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+arXiv:2301.00781v1  [math.QA]  2 Jan 2023
+FUSED K-OPERATORS AND THE q-ONSAGER ALGEBRA
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Abstract. We study universal solutions to reflection equations with a spectral parameter, so-called K-
+operators, within a general framework of universal K-matrices – an extended version of the approach intro-
+duced by Appel-Vlaar. Here, the input data is a quasi-triangular Hopf algebra H, its comodule algebra B
+and a pair of consistent twists. In our setting, the universal K-matrix is an element of B ⊗ H satisfying
+certain axioms, and we mostly consider the case H = LUqsl2, the quantum loop algebra for sl2, and B = Aq,
+the alternating central extension of the q-Onsager algebra. Considering tensor products of evaluation rep-
+resentations of LUqsl2 in “non-semisimple” cases, the new set of axioms allows us to introduce and study
+fused K-operators of spin-j; in particular, to prove that for all j ∈ 1
+2N they satisfy the spectral-parameter
+dependent reflection equation. We provide their explicit expression in terms of elements of the algebra Aq
+for small values of spin-j. The precise relation between the fused K-operators of spin-j and evaluations of a
+universal K-matrix for Aq is conjectured based on supporting evidences. Independently, we study K-operator
+solutions of the twisted intertwining relations associated with the comodule algebra Aq, and expand them in
+the Poincar´e-Birkhoff-Witt basis of Aq. With a reasonably general ansatz, we found a unique solution for
+first few values of j which agrees with the fused K-operators, as expected. We conjecture that in general such
+solutions are uniquely determined and match with the expressions of the fused K-operators.
+MSC: 81R50; 81R10; 81U15.
+Keywords: Reflection equation; Universal K-matrix; Fusion for K-operators; q-Onsager algebra
+Contents
+1.
+Introduction
+2
+1.1.
+Background
+2
+1.2.
+Goal and main results
+4
+2.
+Universal R and K matrices
+6
+2.1.
+Universal R-matrix
+6
+2.2.
+Comodule algebras and twist pairs
+7
+2.3.
+Universal K-matrix
+8
+2.4.
+Examples of (H, B)
+10
+3.
+Tensor product representations of LUqsl2 and sub-representations
+13
+3.1.
+Analysis of the tensor product representation of LUqsl2
+14
+3.2.
+The intertwining maps E(j+ 1
+2 ) and F(j+ 1
+2 )
+15
+3.3.
+The maps ¯E(j− 1
+2 ) and ¯F(j− 1
+2 )
+17
+3.4.
+Additional properties
+19
+4.
+Spin-j L- and K-operators
+20
+4.1.
+Spin-j L-operators
+20
+4.2.
+Spin-j K-operators
+28
+4.3.
+Comodule algebra structure using K-operators
+30
+5.
+Fused K-operators for Aq
+32
+5.1.
+The fundamental K-operator for Aq
+32
+5.2.
+Fused K-operators for Aq
+34
+1
+
+2
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+5.3.
+Unitarity and invertibility properties
+37
+5.4.
+Examples of fused K-operators
+38
+5.5.
+Evaluated coaction of fused K-operators
+42
+5.6.
+Twisted intertwining relations for fused K-operators
+44
+6.
+Fused K-operators and the universal K-matrix for Aq
+45
+6.1.
+Supporting evidences
+45
+6.2.
+Comments
+49
+7.
+K-operators and the PBW basis
+51
+7.1.
+Spin- 1
+2 K-operator
+52
+7.2.
+Spin-1 fused K-operator
+52
+8.
+Summary and Outlook
+54
+Appendix A.
+Quantum algebras
+56
+Appendix B.
+Ordering relations for Aq
+57
+Appendix C.
+The universal R-matrix
+57
+C.1.
+Root vectors
+57
+C.2.
+Khoroshkin-Tolstoy construction
+58
+C.3.
+Evaluation of the universal R-matrix.
+58
+C.4.
+The spin- 1
+2 L-operator L( 1
+2 )(u)
+61
+References
+62
+1. Introduction
+1.1. Background. In the context of quantum integrable systems, the R- and K-matrices are the basic in-
+gredients for the construction of the monodromy matrix (or its double row version) leading to the generating
+function for mutually commuting quantities, the so-called transfer matrix. Here, the formal variable of the
+generating function is called ‘spectral parameter’, denoted by u. By definition, the R-matrix is a solution
+of the Yang-Baxter equation with this spectral parameter, whereas the K-matrix satisfies a reflection equa-
+tion, also known under the name boundary Yang-Baxter equation. For a triple of finite-dimensional vector
+spaces V (jk), for k = 1, 2, 3, the corresponding Yang-Baxter equation in End(V (j1) ⊗ V (j2) ⊗ V (j3)) takes the
+form [Ya67, Ba72]:
+(1.1)
+R(j1,j2)
+12
+(u1/u2)R(j1,j3)
+13
+(u1/u3)R(j2,j3)
+23
+(u2/u3) = R(j2,j3)
+23
+(u2/u3)R(j1,j3)
+13
+(u1/u3)R(j1,j2)
+12
+(u1/u2) ,
+where uk are the spectral parameters, and R(jn,jm)
+nm
+(u) are the R-matrices on corresponding products of
+spaces1. Given a R-matrix R(j1,j2)(u) satisfying (1.1), the reflection equation in End(V (j1) ⊗ V (j2)) is given
+by [Sk88]
+(1.2)
+R(j1,j2)
+12
+(u1/u2)K(j1)
+1
+(u1)R(j2,j1)
+21
+(u1u2)K(j2)
+2
+(u2) = K(j2)
+2
+(u2)R(j1,j2)
+12
+(u1u2)K(j1)
+1
+(u1)R(j2,j1)
+21
+(u1/u2) ,
+where
+(1.3)
+R(j2,j1)
+21
+(u) = P(j2,j1)R(j2,j1)(u)P(j1,j2).
+Along the years, several examples of solutions to (1.1) and (1.2) have been obtained for the case of 2 or
+3-dimensional spaces V (jk), see e.g. [ZZ79, ZF80, GZ93, VG93, IOZ96]. Interestingly, these R-matrices can
+1We use the standard notations:
+R(j1,j2)
+12
+(u) = R(j1,j2)(u) ⊗ I,
+R(j2,j3)
+23
+(u) = I ⊗ R(j2,j3)(u),
+R(j1,j3)
+13
+(u) = (P(j2,j1) ⊗ I)(I ⊗ R(j1,j3)(u))(P(j1,j2) ⊗ I),
+where R(j1,j2)(u) ∈ End(V (j1) ⊗ V (j2)) and P(j1,j2) flips V (j1) ⊗ V (j2) to V (j2) ⊗ V (j1) and it acts as identity on V (j3).
+
+FUSED K-OPERATORS FOR Aq
+3
+be interpreted as intertwining operators for the underlying action of the quantum affine algebra Uq �sl2 on the
+tensor product of so-called evaluation representations of spin- 1
+2, or briefly these are spin- 1
+2 solutions. Similarly,
+the spin- 1
+2 expressions of the K-matrix are interpreted as ‘twisted’ intertwiners (with respect to the spectral
+parameter) for the action of a coideal subalgebra of the quantum affine algebra.
+For higher values of spins jk, constructing R- and K-matrices by brute force is increasingly complicated.
+To circumvent this problem, two different methods have been proposed that are summarized as follows (both
+methods use the underlying action of the quantum affine algebra):
+(i) The R- and K-matrices for higher spins are derived using a fusion procedure. This procedure for the
+R-matrix has been originally developed in [Ka79, KRS81, KR87], and for the K-matrix in [MN92].
+For a more recent approach, see [RSV14, BLN15, NP15]. Starting from R- and K-matrix solutions of
+spin- 1
+2, the fused R- and K-matrices of spin-j are obtained inductively by tensoring (or “fusing”) the
+fundamental representations of Uqsl2 and projecting onto the highest spin sub-representation.
+(ii) The idea is to demand that R-matrices for higher spins satisfy a set of intertwining relations with
+respect to the quantum affine algebra action on V (j1) ⊗ V (j2), given by its coproduct. And similarly
+for the K-matrix, it should satisfy a twisted intertwining relation for the action of a certain coideal
+subalgebra, or more generally a comodule algebra, of the quantum affine algebra. According to the
+representation chosen, the twisted intertwining relations lead to a set of recurrence relations for the
+(scalar) entries of the R- and K-matrices. This technique has been initiated in [KR83] for the R-matrix
+and in [MN97, DM01] for the K-matrix where the coideal subalgebra is now known as the q-Onsager
+algebra Oq [T99, B04]. For more general quantum symmetric pairs, see [AV22].
+For instance, for the R-matrix associated with Uq �sl2 that enjoys P-symmetry [RSV14]
+(1.4)
+R(j2,j1)
+21
+(u) = R(j1,j2)(u) ,
+the corresponding fused solutions using (i) have been constructed in [KS82]; see [KR83] for the implementation
+of (ii). For the K-matrix K(j)(u) associated with Oq (a coideal subalgebra of Uq �sl2), the method (i) for spin-1
+was implemented in [IOZ96], while the method (ii) for an arbitrary spin was studied in [DN02].
+It is well-known [Dr86] that the Yang-Baxter equation (1.1) can be derived from the more general setting
+of Yang-Baxter algebras and the universal Yang-Baxter equation, see [DF93, JLM19a, JLM19b] and a review
+in Section 2.1, by specialization to the finite-dimensional representations V (jk). Namely, R-matrix solutions
+of (1.1) are obtained by specializing L-operators of the form
+(1.5)
+�L(j)(u) ∈ H ⊗ End(V (j))
+that satisfy an equation in H ⊗ End(V (j1) ⊗ V (j2)):
+(1.6)
+R(j1,j2)(u/v)�L(j1)
+1
+(u)�L(j2)
+2
+(v) = �L(j2)
+2
+(v)�L(j1)
+1
+(u)R(j1,j2)(u/v) .
+Here H is assumed to be any quantum affine algebra.
+Furthermore, the L-operators themselves can be
+obtained by specializing the universal R-matrix R ∈ H ⊗ H (if it exists). For instance, for Uq �sl2 the universal
+R-matrix is known in terms of root vectors [KT92a, Da98]. In this situation, the previously existing methods
+(i) and (ii) find a natural interpretation within the framework of the universal R-matrix, they just follow
+from the universal R-matrix axioms (R1)-(R3) as reviewed in Section 2.1. As expected, expressions for the
+R-matrices previously derived in the literature using (i) and (ii) match, up to a scalar function, with those
+derived through the specializations of the universal R-matrix.
+For the reflection equation, it is also expected that (1.2) can be derived from the more general setting of
+reflection algebras [Sk88]. By analogy with (1.5) and (1.6), in this case we introduce the K-operators
+(1.7)
+K(j)(u) ∈ B ⊗ End(V (j)) ,
+
+4
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+where B is a comodule algebra over H, or in particular a coideal subalgebra in H, that satisfy an equation in
+B ⊗ End(V (j1) ⊗ V (j2)):
+(1.8)
+R(j1,j2)
+12
+(u1/u2)K(j1)
+1
+(u1)R(j2,j1)
+21
+(u1u2)K(j2)
+2
+(u2) = K(j2)
+2
+(u2)R(j1,j2)
+12
+(u1u2)K(j1)
+1
+(u1)R(j2,j1)
+21
+(u1/u2) .
+In the case of P-symmetric R-matrices, the K-operators for H = LUqsl2 will be studied in Section 4.
+Furthermore, it is expected that these K-operators K(j)(u) can arise from a universal K-matrix satisfying a
+universal reflection equation. The concept of universal K-matrix is not new [CG92, KSS92, DKM02]. In recent
+years, an important progress on this concept has been made, see [BKo15, Ko17, AV20]. In [BKo15, Ko17]
+universal K-matrices k ∈ H and K ∈ B ⊗ H, respectively, are defined for quantum symmetric pairs (H, B),
+these are certain class of right coideal subalgebras B in a Hopf algebra H. In order to study solutions of the
+spectral parameter-dependent reflection equation (1.2), a twisted version k ∈ H of the universal K-matrix
+defined in [BKo15] is introduced in [AV20]. And in this paper, to produce K-operator solutions of (1.8), we
+introduce a twisted version K ∈ B⊗H of the universal K-matrix from [Ko17] via a new set of axioms (K1)-(K3)
+in Section 2.3. Note that the notion of twist is important here because, as it will be discussed in Section 4, it
+allows the derivation of (1.8) from a ψ-twisted reflection equation satisfied by K, see Proposition 2.11.
+Despite the knowledge of root vectors for a class of comodule algebras over Uq �sl2, including the q-Onsager
+algebra Oq [BK17, LW20], an explicit expression of the universal K-matrix in terms of root vectors is not
+available yet, neither their existence results. However, examples of K-operators have previously appeared in
+the literature2, and in the fundamental case only: for j = 1
+2 in (1.7) and B = Aq known as the alternating
+central extension of Oq [T21a], see [BS09]; for fundamental representations of a few higher rank generalizations
+of Oq, see [BF11]. It is thus natural to investigate further, along the directions (i) or (ii), the spectral parameter
+dependent K-operators for arbitrary spin representations.
+A construction of arbitrary spin K-operators is not only important for a better understanding of the
+universal K-matrix formalism, but also because of applications in quantum integrable systems. In the tensor
+product (or spin-chain) representations of the algebra Aq, the K-operators (also known as Sklyanin’s operators)
+are the basic ingredients in the construction of mutually commuting quantities, like the transfer matrices.
+The algebra Aq in a certain sense governs all known integrable boundary conditions through its quotients and
+degenerate cases [BB16]. For example, the q-Onsager quotients appear in the open XXZ chains with the general
+non-diagonal boundary terms [BK05b]: the K-operator in this spin chain is essentially the “double-row”
+monodromy matrix whose entries are the representations of the q-Onsager currents, while the corresponding
+transfer matrix is the image of an abelian subalgebra of Aq. In this context, many properties of the integrable
+models can be studied even before specializations to the spin-chain representations of Aq, and so the K-
+operators can be used in the representation-independent analysis of the related integrable models. This is
+discussed more in Section 8 and in our forthcoming paper [LBG23].
+1.2. Goal and main results. The purpose of this paper is to construct fused K-operators with a spectral
+parameter for a class of comodule algebras B, and show how they relate to specializations of a universal K-
+matrix. This universal K-matrix satisfies a twisted universal reflection equation while the standard reflection
+equations (1.8) and (1.2) are derived via specializations to finite-dimensional representations. We make focus
+on explicit examples associated with the quantum loop algebra H = LUqsl23 and its comodule algebra
+B = Aq. In order to construct K-operators for arbitrary spin representations, we develop the directions (i)
+and (ii) further to the case of K-operators (1.7) based on our new set of axioms (K1)-(K3), and apply them
+in the case of B = Aq. The main results are the following:
+2There are also K-operators that might not fit our setting. For instance, the K-operators associated with a q-oscillator algebra
+or the Askey-Wilson algebra were constructed in [BK02, B04] respectively. However, we are not aware of any comodule algebra
+structure for these algebras.
+3Recall that the quantum loop algebra is Uq �sl2 with zero central charge. The results presented here can be extended to the
+choice H = Uq �sl2 with the same comodule algebra Aq. However, for simplicity of exposition we consider only the quantum loop
+algebra case.
+
+FUSED K-OPERATORS FOR Aq
+5
+• Method (i): Using the axiom (K2) and the detailed analysis of tensor product representation of LUqsl2,
+we derive fused K-operators for arbitrary values of spin-j, starting from a spin- 1
+2 K-operator, see Definition 5.6.
+One of our main results is that they satisfy the reflection equation (1.8), see Theorem 5.7. For j = 1, 3
+2, we
+give explicit expressions of the fused K-operators in terms of generating functions of Aq.
