Highest ℓ-Weight Representations and Functional Relations

We discuss highest ℓ-weight representations of quantum loop algebras and the corresponding functional relations between integrability objects. In particular, we compare the prefundamental and q-oscillator representations of the positive Borel subalgebras of the quantum group Uq(L(sll₊₁)) for arbitra...

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Date:	2017
Main Authors:	Nirov, K.S., Razumov, A.V.
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Language:	English
Published:	Інститут математики НАН України 2017
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Cite this:	Highest ℓ-Weight Representations and Functional Relations / K.S. Nirov, A.V. Razumov // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 40 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1486442025-02-09T20:29:27Z Highest ℓ-Weight Representations and Functional Relations Nirov, K.S. Razumov, A.V. We discuss highest ℓ-weight representations of quantum loop algebras and the corresponding functional relations between integrability objects. In particular, we compare the prefundamental and q-oscillator representations of the positive Borel subalgebras of the quantum group Uq(L(sll₊₁)) for arbitrary values of l. Our article has partially the nature of a short review, but it also contains new results. These are the expressions for the L-operators, and the exact relationship between different representations, as a byproduct resulting in certain conclusions about functional relations. This paper is a contribution to the Special Issue on Recent Advances in Quantum Integrable Systems. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2016.html. We are grateful to H. Boos, F. G¨ohmann and A. Kl¨umper for discussions. This work was supported in part by the Deutsche Forschungsgemeinschaft in the framework of the research group FOR 2316, by the DFG grant KL 645/10-1, and by the RFBR grants # 14-01-91335 and # 16-01-00473. Kh.S.N. is grateful to the RAQIS’16 Organizers for the invitation and hospitality during the Conference “Recent Advances in Quantum Integrable Systems”, August 22–26, 2016, at the University of Geneva. 2017 Article Highest ℓ-Weight Representations and Functional Relations / K.S. Nirov, A.V. Razumov // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 40 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 16T25; 17B10 DOI:10.3842/SIGMA.2017.043 https://nasplib.isofts.kiev.ua/handle/123456789/148644 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We discuss highest ℓ-weight representations of quantum loop algebras and the corresponding functional relations between integrability objects. In particular, we compare the prefundamental and q-oscillator representations of the positive Borel subalgebras of the quantum group Uq(L(sll₊₁)) for arbitrary values of l. Our article has partially the nature of a short review, but it also contains new results. These are the expressions for the L-operators, and the exact relationship between different representations, as a byproduct resulting in certain conclusions about functional relations.
format	Article
author	Nirov, K.S. Razumov, A.V.
spellingShingle	Nirov, K.S. Razumov, A.V. Highest ℓ-Weight Representations and Functional Relations Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Nirov, K.S. Razumov, A.V.
author_sort	Nirov, K.S.
title	Highest ℓ-Weight Representations and Functional Relations
title_short	Highest ℓ-Weight Representations and Functional Relations
title_full	Highest ℓ-Weight Representations and Functional Relations
title_fullStr	Highest ℓ-Weight Representations and Functional Relations
title_full_unstemmed	Highest ℓ-Weight Representations and Functional Relations
title_sort	highest ℓ-weight representations and functional relations
publisher	Інститут математики НАН України
publishDate	2017
url	https://nasplib.isofts.kiev.ua/handle/123456789/148644
citation_txt	Highest ℓ-Weight Representations and Functional Relations / K.S. Nirov, A.V. Razumov // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 40 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT nirovks highestlweightrepresentationsandfunctionalrelations AT razumovav highestlweightrepresentationsandfunctionalrelations
first_indexed	2025-11-30T12:17:05Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 043, 31 pages Highest `-Weight Representations and Functional Relations Khazret S. NIROV †‡ and Alexander V. RAZUMOV § † Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Ave. 7a, 117312 Moscow, Russia ‡ Mathematics and Natural Sciences, University of Wuppertal, 42097 Wuppertal, Germany E-mail: nirov@uni-wuppertal.de § Institute for High Energy Physics, NRC “Kurchatov Institute”, 142281 Protvino, Moscow region, Russia E-mail: Alexander.Razumov@ihep.ru Received March 01, 2017, in final form June 06, 2017; Published online June 17, 2017 https://doi.org/10.3842/SIGMA.2017.043 Abstract. We discuss highest `-weight representations of quantum loop algebras and the corresponding functional relations between integrability objects. In particular, we compare the prefundamental and q-oscillator representations of the positive Borel subalgebras of the quantum group Uq(L(sll+1)) for arbitrary values of l. Our article has partially the nature of a short review, but it also contains new results. These are the expressions for the L-operators, and the exact relationship between different representations, as a byproduct resulting in certain conclusions about functional relations. Key words: quantum loop algebras; Verma modules; highest `-weight representations; q-os- cillators 2010 Mathematics Subject Classification: 17B37; 16T25; 17B10 1 Introduction The use of highest `-weights and highest `-weight vectors allows one to properly refine the spec- tral data about highest weight representations in the same way as the generalized eigenvalues and eigenvectors do for the eigenvalue problems. The corresponding notion proved especially useful in the classification of irreducible finite-dimensional [13, 14] and infinite-dimensional [26, 35] representations of quantum loop algebras and their Borel subalgebras. For quantum affine al- gebras and their Borel subalgebras, the related category of representations was studied in [25] and [26], respectively. The study of different representations of quantum groups in application to quantum in- tegrable systems received new impetus from the remarkable papers by Bazhanov, Lukyanov and Zamolodchikov [3, 4, 5]. In general terms, in the approach advanced in these papers, the investigation of quantum integrable systems is reduced to the study of representations of the cor- responding quantum groups. More specifically, the method is based on the universal R-matrix. By definition, it is an element of the tensor product of two copies of the quantum group, and one calls the representation spaces for the first and second factors of this tensor product the auxiliary and quantum spaces, respectively. Here, a representation of the quantum group in the auxiliary space gives an integrability object which is either a monodromy- or a transfer-type operator. A representation in the quantum space defines then a physical model. For example, This paper is a contribution to the Special Issue on Recent Advances in Quantum Integrable Systems. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2016.html mailto:nirov@uni-wuppertal.de mailto:Alexander.Razumov@ihep.ru https://doi.org/10.3842/SIGMA.2017.043 http://www.emis.de/journals/SIGMA/RAQIS2016.html 2 Kh.S. Nirov and A.V. Razumov it can be a low-dimensional quantum field theory as in [2, 3, 4, 5] or a spin-chain model as in [9, 10, 36]. The integrability objects satisfy functional relations as a consequence of the cha- racteristics of the representations of the quantum group in the auxiliary and quantum spaces. In fact, such functional relations can be derived in a universal form, fixing representations of the quantum group only in the auxiliary space and being thus independent of the representations in the quantum space. We would like to refer to the paper [10] for more details. Now, it is relevant to point out that the universalR-matrix is actually an element of (a comple- tion of) the tensor product of the positive and negative Borel subalgebras of the initial quantum group. This means inter alia that given a representation of the whole quantum group, one can obtain representations for the construction of integrability objects by restricting it to the cor- responding Borel subalgebras. This way does work for the monodromy and transfer operators. However, there are representations of the Borel subalgebras which cannot be obtained by such a simple restriction. In particular, one obtains such representations mapping the positive Borel subalgebra to a q-oscillator algebra and using representations of the latter. This method was proposed in [4, 5] for the construction of CFT analogs of the Baxter’s Q-operators. Such repre- sentations can be deduced from those ones employed earlier for the construction of monodromy and transfer operators by a certain degeneration procedure [2, 8, 9, 10, 36]. This relationship between representations ascertains that the corresponding integrability objects are involved in nontrivial functional equations. In [26], Hernandez and Jimbo studied inductive limits of the Kirillov–Reshetikhin modules and obtained new simple infinite-dimensional representations of the Borel subalgebras of quan- tum loop algebras. These are highest `-weight modules characterized by highest `-weights of simplest possible form. Later on, in [22], based on the