αk[αk]−1−λ [1− αk]λ−σ12 × Φ((1− α1)zL + α1z1, (1− α2)z1 + α2z2, . . . , (1− αL)zL−1 + αLzL), which is instructive to compare with the expression for the Baxter Q-operator for su(1, 1) spin chain [39]. 5 Summary In this paper we developed an approach of constructing the solutions of the Yang–Baxter equa- tion for the principal series representations of SL(N, C). We obtained the R-operator as the product of elementary operators Uk, k = 1, . . . , 2N − 1. The latter, except for the operator UN , are the intertwining operators for the principal series representations of SL(N, C). The opera- tor UN 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 Uk satisfy the same commutation relations as the operators of the elementary permutation, Pk. In other words, they define the representation of the permutation group SN . It means that any two operators constructed from the operator Uk 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 rep- resentations of sl(N) algebra we figured out which properties of the factorizing operators are vital to proving of the Yang–Baxter equation. The operators Rk play the fundamental role in constructing the Baxter operators. 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Gen., 1999, V.32, 5299–5316, solv-int/9902015. http://arxiv.org/abs/math.QA/0507252 http://arxiv.org/abs/hep-th/0303056 http://arxiv.org/abs/nlin.SI/0512047 http://arxiv.org/abs/nlin.SI/06040180 http://arxiv.org/abs/solv-int/9902015 1 Introduction 2 Principal series representations of the group SL(N,C) 2.1 Generators and right shifts 2.2 Intertwining operators 3 R-operator 3.1 Factorized ansatz for R-matrix 3.2 Exchange operator 3.3 Permutation group and the star-triangle relation 4 Factorizing operators and Baxter Q-operators 4.1 Integral representation 4.2 SL(2,C) magnet 5 Summary

R-matrix and Baxter Q-operators for the noncompact SL(N,C) invariant spin chain

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