+• Method (ii): K-operators that solve the twisted intertwining relations for B = Aq are considered in
+Section 7. For j = 1
+2, 1 and with a reasonably general ansatz of K(j)(u), solutions of the twisted intertwining
+relations are obtained using a Poincar´e-Birkhoff-Witt (PBW) basis of Aq. They are found to be uniquely
+determined (up to an overall factor), and they match with the fundamental K-operator and the spin-1 fused
+K-operator. Then, it is conjectured that the twisted intertwining relations determine uniquely the K-operators
+for any j and that they match with the fused K-operators constructed through the method (i).
+• The interpretation of the fused K-operators as specializations of a universal K-matrix is studied, and an
+explicit conjecture on the relation between them is formulated in Section 6.
+The text is organized as follows. In Section 2, the formalism of universal R- and K-matrix, Hopf algebra and
+almost cylindrical bialgebras are reviewed following [Dr86, Ko17, AV20]. For our purpose, we consider a mild
+modification of the universal K-matrix axioms from [AV20], see Definition 2.8, in order to handle K-operators
+of the form (1.7). In our framework, they are obtained as evaluations of the universal K-matrix K ∈ B ⊗ H
+(if it exists) satisfying a ψ-twisted reflection equation, see Proposition 2.11. An importance is given for the
+choice of ψ, an automorphism of H that forms a twist pair together with a Drinfeld twist J ∈ H ⊗ H, see
+Definition 2.6. Indeed, for ψ = η defined in (2.34) and J = 1 ⊗ 1, the specialization of the ψ-twisted reflection
+equation leads to the reflection equation (1.8). In this section, we also review basic definitions of the quantum
+loop algebra H = LUqsl2, the q-Onsager algebra B = Oq and its alternating central extension B = Aq.
+In Sections 3-5, the method (i) is considered. Namely, the first part of Section 3 is devoted to a detailed
+analysis of the tensor product of evaluation representations of LUqsl2. Sub-representations arising from the
+two-fold tensor product representation of LUqsl2 for special values of the evaluation parameters and various
+intertwining operators are constructed. In Section 4, we assume the existence of a universal K-matrix K for a
+comodule algebra B and a certain twist pair. Spin-j L-operators and K-operators are defined as evaluations
+of R and K, respectively; see Definitions 4.1 and 4.13. Considering the sub-representation associated with the
+‘fusion’ ( 1
+2, j) → (j + 1
+2) of evaluation representations, it leads to a relation satisfied by the spin-j L- and K-
+operators, see Propositions 4.4 and 4.14. The sub-representation corresponding to ‘reduction’ ( 1
+2, j) → (j − 1
+2)
+leads similarly to Propositions 4.7 and 4.16. At the end of Section 4, the comodule algebra structure of B
+is described in terms of the spin- 1
+2 K-operator and the Ding-Frenkel L-operators. In Section 5, we consider
+the choice of the comodule algebra B = Aq.
+However, in that section we do not assume the existence
+of a universal K-matrix for Aq. We introduce fused K-operators of spin-j in Definition 5.6 based on the
+fundamental K-operator (5.4) from the reflection algebra presentation of Aq, see Theorem 5.1. We prove in
+Theorem 5.7 that they satisfy the reflection equation (1.8). Explicit expressions of the fused K-operators for
+j = 1, 3
+2 are derived in Section 5.4. We also give compact expressions for the fused R-matrices and the fused
+K-operators in (5.52), (5.53), solely in terms of the fundamental R-matrix and K-operator. In Section 6, the
+precise relation between the spin-j K-operators and the fused K-operators leads to Conjecture 1, with a few
+supporting evidences discussed.
+In Section 7, the method (ii) is considered, and K-operator solutions of the twisted intertwining relations in
+Proposition 5.17 are investigated without any additional assumption. In Section 8, we give a brief summary of
+our results and discuss a few perspectives of applications of the K-operator formalism to quantum integrable
+systems.
+In Appendix A, we recall the basics about the quantum algebras LUqsl2 and Uqsl2. In Appendix B, we
+give ordering relations for the generating functions of Aq. In Appendix C, we adapt the universal R-matrix
+
+6
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+constructed by Khoroshkin-Tolstoy in [KT92a] to our conventions. The corresponding Ding-Frenkel type L-
+operators L+(u), [L−(u)]−1 and the spin- 1
+2 L-operator L( 1
+2 )(u) are computed by evaluation of the universal
+R-matrix.
+Notations. We denote the set of natural numbers by N = {0, 1, 2, . . .} and the positive integers by
+N+ = {1, 2, . . .}.
+All algebras are considered over the field of complex numbers C, if not stated otherwise. Though the results
+till Section 2.4 are valid also over general fields, and many results in Section 5 can be directly generalized to
+algebraically closed fields of zero characteristic, we fix for simplicity the ground field to be C.
+Let q ∈ C∗, and we assume in this paper that q is not a root of unity. The q-commutator is
+�
+X, Y
+�
+q = qXY − q−1Y X
+and
+�
+X, Y
+�
+=
+�
+X, Y
+�
+1 = XY − Y X. We denote the q-numbers by [n]q = (qn − q−n)/(q − q−1).
+We denote by I2j the 2j × 2j identity matrix. We also use Pauli matrices:
+(1.9)
+σ+ =
+�0 1
+0 0
+�
+,
+σ− =
+�0 0
+1 0
+�
+,
+σx =
+�0 1
+1 0
+�
+,
+σy =
+�0 −i
+i 0
+�
+,
+σz =
+�1 0
+0 −1
+�
+.
+2. Universal R and K matrices
+Firstly, we recall the definition of a quasi-triangular Hopf algebra H with the associated universal R-matrix
+that satisfies the universal Yang-Baxter equation. Then, inspired by the works [Ko17] and [AV20], we define
+in Section 2.3 a universal K-matrix associated with H, a pair of its consistent twists (ψ, J), and its comodule
+algebra B – this is an element in B ⊗ H that satisfies a universal reflection equation (also called ψ-twisted
+reflection equation), see Proposition 2.11. We also recall the examples of H and B that are considered in
+this paper: the quantum loop algebra H = LUqsl2 and its comodule algebra the q-Onsager algebra B = Oq
+as well as the alternating central extension B = Aq ∼= Oq ⊗ C[Z]. The corresponding K-operators and their
+relation with the universal K-matrix will be studied in Sections 5 and 6.
+2.1. Universal R-matrix. Let H be a Hopf algebra with coproduct ∆: H → H ⊗ H, the counit ǫ: H → C
+and the antipode S : H → H, which are subject to consistency conditions4. We denote the opposite coproduct
+∆op = p ◦ ∆, where p is the permutation operator5. Here we use the notation R12 = R ⊗ 1, R23 = 1 ⊗ R,
+R13 = p23 ◦ R12.
+Definition 2.1 ([Dr86]). For a Hopf algebra H, an invertible element R ∈ H ⊗H is called universal R-matrix
+if it satisfies
+R∆(x) = ∆op(x)R ,
+∀x ∈ H ,
+(R1)
+(∆ ⊗ id)(R) = R13R23 ,
+(R2)
+(id ⊗ ∆)(R) = R13R12 .
+(R3)
+If such R exists, then the pair (H, R) is called a quasi-triangular Hopf algebra.
+We note that the universal R-matrix necessarily satisfies
+(S ⊗ id)(R) = R−1 = (id ⊗ S)(R) ,
+(2.1)
+(ǫ ⊗ id)(R) = 1 = (id ⊗ ǫ)(R) .
+(2.2)
+4We refer to [CP95, Chap. 4] for the axioms of a Hopf algebra.
+5Here we define p(x ⊗ y) = y ⊗ x, for all x,y in H.
+
+FUSED K-OPERATORS FOR Aq
+7
+Using the relations (R1)–(R3) one can show that the universal R-matrix satisfies the universal Yang-Baxter
+equation (without evaluation parameter):
+(2.3)
+R12R13R23 = R23R13R12 .
+It is well-known that the universal R-matrix coming from a quasi-triangular Hopf algebra gives a way to
+generate R-matrices on tensor product of representations, via evaluations as we will see in Section 4 [Dr86].
+2.2. Comodule algebras and twist pairs. Appel and Vlaar introduced the notion of an almost cylindrical
+bialgebra [AV20, Def. 2.6] which is a quasi-triangular bialgebra with a universal solution of a twisted reflection
+equation. This approach enables to study solutions of the parameter-dependent reflection equation for the
+case of finite-dimensional representations of quantum affine algebras. In this section, first we review known
+results of [AV20]. Then, inspired by6 [AV20, Prop.2.7], a universal K-matrix satisfying a universal reflection
+equation is introduced.
+The following is an extension of [AV20, Def. 2.2] from bialgebras to Hopf algebras.
+Definition 2.2. Let (H, R) be a quasi-triangular Hopf algebra and ψ: H → H an algebra automorphism.
+The ψ-twisting of (H, R) is the quasi-triangular Hopf algebra (Hψ, Rψψ) obtained from (H, R) by pullback
+through ψ, i.e. Hψ is the Hopf algebra with same multiplication, new coproduct, counit and antipode7:
+(2.4)
+∆ψ = (ψ ⊗ ψ) ◦ ∆ ◦ ψ−1,
+ǫψ = ǫ ◦ ψ−1,
+Sψ = ψ ◦ S ◦ ψ−1 ,
+and the universal R-matrix is given by
+(2.5)
+Rψψ = (ψ ⊗ ψ)(R) .
+In what follows, we also use the opposite version of the ψ-twisting of (H, R). Let Hcop be the Hopf algebra
+with the coproduct ∆op, the antipode S−1 and the other structure maps are the same as H. Let Hcop,ψ be
+the ψ-twisting of Hcop.
+Lemma 2.3. The pair (Hcop,ψ, Rψψ
+21 ) is a quasi-triangular Hopf algebra.
+Proof. We need to show that the following relations hold:
+Rψψ
+21 ∆op,ψ(x) = ∆ψ(x)Rψψ
+21 ,
+∀x ∈ H ,
+(2.6)
+(∆op,ψ ⊗ id)(Rψψ
+21 ) = Rψψ
+31 Rψψ
+32 ,
+(2.7)
+(id ⊗ ∆op,ψ)(Rψψ
+21 ) = Rψψ
+31 Rψψ
+21 .
+(2.8)
+Note that as corollaries of (R1)–(R3) we have:
+∆(x)R21 = R21∆op(x) ,
+∀x ∈ H ,
+(2.9)
+(∆op ⊗ id)(R21) = R31R32 ,
+(2.10)
+(id ⊗ ∆op)(R21) = R31R21 .
+(2.11)
+The relation (2.6) is obtained by applying (ψ ⊗ ψ) to (2.9)
+(ψ ⊗ ψ)[∆(x)R21] = (ψ ⊗ ψ)[R21∆op(x)]
+[(ψ ⊗ ψ)∆(ψ−1ψ(x))]Rψψ
+21 = Rψψ
+21 [(ψ ⊗ ψ)∆op(ψ−1ψ(x))]
+∆ψ(ψ(x))Rψψ
+21 = Rψψ
+21 ∆op,ψ(ψ(x)) .
+Next, rewrite (2.10) as
+(∆op ⊗ id)((ψ−1 ⊗ ψ−1)Rψψ
+21 ) = R31R32 ,
+6We extend their definition.
+7It is enough to verify that ǫψ ◦ Sψ = ǫψ.
+
+8
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+(∆op ◦ ψ−1 ⊗ ψ−1)(Rψψ
+21 ) = R31R32 ,
+and apply (ψ ⊗ ψ ⊗ ψ) to get
+(∆op,ψ ⊗ id)(Rψψ
+21 ) = Rψψ
+31 Rψψ
+32 ,
+and (2.11) is obtained similarly.
+□
+To introduce the concept of universal K-matrix, we also need another type of twists, called Drinfeld twists:
+Definition 2.4 ([Dr86, Dr89a]). A Drinfeld twist of a Hopf algebra H is an invertible element J ∈ H ⊗ H
+satisfying the property
+(2.12)
+(ǫ ⊗ id)(J) = 1 = (id ⊗ ǫ)(J)
+and the cocycle identity
+(2.13)
+(J ⊗ 1)(∆ ⊗ id)(J) = (1 ⊗ J)(id ⊗ ∆)(J) .
+Example 2.5. In the literature, there are two natural choices of Drinfeld twists [AV20]: J = 1 ⊗ 1 and J =
+R21R. It is straightforward to check using the universal R-matrix axioms (R1)–(R3) that (2.12) and (2.13)
+indeed hold for both of them.
+Given a Drinfeld twist J one obtains a new quasi-triangular Hopf algebra (HJ, RJ) with the coprod-
+uct [Dr89b]
+(2.14)
+∆J(x) = J∆(x)J−1 ,
+∀x ∈ H ,
+and the universal R-matrix
+(2.15)
+RJ = J21RJ−1 .
+Definition 2.6 ([AV20]). Let (H, R) be a quasi-triangular algebra. A twist pair (ψ, J) is the datum of an
+algebra automorphism ψ: H → H and a Drinfeld twist J ∈ H ⊗ H such that (Hcop,ψ, Rψψ
+21 ) = (HJ, RJ), i.e.
+such that ǫψ = ǫ,
+(2.16)
+∆op,ψ(x) = J∆(x)J−1 ,
+∀x ∈ H ,
+Rψψ
+21 = J21RJ−1 ,
+where
+(2.17)
+∆op,ψ = (ψ ⊗ ψ) ◦ ∆op ◦ ψ−1.
+Definition 2.7. B is a right comodule algebra over a Hopf algebra H if there exists an algebra map δ: B →
+B ⊗ H, which we call right coaction, such that the coassociativity and counital conditions hold
+(2.18)
+(id ⊗ ∆) ◦ δ = (δ ⊗ id) ◦ δ ,
+(id ⊗ ǫ) ◦ δ = id .
+2.3. Universal K-matrix. Let (ψ, J) be a twist pair for a Hopf algebra H and B is a right comodule algebra
+over H. Inspired by [AV20] we define a universal K-matrix K ∈ B ⊗ H. Here we use the notation K12 = K⊗ 1,
+K13 = p23 ◦ K12.
+Definition 2.8. We say that K ∈ B ⊗ H is universal K-matrix if the following relations hold for all b ∈ B:
+Kδ(b) = δψ(b)K ,
+with δψ = (id ⊗ ψ) ◦ δ ,
+(K1)
+(δ ⊗ id)(K) = (Rψ)32K13R23 ,
+(K2)
+(id ⊗ ∆)(K) = J−1
+23 K13Rψ
+23K12 ,
+(K3)
+where
+(2.19)
+Rψ = (ψ ⊗ id)(R) .
+
+FUSED K-OPERATORS FOR Aq
+9
+We note that δψ defines a coalgebra structure on B over Hψ. Therefore, by (K1) K intertwines two actions
+of B on B ⊗ H, given by δ and δψ respectively. By analogy with (R1), we call (K1) the twisted intertwining
+relation. We now make several remarks concerning this definition.
+Remark 2.9. From the axioms (K2)–(K3) of the universal K-matrix, we get some relations on the level of
+the algebra.
+(i) We provide a consistency check of the axioms (K2) and (K3). From coassociativity property (2.18),
+we have:
+(2.20)
+(id ⊗ ∆ ⊗ id) ◦ (δ ⊗ id)(K) = (δ ⊗ id ⊗ id) ◦ (δ ⊗ id)(K) .
+This relation is checked using (R2) and (K2). Indeed, the l.h.s. equals (Rψ)43(Rψ)42K14R24R34 where
+we used (∆⊗id)((Rψ)21) = (Rψ)32(Rψ)31, while the r.h.s. gives the same expression using twice (K2).