notion of q-characters, generalized Bax- ter’s TQ-relations were given an interpretation as of relations in the Grothendieck ring of the category O from [26]. Just as in [22], we call the above highest `-weight representations prefun- damental . In the paper [11], we found the `-weights and the corresponding `-weight vectors for the finite- and infinite-dimensional representations of the quantum loop algebra Uq(L(sll+1)) for l = 1 and 2 constructed through Jimbo’s evaluation representations. We also found there the `-weights and the `-weight vectors for the q-oscillator representations of the positive Borel subalgebras of the same quantum groups. The work [11] showed how the q-oscillator and prefundamental representations are explicitly related. However, the consideration of these cases, with l = 1 and l = 2, did not allow for a direct generalization to the arbitrary higher ranks. Quite recently, based on the paper [37], we have considered the general case with arbitrary l and obtained the `-weights and the corresponding `-weight vectors for q-oscillator representations of the positive Borel subalgebra of Uq(L(sll+1)) [12]. Here we use the notations and calculations of [12]. The article has partially the nature of a short review, but it also contains new results (see Sections 6 and 7). In Section 2, we recall the quantum groups in general in order to introduce the universal R-matrix as the main tool. Applying to it different representations, we define the universal integrability objects and discuss their basic properties. In Section 3, we specify the general notion of quantum groups to the quantum group of the general linear Lie algebra of arbitrary rank. Here we discuss its highest weight representations. In Section 4, we describe the quantum group of the untwisted loop algebra of the special linear Lie algebra of arbitrary rank. It is traditional to call this object a quantum loop algebra. We construct representations of this algebra using the corresponding Jimbo’s evaluation homomorphism. Besides, we define the Borel subalgebras of the quantum loop algebra and, following our paper [37], describe their representations. In Section 5, we recall necessary data on the highest `-weight representations with rational `-weights. As the respective basic examples of our special interest, we discuss the prefundamental and q-oscillator representations of the positive Borel subalgebra of the quantum loop algebra under consideration. In Section 6, we describe the symmetry transformations which Highest `-Weight Representations and Functional Relations 3 allow us to construct more q-oscillator representations of the Borel subalgebra. In Section 7, we present the highest `-weights for the q-oscillator representations from the preceding section and discuss explicit relations between them. These relations reproduce the defining characteristics of the functional relations between the universal integrability objects. Our results establish a direct connection between the q-oscillator and prefundamental representations. We conclude with some remarks. To subsequently define the quantum groups, we introduce the corresponding deformation parameter. Here we determine a nonzero complex number ~, such that q = exp ~ is not a root of unity. With such a deformation parameter q, the quantum groups under consideration are treated as C-algebras. Besides, we assume that qν = exp(~ν), ν ∈ C. For the q-numbers and q-factorials we use the traditional notations [n]q = qn − q−n q − q−1 , n ∈ Z, and [n]q! = n∏ m=1 [m]q, n ∈ N, [0]q! = 1, respectively. We also use the convenience of the notation κq = q − q−1. 2 Quantum groups and integrability objects Following Drinfeld [17, 18] and Jimbo [27], we treat a quantum group G as a one-parameter deformation of the universal enveloping algebra of a Lie algebra g. Hence the usual notation for G as Uq(g), where q is the mentioned deformation parameter. The nature of the quantum group can essentially depend on the specification of this parameter, see the books [14, 29, 20] for a discussion of the point. The quantum group is defined as a Hopf algebra with respect to appropriate co-multiplication ∆, antipode S and co-unit ε. It is also a Hopf algebra with respect to the opposite co-multiplication ∆op = Π ◦∆, where the permutation operator is defined by Π(a⊗ b) = b⊗ a, a, b ∈ G. The quantum group is a quasitriangular Hopf algebra. It means that there exists the so-called universal R-matrix R being an element of the completed tensor product of two copies of the quantum group and relating the co-multiplication and the opposite one as ∆op(a) = R∆(a)R−1, a ∈ G, R ∈ G ⊗ G, and satisfying the following relations: (∆⊗ id)(R) = R13R23, (id⊗∆)(R) = R13R12, where the indices have the standard meaning. The above relations lead to the following equation for the universal R-matrix: R12R13R23 = R23R13R12 4 Kh.S. Nirov and A.V. Razumov called the Yang–Baxter equation. It is defined in the tensor cube of the quantum group. How- ever, it is important to note that the universal R-matrix belongs to the completed tensor product of the positive and the negative Borel subalgebras of the quantum group, R ∈ B+ ⊗ B− ⊂ G ⊗ G. This fact has profound implications in the theory of quantum integrable systems. First of all, it means that the Yang–Baxter equation is actually defined not in the full tensor cube of the quantum group, but in the space B+ ⊗G ⊗B−. Secondly, it allows one to consider integrability objects, such as monodromy- and transfer-type operators, having essentially different nature. To be more specific, let us describe how the integrability objects associated with the quantum group G and its Borel subalgebra B+ arise in general. We refer the reader to [8] for more details. With the help of a group-like element t, by definition satisfying the relation ∆(t) = t⊗ t, we obtain from the Yang–Baxter equation the equation( R13t1 )( R23t2 ) = ( R12 )−1(R23t2 )( R13t1 )( R12 ) . (2.1) Let ϕ be a representation of G in a vector space V . We define the monodromy-type operator Mϕ associated with this representation as Mϕ = (ϕ⊗ id)(R) and see that it is an element of End(V ) ⊗ B−. Next we define the corresponding transfer-type operator Tϕ as Tϕ = (trV ⊗ id)(Mϕ(ϕ(t)⊗ 1)) = ((trV ◦ϕ)⊗ id)(R(t⊗ 1)). Here 1 is the unit element of G, and we assume that t is such that the trace over the representation space V is well-defined. It is clear that Tϕ belongs to the negative Borel subalgebra B− ⊂ G. We see that to define these integrability objects, Mϕ and Tϕ, one starts with a representation of the whole quantum group G, but one then uses only its restriction to the positive Borel subalgebra B+. Moreover, one can define in this way different transfer-type operators associated with representations ϕ1 and ϕ2 of G and see directly from (2.1) that they commute, Tϕ1Tϕ2 = Tϕ2Tϕ1 . This is the primary indication of the integrability of models which can be associated with G. One can also consider parameterized representations of the quantum group and arrive at the commutativity of the transfer-type operators for different values of the corresponding parame- ters. To have more integrability objects, one considers representations which are not restrictions of a representation of G to B+ or B−, and which cannot be extended from the Borel subalgebras to a representation of the full quantum group. Let ρ be such a representation of B+ in a vector space W . We introduce a monodromy-type operator Lρ associated with this representation, Lρ = (ρ⊗ id)(R), being an element of End(W )⊗B−. The corresponding Q-operator Qρ is then an element of B− defined as Qρ = (trW ⊗ id)(Lρ(ρ(t)⊗ 1)) = ((trW ◦ρ)⊗ id)(R(t⊗ 1)). Highest `-Weight Representations and Functional Relations 5 Here again, 1 is the unity of G, and the group-like element t is such that the trace over the representation space W is well-defined. Using equation (2.1), one can show that QρTϕ = TϕQρ. Appropriate representations of such kind to be used for ρ, the so-called q-oscillator representa- tions, were considered for the first time in [4, 5] and [2], where integrable structures of conformal quantum field theories were investigated. One can show that also the Q-operators commute for different representations ρ1 and ρ2. However, this commutativity and other nontrivial relations between the transfer-type integrability objects do not follow simply from the Yang–Baxter equa- tion (2.1) anymore. For some partial cases the commutativity was proved exploring details on the tensor products of the respective representations in the papers [2, 8, 9, 10, 36]. The proof for the general case was given in the paper [22, Section 5.2]. It is convenient to use a more general definition of monodromy-type operators. Here the mappings ϕ and ρ are homomorphisms from G, or B+, to some algebra with a relevant set of representations. One constructs integrability objects with such ϕ and ρ and then apply to them appropriate representations. We note finally that the integrability objects above have been introduced in such a way that only a representation of the quantum group in the auxiliary space was fixed, and no representation in the quantum space was chosen. In this sense, they are model independent. For this reason we call the above monodromy- and transfer-type operators the universal integrability objects. To obtain the corresponding integrability objects for specific models, one has to fix a representation of the quantum group in the quantum space. Further, we need to specify the quantum group G = Uq(g) and its Borel subalgebras. Actually, we will consider two cases with the Lie algebra g being the general