+The counital property in (2.18) is checked using (K3) and (2.2) as follows:
+(2.21)
+(id ⊗ ǫ ⊗ id) ◦ (δ ⊗ id)(K) = (id ⊗ ǫ ⊗ id)((Rψ)32K13R23) = K .
+(ii) Recall that the coproduct and the counit of a Hopf algebra satisfy
+(2.22)
+(ǫ ⊗ id) ◦ ∆ = id = (id ⊗ ǫ) ◦ ∆ .
+Then, using (K3), (2.2), (2.12), and the fact that ǫ is an algebra homomorphism we obtain:
+(2.23)
+K = [(id ⊗ id ⊗ ǫ) ◦ (id ⊗ ∆)](K) = (id ⊗ id ⊗ ǫ)(J−1
+23 K13Rψ
+23K12) = ([(id ⊗ ǫ)(K)] ⊗ 1) K .
+Applying again (id ⊗ ǫ) on (2.23), we find that (id ⊗ ǫ)(K) is an idempotent. If in addition, K is
+invertible, it follows:
+(2.24)
+(id ⊗ ǫ)(K) = 1 ,
+which is the analogue of (2.2) for the universal R-matrix.
+Remark 2.10. It is easy to check that (id, R−1
+21 ) is a twist pair. In particular, one finds that (K1)–(K3), for
+ψ = id and J = R−1
+21 , correspond to the axioms for the universal K-matrix defined in [Ko17, Def. 2.7].
+Now, assume that B is a right coideal subalgebra of H, that is δ = ∆|B, then define:
+K = (ǫ ⊗ id)(K) ,
+K ∈ H .
+Then, applying the counit on the first tensor factor of the universal K-matrix in (K1)–(K3) yields:
+K b = ψ(b)K ,
+∀b ∈ B ,
+(2.25)
+K = (Rψ)21K2R ,
+(2.26)
+∆|B(K) = J−1K2RψK1 .
+(2.27)
+Recall that in [AV20] one-component universal K-matrices k ∈ H are considered.
+The formulas (2.25)
+and (2.27), with the identification K = k, correspond respectively to the defining relations of the so-called
+cylindrically invariant subalgebra B and the almost cylindrical pair (H, R), see [AV20, Def. 2.6].
+We now derive a universal reflection equation based on the axioms (K1)–(K3) of the universal K-matrix.
+Proposition 2.11. Let (ψ, J) be a twist pair.
+The universal K-matrix satisfies the ψ-twisted reflection
+equation
+(2.28)
+K12(Rψ)32K13R23 = Rψψ
+32 K13Rψ
+23K12 .
+
+10
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Proof. Multiply (K2) on the left by K12 to get
+(2.29)
+K12[(δ ⊗ id)(K)] = K12(Rψ)32K13R23 .
+Then, using (K1), the l.h.s. of this equation equals
+(2.30)
+[(id ⊗ ψ ⊗ id) ◦ (δ ⊗ id)(K)]K12 = Rψψ
+32 K13Rψ
+23K12 ,
+where we used again (K2). Equating (2.29) with (2.30), the equation (2.28) follows.
+□
+Note that the ψ-twisted reflection equation can also be derived from (K3). Multiply (K3) on the left by
+R23 then use (R1) to get
+R23[(id ⊗ ∆)(K)] = [(id ⊗ ∆op)(K)]R23
+R23J−1
+23 K13Rψ
+23K12 = J−1
+32 K12(Rψ)32K13R23 .
+(2.31)
+Multiply (2.31) on the left by J32. Since (ψ, J) is a twist pair, by (2.15) we have Rψψ
+32 = J32R23J−1
+23 and (2.28)
+follows.
+2.4. Examples of (H, B). We introduce here two examples of pairs (H, B) for the Hopf algebra H = LUqsl2.
+Namely, the right comodule algebra B is either chosen to be the q-Onsager algebra B = Oq or its alternating
+central extension B = Aq.
+2.4.1. LUqsl2. We now recall the definition of the quantum loop algebra LUqsl2 corresponding to the presen-
+tation of Chevalley type. We refer the reader to Appendix A for definitions of the quantum algebra Uqsl2 and
+the quantum affine algebra Uq �sl2.
+Definition 2.12. The quantum loop algebra LUqsl2 is the unital associative C-algebra generated by the el-
+ements Ei, Fi, K
+± 1
+2
+i
+; i ∈ {0, 1} which satisfy the defining relations of the Uq �sl2 algebra (A.6)-(A.8) with the
+extra relation
+(2.32)
+K
+1
+2
+0 K
+1
+2
+1 = 1 = K
+1
+2
+1 K
+1
+2
+0 .
+The quantum loop algebra LUqsl2 is a Hopf algebra with the coproduct as in (A.9), counit in (A.10) and
+antipode (A.11) with the substitution K
+1
+2
+0 = K
+− 1
+2
+1
+.
+Note that a presentation of LUqsl2 of Faddeev-Reshetikhin-Taktadjan type is also known [FRT87].
+It
+involves Ding-Frenkel L-operators satisfying the Yang-Baxter algebra [DF93], see details in Section 6.
+The expression of the universal R-matrix associated to quantum affine algebras is well known. Tolstoy
+and Khoroshkin first constructed the universal R-matrix associated with the untwisted affine Lie algebras
+in [KT92a] and they gave its explicit form for Uq �sl2 in [KT92b, eq. (58)]. It is expressed in terms of the root
+vectors of Uq �sl28. We consider here the universal R-matrix R associated to the quotient LUqsl2, see its explicit
+form in our conventions in Appendix C. In the present paper, it is important to note that the convention for
+the coproduct we use is different to the one used in [KT92a]. The two coproducts and corresponding universal
+R-matrices are related via the automorphism ν from (A.12):
+(2.33)
+∆TK = (ν ⊗ ν) ◦ ∆ ◦ ν−1 ,
+(ν−1 ⊗ ν−1)(RTK) = R ,
+where RTK denotes the image of the universal R-matrix given in [KT92a] under the map Uq �sl2 → LUqsl2.
+8Strictly speaking, Uq �sl2 is not quasi-triangular because the universal R-matrix R is an element of an h-adic topology
+completion of the tensor product Uq �sl2 ⊗ Uq �sl2, treating q = eh as a formal power series in h, and it has the form of a product
+over an infinite set of root vectors. The important point is that R is a well-defined operator on the tensor product of finite-
+dimensional evaluation representations at generic values of the evaluation parameters, though it might be not invertible for some
+special values discussed below in the text. This however does not affect the use of the R- and K-matrix axioms.
+
+FUSED K-OPERATORS FOR Aq
+11
+Let us introduce an automorphism η of LUqsl2:
+(2.34)
+η(E0) = F1 ,
+η(E1) = F0 ,
+η(K
+1
+2
+0 ) = K
+− 1
+2
+1
+,
+η(F1) = E0 ,
+η(F0) = E1 ,
+η(K
+1
+2
+1 ) = K
+− 1
+2
+0
+.
+Below, we work mainly with twist pairs (ψ, J), recall Definition 2.6, where ψ = η and J is as in Example 2.5.
+Example 2.13. For H = LUqsl2 the pairs (ψ, J) = (η, 1 ⊗ 1) or (ψ, J) = (η, R21R) are twist pairs, where η
+is the automorphism (2.34).
+Let us prove the above statement for (ψ, J) = (η, 1 ⊗ 1), that is to verify (2.16). Using the definition of
+∆ given in (A.9), we get ∆op,η = ∆ and therefore the first equation in (2.16) indeed holds. It remains to
+show that Rηη
+21 = R. From Lemma 2.3, the pair (Hcop,η, Rηη
+21) satisfies (2.6)–(2.8) with ψ = η. Recall that the
+solution of (R1)–(R3) of the form [KT92b, eq. (42)] is unique [KT92b, Theorem 7.1], and is given by [KT92b,
+eq. (58)]. We then note that Rηη
+21 is of the same form [KT92b, eq. (42)]. Because we have ∆op,η = ∆, then the
+equations (2.6)–(2.8) for Rηη
+21 become simply (R1)–(R3) with the substitution R → Rηη
+21. Therefore, as Rηη
+21
+satisfies the same equations as R, it follows from the uniqueness that they are equal and (η, 1 ⊗ 1) is thus a
+twist pair. Similarly, using ∆R21R = ∆ and RR21R = R, one shows that (η, R21R) is also a twist pair.
+2.4.2. Evaluation map. In further sections, we will use the so-called evaluation map to study the tensor
+product representation of the algebra LUqsl2. Recall the algebra Uqsl2 defined in Appendix A. First, consider
+the algebra map
+(2.35)
+ϕ: LUqsl2 → Uqsl2
+defined by the equations:
+(2.36)
+ϕ(E0) = F ,
+ϕ(F0) = E ,
+ϕ(K
+1
+2
+0 ) = K− 1
+2 ,
+ϕ(E1) = E ,
+ϕ(F1) = F ,
+ϕ(K
+1
+2
+1 ) = K
+1
+2 .
+Let φu be the Z–gradation automorphism of LUqsl2
+(2.37)
+φu : LUqsl2 → LUqsl2 ,
+where u ∈ C∗. It is defined by
+(2.38)
+φu(E0) = u−1E0 ,
+φu(F0) = uF0 ,
+φu(K
+1
+2
+0 ) = K
+1
+2
+0 ,
+φu(E1) = u−1E1 ,
+φu(F1) = uF1 ,
+φu(K
+1
+2
+1 ) = K
+1
+2
+1 .
+Then the evaluation map evu9
+(2.39)
+evu : LUqsl2 → Uqsl2
+is defined by the composition
+(2.40)
+evu = ϕ ◦ φu .
+Its action on the generators of LUqsl2 are obtained from (2.36) and (2.38).
+9For more general evaluation map, see for instance [BGKNR12, eq. (4.32)]. In this paper we set s0 = s1 = −1.
+
+12
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+2.4.3. Comodule algebras Aq and Oq. Consider H = LUqsl2. Our first example of comodule algebra is the
+q-Onsager algebra Oq that originally appeared in the context of P- and Q-polynomial scheme [T99] and later
+on in quantum integrable systems [B04]. Different types of presentations for Oq are known. The original one
+is given in terms of two generators W0, W1 satisfying the so-called q-Dolan-Grady relations:
+[W0, [W0, [W0, W1]q]q−1] = ρ[W0, W1] ,
+(2.41)
+[W1, [W1, [W1, W0]q]q−1] = ρ[W1, W0] ,
+(2.42)
+where
+(2.43)
+ρ = k+k−(q + q−1)2 ,
+with k± ∈ C∗.
+Two other presentations given in terms of an infinite number of PBW type generators and relations are
+known: one of Lusztig’s type in terms of real and imaginary root vectors {Bnδ, Bnδ+αi|i = 0, 1}n∈N with the
+identification W0 = Bα0, W1 = Bα1 [BK17] (see also [T17, LW20]), another one of alternating type in terms
+of ‘alternating generators’ {W−k, Wk+1, Gk+1}k∈N [BB17, T21a].
+For Oq, the coaction map δ: Oq → Oq ⊗ LUqsl2 is such that [BB12]:
+δ(W0) = 1 ⊗
+�
+k+q
+1
+2 E1K
+1
+2
+1 + k−q− 1
+2 F1K
+1
+2
+1
+�
++ W0 ⊗ K1 ,
+(2.44)
+δ(W1) = 1 ⊗
+�
+k+q− 1
+2 F0K
+1
+2
+0 + k−q
+1
+2 E0K
+1
+2
+0
+�
++ W1 ⊗ K0 .
+(2.45)
+Our second and main example is the algebra Aq that first appeared10 in [BS09]. It was understood later on
+that Aq is isomorphic to the central extension of the q-Onsager algebra [BB17, T21a], namely to Oq ⊗ C[Z].
+Here, we recall a so-called compact presentation of this algebra.
+Definition 2.14 (see11 [T21b]). Aq is an associative algebra over C generated by W0, W1, {Gk+1}k∈N subject
+to the following defining relations:
+[W0, [W0, [W0, W1]q]q−1] = ρ[W0, W1] ,
+[W1, [W1, [W1, W0]q]q−1] = ρ[W1, W0] ,
+[W1, G1] = [W1, [W1, W0]q] ,
+[G1, W0] = [[W1, W0]q, W0] ,
+[W1, Gk+1] = ρ−1[W1, [W1, [W0, Gk]q]q] ,
+k ≥ 1 ,
+[Gk+1, W0] = ρ−1[[[Gk, W1]q, W0]q, W0] ,
+k ≥ 1 ,
+[Gk+1, Gℓ+1] = 0 ,
+k, ℓ ∈ N ,
+where ρ is given by (2.43).
+Remark 2.15. Similarly to Oq, Aq admits a presentation in terms of PBW alternating type generators
+{W−k, Wk+1, Gk+1, ˜Gk+1}k∈N. The precise relationship between the alternating generators of Aq and both
+the alternating generators and the root vectors of the q-Onsager algebra Oq is studied in details in [T21c]. In
+particular, one has
+(2.46)
+W0 = W0 ⊗ 1 ,
+W1 = W1 ⊗ 1 ,
+where we used the isomorphism Aq ∼= Oq ⊗ C[Z] discussed before Definition 2.14.
+10Quotients of Aq previously appeared in the context of the open XXZ spin- 1
+2 chain [BK05a].
+11The defining relations of Aq given in Definition 2.14 coincide with the compact presentation of Aq given in [T21b, Prop. 12.1]
+for the identification W0 �→ W0, W1 �→ W1, Gk+1 �→ Gk+1, ρ �→ −(q2 − q−2)2, for some q ∈ C∗.
+
+FUSED K-OPERATORS FOR Aq
+13
+The algebra Aq admits also a presentation of Faddeev-Reshetikhin-Taktadjan type where the defining
+relations take the form of a reflection algebra satisfied by a K-operator [BS09]. In the framework of this
+presentation, we discuss the coaction map δ: Aq → Aq ⊗ LUqsl2 in Section 6. The corresponding coaction of
+the fundamental generators W0, W1 takes the form (2.44), (2.45), with the substitution W0 → W0, W1 → W1.
+The coaction of Gk+1 is more involved, and its explicit form is not needed for our purpose. In this paper, it
+will be sufficient to consider the evaluation of the coaction map δ, denoted δw. For Aq it is defined as:
+(2.47)
+δw = (id ⊗ evw) ◦ δ: Aq → Aq ⊗ Uqsl2 .
+The proof of the following proposition is postponed to the end of Section 5.
+Proposition 2.16. The evaluated coaction map δw : Aq → Aq ⊗ Uqsl2 is such that:
+δw(W0) = 1 ⊗ (k+q
+1
+2 w−1EK
+1
+2 + k−q− 1
+2 wFK
+1
+2 ) + W0 ⊗ K ,
+(2.48)
+δw(W1) = 1 ⊗ (k+q− 1
+2 wEK− 1
+2 + k−q
+1
+2 w−1FK− 1
+2 ) + W1 ⊗ K−1 ,
+(2.49)
+δw(Gk+1) = k−
+k+
+(q − q−1)2
+q + q−1 (Gk − (q + q−1)
+ρ
+�
+W0,
+�
+W0, Gk
+�
+q
+�
+) ⊗ F 2 −
+Gk
+q + q−1 ⊗ (w2K−1 + w−2K)
+(2.50)
++ Gk+1 ⊗ 1 +
+(q − q−1)
+k+(q + q−1)
+�
+q− 1
+2 w−1�
+W0, Gk+1
+�
+q ⊗ FK
+1
+2 + q
+1
+2 w
+�
+Gk+1, W1
+�
+q ⊗ FK− 1
+2 �
+,
+with initial condition:
+Gk
+��
+k=0 = k+k−
+(q + q−1)2
+q − q−1
+.
+3. Tensor product representations of LUqsl2 and sub-representations
+In principle, the evaluations of a universal R- or K-matrix lead to a L-operator (and R-matrix) or K-operator
+(and K-matrix), respectively. Although the root vectors of Oq are known [BK17] as well as their relations
+with the alternating generators of Aq [T21c], the universal K-matrix K ∈ Aq ⊗ LUqsl2 for the twist pair
+(ψ, J) = (η, 1 ⊗ 1), where η is defined in (2.34), (or any other twist pair) is not known, even its existence is an
+open problem. In further sections we give evidences on the existence of such universal K-matrix by considering
+its relation with K-operators that are independently constructed using a fusion procedure. This construction
+is based on the analysis of the tensor product of evaluation representations of LUqsl2. The reducibility criteria
+in terms of ratios of the evaluation parameters for these tensor products are known [CP91, Sect. 4.9]. In this
+section, we study the sub-quotient structure of these tensor products in more details, and construct explicitly
+the corresponding intertwining operators. Using them, in further sections fused L- and K-operators for any
+spin-j are built from the fundamental L- and K-operators.
+First, recall that finite-dimensional irreducible representations of Uqsl2 are labelled by a non-negative
+integer or half-integer j, with the dimension of the representation being 2j + 1. Let V (j) denotes the (2j + 1)-
+dimensional space spanned by |j, m⟩ with m ∈ {−j, −j + 1, . . ., j − 1, j}, then
+(3.1)
+E |j, m⟩ = Aj,m |j, m + 1⟩ ,
+F |j, m⟩ = Bj,m |j, m − 1⟩ ,
+K± 1
+2 |j, m⟩ = q±m |j, m⟩ ,
+with
+(3.2)
+Aj,m =
+�
+[j − m]q [j + m + 1]q ,
+Bj,m =
+�
+[j + m]q [j − m + 1]q .
+Let πj be the representation map of Uqsl2:
+(3.3)
+πj : Uqsl2 → End(C2j+1) .
+Now, given u ∈ C∗, we then define the evaluation representations πj
+u : LUqsl2 → End(C2j+1) by
+(3.4)
+πj
+u = πj ◦ evu ,
+
+14
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+where evu is defined in (2.40). We study now tensor products of these representations
+(3.5)
+(π
+1
+2u1 ⊗ πj
+u2) ◦ ∆: LUqsl2 → End(C2
+u1 ⊗ C2j+1
+u2
+) ,
+and look at special points in the evaluation parameters space so that a proper sub-representation emerges.
+The strategy is the following: we first construct basis vectors {wk}, {vℓ} for the decomposition with respect
+to the subalgebra generated by {E1, F1, K1}.
+Then, we study the action of {E0, F0, K0} on these basis
+vectors. We find that there are only two ratios of evaluation parameters up to a sign when we get a proper
+sub-representation, and we also construct explicitly the corresponding intertwining maps.
+3.1. Analysis of the tensor product representation of LUqsl2. Consider first the subalgebra generated
+by {E1, F1, K1} and construct basis vectors {wk}, {vℓ}, where k = 0, 1, . . ., 2j + 1, ℓ = 0, 1, . . ., 2j − 1 and
+j ∈ 1
+2N+, corresponding to the tensor product decomposition C2 ⊗ C2j+1 = C2j+2 ⊕ C2j. We denote by w0
+and v0 the highest weight vectors of the corresponding spins-(j + 1
+2) and (j − 1
+2). These are defined by the
+relations
+[(π
+1
+2u1 ⊗ πj
+u2)∆(E1)]w0 = 0 ,
+[(π
+1
+2u1 ⊗ πj
+u2)∆(K1)]w0 = q2j+1w0 ,
+(3.6)
+[(π
+1
+2u1 ⊗ πj
+u2)∆(E1)]v0 = 0 ,
+[(π
+1
+2u1 ⊗ πj
+u2)∆(K1)]v0 = q2j−1v0 .
+(3.7)
+Solutions to these equations are uniquely determined, up to a scalar, by
+(3.8)
+w0 = |↑⟩ ⊗ |j, j⟩ ,
+v0 = |↑⟩ ⊗ |j, j − 1⟩ − u1
+u2
+q−j− 1
+2 Aj,j−1 |↓⟩ ⊗ |j, j⟩ ,
+with |↑⟩ = | 1
+2, 1
+2⟩ , |↓⟩ = | 1
+2, − 1
+2⟩, and where Aj,m is given in (3.2). The other basis vectors are constructed via
+the action of F1:
+(3.9)
+wk =
+�
+(π
+1
+2u1 ⊗ πj
+u2)∆(F1)
+�k
+w0 ,
+vℓ =
+�
+(π
+1
+2u1 ⊗ πj
+u2)∆(F1)
+�ℓ
+v0 .
+Now, we study the action of the generators E0 and F0 on these basis vectors:
+(3.10)
+�
+(π
+1
+2u1 ⊗ πj
+u2)∆(E0)
+�
+wk = g1,k(u1, u2)wk+1 + g2,k(u1, u2)vk ,
+�
+(π
+1
+2u1 ⊗ πj
+u2)∆(F0)
+�
+wk = h1,k(u1, u2)wk−1 + h2,k(u1, u2)vk−2 ,
+�
+(π
+1
+2u1 ⊗ πj
+u2)∆(E0)
+�
+vℓ = ˜g1,ℓ(u1, u2)vℓ+1 + ˜g2,ℓ(u1, u2)wℓ+2 ,
+�
+(π
+1
+2u1 ⊗ πj
+u2)∆(F0)
+�
+vℓ = ˜h1,ℓ(u1, u2)vℓ−1 + ˜h2,ℓ(u1, u2)wℓ .
+In particular we have
+g2,k(u1, u2) = q2j+ 1
+2 (u2
+1 − u2
+2q−2j−1)
+�
+[2j]q
+u2
+1u2 [2j + 1]q
+,
+˜g2,ℓ(u1, u2) =
+u2
+1 − u2
+2q2j+1
+q2j+ 1
+2 u2
+1u3
+2
+�
+[2j]q[2j + 1]q
+,
+(3.11)
+h2,k(u1, u2) = (u2
+1 − u2
+2q−2j−1)u2q2j+ 1
+2 �
+[2j]q[k]q[k − 1]q
+[2j + 1]q
+,
+˜h2,ℓ(u1, u2) ∝ (u2
+1 − u2
+2q2j+1) ,
+(3.12)
+we omit the expressions for the coefficients with index 1,k and 1,ℓ because there is no value for u1/u2 such
+that they vanish. Note that the coefficients with index 2,k and 2,ℓ are the ones which mix the two spin
+components. For generic values of u1/u2 the coefficients in the r.h.s. of (3.10) do not vanish. The action
+of the generators is depicted in the diagram below, where we indicate the action of Fi and Ei. The red
+(resp. blue) arrows correspond to the action of F0 (resp. E0). The branching points correspond to the linear
+combinations from (3.10).
+
+FUSED K-OPERATORS FOR Aq
+15
+w0
+w1
+w2
+. . .
+w2j−1
+w2j
+w2j+1
+v0
+v1
+. . .
+v2j−2
+v2j−1
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+Figure 1. Action of Ei, Fi on wk, vℓ for generic values of u1 and u2.
+From the expression of the coefficients (3.11) and (3.12), one finds that g2,k(u1, u2) = h2,k(u1, u2) = 0 iff
+u1/u2 = ±q−j− 1
+2 and ˜g2,ℓ(u1, u2) = ˜h2,ℓ(u1, u2) = 0 iff u1/u2 = ±qj+ 1
+2 . Bold arrows in Figure 1 are used to
+emphasize that they disappear after fixing the ratio u1/u2 = ±q−j− 1
+2 . However we do not have simultaneously
+these four coefficients equal to zero so we do not have a direct sum decomposition. Instead, by fixing the ratio
+u1/u2 to the special values we have a sub-representation as depicted below by the dotted rectangles.
+w0
+w1
+w2
+. . .
+w2j−1
+w2j
+w2j+1
+v0
+v1
+. . .
+v2j−2
+v2j−1
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+u1/u2 = ±q−j− 1
+2 .
+w0
+w1
+w2
+. . .
+w2j−1
+w2j
+w2j+1
+v0
+v1
+. . .
+v2j−2
+v2j−1
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+E1
+F1
+E0
+F0
+u1/u2 = ±qj+ 1
+2 .
+Figure 2. Action of Ei, Fi on wk, vℓ for fixed values of u1/u2.
+We thus find that fixing u1/u2 = ±q−j− 1
+2 (resp. u1/u2 = ±qj+ 1
+2 ) gives a spin-(j + 1
+2) (resp. spin-(j − 1
+2))
+sub-representation.
+3.2. The intertwining maps E(j+ 1
+2 ) and F(j+ 1
+2 ). We now study the spin-(j + 1
+2) sub-representation when
+u1/u2 = q−j− 1
+2 , and construct the corresponding intertwining operator explicitly. Introduce the two linear
+operators E(j+ 1
+2 ) and F(j+ 1
+2 ) for any j ∈ 1
+2N:
+E(j+ 1
+2 ) : C2j+2
+u
+→ C2
+u1 ⊗ C2j+1
+u2
+,
+(3.13)
+F(j+ 1
+2 ) : C2
+u1 ⊗ C2j+1
+u2
+→ C2j+2
+u
+,
+(3.14)
+
+16
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+given by:
+(3.15)
+E(j+ 1
+2 ) =
+4j+2
+�
+a=1
+2j+2
+�
+b=1
+E
+(j+ 1
+2 )
+a,b
+E(2j,j)
+a,b
+,
+F(j+ 1
+2 )E(j+ 1
+2 ) = I2j+2 ,
+where E
+(j+ 1
+2 )
+a,b
+are certain scalars, E(j1,j2)
+a,b
+denotes the matrix of dimension (2j1 + 2) × (2j2 + 2) with 1 at
+position (a, b) and 0 otherwise. Here, we choose the bases of the source {|j + 1
+2, m⟩} with m = j + 1
+2, j − 1
+2,
+. . ., −j + 1
+2, −j − 1
+2 and the target {|↑⟩ ⊗ |j, j⟩, . . ., |↑⟩ ⊗ |j, −j⟩, |↓⟩ ⊗ |j, j⟩, . . ., |↓⟩ ⊗ |j, −j⟩}.
+We now calculate the coefficients E
+(j+ 1
+2 )
+a,b
+from (3.15) provided E(j+ 1
+2 ) is a LUqsl2-intertwiner for the con-
+ditions u1/u2 = q−j− 1
+2 and u2 = uq
+1
+2 , that was found in the previous subsection for j ∈ 1
+2N+. First of all for
+j = 0, we have for any u that π0
+u = ǫ, where the counit is defined in (A.10), i.e. the trivial representation of
+LUqsl2. Then identifying C2 ⊗ C with C2, it follows from (3.13), (3.14) and (3.15) that E( 1
+2 ) = F( 1
+2 ) = I2.
+Lemma 3.1. Let u1/u2 = q−j− 1
+2 and u2 = uq
+1
+2 , then the map E(j+ 1
+2 ) in (3.15) is a LUqsl2-intertwiner
+(3.16)
+E(j+ 1
+2 )(π
+j+ 1
+2
+u
+)(x) = (π
+1
+2
+uq−j ⊗ πj
+uq
+1
+2 )(∆(x))E(j+ 1
+2 ) ,
+∀x ∈ LUqsl2 ,
+if and only if its entries are given for any j ∈ 1
+2N+ by
+(3.17)
+E
+(j+ 1
+2 )
+1,1
+= 1 ,
+E
+(j+ 1
+2 )
+1+n,1+n =
+n−1
+�
+p=0
+Bj,j−p
+Bj+ 1
+2 ,j+ 1
+2 −p
+,
+E
+(j+ 1
+2 )
+2j+1+m,1+m = [m]q
+E
+(j+ 1
+2 )
+m,m
+Bj+ 1
+2 ,j+ 3
+2 −m
+,
+where n = 1, 2, . . ., 2j, m = 1, 2, . . ., 2j + 1 and Bj,m is given in (3.2), and all the other entries are zero.
+Proof. Recall that for u1 = u2q−j− 1
+2 we have a spin-(j + 1
+2) sub-representation as depicted in left-top of
+Figure 2, and therefore we also have a corresponding interwining operator that we are going to construct. In
+this case, the sub-representation has the basis {wk | 0 ≤ k ≤ 2j + 1} given in (3.9). Let us introduce a basis
+{w′
+k | 0 ≤ k ≤ 2j +1} in the source of the map E(j+ 1
+2 ): we define w′
+0 = |j + 1
+2, j + 1
+2⟩ and w′
+k = [π
+j+ 1
+2
+u
+(F1)]kw′
+0.
+The intertwining property reads
+(3.18)
+E(j+ 1
+2 ) �
+π
+j+ 1
+2
+u
+(x)
+�
+w′
+k =
+�
+(π
+1
+2u1 ⊗ πj
+u2)(∆(x))
+�
+E(j+ 1
+2 )(w′
+k) ,
+∀x ∈ LUqsl2 .
+This equation for x = K0, K1 and k = 0 gives E(j+ 1
+2 )(w′
+0) = w0. Then for k = 0 and x = (F1)n, with n = 1,
+2, . . ., 2j + 1, one shows
+(3.19)
+E(j+ 1
+2 )(w′
+n) = wn ,
+for all n.
+Using this, we then check directly that (3.18) indeed holds now for all k and x = F1 and x = E1. It remains
+to check that (3.18) is satisfied for the action of E0 and F0. By straightforward calculations using (2.38), (3.4)
+and (A.9), one gets:
+�
+π
+j+ 1
+2
+u
+(E0)
+�
+w′
+k = u−2w′
+k+1 ,
+�
+π
+j+ 1
+2
+u
+(F0)
+�
+w′
+k = u2w′
+k−1 ,
+(3.20)
+�
+(π
+1
+2u1 ⊗ πj
+u2)∆(E0)
+�
+wk = qu−2
+2 wk+1 ,
+�
+(π
+1
+2u1 ⊗ πj
+u2)∆(F0)
+�
+wk = q−1u2
+2wk−1 .
+(3.21)
+Using (3.20) and (3.21), one finds that (3.18) holds for u2 = uq
+1
+2 . Finally, we get the matrix elements of
+E(j+ 1
+2 ). The basis vectors read explicitly:
+wk = uk
+�k−1
+�
+p=0
+Bj,j−p |↑⟩ ⊗ |j, j − k⟩ + [k]q
+k−2
+�
+p=0
+Bj,j−p |↓⟩ ⊗ |j, j − k + 1⟩
+�
+,
+
+FUSED K-OPERATORS FOR Aq
+17
+w′
+k = uk
+k−1
+�
+p=0
+Bj+ 1
+2 ,j+ 1
+2 −p |j + 1
+2, j + 1
+2 − k⟩ ,
+and we set
+n
+�
+p=0
+Bj,j−p = 1 if n is negative. Then solving (3.19) for E(j+ 1
+2 ) in (3.15), one gets (3.17) (recall that
+the basis used in (3.15) is |j + 1
+2, m⟩ with m = j + 1
+2, j − 1
+2, . . ., −j + 1
+2, −j − 1
+2, and not w′
+k).