linear Lie algebra gll+1 and the loop algebra L(sll+1). Recall that the deformation parameter q is always supposed to be an exponential of a complex number ~, such that q is not a root of unity. It allows one to treat Uq(g) as a unital associative C-algebra obtained by the q-deformation of the universal enveloping algebra of the Lie algebra g. 3 Quantum group Uq(gll+1) and its representations Let kl+1 be the standard Cartan subalgebra of the Lie algebra gll+1 and 4 be the root system of gll+1 respective to kl+1. Denote by {αi ∈ k∗l+1 \| i = 1, . . . , l} the corresponding set of simple roots. Then, with Ki, i = 1, . . . , l + 1, forming the standard basis of kl+1, we have 〈αj ,Ki〉 = cij , where cij = δij − δi,j+1. The general linear Lie algebra gll+1 is generated by 2l Chevalley generators Ei, Fi, i = 1, . . . , l, and by l+1 Cartan elements Ki, together satisfying well-known defining relations supplemented also with the Serre relations. For the total root system we have 4 = 4+ t4−, where 4+ is the set of positive roots which are all of the form αij = j−1∑ k=i αk, 1 ≤ i < j ≤ l + 1, 6 Kh.S. Nirov and A.V. Razumov and we also have αi = αi,i+1, and 4− = −4+ is the set of negative roots. The restriction to the special linear Lie algebra sll+1 is obtained by setting Hi = Ki −Ki+1, i = 1, . . . , l, as the generators of the standard Cartan subalgebra hl+1 of sll+1 and keeping Ei and Fi as the corresponding Chevalley generators. The positive and negative roots of sll+1 are the restrictions of αij and −αij to hl+1, respectively. Then we have 〈αj , Hi〉 = aij , where aij = cij − ci+1,j are the entries of the Cartan matrix of sll+1. As usual the fundamental weights ωi ∈ h∗, i = 1, . . . , l, are defined by the relations 〈ωi, Hj〉 = δij , j = 1, . . . , l. The quantum group Uq(gll+1) is generated by the elements Ei, Fi, i = 1, . . . , l, qX , X ∈ kl+1, satisfying the defining relations q0 = 1, qX1qX2 = qX1+X2 , (3.1) qXEiq −X = q〈αi,X〉Ei, qXFiq −X = q−〈αi,X〉Fi, (3.2) [Ei, Fj ] = δij qKi−Ki+1 − q−Ki+Ki+1 q − q−1 , (3.3) and the Serre relations EiEj = EjEi, FiFj = FjFi, \|i− j\| ≥ 2, E2 i Ei±1 − [2]qEiEi±1Ei + Ei±1E 2 i = 0, F 2 i Fi±1 − [2]qFiFi±1Fi + Fi±1F 2 i = 0. The set of the elements of the form qX is parameterized by the Cartan subalgebra kl+1. The quantum group Uq(sll+1) is generated by the same generators as Uq(gll+1), only that the gen- erators qX of Uq(sll+1) are parameterized by the Cartan subalgebra hl+1. The generators of Uq(sll+1) fulfil the same relations as of Uq(gll+1), with one exception that (3.3) takes now the form [Ei, Fj ] = δij qHi − q−Hi q − q−1 . We assume everywhere that qX+ν = qνqX , [X + ν]q = qX+ν − q−X−ν q − q−1 , X ∈ kl+1, ν ∈ C. Both quantum groups, Uq(gll+1) and Uq(sll+1), are Hopf algebras with respect to the co- multiplication, antipode and co-unit defined as follows: ∆(qX) = qX ⊗ qX , ∆(Ei) = Ei ⊗ 1 + qHi ⊗ Ei, ∆(Fi) = Fi ⊗ q−Hi + 1⊗ Fi, Highest `-Weight Representations and Functional Relations 7 S(qX) = q−X , S(Ei) = −q−HiEi, S(Fi) = −FiqHi , ε(qX) = 1, ε(Ei) = 0, ε(Fi) = 0. Although these relations are not used in this paper, we note that the Hopf algebra structure is crucial for the quantum integrable systems associated with these quantum groups. The quantum group Uq(gll+1) possesses a Poincaré–Birkhoff–Witt basis. To construct it, one needs an appropriate definition of the root vectors. We first introduce a Q-gradation of Uq(gll+1) with respect to the root lattice of gll+1. The latter is the abelian group Q = l⊕ i=1 Zαi, and Uq(gll+1) becomes Q-graded if we assume that Ei ∈ Uq(gll+1)αi , Fi ∈ Uq(gll+1)−αi , qX ∈ Uq(gll+1)0 for all i = 1, . . . , l and X ∈ kl+1. An element a of Uq(gll+1) is called a root vector corresponding to the root γ of gll+1 if a ∈ Uq(gll+1)γ . In the case under consideration, this is equivalent to the relations qXaq−X = q〈γ,X〉a, X ∈ kl+1. The Chevalley generators Ei and Fi are obviously root vectors corresponding to the roots αi and −αi. Now we define the whole set of linearly independent root vectors. We start introducing the set Λl = {(i, j) ∈ N× N \| 1 ≤ i < j ≤ l + 1} and define the elements Eij and Fij , (i, j) ∈ Λl, according to Jimbo [28], Ei,i+1 = Ei, i = 1, . . . , l, Eij = Ei,j−1Ej−1,j − qEj−1,jEi,j−1, j − i > 1, Fi,i+1 = Fi, i = 1, . . . , l, Fij = Fj−1,jFi,j−1 − q−1Fi,j−1Fj−1,j , j − i > 1. The elements Eij are the root vectors corresponding to the positive roots αij , and the ele- ments Fij are the root vectors corresponding to the negative roots −αij . The Cartan–Weyl generators of Uq(gll+1) are the elements qX , X ∈ kl+1, and Eij , Fij . The Poincaré–Birkhoff– Witt basis of Uq(gll+1) is formed by the ordered monomials constructed from the Cartan–Weyl generators. To define such monomials explicitly, let us impose the lexicographic order on the set Λl, which means that (i, j) < (m,n) if i < m, or if i = m and j < n.1 Then, a respectively ordered monomial being an appropriate Poincaré–Birkhoff–Witt basis element can be taken in the form Fi1j1 · · ·FirjrqXEm1n1 · · ·Emsns , (3.4) where (i1, j1) ≤ · · · ≤ (ir, jr), (m1, n1) ≤ · · · ≤ (ms, ns) and X is an arbitrary element of kl+1. The monomials of the same form with X ∈ hl+1 form a Poincaré–Birkhoff–Witt basis of Uq(sll+1). By definition of the Poincaré–Birkhoff–Witt basis, any monomial can be given by a sum of ordered monomials of the form (3.4). To find an ordered form of a given monomial, one must 1Note that in [37] we used the co-lexicographic ordering. If we define an ordering of the positive roots according to the co-lexicographic order on Λl, we would have a normal ordering in the sense of [1, 33]. Here, in contrast, we use the lexicographic order and obtain a different realization of the normal ordering of positive roots. 8 Kh.S. Nirov and A.V. Razumov be able to reorder, if necessary, the constituent elements qX , Eij and Fij , and the process of reordering requires certain relations between these elements. All such relations were derived in [40]. In a recent paper [37], we adopted those relations in a suitable for our definitions form and used them to obtain the defining relations of the Verma Uq(gll+1)-module. We denote by Ṽ λ the Verma Uq(gll+1)-module corresponding to the highest weight λ ∈ k∗l+1. Here, for the highest weight vector vλ we have the defining relations Eiv λ = 0, i = 1, . . . , l, qXvλ = q〈λ,X〉vλ, X ∈ kl+1, λ ∈ k∗l+1. As usual, the highest weight is identified with its components respective to the basis of kl+1, λi = 〈λ,Ki〉. The representation of Uq(gll+1) corresponding to Ṽ λ is denoted by π̃λ. The structure and properties of Ṽ λ and π̃λ for l = 1 and l = 2 are considered in much detail in our papers [8, 9, 10, 11, 36]. The case of general l was studied in our recent paper [37]. Here we shortly recall the corresponding results from [37]. Let us denote by m the l(l+ 1)/2-tuple of non-negative integers mij , arranged in the lexico- graphic order of (i, j) ∈ Λl. Explicitly we have m = (m12,m13, . . . ,m1,l+1, . . . ,mi,i+1,mi,i+2, . . . ,mi,l+1, . . . ,ml,l+1). The vectors vm = Fm12 12 Fm13 13 · · ·Fm1,l+1 1,l+1 · · ·F mi,i+1 i,i+1 · · ·F mi,i+2 i,i+2 · · ·F mi,l+1 i,l+1 · · ·F ml,l+1 l,l+1 v0, where for consistency v0 denotes the highest-weight vector vλ, form a basis of Ṽ λ. Note that in [37] the integers mij were arranged in the co-lexicographic order, but the basis vectors vm for both orderings, the lexicographic and co-lexicographic ones, coincide, and the defining module relations do not distinguish the choice between these orderings. The Uq(gll+1)-module defining relations from [37] are as follows: qνKivm = q ν ( λi+ i−1∑ k=1 mki− l+1∑ k=i+1 mik ) vm, i = 1, . . . , l + 1, (3.5) Fi,i+1vm = q − i−1∑ k=1 (mki−mk,i+1) vm+εi,i+1 + i−1∑ j=1 q − j−1∑ k=1 (mki−mk,i+1) [mji]qvm−εji+εj,i+1 , Ei,i+1vm = [ λi − λi+1 − l+1∑ j=i+2 (mij −mi+1,j)−mi,i+1 + 1 ] q [mi,i+1]qvm−εi,i+1 + q λi−λi+1−2mi,i+1− l+1∑ j=i+2 (mij−mi+1,j) i−1∑ j=1 q i−1∑ k=j+1 (mki−mk,i+1) [mj,i+1]qvm−εj,i+1+εji − l+1∑ j=i+2 q −λi+λi+1−2+ l+1∑ k=j (mik−mi+1,k) [mij ]qvm−εij+εi+1,j , where i = 1, . . . , l in last two equations. Here and below m + νεij means shifting by ν the entry mij in the l(l + 1)/2-tuple m. In what follows, we will also need the action of the root vectors F1,l+1 on the basis vectors vm. This is given by the equation F1,l+1vm = q l∑ i=2 m1i vm+ε1,l+1 . Highest `-Weight Representations and Functional Relations 9 The reduction to the special linear case from the general linear one can obviously be obtained by replacing equation (3.5) by qνHivm = q ν [ λi−λi+1+ i−1∑ k=1 (mki−mk,i+1)−2mi,i+1− l+1∑ k=i+2 (mik−mi+1,k) ] vm. It is clear that Ṽ λ and π̃λ are infinite-dimensional for the general weights λ ∈ k∗l+1. However, if all the differences λi−λi+1, i = 1, . . . , l, are non-negative integers, there is a maximal submodule, such that the respective quotient module is finite-dimensional. This quotient is then denoted by V λ and the corresponding representation is denoted by πλ. 4 Quantum loop algebra Uq(L(sll+1)) and its representations 4.1 Cartan–Weyl data It is convenient to introduce the sets I = {1, . . . , l} and Î = {0, 1, . . . , l}. We use notations adopted by Kac in his book [30]. Thus, L(sll+1) means the loop algebra of sll+1, L̃(sll+1) its standard extension by a one-dimensional center Cc, and L̂(sll+1) the Lie algebra obtained from L̃(sll+1) by adding a natural derivation d. The Cartan subalgebra ĥl+1 of L̂(sll+1) is ĥl+1 = hl+1 ⊕ Cc⊕ Cd. Denote by hi, i ∈ I, the generators of the standard Cartan subalgebra of sll+1 considered as a subalgebra of L̂(sll+1). Introducing an additional Kac–Moody generator h0 = c− ∑ i∈I hi we obtain ĥl+1 = (⊕ i∈Î Chi ) ⊕ Cd. We