+□
+We now give an expression of F(j+ 1
+2 ) which is a pseudo-inverse of E(j+ 1
+2 ). It takes the form:
+(3.22)
+F(j+ 1
+2 ) =
+2j+2
+�
+a=1
+4j+2
+�
+b=1
+F
+(j+ 1
+2 )
+a,b
+E(j,2j)
+a,b
+,
+where F
+(j+ 1
+2 )
+a,b
+are scalars. The solution of F(j+ 1
+2 )E(j+ 1
+2 ) = I2j+2 is not unique. For instance, we fix the entries
+of F(j+ 1
+2 ) for n = 2, 3, . . ., 2j + 1 as follows:
+F
+(j+ 1
+2 )
+1,1
+= 1 ,
+F
+(j+ 1
+2 )
+n,n+2j =
+E
+(j+ 1
+2 )
+n+2j,n
+(E
+(j+ 1
+2 )
+n,n
+)2 + (E
+(j+ 1
+2 )
+n+2j,n)2 ,
+(3.23)
+F
+(j+ 1
+2 )
+2j+2,4j+2 = (E
+(j+ 1
+2 )
+4j+2,2j+2)−1 ,
+F
+(j+ 1
+2 )
+n,n
+= 1 − F
+(j+ 1
+2 )
+n,n+2jE
+(j+ 1
+2 )
+n+2j,n
+E
+(j+ 1
+2 )
+n,n
+,
+(3.24)
+and all other entries are zero. This choice is important because it allows the factorization of the R-matrix as
+in Lemma 3.5 below. We finally note that any pseudo-inverse of E(j+ 1
+2 ), in particular the one given above, is
+not a LUqsl2-intertwiner because the sub-representation involved is not a direct summand, recall the structure
+in Fig. 2.
+3.3. The maps ¯E(j− 1
+2 ) and ¯F(j− 1
+2 ). We now study the spin-(j − 1
+2) sub-representation when u1/u2 = qj+ 1
+2 .
+Introduce the two maps ¯E(j− 1
+2 ) and ¯F(j− 1
+2 ) for any j ∈ 1
+2N+:
+¯E(j− 1
+2 ) : C2j
+u → C2
+u1 ⊗ C2j+1
+u2
+,
+(3.25)
+¯F(j− 1
+2 ) : C2
+u1 ⊗ C2j+1
+u2
+→ C2j
+u ,
+(3.26)
+given by:
+(3.27)
+¯E(j− 1
+2 ) =
+4j+2
+�
+a=1
+2j
+�
+b=1
+¯E
+(j− 1
+2 )
+a,b
+E(2j,j−1)
+a,b
+,
+¯F(j− 1
+2 ) ¯E(j− 1
+2 ) = I2j ,
+where ¯E
+(j− 1
+2 )
+a,b
+are certain scalars. The bases of the source and the target of ¯E(j− 1
+2 ) are respectively {|j − 1
+2, m⟩}
+with m = j − 1
+2, j − 3
+2, . . ., −j + 3
+2, −j + 1
+2 and {|↑⟩ ⊗ |j, j⟩ , . . . , |↑⟩ ⊗ |j, −j⟩ , |↓⟩ ⊗ |j, j⟩ , . . . , |↓⟩ ⊗ |j, −j⟩}.
+Lemma 3.2. Let u1/u2 = qj+ 1
+2 and u2 = uq
+1
+2 , then the map ¯E(j− 1
+2 ) in (3.27) is a LUqsl2-intertwiner
+(3.28)
+¯E(j− 1
+2 )(π
+j− 1
+2
+u
+)(x) = (π
+1
+2
+uqj+1 ⊗ πj
+uq
+1
+2 )(∆(x)) ¯E(j− 1
+2 ) ,
+∀x ∈ LUqsl2 ,
+if and only if its entries are given for any j ∈ 1
+2N+ by
+(3.29)
+¯E
+(j− 1
+2 )
+2,1
+= 1 ,
+¯E
+(j− 1
+2 )
+2+n,1+n =
+n−1
+�
+p=0
+Bj,j−p−1
+Bj− 1
+2 ,j− 1
+2 −p
+,
+¯E
+(j− 1
+2 )
+2j+2+m,1+m = [m − 2j]q
+Bj,j−m
+¯E
+(j− 1
+2 )
+2+m,1+m ,
+where n = 1, 2, . . ., 2j − 1, m = 0, 1, . . ., 2j − 1 and Bj,m is given in (3.2), and all the other entries are zero.
+
+18
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Proof. Recall that for u1 = u2qj+ 1
+2 we have a spin-(j − 1
+2) sub-representation as depicted in right-bottom
+of Figure 2.
+We also have a corresponding intertwining operator ¯E(j− 1
+2 ) that we now construct.
+In this
+case, the sub-representation has the basis {vℓ | 0 ≤ ℓ ≤ 2j − 1} given in (3.9).
+Let us introduce a basis
+{v′
+k | 0 ≤ k ≤ 2j − 1} in the source of the map ¯E(j− 1
+2 ): we define v′
+0 = |j − 1
+2, j − 1
+2⟩ and v′
+k = [π
+j− 1
+2
+u
+(F1)]kv′
+0.
+Analogously to the proof of Lemma 3.1, one shows ¯E(j− 1
+2 )(v′
+ℓ) = vℓ. Then, one finds the constraint on the
+ratio u/u2 and the coefficients (3.29) are obtained by solving the latter equation for ¯E(j− 1
+2 ) in (3.27).
+□
+We now give expression of ¯F(j− 1
+2 ) which is a pseudo-inverse of ¯E(j− 1
+2 ). It takes the form:
+(3.30)
+¯F(j− 1
+2 ) =
+2j
+�
+a=1
+4j+2
+�
+b=1
+¯F
+(j− 1
+2 )
+a,b
+E(j−1,2j)
+a,b
+,
+where ¯F
+(j− 1
+2 )
+a,b
+are scalars. The solution of ¯F(j− 1
+2 ) ¯E(j− 1
+2 ) = I2j is not unique. Similarly to the fusion case
+above, we fix the entries of ¯F(j− 1
+2 ) for n = 1, 2, . . ., 2j this way:
+(3.31)
+¯F
+(j− 1
+2 )
+n,n+2j+1 =
+¯E
+(j− 1
+2 )
+n+2j+1,n
+( ¯E
+(j− 1
+2 )
+n+1,n)2 + ( ¯E
+(j− 1
+2 )
+n+2j+1,n)2 ,
+¯F
+(j− 1
+2 )
+n,n+1 =
+1 − ¯F
+(j− 1
+2 )
+n,n+2j+1 ¯E
+(j− 1
+2 )
+n+2j+1,n
+¯E
+(j− 1
+2 )
+n+1,n
+,
+and all other entries are zero. We note that, similarly to the previous case, ¯F(j− 1
+2 ) is not an intertwiner.
+In summary, imposing some conditions on the ratio of evaluation parameters as in Lemmas 3.1 and 3.2, the
+tensor product representation of LUqsl2 admits a non-trivial sub-representation either of spin-(j + 1
+2) or of
+spin-(j − 1
+2). And we have constructed intertwining operators E(j+ 1
+2 ) : C2 ⊗ C2j+1 → C2j+2 and ¯E(j− 1
+2 ) : C2 ⊗
+C2j+1 → C2j, and their pseudo-inverses F(j+ 1
+2 ) and ¯F(j− 1
+2 ), respectively.
+In what follows, we will need action on tensor product using opposite coproduct. For this new action12,
+the corresponding intertwining operators appear at different evaluation parameters.
+Remark 3.3. Consider the opposite coproduct ∆op = p ◦ ∆ with the definition for ∆ in (A.9). Using now the
+action of LUqsl2 on the tensor product given by ∆op, we repeat the sub-representation analysis from Section 3.1
+using the corresponding basis { ˜wk | 0 ≤ k ≤ 2j + 1} and {˜vℓ | 0 ≤ ℓ ≤ 2j − 1} defined by
+˜wk =
+�
+(π
+1
+2u1 ⊗ πj
+u2)∆op(F1)
+�k
+˜w0 ,
+˜vℓ =
+�
+(π
+1
+2u1 ⊗ πj
+u2)∆op(F1)
+�ℓ
+˜v0 ,
+with ˜w0 = w0 in (3.8), and ˜v0 is the solution of (3.7) with the substitution ∆ → ∆op
+˜v0 = |↑⟩ ⊗ |j, j − 1⟩ − u1
+u2
+qj+ 1
+2 Aj,j−1 |↓⟩ ⊗ |j, j⟩ .
+Then, we find that the conditions on the evaluations parameter are different. Indeed, we have for the spin-
+(j + 1
+2) sub-representation:
+(3.32)
+u1/u2 = ±qj+ 1
+2 ,
+u = u2q
+1
+2 ,
+and for the spin-(j − 1
+2) sub-representation:
+(3.33)
+u1/u2 = ±q−j− 1
+2 ,
+u = u2q
+1
+2 .
+However, it leads to the intertwining operator E(j+ 1
+2 ) (resp. ¯E(j− 1
+2 )) with the same matrix elements as in (3.17)
+(resp. in (3.29)). Indeed, the matrix elements are invariant under the replacement of q by q−1. Then, the
+12Recall that for a bialgebra H, we can define another bialgebra Hcop with the coproduct ∆op. Therefore, ∆op also defines
+an action of the algebra H on the tensor product of H-modules.
+
+FUSED K-OPERATORS FOR Aq
+19
+corresponding intertwining properties for the conditions (3.32), (3.33) and the choice of the positive sign read
+respectively for all x ∈ LUqsl2:
+E(j+ 1
+2 )(π
+j+ 1
+2
+u
+)(x) = (π
+1
+2
+uqj ⊗ πj
+uq− 1
+2 )(∆op(x))E(j+ 1
+2 ) ,
+(3.34)
+¯E(j− 1
+2 )(π
+j− 1
+2
+u
+)(x) = (π
+1
+2
+uq−j−1 ⊗ πj
+uq− 1
+2 )(∆op(x)) ¯E(j− 1
+2 ) .
+(3.35)
+Remark 3.4. Let us mention that in the literature there are different conventions for the coproduct of LUqsl2.
+For example, consider the coproduct in [BGKNR10]
+(3.36)
+∆TK(Ei) = Ei ⊗ 1 + K−1
+i
+⊗ Ei ,
+∆TK(Fi) = Fi ⊗ Ki + 1 ⊗ Fi ,
+with the representation of Uqsl2
+(3.37)
+E |j, m⟩ = |j, m + 1⟩ ,
+F |j, m⟩ = [j + m]q [j − m + 1]q |j, m − 1⟩ ,
+K± 1
+2 |j, m⟩ = q±m |j, m⟩ .
+For the coproduct (3.36), the values of u, u1 and u2 coincide with Lemma 3.1. However, the expressions of
+E(j+ 1
+2 ), F(j+ 1
+2 ) are different.
+3.4. Additional properties. We conclude this section with a few observations on relations between the
+intertwining operator E(j+ 1
+2 ), its pseudo-inverse F(j+ 1
+2 ) and the R-matrix. In the literature the expression of
+the R-matrix R( 1
+2 ,j)(u) ∈ End(C2 ⊗ C2j+1) is known [KR83, DN02]. It reads:
+(3.38)
+R( 1
+2 ,j)(u) =
+2j−2
+�
+k=0
+c(uqj− 1
+2 −k)×
+�
+(q − q−1)
+�
+σ+ ⊗ πj(F) + σ− ⊗ πj(E)
+�
++ uq
+1
+2 (I4j+2+σz⊗πj(H)) − u−1q− 1
+2 (I4j+2+σz⊗πj(H))�
+,
+where πj(E), πj(F) are given in (3.1),
+�
+πj(H)
+�
+mn = 2(j + 1 − n)δm,n, with m, n = 1, 2, . . ., 2j + 1, and
+where we use the scalar function
+(3.39)
+c(u) = u − u−1 .
+Note that this R-matrix satisfies the unitarity property
+(3.40)
+R( 1
+2 ,j)(u)R( 1
+2 ,j)(u−1) =
+�2j−1
+�
+k=0
+c(uqj+ 1
+2 −k)c(u−1qj+ 1
+2 −k)
+�
+I4j+2 .
+Let H(j+ 1
+2 ) and ¯H(j− 1
+2 ) be invertible diagonal matrices given by:
+H(j+ 1
+2 ) = Diag(H
+(j+ 1
+2 )
+1
+, H
+(j+ 1
+2 )
+2
+, . . . , H
+(j+ 1
+2 )
+2j+2 ) ,
+(3.41)
+¯H(j− 1
+2 ) = Diag( ¯
+H
+(j− 1
+2 )
+1
+, ¯H
+(j− 1
+2 )
+2
+, . . . , ¯H
+(j− 1
+2 )
+2j
+) ,
+(3.42)
+where H
+(j+ 1
+2 )
+m
+and ¯H
+(j− 1
+2 )
+n
+are scalars.
+Inspired by [BLN15], the R-matrix (3.38) admits two special points for which its rank drops below its
+maximum. Then at these points, the R-matrix decomposes in terms of the intertwining operator E(j+ 1
+2 ) and
+the operator F(j+ 1
+2 ), defined above, as follows:
+
+20
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Lemma 3.5. The R-matrix (3.38) at the point u = qj+ 1
+2 has rank 2j + 2 and decomposes as:
+(3.43)
+R( 1
+2 ,j)(qj+ 1
+2 ) = E(j+ 1
+2 )H(j+ 1
+2 )F(j+ 1
+2 ) ,
+where E(j+ 1
+2 ) is fixed by Lemma 3.1, F(j+ 1
+2 ) is given in (3.22) with (3.23), (3.24) and the entries of H(j+ 1
+2 )
+are
+(3.44)
+H
+(j+ 1
+2 )
+1
+= H
+(j+ 1
+2 )
+2j+2 =
+�2j−2
+�
+k=0
+c(q2j−k)
+�
+(q2j+1 − q−2j−1) ,
+H
+(j+ 1
+2 )
+n
+=
+�2j−2
+�
+k=0
+c(q2j−k)
+�
+(q − q−1)Bj,−j−1+n
+E
+(j+ 1
+2 )
+n,n
+F
+(j+ 1
+2 )
+n,n+2j
+,
+for n = 2, 3, . . ., 2j + 1.
+Proof. Recall that E(j+ 1
+2 ) is given in Lemma 3.1 and its pseudo-inverse F(j+ 1
+2 ) is not unique.
+However,
+imposing (3.43), with H(j+ 1
+2 ) defined in (3.41), fixes both F(j+ 1
+2 ) and H(j+ 1
+2 ) uniquely as we now show.
+Indeed, solving F(j+ 1
+2 )E(j+ 1
+2 ) = I2j+2 imposes (3.24) and F
+(j+ 1
+2 )
+1,1
+= 1. There are still 2j unfixed coefficients
+F
+(j+ 1
+2 )
+n,n+2j, with n = 2, 3, . . ., 2j + 1.
+They are fixed as in (3.23), as well as the entries of H(j+ 1
+2 ), by
+solving (3.43).
+□
+Then, with the decomposition (3.43) and using the pseudo-inverse property F(j+ 1
+2 )E(j+ 1
+2 ) = I2j+2, we have:
+Corollary 3.6. The following relations hold:
+E(j+ 1
+2 )H(j+ 1
+2 ) = R( 1
+2 ,j)(qj+ 1
+2 )E(j+ 1
+2 ) ,
+(3.45)
+H(j+ 1
+2 )F(j+ 1
+2 ) = F(j+ 1
+2 )R( 1
+2 ,j)(qj+ 1
+2 ) ,
+(3.46)
+R( 1
+2 ,j)(qj+ 1
+2 ) = E(j+ 1
+2 )F(j+ 1
+2 )R( 1
+2 ,j)(qj+ 1
+2 ) .
+(3.47)
+We note that Lemma 3.5 and Corollary 3.6 will be used many times in Sections 5 and 6, in particular to
+prove the reflection equation in Theorem 5.7.
+Similarly to Lemma 3.5, for the second special point we have:
+Lemma 3.7. The R-matrix (3.38) at the point u = q−j− 1
+2 has rank 2j and is decomposed as:
+(3.48)
+R( 1
+2 ,j)(q−j− 1
+2 ) = ¯E(j− 1
+2 ) ¯H(j− 1
+2 ) ¯F(j− 1
+2 ) ,
+where ¯E(j− 1
+2 ) is fixed by Lemma 3.2, ¯F(j− 1
+2 ) is given in (3.30) with (3.31) and the entries of ¯H(j− 1
+2 ) are
+(3.49)
+¯H
+(j− 1
+2 )
+n
+=
+�2j−2
+�
+k=0
+c(q−k−1)
+�
+(q − q−1)Bj,−j+n
+¯E
+(j− 1
+2 )
+n+2j+1,n ¯F
+(j− 1
+2 )
+n,n+1
+,
+for n = 1, 2, . . ., 2j.