identify the space h∗l+1 with the subspace of ĥ∗l+1 formed by the elements γ ∈ ĥ∗l+1 satisfying the equations 〈γ, c〉 = 0, 〈γ, d〉 = 0. We also denote h̃l+1 = hl+1 ⊕ Cc. Similarly as above, we denote the generators of the standard Cartan subalgebra of sll+1 consid- ered as a subalgebra of h̃l+1 by hi. Then we can write h̃l+1 = (⊕ i∈I Chi ) ⊕ Cc = ⊕ i∈Î Chi. In fact, below we use the notation hi even for the generators of the standard Cartan subalgebra of sll+1 itself. This never leads to misunderstanding. We identify the space h∗l+1 with the subspace of h̃∗l+1 which consists of the elements γ̃ ∈ h̃∗l+1 subject to the condition 〈γ̃, c〉 = 0. (4.1) 10 Kh.S. Nirov and A.V. Razumov Here and everywhere below we mark such elements by a tilde. Explicitly the identification is performed as follows. The element γ̃ ∈ h̃∗l+1 satisfying (4.1) is identified with the element γ ∈ h∗l+1 defined by the equations 〈γ, hi〉 = 〈γ̃, hi〉, i ∈ I. In the opposite direction, given an element γ ∈ h∗l+1, we identify it with the element γ̃ ∈ h̃∗l+1 determined by the relations 〈γ̃, h0〉 = − ∑ i∈I 〈γ, hi〉, 〈γ̃, hi〉 = 〈γ, hi〉, i ∈ I. It is clear that γ̃ satisfies (4.1). The simple roots αi ∈ ĥ∗l+1, i ∈ Î, of the Lie algebra L̂(sll+1) are defined by the relations 〈αi, hj〉 = aji, i, j ∈ Î , 〈α0, d〉 = 1, 〈αi, d〉 = 0, i ∈ I. Here aij , i, j ∈ Î, are the entries of the extended Cartan matrix of sll+1. The full system 4̂+ of positive roots of the Lie algebra L̂(sll+1) is related to the system 4+ of positive roots of sll+1 as 4̂+ = {γ + nδ \| γ ∈ 4+, n ∈ Z+} ∪ {nδ \|n ∈ N} ∪ {(δ − γ) + nδ \| γ ∈ 4+, n ∈ Z+}, where δ = ∑ i∈Î αi is the minimal positive imaginary root. We note here that α0 = δ − ∑ i∈I αi = δ − θ, where θ is the highest root of sll+1. The system of negative roots 4̂− is 4̂− = −4̂+, and the full system of roots is 4̂ = 4̂+ t 4̂− = {γ + nδ \| γ ∈ 4, n ∈ Z} ∪ {nδ \|n ∈ Z \ {0}}. The set formed by the restriction of the simple roots αi to h̃l+1 is linearly dependent, as the restriction of δ on h̃l+1 evidently vanishes. This is exactly why we pass from L̃(sll+1) to L̂(sll+1). A non-degenerate symmetric bilinear form on ĥl+1 is fixed by the equations (hi \|hj) = aij , (hi \| d) = δi0, (d \| d) = 0, where i, j ∈ Î. For the corresponding symmetric bilinear form on ĥ∗l+1 one has (αi \|αj) = aij . This relation implies that (δ \|αij) = 0, (δ \| δ) = 0 for all 1 ≤ i < j ≤ l + 1. Highest `-Weight Representations and Functional Relations 11 To define the quantum loop algebra Uq(L(sll+1)), it is reasonable to start with the quantum group Uq(L̂(sll+1)). The latter is generated by the elements ei, fi, i ∈ Î, and qx, x ∈ ĥl+1, subject to the relations q0 = 1, qx1qx2 = qx1+x2 , (4.2) qxeiq −x = q〈αi,x〉ei, qxfiq −x = q−〈αi,x〉fi, (4.3) [ei, fj ] = δij qhi − q−hi q − q−1 , (4.4) 1−aij∑ k=0 (−1)ke (1−aij−k) i eje (k) i = 0, 1−aij∑ k=0 (−1)kf (1−aij−k) i fjf (k) i = 0, (4.5) where e (n) i = eni /[n]q!, f (n) i = fni /[n]q!, and the indices i and j in the Serre relations (4.5) are distinct. Uq(L̂(sll+1)) is a Hopf algebra with respect to the co-multiplication, antipode and co-unit defined as ∆(qx) = qx ⊗ qx, ∆(ei) = ei ⊗ 1 + qhi ⊗ ei, ∆(fi) = fi ⊗ q−hi + 1⊗ fi, S(qx) = q−x, S(ei) = −q−hiei, S(fi) = −fiqhi , ε(qx) = 1, ε(ei) = 0, ε(fi) = 0. The quantum group Uq(L̂(sll+1)) does not have any finite-dimensional representations with a nontrivial action of the element qνc [13, 14]. In contrast, the quantum loop algebra Uq(L(sll+1)) possesses, apart from the infinite-dimensional representations, also nontrivial finite-dimensional representations. Therefore, we proceed to the quantum loop algebra Uq(L(sll+1)). First, we define the quantum group Uq(L̃(sll+1)) as a Hopf subalgebra of Uq(L̂(sll+1)) generated by the elements ei, fi, i ∈ Î, and qx, x ∈ h̃l+1, with relations (4.2)–(4.5) and the above Hopf algebra structure. Second, the quantum loop algebra Uq(L(sll+1)) is defined as the quotient algebra of Uq(L̃(sll+1)) by the two-sided Hopf ideal generated by the elements of the form qνc − 1 with ν ∈ C×. It is convenient to treat the quantum loop algebra Uq(L(sll+1)) as a complex alge- bra with the same generators as Uq(L̃(sll+1)), but satisfying, additionally to relations (4.2)–(4.5), also the relations qνc = 1, ν ∈ C×. For the quantum group under consideration one can also define the root vectors and construct a Poincaré–Birkhoff–Witt basis. This basis is used, in particular, to relate two realizations of the quantum loop algebra. To define the root vectors, we introduce the root lattice of L̂(sll+1). This is the abelian group Q̂ = ⊕ i∈Î Zαi. The algebra Uq(L(sll+1)) becomes Q̂-graded if we assume ei ∈ Uq(L(sll+1))αi , fi ∈ Uq(L(sll+1))−αi , qx ∈ Uq(L(sll+1))0 for any i ∈ Î and x ∈ h̃. Then, an element a of Uq(L(sll+1)) is called a root vector corre- sponding to a root γ of L̂(sll+1) if a ∈ Uq(L(sll+1))γ . The generators ei and fi are root vectors corresponding to the roots αi and −αi. 12 Kh.S. Nirov and A.V. Razumov Now we obtain linearly independent root vectors corresponding to the roots from 4̂. We use here the procedure of Khoroshkin and Tolstoy [31, 39] as the most suitable for the pur- pose. The root vectors, together with the elements qx, x ∈ h̃, are the Cartan–Weyl generators of Uq(L(sll+1)). We endow 4̂+ with an order ≺ in the following way. First we assume that imaginary roots follow each other in any order. Then we additionally assume that α+ kδ ≺ mδ ≺ (δ − β) + nδ (4.6) for any α, β ∈ 4+ and k,m, n ∈ Z+. Finally we impose a normal order in the sense of [1, 33] on the system of real positive roots from 4+. We specify this normal order as described, for example, in [34]. It is clear from (4.6) that it is sufficient to define the ordering separately for the roots α+kδ and (δ−β)+nδ, where α, β ∈ 4+. We assume that αij+rδ ≺ αmn+sδ if i < m, or if i = m and r < s, or if i = m, r = s and j < n. Similarly, (δ − αij) + rδ ≺ (δ − αmn) + sδ if i > m, or if i = m and r > s, or if i = m, r = s and j < n. The restriction of this ordering to 4+ gives the lexicographic ordering described in the preceding Section 3. The root vectors can be defined inductively. We start with the root vectors corresponding to the simple roots, which are nothing but the generators of Uq(L(sll+1)), eδ−θ = e0, eαi = ei, fδ−θ = f0, fαi = fi, i ∈ I. As usual, a root vector corresponding to a positive root γ is denoted by eγ , and a root vector corresponding to a negative root −γ is denoted by fγ . Let a root γ ∈ 4̂+ be such that γ = α+β for some α, β ∈ 4̂+. For definiteness, we assume that α ≺ γ ≺ β, and there are no other roots α′ � α and β′ ≺ β such that γ = α′+β′. Then, if the root vectors eα, eβ and fα, fβ are already defined, we put [31, 39] eγ = [eα, eβ]q, fγ = [fβ, fα]q, where the q-commutator [ , ]q is defined by the relations [eα, eβ]q = eαeβ − q−(α\|β)eβeα, [fα, fβ]q = fαfβ − q(α\|β)fβfα with ( \| ) standing for the symmetric bilinear form on ĥ∗. Next we define root vectors corresponding to the roots αij and −αij . The root vectors eαi,i+1 and fαi,i+1 corresponding to the roots αi,i+1 = αi and −αi,i+1 = −αi are already given. The higher root vectors for the positive and negative composite roots can be defined by the relations eαij = [eαi,i+1 , eαi+1,j ]q = eαi,i+1eαi+1,j − qeαi+1,jeαi,i+1 and fαij = [fαi+1,j , fαi,i+1 ]q = fαi+1,jfαi,i+1 − q−1fαi,i+1fαi+1,j , respectively. These definitions uniquely give eαij = [eαi , . . . [eαj−2 , eαj−1 ]q . . .]q, fαij = [. . . [fαj−1 , fαj−2 ]q . . . , fαi ]q. In general, we begin with the simple root αj−1 = αj−1,j and sequentially append necessary simple roots from the left to obtain the final root αij . We can certainly start with any simple root αk with i < k < j and go by adding the appropriate simple roots from the left or from the right in arbitrary order. However, the resulting root vector will always be the same. Highest `-Weight Representations and Functional Relations 13 Further, we proceed to the roots of the form δ−αij and −(δ−αij). First, we note that the root vectors eδ−θ and fδ−θ corresponding to the roots δ− θ = δ−α1,l+1 and −(δ− θ) = −(δ−α1,l+1) are already given. Then, we define inductively eδ−αij = [eαi−1,i , eδ−αi−1,j ]q = eαi−1,ieδ−αi−1,j − qeδ−αi−1,j eαi−1,i , (4.7) fδ−αij = [fδ−αi−1,j , fαi−1,i ]q = fδ−αi−1,j fαi−1,i − q−1fαi−1,ifδ−αi−1,j (4.8) if i > 1, and eδ−α1j = [eαj,j+1 , eδ−α1,j+1 ]q = eαj,j+1eδ−α1,j+1 − qeδ−α1,j+1 eαj,j+1 , (4.9) fδ−α1j = [fδ−α1,j+1 , fαj,j+1 ]q = fδ−α1,j+1 fαj,j+1 − q−1fαj,j+1fδ−α1,j+1 (4.10) for j < l + 1. The inductive rules (4.7), (4.8) and (4.9), (4.10) uniquely lead to the expressions eδ−αij = [eαi−1 , . . . [eα1 , [eαj , . . . [eαl , eδ−θ]q . . .]q]q . . .]q, (4.11) fδ−αij = [. . . [[. . . [fδ−θ, fαl ]q, . . . fαj ]q, fα1 ]q, . . . fαi−1 ]q. (4.12) Generally speaking, we begin with the highest root θ and sequentially subtract redundant simple roots first from the right and then from the left. We can arbitrarily interchange subtractions from the left and from the right, but the result will be the same. Indeed, there is another obvious possibility to write relations (4.11), (4.12), namely eδ−αij = [eαj , . . . [eαl , [eαi−1 , . . . [eα1 , eδ−θ]q . . .]q]q . . .]q, fδ−αij = [. . . [[. . . [fδ−θ, fα1 ]q, . . . fαi−1 ]q, fαl ]q, . . . fαj ]q. Finally, for j = i + 1, so that αij at the left hand side of relations (4.11), (4.12) means any of the simple roots αi,i+1 = αi, i ∈ I, we obtain eδ−α1 = [eα2 , [eα3 , . . . [eαl , eδ−θ]q . . .]q]q, (4.13) eδ−αi = [eαi−1 , . . . [eα1 , [eαi+1 , . . . [eαl , eδ−θ]q . . .]q]q . . .]q, i = 2, . . . , l − 1, (4.14) eδ−αl = [eαl−1 , . . . [eα2 , [eα1 , eδ−θ]q]q . . .]q (4.15) and, similarly, fδ−α1 = [[. . . [fδ−θ, fαl ]q, . . . fα3 ]q, fα2 ]q, (4.16) fδ−αi = [. . . [[. . . [fδ−θ, fαl ]q, . . . fαi+1 ]q, fα1 ]q, . . . fαi−1 ]q, i = 2, . . . , l − 1, (4.17) fδ−αl = [. . . [[fδ−θ, fα1 ]q, fα2 ]q, . . . fαl−1 ]q. (4.18) It is clear that (4.14) and (4.17) can also be written equivalently as eδ−αi = [eαi+1 , . . . [eαl , [eαi−1 , . . . [eα1 , eδ−θ]q . . .]q]q . . .]q, fδ−αi = [. . . [[. . . [fδ−θ, fα1 ]q, . . . fαi−1 ]q, fαl ]q, . . . fαi+1 ]q. When the root vectors corresponding to all the roots αij and δ − αij with αij ∈ 4+ are defined, we can continue by adding imaginary roots nδ. The root vectors corresponding to the imaginary roots are additionally labelled by the positive roots γ ∈ 4+ of sll+1 and are given by the relations e′δ,γ = [eγ , eδ−γ ]q, f ′δ,γ = [fδ−γ , fγ ]q. The remaining higher root vectors are defined iteratively by [31, 39] eγ+nδ = ([2]q) −1[eγ+(n−1)δ, e ′ δ,γ ]q, fγ+nδ = ([2]q) −1[f ′δ,γ , fγ+(n−1)δ]q, 14 Kh.S. Nirov and A.V. Razumov e(δ−γ)+nδ = ([2]q) −1[e′δ,γ , e(δ−γ)+(n−1)δ]q, f(δ−γ)+nδ = ([2]q) −1[f(δ−γ)+(n−1)δ, f ′ δ,γ ]q, e′nδ,γ = [eγ+(n−1)δ, eδ−γ ]q, f ′nδ,γ = [fδ−γ , fγ+(n−1)δ]q, where we use that (αij \|αij) = 2 for all i and j. It is worth to note that, among all imaginary root vectors e′nδ,γ and f ′nδ,γ only the root vectors e′nδ,αi and f ′nδ,αi , i ∈ I, are independent and required for the construction of the Poincaré–Birkhoff–Witt basis. Besides, there is another set of useful root vectors introduced by the functional equations −κqeδ,γ(u) = log(1− κqe′δ,γ(u)), (4.19) κqfδ,γ(u−1) = log(1 + κqf ′ δ,γ(u−1)), (4.20) where the generating functions e′δ,γ(u) = ∞∑ n=1 e′nδ,γu n, eδ,γ(u) = ∞∑ n=1 enδ,γu n, f ′δ,γ(u−1) = ∞∑ n=1 f ′nδ,γu −n, fδ,γ(u−1) = ∞∑ n=1 fnδ,γu −n are defined as formal power series. The unprimed imaginary root vectors arise, for example, in formulas for the universal R-matrix of quantum affine algebras [31, 39]. 4.2 Drinfeld’s second realization Drinfeld realized Uq(L(sll+1)) also in a different way [18, 19], as an algebra generated by ξ±i,n, i ∈ I, n ∈ Z, qx, x ∈ h, and χi,n, i ∈ I, n ∈ Z \ {0}. These generators satisfy the defining relations q0 = 1, qx1qx2 = qx1+x2 , [χi,n, χj,m] = 0, qxχj,n = χj,nq x, qxξ±i,nq −x = q±〈αi,x〉ξ±i,n, [χi,n, ξ ± j,m] = ± 1 n [naij ]qξ ± j,n+m, ξ±i,n+1ξ ± j,m − q ±aijξ±j,mξ ± i,n+1 = q±aijξ±i,nξ ± j,m+1 − ξ ± j,m+1ξ ± i,n, [ξ+ i,n, ξ − j,m] = δij φ+ i,n+m − φ − i,n+m q − q−1 . Besides, there are the Serre relations. However, their explicit form is not relevant, and so, we do not put them here. In the above relations, aij are the entries of the Cartan matrix of sll+1. The quantities φ±i,n, i ∈ I, n ∈ Z, are given by the formal power series ∞∑ n=0 φ±i,±nu ±n = q±hi exp ( ±κq ∞∑ n=1 χi,±nu ±n ) , (4.21) where the conditions φ+ i,n = 0, n < 0, φ−i,n = 0, n > 0 are assumed. There is an isomorphism of the two realizations of the quantum loop algebras. In the case under consideration, the generators of the Drinfeld’s second realization are connected with the Highest `-Weight Representations and Functional Relations 15 Cartan–Weyl generators as follows [31, 32]. The generators qx in the Drinfeld–Jimbo’s and Drinfeld’s second realizations are the same, with an important exception that in the first case x ∈ h̃, and in the second case x ∈ h ⊂ h̃. For the generators ξ±i,n and χi,n of the Drinfeld’s second realization one has explicitly ξ+ i,n = { (−1)nieαi+nδ, n ≥ 0, −(−1)niq−hif(δ−αi)−(n+1)δ, n < 0, (4.22) ξ−i,n = { −(−1)(n+1)ie(δ−αi)+(n−1)δq hi , n > 0, (−1)nifαi−nδ, n ≤ 0, (4.23) χi,n = { −(−1)nienδ,αi , n > 0, −(−1)nif−nδ,αi , n < 0. (4.24) As follows from (4.19), (4.20), (4.21) and (4.24), φ+ i,n = { −(−1)niκqq hie′nδ,αi , n > 0, qhi , n = 0, φ−i,n = { q−hi , n = 0, (−1)niκqq −hif ′−nδ,αi , n < 0. Introducing the generating functions φ+ i (u) and φ−i (u) by the formal power series φ+ i (u) = ∞∑ n=0 φ+ i,nu n, φ−i (u−1) = ∞∑ n=0 φ−i,−nu −n, we obtain φ+ i (u) = qhi ( 1− κqe′δ,αi ( (−1)iu )) , (4.25) φ−i (u−1) = q−hi ( 1 + κqf ′ δ,αi ( (−1)iu−1 )) . (4.26) We refer also to [6], where this isomorphism between two realizations of the untwisted quan- tum loop algebra is established by means of a different approach. Besides, in [15] for more general case of twisted affine quantum algebras it was shown that the relation between the two realizations of the quantum group, defined as a C(q)-algebra, is given by a surjective homo- morphism from the Drinfeld’s second realization to the Drinfeld–Jimbo’s realization, and later, in [16], this surjective homomorphism was shown to be injective, thus proving the isomorphism between the two realizations. 4.3 Jimbo’s homomorphism Highest weight representations of the quantum loop algebra Uq(L(sll+1)) are based on the eva- luation homomorphism ε from Uq(L(sll+1)) to Uq(gll+1) [28]. It is defined by the relations ε(qνh0) = qν(Kl+1−K1), ε(qνhi) = qν(Ki−Ki+1), ε(e0) = F1,l+1q K1+Kl+1 , ε(ei) = Ei,i+1, ε(f0) = E1,l+1q −K1−Kl+1 , ε(fi) = Fi,i+1, where i takes all integer values from 1 to l. Thus, if π is a representation of Uq(gll+1), then the composition π ◦ ε gives a representation of Uq(L(sll+1)). In quantum integrable systems, one considers families of representations parameterized by the so-called spectral parameters. We introduce a spectral parameter by means of the mappings 16 Kh.S. Nirov and A.V. Razumov Γζ : Uq(L(sll+1)) → Uq(L(sll+1)), ζ ∈ C×, defined explicitly by the following action on the generators: Γζ(q x) = qx, Γζ(ei) = ζsiei, Γζ(fi) = ζ−sifi. Here, si are arbitrary integers, and it is convenient to denote their total sum by s. Further, given any representation ϕ of Uq(L(sll+1)), we define the corresponding family ϕζ of representations as ϕζ = ϕ ◦ Γζ . We are interested in the representations (ϕ̃λ)ζ and (ϕλ)ζ related to infinite- and finite-dimensi- onal representations π̃λ and πλ of Uq(gll+1). They are defined as( ϕ̃λ ) ζ = π̃λ ◦ ε ◦ Γζ , ( ϕλ ) ζ = πλ ◦ ε ◦ Γζ . Slightly abusing notation we denote the corresponding Uq(L(sll+1))-modules by Ṽ λ and V λ. The defining relations for these modules were obtained in [37] and are as follows: qνh0vm = q ν [ λl+1−λ1+ l∑ i=2 (m1i+mi,l+1)+2m1,l+1 ] vm, qνhivm = q ν [ λi−λi+1+ i−1∑ k=1 (mki−mk,i+1)−2mi,i+1− l+1∑ k=i+2 (mik−mi+1,k) ] vm, e0vm = ζs0q λ1+λl+1+ l∑ i=2 mi,l+1 vm+ε1,l+1 , eivm = ζsi [ λi − λi+1 − l+1∑ j=i+2 (mij −mi+1,j)−mi,i+1 + 1 ] q [mi,i+1]qvm−εi,i+1 + ζsiq λi−λi+1−2mi,i+1− l+1∑ j=i+2 (mij−mi+1,j) i−1∑ j=1 q i−1∑ k=j+1 (mki−mk,i+1) [mj,i+1]qvm−εj,i+1+εji − ζsi l+1∑ j=i+2 q −λi+λi+1−2+ l+1∑ k=j (mik−mi+1,k) [mij ]qvm−εij+εi+1,j , fivm = ζ−siq − i−1∑ j=1 (mji−mj,i+1) vm+εi,i+1 + ζ−si i−1∑ j=1 q − j−1∑ k=1 (mki−mk,i+1) [mji]qvm−εji+εj,i+1 , where i ∈ I. To complete the defining Uq(L(sll+1))-module relations, one also needs an expres- sion for f0vm, see [37], but its explicit form is not used here and we omit it. Twisting (ϕ̃λ)ζ and (ϕλ)ζ by the automorphisms of Uq(L(sll+1)), we can construct more representations of this quantum loop algebra. There are two automorphisms which can be used for the purpose. They are defined by the relations σ(qνhi) = qνhi+1 , σ(ei) = ei+1, σ(fi) = fi+1, i ∈ Î , where we use the identification qνhl+1 = qνh0 , el+1 = e0, fl+1 = f0, and τ(qh0) = qh0 , τ(qhi) = qhl−i+1 , i ∈ I, τ(e0) = e0, τ(ei) = el−i+1, τ(f0) = f0, τ(fi) = fl−i+1, i ∈ I. Highest `-Weight Representations and Functional Relations 17 Here we have σl+1 = id and τ2 = id. We note that the transfer operators related to the twisting of (ϕ̃λ)ζ and (ϕλ)ζ by any powers of σ differ from each other only by permutations of the components of the highest weight λ ∈ k∗l+1, see, for example, [9]. However, considering representations of the Borel subalgebras of Uq(L(sll+1)), we can use the automorphism σ to obtain new interesting representations. Also the twisting of (ϕ̃λ)ζ and (ϕλ)ζ by τ leads to actually different representations of Uq(L(sll+1)) and its Borel subalgebras. 4.4 Uq(b+)-modules The quantum loop algebra Uq(L(sll+1)) has two Borel subalgebras, the positive and the nega- tive ones, denoted by Uq(b+) and Uq(b−), respectively. Representations of these standard Borel subalgebras are what is usually required for the application in quantum integrable systems. In terms of the generators of the Drinfeld–Jimbo’s realization of the quantum loop algebra, the Borel subalgebras are defined in the following simple way. The positive Borel subalgebra is the subalgebra generated by ei, i ∈ Î, and qx, x ∈ h̃, and the negative Borel subalgebra is the subal- gebra generated by fi, i ∈ Î, and qx, x ∈ h̃. It is important that the Borel subalgebras are Hopf subalgebras of Uq(L(sll+1)). The description of Uq(b+) and Uq(b−) in terms of the generators of the Drinfeld’s second realization of Uq(L(sll+1)) is more intricate. Based on (4.22)–(4.24), we note that Uq(b+) contains the Drinfeld generators ξ+ i,n, ξ−i,m, χi,m with i ∈ I, n ≥ 0 and m > 0, while Uq(b−) contains the Drinfeld generators ξ−i,n, ξ+ i,m, χi,m with i ∈ I, n ≤ 0 and m < 0. Since the positive and negative Borel subalgebras are related by the quantum Chevalley involution, we restrict ourselves to the consideration of Uq(b+) only. Restricting any representation of Uq(L(sll+1)) to Uq(b+) one comes to a representation of Uq(b+). In particular, one can consider the restriction of (ϕ̃λ)ζ and (ϕλ)ζ . The correspon- ding Uq(b+)-module relations are obtained by singling out the expressions for qνhivm and eivm, i ∈ Î, from the Uq(L(sll+1))-module relations in Section 4.3. The representations of type ϕ introduced in Section 2, specified later as (ϕ̃λ)ζ and (ϕλ)ζ , are used for the construction of the transfer operators. The representations of type ρ from Section 2 used for the construction of the Q-operators are very different. For the quantum integrable