+4. Spin-j L- and K-operators
+In this section, we define spin-j L- and K-operators as evaluations of universal R- and K-matrices for
+H = LUqsl2 and B a comodule algebra for a certain twist pair. Using the intertwining operators studied
+in the previous section, we show that the spin-j L- and K-operators satisfy certain properties named as
+‘fusion’ and ‘reduction’. At the end of this section, the comodule algebra structure is characterized within the
+framework of the spin-j K-operators and the Ding-Frenkel L-operators.
+4.1. Spin-j L-operators.
+
+FUSED K-OPERATORS FOR Aq
+21
+4.1.1. Evaluation of the universal R-matrix. The universal R-matrix R is an element of a completion of
+Uq �sl2 ⊗ Uq �sl2 having the form of a product of infinite series over root vectors [KT92a, Theorem 1], see the
+expression in our conventions in Appendix C. These infinite series converge on finite-dimensional evaluation
+representations and therefore we have well-defined L-operators13:
+Definition 4.1. For j ∈ 1
+2N:
+(4.1)
+L(j)(u1/u2) = (evu1 ⊗ πj
+u2)(R)
+∈ Uqsl2 ⊗ End(C2j+1) .
+We call L(j)(u) the spin-j L-operator.
+Considering u as a formal variable, we will see below that L(j)(u) is in Uqsl2[[u−1]] ⊗ End(C2j+1). Note
+that L(0)(u) = 1 by (2.2).
+Evaluating the first component of L(j)(u) on a finite-dimensional representation of Uqsl2 we get the R-
+matrix. For any spin j1, j2, we denote the R-matrix by
+R(j1,j2)(u1/u2) = (πj1
+u1 ⊗ πj2
+u2)(R)
+(4.2)
+= (πj1 ⊗ id)(L(j2)(u1/u2)) .
+Recall the L-operator satisfies the RLL equation. Indeed, applying (πj1
+u−1
+1
+⊗ πj2
+u−1
+2
+⊗ evu3=1) ◦ p13 to (2.3), and
+noticing from (4.1) that we have14 L(j)(u1/u2) = (πj
+u2 ⊗ evu1)(R21), one finds15
+(4.3)
+R(j1,j2)
+12
+(u1/u2)L(j1)
+1
+(u1)L(j2)
+2
+(u2) = L(j2)
+2
+(u2)L(j1)
+1
+(u1)R(j1,j2)
+12
+(u1/u2) .
+Recall also that the R-matrix satisfies the Yang-Baxter equation. It is found by applying (πj1
+u1 ⊗ πj2
+u2 ⊗ πj3
+u3=1)
+to (2.3):
+(4.4)
+R(j1,j2)
+12
+(u1/u2)R(j1,j3)
+13
+(u1)R(j2,j3)
+23
+(u2) = R(j2,j3)
+23
+(u2)R(j1,j3)
+13
+(u1)R(j1,j2)
+12
+(u1/u2) .
+Note that the explicit computation of the spin- 1
+2 L-operator L( 1
+2 )(u) and the R-matrix for j1 = j2 = 1
+2 as
+the evaluation of the universal R-matrix can be found in [BGKNR12, eqs. (4.62), (4.53)], see Appendix C.4
+for the derivation in our conventions. For j = 1
+2 in (4.1), the L-operator is given by:
+(4.5)
+L( 1
+2 )(u) = µ(u)L( 1
+2 )(u) ,
+with
+L( 1
+2 )(u) =
+�uq
+1
+2 K
+1
+2 − u−1q− 1
+2 K− 1
+2
+(q − q−1)F
+(q − q−1)E
+uq
+1
+2 K− 1
+2 − u−1q− 1
+2 K
+1
+2
+�
+,
+where the ‘normalization’ µ(u) is a power series in u and the coefficients are central elements of Uqsl2. It is
+given by
+(4.6)
+µ(u) = u−1q− 1
+2 eΛ(u−2q−1) ,
+where Λ(u) is the power series (C.34) with the coefficients Ck ∈ Uqsl2 which are certain polynomials in the
+Casimir element, see (C.13) and (C.14) for more details. Therefore, L( 1
+2 )(u) is in Uqsl2[[u−1]] ⊗ End(C2).
+13Note that the L-operator L(j)(u) is the evaluated version of �L(j)(u) introduced in (1.5), or compare with the Ding-Frenkel
+L-operators in Section 4.3.
+14Although elements in Uqsl2 ⊗ End(C2j+1) and End(C2j+1) ⊗ Uqsl2 are related through a flip of the first and second space,
+both can be seen as (2j + 1) × (2j + 1) matrices with entries in Uqsl2.
+15The RLL equation belongs to Uqsl2 ⊗ End(C2j1+1) ⊗ End(C2j2+1). Strictly speaking, the L-operator should be written as
+L(j)
+0i (u) but here we use standard notation L(j)
+i
+(u) where we omit the label 0 corresponding to Uqsl2.
+
+22
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Remark 4.2. We notice that the expression for L( 1
+2 )(u) in (4.5) differs from the one in [BGKNR12, eq. (4.62)]
+due to different conventions on the coproduct. The two L-operators are related as follows. Recall that the auto-
+morphism (2.33) relates the universal R-matrix corresponding to our coproduct with the universal R-matrix
+of [KT92a], the one used by [BGKNR12]. Then (4.1) for j = 1
+2 reads
+(4.7)
+L( 1
+2 )(u1/u2) = (id ⊗ π
+1
+2 ) ◦ (evu1 ⊗ evu2) ◦ (ν−1 ⊗ ν−1)(RT K) ,
+where ν is given in (A.12). Now, consider the automorphism τ : Uqsl2 → Uqsl2 defined by
+(4.8)
+τ(E) = q− 1
+2 EK− 1
+2 ,
+τ(F) = q− 1
+2 FK
+1
+2 ,
+τ(K± 1
+2 ) = K± 1
+2 .
+Noting that it satisfies the property evu ◦ ν−1(x) = τ ◦ evuq− 1
+2 (x) for all x in LUqsl2, then (4.7) becomes
+(4.9)
+L( 1
+2 )(u1/u2) = (id ⊗ π
+1
+2 ) ◦ (τ ⊗ τ) ◦ (evu1q− 1
+2 ⊗ evu2q− 1
+2 )(RT K) .
+Example 4.3. For j1 = j2 = 1
+2 in (4.2), the corresponding R-matrix is given by:
+(4.10)
+R( 1
+2 , 1
+2 )(u) = π
+1
+2 (µ(u))R( 1
+2 , 1
+2 )(u) ,
+with
+R( 1
+2 , 1
+2 )(u) =
+
+
+
+
+c(uq)
+0
+0
+0
+0
+c(u) c(q)
+0
+0
+c(q) c(u)
+0
+0
+0
+0
+c(uq)
+
+
+
+ ,
+where c(u) is given in (3.39) and with
+(4.11)
+π
+1
+2 (µ(u)) = u−1q− 1
+2 exp
+� ∞
+�
+k=1
+q2k + q−2k
+1 + q2k
+u−2k
+k
+�
+,
+where we used the evaluation of the coefficients Ck of Λ(u) given in (C.34), see [BGKNR12, eq. (4.59)]16.
+Note that R( 1
+2 , 1
+2 )(u) coincides with the expression in (3.38) for j = 1
+2.
+We now recall a special central element in Uqsl2, called the quantum determinant γ(u). It is given by [Sk88]:
+(4.12)
+γ(u) = tr12
+�
+P−
+12L
+( 1
+2 )
+1
+(u)L
+( 1
+2 )
+2
+(uq)
+�
+= u2q2 + u−2q−2 − C ,
+where tr12 stands for the trace over V1 ⊗ V2 and where C is the Casimir element of Uqsl2 defined in (A.5).
+Here, as usual, the permutation matrix P12 ≡ P with P = R( 1
+2 , 1
+2 )(1)/(q − q−1) for the R-matrix (4.10) and
+P−
+12 = (1 − P)/2.
+By straightforward calculations, one finds that the L-operator L( 1
+2 )(u) given in (4.5) satisfies a unitarity
+property:
+(4.13)
+L( 1
+2 )(u−1)L( 1
+2 )(u) = L( 1
+2 )(u)L( 1
+2 )(u−1) = c(u)I2 ,
+with c(u) = −γ(uq−1)µ(u)µ(u−1) ,
+where the quantum determinant γ(u) is given in (4.12). Note that c(u) is invariant by the inversion of its
+argument, i.e. c(u) = c(u−1).
+16For any spin-j, the evaluation is given by
+πj(µ(u)) = u−1q− 1
+2 exp
+� ∞
+�
+k=1
+qk(2j+1) + q−k(2j+1)
+1 + q2k
+u−2k
+k
+�
+.
+
+FUSED K-OPERATORS FOR Aq
+23
+4.1.2. L-operators and fusion ( 1
+2, j) → (j + 1
+2). We study a so-called fusion relation for L-operators that
+relates L(j+ 1
+2 )(u) to L(j)(u) and L( 1
+2 )(u). For this, we evaluate the equation (R3) on the second and third
+tensor components for a special choice of evaluation parameters. Fix u1/u2 = q−j− 1
+2 to get a spin-(j + 1
+2)
+sub-representation in the tensor product (3.5) of evaluation representations of LUqsl2 as depicted in the left-
+top of Figure 2. The corresponding intertwining operator E(j+ 1
+2 ) is fixed by Lemma 3.1 and its pseudo-inverse
+F(j+ 1
+2 ) is given in (3.22) with (3.23), (3.24). Then, inserting the product F(j+ 1
+2 )E(j+ 1
+2 ) = I2j+2 and using
+the intertwining property (3.16), fusion relations satisfied by L-operators and R-matrices are exhibited in the
+next proposition (where we use the notation ⟨12⟩ to indicate which spaces are fused, that is to say, where the
+intertwiner acts).
+Proposition 4.4. The L-operators (4.1) satisfy for j ∈ 1
+2N:
+(4.14)
+L(j+ 1
+2 )(u) = F
+(j+ 1
+2 )
+⟨12⟩
+L(j)
+2 (uq− 1
+2 )L
+( 1
+2 )
+1
+(uqj)E
+(j+ 1
+2 )
+⟨12⟩
+.
+Proof. By definition of the L-operator we have
+(4.15)
+L(j+ 1
+2 )(w/u) = (evw ⊗ π
+j+ 1
+2
+u
+)(R) .
+Using the pseudo-inverse property F(j+ 1
+2 )E(j+ 1
+2 ) = I2j+2, we get
+L(j+ 1
+2 )(w/u) = (1 ⊗ F(j+ 1
+2 )E(j+ 1
+2 ))
+�
+(evw ⊗ π
+j+ 1
+2
+u
+)(R)
+�
+= (1 ⊗ F(j+ 1
+2 ))
+�
+(evw ⊗ π
+1
+2
+uq−j ⊗ πj
+uq
+1
+2 )(id ⊗ ∆)(R)
+�
+(1 ⊗ E(j+ 1
+2 ))
+= (1 ⊗ F(j+ 1
+2 ))
+�
+(evw ⊗ π
+1
+2
+uq−j ⊗ πj
+uq
+1
+2 )(R13R12)
+�
+(1 ⊗ E(j+ 1
+2 ))
+= (1 ⊗ F(j+ 1
+2 ))L(j)
+2 (q− 1
+2 w/u)L
+( 1
+2 )
+1
+(qjw/u)(1 ⊗ E(j+ 1
+2 )) .
+The second equality is obtained using the intertwining property (3.16):
+(4.16)
+(1 ⊗ E(j+ 1
+2 ))
+�
+(id ⊗ π
+j+ 1
+2
+u
+)(R)
+�
+=
+�
+(id ⊗ π
+1
+2
+uq−j ⊗ πj
+uq
+1
+2 ) ◦ (id ⊗ ∆)(R)
+�
+(1 ⊗ E(j+ 1
+2 )) .
+Then, the third equality is due to (R3) and the last one follows by definition of the L-operator.
+□
+From Proposition 4.4, we see that L(j)(u) for j ∈ 1
+2N+ is in Uqsl2[[u−1]] ⊗ End(C2j+1).
+Proposition 4.5. The R-matrices (4.2) satisfy
+(4.17)
+R(j1,j2)(u) = F(j1)
+⟨12⟩R
+( 1
+2 ,j2)
+13
+(uq−j1+ 1
+2 )R
+(j1− 1
+2 ,j2)
+23
+(uq
+1
+2 )E(j1)
+⟨12⟩ ,
+where
+(4.18)
+R( 1
+2 ,j+ 1
+2 )(u) = F
+(j+ 1
+2 )
+⟨23⟩
+R
+( 1
+2 ,j)
+13
+(uq− 1
+2 )R
+( 1
+2 , 1
+2 )
+12
+(uqj)E
+(j+ 1
+2 )
+⟨23⟩
+.
+Proof. First we show (4.18). By definition we have R( 1
+2 ,j2)(u) = (π
+1
+2 ⊗ id)(L(j2)(u)), therefore the application
+of (π
+1
+2 ⊗ id) on (4.14) yields17 (4.18). In order to show (4.17), we fuse the first component of R( 1
+2 ,j2+ 1
+2 )(u)
+similarly to the proof of Proposition 4.4 but using (R2) which gives
+R(j1,j2)(u) = F(j1)
+⟨12⟩E(j1)
+⟨12⟩(πj1
+u ⊗ πj2
+v=1)(R)
+17The shifting of the labels {0, 1, 2} to {1, 2, 3} is due to the convention that the first tensor component of L(j)(u) is labelled
+by 0.
+
+24
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+= F(j1)
+⟨12⟩
+�
+(π
+1
+2
+uq−j1+ 1
+2 ⊗ π
+j1− 1
+2
+uq
+1
+2
+⊗ πj2
+v=1)(R13R23)
+�
+E(j1)
+⟨12⟩ .
+□
+Remark 4.6. Recall that F(j+ 1
+2 ) is not uniquely determined, see Section 3.2. From the construction in the
+proof of Proposition 4.4, it is clear that taking different expressions for F(j+ 1
+2 ) yields the same L-operators
+and R-matrices.
+4.1.3. L-operators and reduction ( 1
+2, j) → (j − 1
+2). We now consider spin-(j − 1
+2) sub-representations in the
+tensor product of evaluation representations of LUqsl2 by fixing u1/u2 = qj+ 1
+2 as depicted in the right-bottom
+of Figure 2. The corresponding intertwining operator ¯E(j− 1
+2 ) is fixed by Lemma 3.2 and its pseudo-inverse
+¯F(j− 1
+2 ) is given in (3.30) with (3.31).
+Proposition 4.7. The L-operators (4.1) satisfy for j ∈ 1
+2N+:
+(4.19)
+L(j− 1
+2 )(u) = ¯F
+(j− 1
+2 )
+⟨12⟩
+L(j)
+2 (uq− 1
+2 )L
+( 1
+2 )
+1
+(uq−j−1) ¯E
+(j− 1
+2 )
+⟨12⟩
+.
+Proof. Fix u1 = u2qj+ 1
+2 and u2 = uq
+1
+2 as in Lemma 3.2.