systems related to Uq(L(sll+1)) the desirable representations of type ρ for the Q-operators can be obtained from (ϕ̃λ)ζ as submo- dules of certain degenerations, sending each difference λi − λi+1, i ∈ I, to positive or negative infinity, see, e.g., [2] and [9, 36] for the particular cases l = 1 and l = 2. The general case with arbitrary l was considered in [37]. There, it was shown, in particular, that the relations qνh0vm = q ν ( 2m1+ l∑ j=2 mj ) vm, (4.27) qνhivm = qν(mi+1−mi)vm, i = 1, . . . , l − 1, (4.28) qνhlvm = q −ν ( 2ml+ l−1∑ i=1 mi ) vm, (4.29) e0vm = q ∑l j=2mjvm+ε1 , (4.30) eivm = −qmi−mi+1−1[mi]qvm−εi+εi+1 , i = 1, . . . , l − 1, (4.31) elvm = −κ−1 q qml [ml]qvm−εl , (4.32) where m denotes the l-tuple of nonnegative integers (m1, . . . ,ml), and m + νεi means the respective shift of mi, define an irreducible representation of Uq(b+). Actually, these are defining relations for a submodule of a degeneration of a shifted Uq(b+)-module [37]. Comparing these relations with the defining relations for the representation (ϕ̃λ)ζ in Section 4.3, we can see the limit relation between the universal transfer operator associated with (ϕ̃λ)ζ and the universal Q- operator associated with the representation given by (4.27)–(4.32). Such limit relations between 18 Kh.S. Nirov and A.V. Razumov the universal transfer and Q-operators for l = 1 and l = 2 were established in [8, 36] and [9], respectively. Relations (4.27)–(4.32) define a representation of type ρ described in Section 2, that is, this representation and its twisting by the automorphisms σ and τ are exactly what we need for the construction of representations for the Q-operators. 5 Highest `-weight representations 5.1 Rational `-weights Here we consider Uq(L(sll+1))-modules in the categoryO only. For the definition of this category, we refer to the original papers [24, 25], and also to the later paper [35] as the most appropriate for our purposes. A very useful tool to analyze these modules is the notion of `-weights and `-weight vectors, see, for example, the papers [21, 23, 35]. One defines an `-weight Ψ as a triple Ψ = (λ,Ψ+,Ψ−), (5.1) where λ ∈ h∗, Ψ+ and Ψ− are l-tuples Ψ+ = (Ψ+ i (u))i∈I , Ψ− = ( Ψ−i ( u−1 )) i∈I of formal series Ψ+ i (u) = ∑ n∈Z+ Ψ+ i,nu n ∈ C[[u]], Ψ+ i (u−1) = ∑ n∈Z+ Ψ+ i,nu −n ∈ C [[ u−1 ]] , such that Ψ+ i,0 = q〈λ,hi〉, Ψ−i,0 = q−〈λ,hi〉. (5.2) For an `-weight vector v of `-weight Ψ we then have qxv = q〈λ̃,x〉v for any x ∈ h̃, and φ+ i,nv = Ψ+ i,nv, φ−i,−nv = Ψ−i,−nv, i ∈ I, n ∈ Z+, or, equivalently, φ+ i (u)v = Ψ+ i (u)v, φ−i ( u−1 ) v = Ψ−i ( u−1 ) v, i ∈ I. A Uq(L(sll+1))-module V in the category O is called a highest `-weight module with highest `-weight Ψ, if there exists an `-weight vector v ∈ V of `-weight Ψ, such that ξ+ i,nv = 0, i ∈ I, n ∈ Z, and V = Uq(L(sll+1))v. Up to a scalar factor, such a vector v is determined uniquely. It is called the highest `-weight vector of V . Highest `-Weight Representations and Functional Relations 19 If for some non-negative integers pi, i ∈ I, and complex numbers aik, bik, i ∈ I, 0 ≤ k ≤ pi, one has Ψ+ i (u) = aipiu pi + ai,pi−1u pi−1 + · · ·+ ai0 bipiu pi + bi,pi−1upi−1 + · · ·+ bi0 , (5.3) Ψ−i (u−1) = aipi + ai,pi−1u −1 + · · ·+ ai0u −pi bipi + bi,pi−1u−1 + · · ·+ bi0u−pi , (5.4) then one says that the corresponding `-weight Ψ is rational. The numbers aipi , ai0, bipi , bi0 must be nonzero, such that ai0 bi0 = q〈λ,hi〉, aip bip = q−〈λ,hi〉. These equations are equivalent to (5.2). All `-weights of a Uq(L(sll+1))-module in the category O are rational, see [35] and references therein. In fact, for any rational `-weight Ψ there is a simple Uq(L(sll+1))-module L(Ψ) with highest `-weight Ψ. Any simple Uq(L(sll+1))-module in the category O is isomorphic to L(Ψ) for some rational `-weight Ψ. Thus, there is a one-to-one correspondence between the rational `-weights and the equivalence classes of the simple Uq(L(sll+1))-modules in the category O. In general, the rational `-weights, given explicitly by (5.3), (5.4), correspond to infinite-- dimensional Uq(L(sll+1))-modules. For the finite-dimensional modules they have a special form, see [23, Proposition 1]. One defines the product of `-weights Ψ1 = (λ1,Ψ + 1 ,Ψ − 1 ) and Ψ2 = (λ2,Ψ + 2 ,Ψ − 2 ) as the triple Ψ1Ψ2 = ( λ1 + λ2,Ψ + 1 Ψ+ 2 ,Ψ − 1 Ψ−2 ) , where Ψ+ 1 Ψ+ 2 = ( Ψ+ 1i(u)Ψ+ 2i(u) ) i∈I Ψ−1 Ψ−2 = ( Ψ−1i ( u−1 ) Ψ−2i ( u−1 )) i∈I . Given rational `-weights Ψ1 and Ψ2, the submodule of the tensor product L(Ψ1) ⊗ L(Ψ2) generated by the tensor product of the highest `-weight vectors is a highest `-weight Uq(L(sll+1))- module with highest `-weight Ψ1Ψ2. In particular, L(Ψ1Ψ2) is a subquotient of L(Ψ1)⊗L(Ψ2), see [35] and references therein. As in [12], we denote such subquotient as L(Ψ1)⊗L(Ψ2). Note that the operation ⊗ is associative. In the case of the Borel subalgebra Uq(b+) we are left with only two first components of the triple (5.1), and we define an `-weight Ψ+ as a pair Ψ = ( λ,Ψ+ ) , where λ ∈ h∗ and Ψ+ is an l-tuple Ψ+ = (Ψ+ i (u))i∈I of formal series Ψ+ i (u) = ∑ n∈Z+ Ψ+ i,nu n ∈ C[[u]], such that Ψ+ i,0 = q〈λ,hi〉. 20 Kh.S. Nirov and A.V. Razumov For an `-weight vector v of `-weight Ψ one has qxv = q〈λ̃,x〉v for any x ∈ h̃, and φ+ i,nv = Ψ+ i,nv, i ∈ I, n ∈ Z+, or, equivalently, φ+ i (u)v = Ψ+ i (u)v, i ∈ I. A Uq(b+)-module W in the category O is called a highest `-weight module with highest `-weight Ψ if there exists an `-weight vector v ∈W of `-weight Ψ, such that ξ+ i,nv = 0, i ∈ I, n ∈ Z+, and W = Uq(b+)v. Such a vector v is unique up to a scalar factor, and it is called the highest `-weight vector of W . An `-weight Ψ of a Uq(b+)-module is said to be rational, if for some non-negative integers pi, qi, i ∈ I, and complex numbers air, bis, i ∈ I, 0 ≤ r ≤ pi, 0 ≤ s ≤ qi, one has Ψ+ i (u) = aipiu pi + ai,pi−1u pi−1 + · · ·+ ai0 biqiu qi + bi,qi−1uqi−1 + · · ·+ bi0 , (5.5) where ai0 bi0 = q〈λ,hi〉. All `-weights of a Uq(b+)-module in the category O are rational, see [26] and references therein. For any rational `-weight Ψ there is a simple Uq(b+)-module L(Ψ) with highest `- weight Ψ. This module is unique up to isomorphism. Any simple Uq(b+)-module is isomorphic to L(Ψ) for some `-weight Ψ. Thus, there is a one-to-one correspondence between the rational `-weights and the equivalence classes of the simple Uq(b+)-modules in the category O. Similarly as in the case of Uq(L(sll+1))-modules, the general rational `-weights (5.5) corre- spond to infinite-dimensional Uq(b+)-modules. For the description of finite-dimensional modules we refer to [22, Remark 3.11]. Given rational `-weights Ψ1 and Ψ2, the submodule of the tensor product L(Ψ1) ⊗ L(Ψ2) generated by the tensor product of the highest `-weight vectors is a highest `-weight Uq(b+)- module with highest `-weight Ψ1Ψ2. In particular, L(Ψ1Ψ2) is a subquotient of L(Ψ1)⊗L(Ψ2) denoted as L(Ψ1)⊗ L(Ψ2). We have already noted in Section 4.1 a special role of the higher root vectors e′nδ,αi and f ′nδ,αi . Besides, we see from equations (4.25), (4.26) and the definition of the highest `-weight repre- sentations that only the root vectors e′nδ,αi , f ′nδ,αi are used to determine the highest `-weight vectors and highest `-weights. Relative to the Borel subalgebra Uq(b+), it means that only the root vectors e′nδ,αi are used for the corresponding highest `-weight vectors and highest `-weights. Highest `-Weight Representations and Functional Relations 21 5.2 Prefundamental and q-oscillator representations The first example of the highest `-weight representations of the Borel subalgebras is given by the prefundamental representations [26]. They are defined by simple highest `-weight Uq(b+)- modules L±i,a with the highest `-weights (λi,a, (Ψ ± i,a) +) of the simplest nontrivial form with λi,a = 0, (Ψ±i,a) + = (1, . . . , 1 i−1 , (1− au)±1, 1, . . . , 1 l−i ), i ∈ I, a ∈ C×. Also the one-dimensional representation with the highest `-weight Ψξ = (λξ, (Ψξ) +) defined as λξ = ξ, (Ψξ) + = ( q〈ξ,h1〉, . . . q〈ξ,hl〉 ) (5.6) is treated as a prefundamental representation. The corresponding Uq(b+)-module is denoted by Lξ. We see that, in the case under consideration, there are actually 2l really different prefundamental representations. It is relevant to recall here the notion of a shifted Uq(b+)-module. Let V be a Uq(b+)-module in the category O. Given an element ξ ∈ h∗, the shifted Uq(b+)-module V [ξ] is defined as follows. If ϕ is the representation of Uq(b+) corresponding to the module V and ϕ[ξ] is the representation corresponding to the module V [ξ], then ϕ[ξ](ei) = ϕ(ei), i ∈ I, ϕ[ξ](qx) = q〈ξ̃,x〉ϕ(qx), x ∈ h̃. It is clear that the module V [ξ] is in the category O and is isomorphic to V ⊗Lξ. Any Uq(b+)- module in the category O is a subquotient of a tensor product of prefundamental representa- tions [26]. The second example of the highest `-weight representations of Uq(b+) is provided by the q-oscillator representations. The q-oscillator algebra Oscq is a unital associative C-algebra with generators b†, b, qνN , ν ∈ C, satisfying the relations q0 = 1, qν1Nqν2N = q(ν1+ν2)N , qνNb†q−νN = qνb†, qνNbq−νN = q−νb, b†b = qN − q−N q − q−1 , bb† = qqN − q−1q−N q − q−1 . We use two representations of Oscq. First, let W+ denote the free vector space generated by the set {v0, v1, . . .}. The relations qνNvm = qνmvm, b†vm = vm+1, bvm = [m]qvm−1, where it is assumed that v−1 = 0, endow W+ with the structure of an Oscq-module. The corresponding representation of Oscq is denoted by χ+. Second, let W− denote the free vector space generated by the set {v0, v1, . . .}. The relations qνNvm = q−ν(m+1)vm, bvm = vm+1, b†vm = −[m]qvm−1, where it is assumed that v−1 = 0, endow the vector space W− with the structure of an Oscq- module. The corresponding representation of Oscq by χ−. In the case under consideration, we need the tensor product of l copies of the q-oscillator algebra, Oscq ⊗ . . .⊗Oscq = (Oscq) ⊗l. Here we introduce the notation bi = 1⊗ · · · ⊗ 1⊗ b⊗ 1⊗ · · · ⊗ 1, b†i = 1⊗ · · · ⊗ 1⊗ b† ⊗ 1⊗ · · · ⊗ 1, qνNi = 1⊗ · · · ⊗ 1⊗ qνN ⊗ 1⊗ · · · ⊗ 1, where b, b† and qνN take only the i-th place of the respective tensor products. 22 Kh.S. Nirov and A.V. Razumov As was shown in [37], the mapping ρ : Uq(b+)→ (Oscq) ⊗l defined by the relations ρ(qνh0) = q ν ( 2N1+ l∑ j=2 Nj ) , ρ(e0) = b†1q l∑ j=2 Nj , (5.7) ρ(qνhi) = qν(Ni+1−Ni), ρ(ei) = −bib † i+1q Ni−Ni+1−1, (5.8) ρ(qνhl) = q −ν ( 2Nl+ l−1∑ j=1 Nj ) , ρ(el) = −κ−1 q blq Nl , (5.9) where i = 1, . . . , l − 1, is a homomorphism from the Borel subalgebra Uq(b+) to the respective tensor power of the q-oscillator algebra. Indeed, relations (5.7)–(5.9) give an obvious interpre- tation of the Uq(b+)-module relations (4.27)–(4.32) in terms of the q-oscillators. To get further a representation of Uq(b+), one takes the composition of a representation of (Oscq) ⊗l with the mapping ρ. 6 Automorphisms and further representations Fixing a finite-dimensional representation of the quantum loop algebra in the quantum space, we can construct explicitly the monodromy- and L-operators corresponding to the homomorphisms ε ◦ Γζ and ρ ◦ Γζ , respectively. Here, ε is the Jimbo’s homomorphism defined in Section 4.3 and ρ is defined in the preceding section as a homomorphism of Uq(b+) to the q-oscillator alge- bra [37]. Besides, Γζ from Section 4.3 is the grading automorphism of Uq(L(sll+1)) introducing the spectral parameter. Let the finite-dimensional representation in the quantum space be the first fundamental representation (ϕ(1,0,...,0))η, so that the monodromy operator is given by the expression M(ζ\|η) = ( εζ ⊗ ( ϕ(1,0,...,0) ) η ) (R). It is clear that M(ζ\|η) ∈ Uq(gll+1)⊗End(Cl+1) for any ζ, η ∈ C×. It follows from the structure of the universal R-matrix that M(ζν\|ην) = M(ζ\|η). Therefore, one can write M(ζ\|η) = M ( ζη−1\|1 ) = M ( ζη−1 ) , where M(ζ) = M(ζ\|1). Identifying End(Cl+1) with Matl+1(C), one can represent M(ζ) as M(ζ) = l+1∑ i,j=1 M(ζ)ij ⊗ Eij . Here M(ζ)ij ∈ Uq(gll+1), and Eij ∈ Matl+1(C) are the standard matrix units.2 We denote by M(ζ) the matrix with the matrix entries M(ζ)ij . Generalizing the results of the papers [36] and [38], we see that this matrix has the form M(ζ) = qK/(l+1)eF (ζs)M̃(ζ), where F (ζ) is a transcendental function of ζ, while the entries of M̃(ζ) are rational functions. We use the notation K = l+1∑ i=1 Ki, s = l∑ i=0 si. 2Do not confuse with generators of Uq(gll+1). Highest `-Weight Representations and Functional Relations 23 The function F (ζ) is defined as follows: F (ζ) = ∑ m∈N Cm [l + 1]qm ζm m , where one has ∑ m∈N Cm ζm m = − log ( 1− l+1∑ k=1 C(k)ζk ) , where the elements C(k), k = 1, . . . , l + 1, are the appropriately normalized quantum Casimir operators of the quantum group Uq(gll+1). For F (ζ), one also has the defining relations l∑ i=0 F ( ql−2iζ ) = − log ( 1− l+1∑ k=1 C(k)ζk ) . The off-diagonal matrix entries M̃(ζ)ij are explicitly given by the relations M̃(ζ)ij = −ζs−sijκqqKiFij , 1 ≤ i < j ≤ l + 1, M̃(ζ)ij = −ζsjiκqEjiq−Kj , 1 ≤ j < i ≤ l + 1, while for the diagonal ones we have M̃(ζ)ii = q−Ki − ζsqKi , i = 1, . . . , l + 1. Here and below we denote sij = j−1∑ k=i sk. Under the automorphism Ei → q1/2Eiq Ki−Ki+1 , Fi → q−1/2q−(Ki−Ki+1)Fi, qνKi → qνKi the matrix M̃(ζ) transforms to the matrix for which M̃(ζ)ij = −ζs−sijκqq(Ki+Kj−1)/2Fij , 1 ≤ i < j ≤ l + 1, M̃(ζ)ij = −ζsjiκqEjiq−(Ki+Kj−1)/2, 1 ≤ j < i ≤ l + 1. The diagonal entries remain the same. Thus, up to a factor belonging to the center of Uq(gll+1), we reproduce the result obtained by Jimbo [28]. In a similar way we define the L-operator L(ζ\|η) = ( ρζ ⊗ ( ϕ(1,0,...,0) ) η ) (R) and denote by L(ζ) the corresponding matrix with the entries in (Oscq) ⊗l. One has L(ζ) = ef(ζs)L̃(ζ), where the transcendental function f is given by the defining equation l∑ j=0 f(q2j−lζ) = − log(1− ζ) 24 Kh.S. Nirov and A.V. Razumov and can explicitly be written as a series f(ζ) = ∑ m∈N 1 [l + 1]qm ζm m . The entries of L̃(ζ) are rational functions. For the entries below and above the diagonal we have L̃(ζ)ij = −ζsjiκqbjb † iq Nj+Nji−Ni+i−j−2, 1 < i− j < l, L̃(ζ)i+1,i = ζsiκqbib † i+1q 2Ni−Ni+1−1, i = 1, . . . , l − 1, L̃(ζ)l+1,i = ζsi,l+1biq Ni+Ni,l+1+l−i, i = 1, . . . , l, and L̃(ζ)i,l+1 = −ζs−si,l+1κqb † iq 2N1i+N1,l+1+Ni+1,l+1+i−1, i = 1, . . . , l, L̃(ζ)ij = 0, i < j < l + 1, respectively. The diagonal elements of L̃(ζ) are L̃(ζ)ii = qNi , i = 1, . . . , l, L̃(ζ)l+1,l+1 = q−N1,l+1 − ζsqN1,l+1+l+1. Here we use the convention Nij = j−1∑ k=i Nk. For the cases l = 1 and l = 2 we refer to the paper [7], where such L-operators were constructed from the universal R-matrix. The monodromy operator M(ζ) and the L-operator L(ζ) satisfy the Yang–Baxter equation with the R-matrix R(ζ) = q−l/(l+1)ef(q−lζs)−f(qlζs)R̃(ζ), where R̃(ζ) = l+1∑ i=1 Eii ⊗ Eii + a(ζs) l+1∑ i,j=1 i 6=j Eii ⊗ Ejj + b(ζs) ∑ i<j ζsijEij ⊗ Eji + ∑ i<j ζs−sijEji ⊗ Eij  and we have denoted a(ζ) = q(1− ζ) 1− q2ζ , b(ζ) = 1− q2 1− q2ζ . Similarly as in Section 4.3 more representations of type ϕ were produced by twisting an initial basic representation (ϕ̃λ)ζ or (ϕλ)ζ by the automorphisms of Uq(L(sll+1)), also more representa- tions of type ρ can be produced from the initial homomorphism ρ (5.7)–(5.9) twisting it by the automorphisms of Uq(b+). The latter can be obtained as the restriction of the automorphisms σ and τ from Uq(L(sll+1)) to Uq(b+) and are explicitly defined as follows: σ(qνhi) = qνhi+1 , σ(ei) = ei+1, i ∈ Î , Highest `-Weight Representations and Functional Relations 25 where the identification qνhl+1 = qνh0 and el+1 = e0 is assumed, and τ(qh0) = qh0 , τ(qhi) = qhl−i+1 , τ(e0) = e0, τ(ei) = el−i+1, i ∈ I. Here we have that σl+1 and τ2 are the identity transformations. Now we define ρa = ρ ◦ σ−a, ρa = ρ ◦ τ ◦ σ−a+1, a = 1, . . . , l + 1, (6.1) and note that ρa in (6.1) can also be written in another form with the help of the relation τ ◦ σ−a+1 = σa−1 ◦ τ = σa−l−2 ◦ τ, a = 1, . . . , l + 1. Then we obtain from relations (5.7)–(5.9) ρa ( qνhi ) = qν(Ni−a+1−Ni−a), i = a+ 1, . . . , l, . . . , l + a− 1, (6.2) ρa ( qνha−1 ) = q −ν ( 2Nl+ l−1∑ j=1 Nj ) , ρa ( qνha ) = q ν ( 2N1+ l∑ j=2 Nj ) , (6.3) ρa(ei) = −bi−ab † i−a+1q Ni−a−Ni−a+1−1, i = a+ 1, . . . , l, . . . , l + a− 1, (6.4) ρa(ea−1) = −κ−1 q blq Nl , ρa(ea) = b†1q l∑ j=2 Nj , (6.5) where a = 1, . . . , l+1, and the index i at the left hand side of (6.2) and (6.4) takes values modulo l + 1. The latter assumption means that the identification qνhl+1 = qνh0 and el+1 = e0 holds. Using tensor products of the representations χ− and χ+, we define the representations θa as θa = (χ− ⊗ · · · ⊗ χ− l−a+1 ⊗χ+ ⊗ · · · ⊗ χ+ a−1 ) ◦ ρa, a = 1, . . . , l + 1. (6.6) These representations are chosen so as to obtain highest `-weight representations. The corre- sponding basis vectors can be defined as v (a) m = bm1 1 · · · b ml−a+1 l−a+1 b †ml−a+2 l−a+2 · · · b†ml l v0, where mi ∈ Z+ for all i = 1, . . . , l and we use the notation m = (m1, . . . ,ml) and v0 = v(0,...,0). For the mappings ρa, a = 1, . . . , l + 1, we obtain the following relations: ρa ( qνhi ) = qν(Na−i−Na−i−1), i = 0, 1, . . . , a− 2, ρa ( qνha−1 ) = q ν ( 2N1+ l∑ j=2 Nj ) , ρa ( qνha ) = q −ν ( 2Nl+ l−1∑ j=1 Nj ) , ρa ( qνhi ) = qν(Nl+a−i+1−Nl+a−i), i = a+ 1, a+ 2, . . . , l, ρa(ei) = −ba−i−1b † a−iq Na−i−1−Na−i−1, i = 0, 1, . . . , a− 2, ρa(ea−1) = b†1q l∑ j=2 Nj , ρa(ea) = −κ−1 q blq Nl , ρa(ei) = −bl+a−ib † l+a−i+1q Nl+a−i−Nl+a−i+1−1, i = a+ 1, a+ 2, . . . , l. Respectively, the homomorphisms θa allowing one to obtain highest `-weight representations are now defined as θa = (χ− ⊗ · · · ⊗ χ− a−1 ⊗χ+ ⊗ · · · ⊗ χ+ l−a+1 ) ◦ ρa, a = 1, . . . , l + 1. 26 Kh.S. Nirov and A.V. Razumov Then the corresponding basis vectors are given by v (a) m = bm1 1 · · · b ma−1 a−1 b†ma a · · · b†ml l v0. The vectors v (a) m and v (a) m are actually `-weight vectors for the representations θa and θa, respectively. Starting from the representations θa and θa we define the families (θa)ζ and (θa)ζ as (θa)ζ = θa ◦ Γζ , (θa)ζ = θa ◦ Γζ . Note here that for a representation ϕ of Uq(L(sll+1)) we have ϕζ(φ + i (u)) = ϕ(φ+ i (ζsu)), ϕζ ( φ−i ( u−1 )) = ϕ ( φ−i ( ζ−su−1 )) . (6.7) If ϕ is a representation of Uq(b+), only the first one of the above two equations is to be considered. The vectors v (a) m and v (a) m are `-weight vectors for the representations (θa)ζ and (θa)ζ as well. We use for the corresponding `-weights the notation given by the equations (θa)ζ(φ + i (u))v (a) m = θa(φ + i (ζsu))v (a) m = Ψ+ i,m,a(u)v (a) m , (θa)ζ(φ + i (u))v (a) m = θa(φ + i (ζsu))v (a) m = Ψ+ i,m,a(u)v (a) m , where the first equation of (6.7) is taken into account. The corresponding elements of h∗ are denoted as λm,a and λm,a. It is worthwhile to note that ρa ( qνhi ) = ρl−a+2 ( qνhl−i+1 ) , ρa(ei) = ρl−a+2(el−i+1) and v̄ (a) m = v (l−a+2) m , a = 1, . . . , l + 1. Applied to the relation between φ+ i (u) and e′nδ,αi in (4.25), this leads us to the conclusion that the `-weights Ψ+ i,m,a(u) are connected with the `-weights Ψ+ i,m,a(u) as Ψ+ i,m,a(u) = Ψ+ l−i+1,m,l−a+2 ( −(−1)lu ) . (6.8) For the elements λm,a we have λm,a = ι(λm,l−a+2), (6.9) where the linear mapping ι : h∗ → h∗ is determined by the relation ι(ωi) = ωl−i+1. We have thus 2(l + 1) different highest `-weight q-oscillator representations.3 And this is actually the number of different L- and respective Q-operators to be considered in the quantum integrable systems associated with the quantum loop algebra Uq(L(sll+1)). 