+With the use of the pseudo-inverse property
+¯F(j− 1
+2 ) ¯E(j− 1
+2 ) = I2j and the interwining property (1 ⊗ ¯E(j− 1
+2 ))(id ⊗ π
+j− 1
+2
+u
+)(R) = [(id ⊗ π
+1
+2
+uqj+1 ⊗ πj
+uq
+1
+2 ) ◦ (id ⊗
+∆)(R)](1 ⊗ ¯E(j− 1
+2 )), the proof is then similar to Proposition 4.4.
+□
+4.1.4. P-symmetry of spin-j L-operators. We now show that the P-symmetry defined in (1.4) holds for any
+j1, j2, for the case of H = LUqsl2. Note that a proof can be found in [RSV14, Lem. 2.1]. Here, we give a
+different proof by showing first a more general relation
+(4.20)
+(evu−1
+1
+⊗ πj
+u−1
+2 )(R) = (evu1 ⊗ πj
+u2)(R21) .
+Recall L(j)(u) has been defined by (4.1) and that it admits a fused expression (4.14), then the above relation
+can be interpreted as the P-symmetry on the level of L-operators. Now, identifying the l.h.s. of (4.20) with
+the spin-j L-operator, the equation (4.20) reads
+L(j)(u2/u1) = (evu1 ⊗ πj
+u2)(R21) .
+(4.21)
+The proof is done by induction on j. It is straightforward to check that (4.21) holds for j = 1
+2, by a calculation
+similar to the evaluation of R in Appendix C. Now assume (4.21) holds for a fixed value of j, we show it holds
+for (j + 1
+2). It is done by identifying the r.h.s. of (4.21) with (4.14). Indeed, by an analysis similar to the
+proof of Proposition 4.4, using the pseudo-inverse property F(j+ 1
+2 )E(j+ 1
+2 ) = I2j+2, (3.34) and (2.11), one gets:
+(evu1 ⊗ π
+j+ 1
+2
+u2
+)(R21) = (1 ⊗ F(j+ 1
+2 ))
+�
+(evu1 ⊗ π
+1
+2
+u2qj ⊗ πj
+u2q− 1
+2 )(R31R21)
+�
+(1 ⊗ E(j+ 1
+2 ))
+= F
+(j+ 1
+2 )
+⟨12⟩
+L
+( 1
+2 )
+2
+(u2q− 1
+2 /u1)L(j)
+1 (u2qj/u1)E
+(j+ 1
+2 )
+⟨12⟩
+,
+(4.22)
+where we used the assumption (4.21) for a fixed j to get the last line. Then, comparing (4.22) with (4.14), one
+finds that indeed (evu1 ⊗ π
+j+ 1
+2
+u2
+)(R21) = L(j+ 1
+2 )(u2/u1). Secondly, by specializing the first tensor component
+of (4.21) on the spin-j1 representation and for j = j2, one obtains
+(4.23)
+R(j1,j2)(u2/u1) = (πj1
+u1 ⊗ πj2
+u2)(R21) .
+Finally, the immediate corollary of the latter relation is the P-symmetry
+(4.24)
+R(j2,j1)
+21
+(u) = R(j1,j2)(u) .
+
+FUSED K-OPERATORS FOR Aq
+25
+Indeed, the R-matrix defined in (1.3) can be interpreted as the evaluation of the flipped universal R-matrix
+(4.25)
+R(j2,j1)
+21
+(u2/u1) = (πj1
+u1 ⊗ πj2
+u2)(R21) ,
+and thus (4.24) holds.
+4.1.5. Fused L-operators and fused R-matrices. A higher spin generalization of L( 1
+2 )(u) from (4.5) is obtained
+as follows. Starting from the fundamental L-operator L( 1
+2 )(u) given in (4.5), for any j ∈ 1
+2N+ define the fused
+L-operators L(j)(u) ∈ Uqsl2 ⊗ End(C2j+1) as:
+(4.26)
+L(j+ 1
+2 )(u) = F
+(j+ 1
+2 )
+⟨12⟩
+L(j)
+2 (uq− 1
+2 )L
+( 1
+2 )
+1
+(uqj)E
+(j+ 1
+2 )
+⟨12⟩
+.
+Although not needed here, it can be proven directly by induction that L(j)(u)’s satisfy the Yang-Baxter
+equation (4.3), where L(j)(u) are replaced by L(j)(u).
+We now give the relations between the spin-j L-
+operators (4.1), obtained by evaluation of the universal R-matrix, and the fused L-operators (4.14).
+Lemma 4.8. The spin-j L-operators and the fused L-operators are related as follows:
+(4.27)
+L(j)(u) = µ(j)(u)L(j)(u) ,
+where
+(4.28)
+µ(j)(u) =
+2j−1
+�
+k=0
+µ(uqj− 1
+2 −k)
+is central in Uqsl2.
+Proof. The relation (4.27) is shown by induction on j using (4.14) and (4.26) and the fact that µ(u) is central
+in Uqsl2. The first step of the induction is given by (4.5).
+□
+Above, we have shown that the L-operators L(j)(u)’s satisfy both fusion and reduction relations. As the
+L-operator is the evaluation of the universal R-matrix (4.1), the expression (4.14) with j replaced by j − 1
+equals (4.19). By the consistency of the fusion and reduction relations (4.14) and (4.19), respectively, one
+gets a functional relation satisfied by the central element µ(u) from (4.6).
+Lemma 4.9. The following relation holds:
+(4.29)
+µ(u)µ(uq)γ(u) = 1 .
+Proof. Comparing L( 1
+2 )(u) = µ(u)L( 1
+2 )(u) with (4.19) for j = 1 it follows:
+(4.30)
+L( 1
+2 )(u) = µ(uq−1)µ(uq−2) ¯F
+( 1
+2 )
+⟨12⟩L(1)
+2 (uq− 1
+2 )L
+( 1
+2 )
+1
+(uq−2) ¯E
+( 1
+2 )
+⟨12⟩ ,
+where we used µ(1)(u) in (4.28). For the r.h.s. of (4.30), after a direct calculation we find that
+¯F
+( 1
+2 )
+⟨12⟩L(1)
+2 (uq− 1
+2 )L
+( 1
+2 )
+1
+(uq−2) ¯E
+( 1
+2 )
+⟨12⟩ = γ(uq−2)L( 1
+2 )(u) .
+Then, since L( 1
+2 )(u) is invertible, the relation (4.29) follows.
+□
+Corollary 4.10. The quantum determinant of the L-operator L( 1
+2 )(u) is such that
+(4.31)
+tr12
+�
+P−
+12L
+( 1
+2 )
+1
+(u)L
+( 1
+2 )
+2
+(uq)
+�
+= 1 .
+
+26
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+We note that Lemma 4.9 provides an independent derivation of18 [BGKNR12, eq. (4.60)]. We do not give
+a general solution to the functional equation (4.29). Solutions for spin-representations of Uqsl2 can be easily
+constructed. For instance, for π
+1
+2 (µ(u)) the ‘minimal’ solution is given by (4.11).
+We now introduce fused R-matrices (by analogy with (4.17) and (4.18))
+R(j1,j2)(u) = F(j1)
+⟨12⟩R
+( 1
+2 ,j2)
+13
+(uq−j1+ 1
+2 )R
+(j1− 1
+2 ,j2)
+23
+(uq
+1
+2 )E(j1)
+⟨12⟩ ,
+(4.32)
+for j1 ≥ 1 and where
+R( 1
+2 ,j+ 1
+2 )(u) = F
+(j+ 1
+2 )
+⟨23⟩
+R
+( 1
+2 ,j)
+13
+(uq− 1
+2 )R
+( 1
+2 , 1
+2 )
+12
+(uqj)E
+(j+ 1
+2 )
+⟨23⟩
+,
+(4.33)
+with R( 1
+2 , 1
+2 )(u) given in the right part of (4.10), and show that (4.33) agrees with (3.38).
+Lemma 4.11. The R-matrices (4.2) and the fused R-matrices (4.32) are related by
+(4.34)
+R(j1,j2)(u) = f (j1,j2)(u)R(j1,j2)(u) ,
+where
+(4.35)
+f (j1,j2)(u) = u−4j1j2q−2j1j2 exp
+� ∞
+�
+k=1
+q2k + q−2k
+1 + q2k
+[2j1]qk[2j2]qk u−2k
+k
+�
+,
+and (4.33) agrees with (3.38).
+Proof. Firstly, we prove (4.34) for j1 = 1
+2. We show by induction on j2 that
+(4.36)
+R( 1
+2 ,j2)(u) = π
+1
+2 (µ(j2)(u)) R( 1
+2 ,j2)(u) ,
+and we identify π
+1
+2 (µ(j2))(u) with f ( 1
+2 ,j2)(u). For j2 = 1
+2, it is given in (4.10). Now, assuming (4.36) holds
+for a fixed value of j2, we show it holds for j2 + 1
+2. Inserting (4.18) and (4.33) in (4.36) for j2 → j2 + 1
+2 and
+using (4.28), one finds that the equality indeed holds. Then, using (4.28) and (4.11) we find that
+(4.37)
+π
+1
+2 (µ(j2))(u) = u−2j2q−j2 exp
+� ∞
+�
+k=1
+q2k + q−2k
+1 + q2k
+[2j2]qk u−2k
+k
+�
+,
+and it coincides with (4.35) for j1 = 1
+2.
+Secondly, we show that (4.33) agrees with (3.38). On one hand, using (4.5) it is straightforward to find that
+(recall the notation in (1.3))
+(4.38)
+R
+(j, 1
+2 )
+21
+(u) = P(j, 1
+2 )[(πj ⊗ id)(L( 1
+2 )(u))]P( 1
+2 ,j)
+is proportional to (3.38). On the other hand, from the P-symmetry in (4.24), one has R
+(j, 1
+2 )
+21
+(u) = R( 1
+2 ,j)(u).
+Therefore, R( 1
+2 ,j)(u) is equally proportional to (3.38), as well as the expression in (4.33), recall (4.36). Finally,
+comparing the matrix entry (1, 1) of (3.38) and (4.33), one finds that they are equal.
+Thirdly, assuming that R(j1,j2)(u) is proportional to R(j1,j2)(u) as in (4.34), we show that f (j1,j2)(u) takes the
+form (4.35). Replacing the R-matrices and the fused R-matrices in (4.34) by (4.17), (4.32), and using (4.36)
+one gets:
+f (j1,j2)(u) = π
+1
+2 (µ(j2)(uq−j1+ 1
+2 ))f (j1− 1
+2 ,j2)(uq
+1
+2 )
+=
+2j1−1
+�
+k=0
+�
+π
+1
+2 (µ(j2)(uqj1− 1
+2 −k))
+�
+,
+(4.39)
+18It is an exponential version of [BGKNR12] with the identification τ → u, eΛ(q−1τs) → µ(u)uq
+1
+2 , s → −2, s0 → −1,
+s1 → −1.
+
+FUSED K-OPERATORS FOR Aq
+27
+where we set f (0,j2)(u) = 1. Finally, using (4.37) in the latter relation, one finds that f (j1,j2)(u) is indeed
+given by (4.35) and so the claim follows.
+□
+4.1.6. Unitarity properties of L-operators. Later in the text, we will need various relations satisfied by the
+L-operators and R-matrices. They are obtained from the action of the quantum loop algebra (on tensor
+products) defined by the opposite coproduct, recall Remark 3.3.
+Lemma 4.12. The following relations hold:
+L(j+ 1
+2 )(u) = F
+(j+ 1
+2 )
+⟨12⟩
+L
+( 1
+2 )
+1
+(uq−j)L(j)
+2 (uq
+1
+2 )E
+(j+ 1
+2 )
+⟨12⟩
+,
+(4.40)
+R(j1,j+ 1
+2 )(u) = F
+(j+ 1
+2 )
+⟨23⟩
+R
+(j1, 1
+2 )
+12
+(uq−j)R(j1,j)
+13
+(uq
+1
+2 )E
+(j+ 1
+2 )
+⟨23⟩
+,
+(4.41)
+R(j+ 1
+2 ,j2)(u) = F
+(j+ 1
+2 )
+⟨12⟩
+R(j,j2)
+23
+(uq− 1
+2 )R
+( 1
+2 , 1
+2 )
+13
+(uqj)E
+(j+ 1
+2 )
+⟨12⟩
+.
+(4.42)
+Proof. Recall the discussion in Remark 3.3 about the action on tensor product of evaluation irreducible
+representations given by the opposite coproduct. Now, combining (3.34) with (∆op ⊗id)(R) = R23R13, which
+is obtained by applying p12 to (R2), then (4.40) is proven similarly to Proposition 4.4. Eq. (4.41) follows
+from (4.40) by specialization. The last relation (4.42) is similarly obtained, using (3.34) and (id ⊗ ∆op)(R) =
+R12R13, coming from the application of p23 to (R3).
+□
+We now study the unitarity property of the spin-1 L-operator, as was done in (4.13) for the spin- 1
+2 case.
+Using the fusion relations (4.14) and (4.40) for j = 1
+2, we get
+L(1)(u)L(1)(u−1) = F(1)
+⟨12⟩L
+( 1
+2 )
+1
+(uq− 1
+2 )L
+( 1
+2 )
+2
+(uq
+1
+2 )E(1)
+⟨12⟩F(1)
+⟨12⟩L
+( 1
+2 )
+2
+(u−1q− 1
+2 )L
+( 1
+2 )
+1
+(u−1q
+1
+2 )E(1)
+⟨12⟩ .
+(4.43)
+Then, we need to use (4.13) to show that (4.43) is proportional to the identity matrix. However, there is an
+unwanted product E(1)F(1) which is removed as follows. Recall Corollary 3.6. Insert H(1)
+⟨12⟩[H(1)
+⟨12⟩]−1 = I3 in
+the r.h.s. of (4.43) and use (3.45), one has:
+L(1)(u)L(1)(u−1) = F(1)
+⟨12⟩L
+( 1
+2 )
+1
+(uq− 1
+2 )L
+( 1
+2 )
+2
+(uq
+1
+2 )E(1)
+⟨12⟩F(1)
+⟨12⟩L
+( 1
+2 )
+2
+(u−1q− 1
+2 )L
+( 1
+2 )
+1
+(u−1q
+1
+2 )R( 1
+2 , 1
+2 )(q)E(1)
+⟨12⟩[H(1)
+⟨12⟩]−1
+= F(1)
+⟨12⟩L
+( 1
+2 )
+1
+(uq− 1
+2 )L
+( 1
+2 )
+2
+(uq
+1
+2 )E(1)
+⟨12⟩F(1)
+⟨12⟩R( 1
+2 , 1
+2 )(q)L
+( 1
+2 )
+1
+(u−1q
+1
+2 )L
+( 1
+2 )
+2
+(u−1q− 1
+2 )E(1)
+⟨12⟩[H(1)
+⟨12⟩]−1
+= F(1)
+⟨12⟩L
+( 1
+2 )
+1
+(uq− 1
+2 )L
+( 1
+2 )
+2
+(uq
+1
+2 )L
+( 1
+2 )
+2
+(u−1q− 1
+2 )L
+( 1
+2 )
+1
+(u−1q
+1
+2 )E(1)
+⟨12⟩ ,
+(4.44)
+where we used the RLL equation given in (4.3) to get the second line, and the property (3.47), the RLL
+equation and (3.45) to get the third line. Finally, using (4.13) and c(u) = c(u−1), we get
+L(1)(u−1)L(1)(u) = L(1)(u)L(1)(u−1) = c(uq
+1
+2 )c(uq− 1
+2 )I3 .
+More generally, by induction one gets the unitarity property for any spin-j:
+(4.45)
+L(j)(u−1)L(j)(u) = L(j)(u)L(j)(u−1) =
+�2j−1
+�
+k=0
+c(uq−j+ 1
+2 +k)
+�
+I2j+1 ,
+where c(u) is given in (4.13). It follows from (4.45) and (4.2) that both R(j1,j2)(u)R(j1,j2)(u−1) and
+R(j1,j2)(u−1)R(j1,j2)(u) are equal and proportional to the identity matrix for any j1 and j2. Because R(j1,j2)(u)
+is proportional to R(j1,j2)(u) due to (4.34), we also have
+(4.46)
+R(j1,j2)(u)R(j1,j2)(u−1) = R(j1,j2)(u−1)R(j1,j2)(u) ∝ I(2j1+1)(2j2+1) .