3However, for l = 1 there are only 2 such representations. Highest `-Weight Representations and Functional Relations 27 7 Highest `-weights and functional relations In our recent paper [12], we have presented the `-weights corresponding to the representations (θa)ζ and (θa)ζ , a = 1, . . . , l + 1. As a consequence, putting m = 0 in those expressions,we obtain the corresponding highest `-weights. For the representation (θa)ζ we have the highest `-weights with λ0,1 = −(l + 1)ω1, (7.1) Ψ+ i,0,1(u) = { q−l−1 ( 1− q−lζsu )−1 , i = 1, 1, i = 2, . . . , l, (7.2) λ0,a = (l − a+ 1)ωa−1 − (l − a+ 2)ωa, (7.3) Ψ+ i,0,a(u) =  1, i = 1, . . . , a− 2, ql−a+1 ( 1− q−l+aζsu ) , i = a− 1, q−l+a−2 ( 1− q−l+a−1ζsu )−1 , i = a, 1, i = a+ 1, . . . , l, (7.4) λ0,l+1 = 0, (7.5) Ψ+ i,0,l+1(u) = { 1, i = 1, . . . , l − 1, 1− qζsu, i = l. (7.6) Then, using (6.8) and (6.9), we obtain from (7.1)–(7.6) the highest `-weights also for the repre- sentations (θa)ζ . They are λ0,1 = 0, Ψ+ i,0,1(u) = { 1 + (−1)lqζsu, i = 1, 1, i = 2, . . . , l, λ0,a = −aωa−1 + (a− 1)ωa, Ψ+ i,0,a(u) =  1, i = 1, . . . , a− 2, q−a ( 1 + (−1)lq−a+1ζsu )−1 , i = a− 1, qa−1 ( 1 + (−1)lq−a+2ζsu ) , i = a, 1, i = a+ 1, . . . , l, λ0,l+1 = − (l + 1)ωl, Ψ+ i,0,l+1(u) = { 1, i = 1, . . . , l − 1, q−l−1 ( 1 + (−1)lq−lζsu )−1 , i = l. The explicit forms of the highest `-weights allow us to conclude that the representations (θl+1)ζ and (θ1)ζ are isomorphic to prefundamental representations, the representations (θ1)ζ and (θl+1)ζ are isomorphic to shifted prefundamental representations, and the other representa- tions (θa)ζ and (θa)ζ with a = 2, . . . , l are isomorphic to subquotients of tensor products of two certain prefundamental representations of Uq(b+). Explicitly we have (θ1)ζ ∼= Lξ1 ⊗ L − 1,q−lζs , (θa)ζ ∼= Lξa ⊗ ( L+ a−1,q−l+aζs ⊗ L− a,q−l+a−1ζs ) , a = 2, . . . , l, (θl+1)ζ ∼= L+ l,qζs , 28 Kh.S. Nirov and A.V. Razumov where the shifts ξa are determined by the equation ξa = (l − a+ 1)ωa−1 − (l − a+ 2)ωa for the representations (θa)ζ , and( θ1 ) ζ ∼= L+ 1,(−1)l+1qζs ,( θa ) ζ ∼= L ξ̄a ⊗ ( L− a−1,(−1)l−1q−a+1ζs ⊗ L+ a,(−1)l+1q−a+1ζs ) , a = 2, . . . , l,( θl+1 ) ζ ∼= L ξ̄l+1 ⊗ L− l,(−1)l+1q−lζs , where the shifts ξ̄a are determined by the equation ξa = − aωa−1 + (a− 1)ωa for the representations (θa)ζ . The operation ⊗ means taking the corresponding subquotients, as introduced in Section 5. We can also reverse the above relations in order to express the prefundamental representations via subquotients of tensor products of highest `-weight q-oscillator representations. We obtain Lξ−i ⊗ L−i,ζs ∼= (θ1)ql+i−1ζs ⊗ (θ2)ql+i−3ζs ⊗ · · · ⊗ (θi)ql−i+1ζs , (7.7) where the elements ξ−i are defined as ξ−i = −2 i−1∑ j=1 ωj − (l − i+ 2)ωi, (7.8) and Lξ+i ⊗ L+ i,ζs ∼= (θi+1)ql−i−1ζs ⊗ (θi+2)ql−i−3ζs ⊗ · · · ⊗ (θl+1)q−l+i−1ζs , (7.9) with the elements ξ+ i given by the equation ξ+ i = (l − i)ωi − 2 j∑ j=i+1 ωj . (7.10) Similar relations can also be written for the representations θa. Actually, we have Lξ+i ⊗ L+ i,ζs ∼= ( θ1 ) (−1)l−1q−iζs ⊗ ( θ2 ) (−1)l−1q2−iζs ⊗ · · · ⊗ ( θi ) (−1)l−1qi−2ζs , (7.11) where the elements ξ+ i are defined as ξ+ i = −2 i−1∑ j=1 ωj + (i− 1)ωi (7.12) and Lξ−i ⊗ L−i,ζs ∼= ( θi+1 ) (−1)l−1qiζs ⊗ ( θi+2 ) (−1)l−1qi+2ζs ⊗ · · · ⊗ ( θl+1 ) (−1)l−1q2l−iζs , (7.13) with the elements ξ−i given by the equation ξ−i = −(i+ 1)ωi − 2 l∑ j=i+1 ωj . (7.14) Highest `-Weight Representations and Functional Relations 29 Now it is clear that, in the case under consideration, the q-oscillator representations could quite be treated as no less fundamental than the prefundamental ones. Indeed, any Uq(b+)- module in the category O can be presented as a shifted subquotient of a tensor product of q-oscillator representations. And it should also be noted that the highest `-weights of the q-oscillator representations are as simple as the highest `-weights of the prefundamental repre- sentations. We denote the Uq(b+)-modules corresponding to the representations θa defined in (6.6) by Wa, a = 1, . . . , l + 1, and consider the Uq(b+)-module (W1)ζ1 ⊗ · · · ⊗ (Wl+1)ζl+1 . Then, the tensor product of the highest `-weight vectors is an `-weight vector of `-weight determined by the functions Ψ+ i (u) = q−2 1− q−l+i+1ζsi+1u 1− q−l+i−1ζsi u , i = 1, . . . , l. And now, let us take the representation ϕ̃λ, λ ∈ k∗, of the whole quantum loop algebra Uq(L(sll+1)) constructed with the help of the Jimbo’s homomorphism [37] and consider its restriction to the Borel subalgebra Uq(b+). We denote this restriction and the correspond- ing Uq(b+)-module again by ϕ̃λ and Ṽ λ. It can be shown4 that the highest `-weight of the Uq(b+)-module (Ṽ λ)ζ is determined by the functions Ψ+ i (u) = qλi−λi+1 1− q2λi+1−i+1ζsu 1− q2λi−i+1ζsu , i = 1, . . . , l. Let ρ denote the half-sum of all positive roots of gll+1. One can show that 〈ρ,Ki〉 = l 2 − i+ 1. We see that if ζi = q2〈λ+ρ,Ki〉/sζ, then the submodule of (W1)ζ1 ⊗ · · · ⊗ (Wl+1)ζl+1 generated by the tensor product of the highest `-weight vectors of (Wa)ζa , a = 1, . . . , l+ 1, is isomorphic to the shifted module (Ṽ λ)ζ [ξ], where the shift ξ is determined by the equation ξ = − l∑ i=1 (λi − λi+1 + 2)ωi. A similar conclusion holds for θa as well. This connection between the highest `-weights reflects the basic functional relation between the universal transfer operator based on the infinite- dimensional representation (ϕ̃λ)ζ and the product of all universal Q-operators based on the q-oscillator representations (θa)ζ at certain values of the spectral parameters. Such relations for l = 1 and l = 2 were proved in the papers [4, 5, 8, 10, 36] and [2, 9], respectively. Besides, comparing (7.7) with (7.13), also taking into account the shifts (7.8) and (7.14), we can relate the integrability objects Qa(ζ) and Qa(ζ) associated with the representations (θa)ζ and (θa)ζ , respectively. Specifically, linear combinations of the products Q1(ζ1) · · · Qi(ζi) are expressed through the product Qi+1(ζi+1) · · · Ql+1(ζl+1), i = 1, . . . , l, at certain values of the spectral parameters ζa, a = 1, . . . , l+1. In the same way, comparing (7.9) with (7.11), also taking into account the shifts (7.10) and (7.12), we can relate the products Q1(ζ1) · · · Qi(ζi) with the products Qi+1(ζi+1) · · · Ql+1(ζl+1), i = 1, . . . , l, at certain values of the spectral parameters ζa, a = 1, . . . , l+ 1. Obviously, such relations between the universal Q-operators are absent if l = 1. For the simplest higher rank case, l = 2, the corresponding relations were proved in [2, 9]. 4Work in progress, to appear elsewhere. 30 Kh.S. Nirov and A.V. Razumov 8 Conclusion We have explicitly related the highest `-weight q-oscillator representations (θa)ζ , (θa)ζ of the Borel subalgebra Uq(b+) of the quantum loop algebra Uq(L(sll+1)) with arbitrary rank l with the shifted prefundamental representations L±i,x from the category O. Thus, not only the q-oscillator representations can be obtained as subquotients of tensor products of the prefundamental rep- resentations, but also the latter can be expressed via appropriate tensor products of the former. We have also demonstrated how the information about `-weights can be used for the construction of functional relations. For all representations (θa)ζ , (θa)ζ there is a basis of the corresponding representation space consisting of `-weight vectors [12]. It is worthwhile noting that for l ≥ 2 one has 2(l + 1) q-oscillator representations and only 2l prefundamental representations. The case of l = 1 is special in the sense that only 2 representations of both kinds are present [11]. Acknowledgements We are grateful to H. Boos, F. Göhmann and A. Klümper for discussions. 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Sci. 25 (1989), 503–520. https://doi.org/10.2977/PRIMS/86 https://arxiv.org/abs/1406.6729 https://doi.org/10.4171/PRIMS/150 https://arxiv.org/abs/1407.0341 https://doi.org/10.1090/surv/058 https://arxiv.org/abs/1609.05724 https://doi.org/10.1215/00127094-3146282 https://arxiv.org/abs/1308.3444 https://doi.org/10.1090/conm/248/03823 https://arxiv.org/abs/math.QA/9810055 https://doi.org/10.1007/s00031-005-1005-9 https://arxiv.org/abs/math.QA/0312336 https://doi.org/10.1112/plms/pdm017 https://arxiv.org/abs/math.QA/0504269 https://doi.org/10.1112/S0010437X12000267 https://arxiv.org/abs/1104.1891 https://doi.org/10.1007/BF00704588 https://doi.org/10.1007/BF00400222 https://doi.org/10.1090/cbms/085 https://doi.org/10.1090/cbms/085 https://doi.org/10.1017/CBO9780511626234 https://doi.org/10.1016/0393-0440(93)90070-U https://arxiv.org/abs/hep-th/9404036 https://doi.org/10.1007/BF01075497 https://arxiv.org/abs/1504.04572 https://doi.org/10.1090/S0002-9947-2014-06039-X https://arxiv.org/abs/1204.2769 https://doi.org/10.1016/j.geomphys.2016.10.014 https://arxiv.org/abs/1412.7342 https://arxiv.org/abs/1610.02901 https://doi.org/10.1088/1751-8113/46/38/385201 https://arxiv.org/abs/1211.3590 https://doi.org/10.1007/BF01077085 https://doi.org/10.1007/BF01077085 https://doi.org/10.2977/prims/1195173355 1 Introduction 2 Quantum groups and integrability objects 3 Quantum group Uq(gll + 1) and its representations 4 Quantum loop algebra Uq(L(sll + 1)) and its representations 4.1 Cartan–Weyl data 4.2 Drinfeld's second realization 4.3 Jimbo's homomorphism 4.4 Uq(b+)-modules 5 Highest -weight representations 5.1 Rational -weights 5.2 Prefundamental and q-oscillator representations 6 Automorphisms and further representations 7 Highest -weights and functional relations 8 Conclusion References

Highest ℓ-Weight Representations and Functional Relations

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