+
+28
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+4.2. Spin-j K-operators. Analogs of spin-j L-operators, that we call spin-j K-operators, are defined as
+evaluation of a universal K-matrix which is assumed to exist for a certain comodule algebra B and a twist
+pair (ψ, J). We show here that they satisfy the reflection equation (1.8) which follows from the evaluation of
+the ψ-twisted reflection equation (2.28). Then, using the intertwining operators constructed in Section 3, we
+propose certain fusion and reduction relations satisfied by the spin-j K-operators.
+4.2.1. Evaluation of the universal K-matrix. From now on, we focus on the case H = LUqsl2 as introduced
+in Section 2.4, without specifying its comodule algebra B. Assume that a universal K-matrix K exists for a
+choice of B and the twist pair (ψ, J) = (η, 1 ⊗ 1) where η is defined in (2.34). Recall that we have two twist
+pairs associated with the automorphism η as seen in Example 2.13. The other choice of twist pair (η, R21R)
+will be discussed at the end of the section.
+Since K ∈ B ⊗ LUqsl2, we can consider its evaluation on the second tensor component:
+Definition 4.13. For j ∈ 1
+2N, introduce
+(4.47)
+K(j)(u) = (id ⊗ πj
+u−1)(K)
+∈ B ⊗ End(C2j+1) .
+We call K(j)(u) the spin-j K-operator.
+Similarly to the case of L-operators, we consider u as a formal variable and assume that K( 1
+2 )(u) is in
+B[[u−1]] ⊗ End(C2).
+By Proposition 2.11, the universal K-matrix K satisfies the ψ-twisted reflection equation (2.28). We now
+show that the evaluation of this equation leads to the reflection equation (1.8). To do so, we need evaluation
+of the ψ-twisted universal R-matrices (2.19). Firstly, note that ψ = η from (2.34) is such that (recall the
+definition in (2.40))
+(4.48)
+evu ◦ η = evu−1 .
+Then the evaluations of the ψ-twisted universal R-matrices read:
+(πj1
+u1 ⊗ πj2
+u2)(Rη) = R(j1,j2)(1/(u1u2)) ,
+(4.49)
+(πj1
+u1 ⊗ πj2
+u2)((Rη)21) = R(j2,j1)
+21
+(1/(u1u2)) ,
+(4.50)
+(πj1
+u1 ⊗ πj2
+u2)((R21)η) = R(j2,j1)
+21
+(u1u2) ,
+(4.51)
+(πj1
+u1 ⊗ πj2
+u2)(Rηη
+21) = R(j2,j1)
+21
+(u1/u2) ,
+(4.52)
+(πj1
+u1 ⊗ πj2
+u2)(Rηη) = R(j1,j2)(u2/u1) ,
+(4.53)
+where R(j2,j1)
+21
+(u) is defined in (1.3). Applying p23 to (2.28) leads to K13(Rη)23K12R32 = Rηη
+23K12(Rη)32K13. Fi-
+nally, applying (id⊗πj1
+u−1
+1 ⊗πj2
+u−1
+2 ) to the latter equation using (4.49)-(4.53), it follows that the K-operator (4.47)
+satisfies the reflection equation19
+(4.54) R(j1,j2)
+12
+(u1/u2)K(j1)
+1
+(u1)R(j2,j1)
+21
+(u1u2)K(j2)
+2
+(u2) = K(j2)
+2
+(u2)R(j1,j2)
+12
+(u1u2)K(j1)
+1
+(u1)R(j2,j1)
+21
+(u1/u2) ,
+for any value of j1, j2.
+Recall that we fixed20 H = LUqsl2 and the twist pair (η, 1 ⊗ 1), then due to the P-symmetry (4.24), the
+relations (4.50)-(4.52) become
+(πj1
+u1 ⊗ πj2
+u2)((Rη)21) = R(j1,j2)(1/(u1u2)) ,
+(4.55)
+(πj1
+u1 ⊗ πj2
+u2)((R21)η) = R(j1,j2)(u1u2) ,
+(4.56)
+19As for the L-operator L(j)(u), the K-operator should be written as K(j)
+0i (u) but here we use standard notation K(j)
+i
+(u)
+where we omit the label 0 corresponding to B.
+20The derivation of (4.54) from the ψ-twisted reflection equation can be generalized to H = LUqsln.
+
+FUSED K-OPERATORS FOR Aq
+29
+(πj1
+u1 ⊗ πj2
+u2)(Rηη
+21) = R(j1,j2)(u1/u2) ,
+(4.57)
+and the reflection equation (4.54) becomes the standard reflection equation
+(4.58) R(j1,j2)(u1/u2)K(j1)
+1
+(u1)R(j1,j2)(u1u2)K(j2)
+2
+(u2) = K(j2)
+2
+(u2)R(j1,j2)(u1u2)K(j1)
+1
+(u1)R(j1,j2)(u1/u2) .
+4.2.2. K-operators and fusion ( 1
+2, j) → (j + 1
+2). We follow here the same approach used for L-operators based
+on sub-representations in the tensor product of evaluation representations of LUqsl2), now considering (K3)
+instead of (R3).
+Recall the intertwining operator E(j+ 1
+2 ) is fixed by Lemma (3.1) and its pseudo-inverse
+F(j+ 1
+2 ) is given in (3.22) with (3.23), (3.24) We now obtain our first main result.
+Proposition 4.14. The K-operators (4.47) satisfy for j ∈ 1
+2N:
+(4.59)
+K(j+ 1
+2 )(u) = F
+(j+ 1
+2 )
+⟨12⟩
+K(j)
+2 (uq− 1
+2 )R( 1
+2 ,j)(u2qj− 1
+2 )K
+( 1
+2 )
+1
+(uqj)E
+(j+ 1
+2 )
+⟨12⟩
+.
+Proof. By definition of the K-operator we have
+K(j+ 1
+2 )(u) = (id ⊗ π
+j+ 1
+2
+u−1 )(K) .
+Using the pseudo-inverse property F(j+ 1
+2 )E(j+ 1
+2 ) = I2j+2, we get
+(id ⊗ π
+j+ 1
+2
+u−1 )(K) = (1 ⊗ F(j+ 1
+2 )E(j+ 1
+2 ))(id ⊗ π
+j+ 1
+2
+u−1 )(K)
+= (1 ⊗ F(j+ 1
+2 ))[(id ⊗ π
+1
+2
+u−1q−j ⊗ πj
+u−1q
+1
+2 ) ◦ (id ⊗ ∆)(K)](1 ⊗ E(j+ 1
+2 ))
+= (1 ⊗ F(j+ 1
+2 ))[(id ⊗ π
+1
+2
+u−1q−j ⊗ πj
+u−1q
+1
+2 )(K13Rη
+23K12)](1 ⊗ E(j+ 1
+2 )) .
+The second equality is obtained using the intertwining property (3.16):
+(4.60)
+(1 ⊗ E(j+ 1
+2 ))(id ⊗ π
+j+ 1
+2
+u−1 )(K) = [(id ⊗ π
+1
+2
+u−1q−j ⊗ πj
+u−1q
+1
+2 ) ◦ (id ⊗ ∆)(K)](1 ⊗ E(j+ 1
+2 )) .
+Then, the third equality is due to (K3) and finally, from the definition of the K-operator (4.47) and the
+evaluation of the twisted universal R-matrix (4.49), the claim follows.
+□
+Using the power series assumption on K( 1
+2 )(u), we see that K(j)(u) is also a formal power series in u−1,
+i.e. it is in B[[u−1]] ⊗ End(C2j+1).
+Remark 4.15. Similarly to Remark 4.6, it is clear from the proof of Proposition 4.14 that the K-operator
+does not depend on the choice of F(j), as long as it satisfies F(j)E(j) = I2j+1.
+4.2.3. K-operators and reduction ( 1
+2, j) → (j − 1
+2). The proof of the following proposition is done similarly to
+the proof of the reduction relation (4.19) for the L-operators, thus we skip it. Recall the intertwining operator
+¯E(j− 1
+2 ) is fixed by Lemma 3.2 and its pseudo-inverse ¯F(j− 1
+2 ) is given in (3.30) with (3.31).
+Proposition 4.16. The K-operators (4.47) satisfy for j ∈ 1
+2N+:
+(4.61)
+K(j− 1
+2 )(u) = ¯F
+(j− 1
+2 )
+⟨12⟩
+K(j)
+2 (uq− 1
+2 )R( 1
+2 ,j)(u2q−j− 3
+2 )K
+( 1
+2 )
+1
+(uq−j−1) ¯E
+(j− 1
+2 )
+⟨12⟩
+.
+Recall that we assumed that the universal K-matrix exists for a given choice of B and the twist pair (η, 1⊗1).
+Therefore, the K-operator for a given spin is unique, that is, similarly to the case of the L-operator, we obtain
+the same operator K(j)(u) either using the fusion for ( 1
+2, j − 1
+2) → j or using the reduction ( 1
+2, j + 1
+2) → j.
+
+30
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Remark 4.17. Consider the opposite coproduct ∆op = p ◦ ∆ with the definition (A.9). It follows from (K3):
+(4.62)
+(id ⊗ ∆op)(K) = K12(Rψ)32K13 .
+Recall that we obtained for ∆op an intertwining relation in (3.34) where E(j+ 1
+2 ) is fixed as in (3.17), see
+Remark 3.3. Thus, we also take F(j+ 1
+2 ) as defined in (3.23), (3.24). Then, using (3.34) and (4.62), we obtain
+a new fusion relation for any j ∈ 1
+2N:
+(4.63)
+K(j+ 1
+2 )(u) = F
+(j+ 1
+2 )
+⟨12⟩
+K
+( 1
+2 )
+1
+(uq−j)R( 1
+2 ,j)(u2q−j+ 1
+2 )K(j)
+2 (uq
+1
+2 )E
+(j+ 1
+2 )
+⟨12⟩
+.
+Similarly, we also have the intertwining relation for ∆op given in (3.33) with ¯E(j− 1
+2 ) fixed as in (3.29), see
+Remark 3.3, and we take ¯F(j− 1
+2 ) as defined in (3.30) with (3.31). Then, using (4.62) and (3.35), we obtain
+a new reduction relation for any j ∈ 1
+2N+:
+(4.64)
+K(j− 1
+2 )(u) = ¯F
+(j− 1
+2 )
+⟨12⟩
+K
+( 1
+2 )
+1
+(uqj+1)R( 1
+2 ,j)(u2qj+ 3
+2 )K(j)
+2 (uq
+1
+2 ) ¯E
+(j− 1
+2 )
+⟨12⟩
+.
+To conclude this section, let us discuss the other choice of Drinfeld twist J = R21R. We first consider the
+evaluation of the twist J = R21R on πj1
+u1 ⊗ πj2
+u2. From (4.45) and using
+πj2(c(u)) = πj2(µ(u)µ(u−1))(q2j2+1 + q−2j2−1 − u2q2 − u−2q−2) ,
+one gets the expression
+(πj1
+u1 ⊗ πj2
+u2)(R21R) =
+2j1−1
+�
+k=0
+�
+πj2�
+µ(q−j1+ 1
+2 −ku2/u1)µ(qj1− 1
+2 −ku1/u2)
+��
+(4.65)
+×
+2j1−1
+�
+k=0
+��
+q2j2+1 + q−2j2−1 − (u2/u1)2q−2j1+1+2k − (u1/u2)2q2j1−1−2k��
+.
+In particular, for j1 = 1
+2, j2 = j and u1/u2 = q±(j+ 1
+2 ), one finds:
+(4.66)
+(π
+1
+2u1 ⊗ πj
+u2)(R21R) = 0 ,
+where we used that µ(u) has no poles, see (4.11).
+If there exists a universal K-matrix for the twist pair (η, R21R), similarly to Propositions 4.14 and 4.16
+one obtains a fusion relation on the corresponding K-operators K
+(j)(u):
+(4.67)
+�
+F
+(j+ 1
+2 )
+⟨12⟩
+[(π
+1
+2
+u−1qj ⊗ πj
+u−1q
+1
+2 ) (R21R)]E
+(j+ 1
+2 )
+⟨12⟩
+�
+K
+(j+ 1
+2 )(u)
+= F
+(j+ 1
+2 )
+⟨12⟩
+K
+(j)
+2 (uq− 1
+2 )R( 1
+2 ,j)(u2qj− 1
+2 )K1(uqj)E(j+ 1
+2 ) .
+However, this relation doesn’t allow us to determine K
+(j+ 1
+2 )(u) because the evaluation of the Drinfeld twist
+J = R21R is zero due to (4.66), and so the l.h.s. of (4.67) is zero too.
+4.3. Comodule algebra structure using K-operators. Given the Hopf algebra H = LUqsl2, it is known
+that the coproduct, antipode and counit can be formulated solely in terms of so-called Ding-Frenkel L-
+operators [DF93]:
+(4.68)
+L+(u) = (id ⊗ π
+1
+2
+u−1)(R) ,
+L−(u) = [(id ⊗ π
+1
+2
+u−1)(R21)]−1 ,
+They are computed in Appendix C, see (C.26) and (C.33). Then, one finds that L±(u) are formal power
+series in u∓1, i.e. L±(u) are in LUqsl2[[u∓1]] ⊗ End(C2). The modes of the entries of L±(u) generate LUqsl2
+and they satisfy the Yang-Baxter algebra relations:
+R( 1
+2 , 1
+2 )(u/v)L±
+1 (u)L±
+2 (v) = L±
+2 (v)L±
+1 (u)R( 1
+2 , 1
+2 )(u/v) ,
+(4.69)
+
+FUSED K-OPERATORS FOR Aq
+31
+R( 1
+2 , 1
+2 )(u/v)L±
+1 (u)L∓
+2 (v) = L∓
+2 (v)L±
+1 (u)R( 1
+2 , 1
+2 )(u/v) ,
+(4.70)
+where R( 1
+2 , 1
+2 )(u) is defined in (4.10). These relations follow from the evaluation of (2.3). The coproduct,
+antipode and counit of LUqsl2 are given, respectively, by21:
+(∆ ⊗ id)(L±(u)) =
+�
+L±(u)
+�
+[1]
+�
+L±(u)
+�
+[2] ,
+(4.72)
+(S ⊗ id)(L±(u)) = (L±(u))−1 ,
+(4.73)
+(ǫ ⊗ id)(L±(u)) = 1 .
+(4.74)
+These relations are easily understood using (R2), (2.1) and (2.2). Indeed, the relation (4.72) is obtained by
+applying (id⊗id⊗π
+1
+2
+u−1) on (R2), and similarly for L−(u) using (π
+1
+2
+u−1 ⊗id⊗id)◦(id⊗∆)(R−1). The relations
+(4.73), (4.74) follow immediately from (2.1), (2.2).
+Consider the subalgebra in B generated by the matrix entries of the K-operator K( 1
+2 )(u). They satisfy the
+reflection equation (4.58) for j1 = j2 = 1
+2. Similarly to the coproduct of LUqsl2 discussed above, the coaction
+for this subalgebra can be expressed in terms of L- and K-operators.
+Proposition 4.18. The coaction map δ: B → B ⊗LUqsl2 restricted to the subalgebra generated by the matrix
+entries of K( 1
+2 )(u) is such that
+(4.75)
+(δ ⊗ id)(K( 1
+2 )(u)) =
+�
+[L−(u−1)]−1�
+[2]
+�
+K( 1
+2 )(u)
+�
+[1]
+�
+L+(u)
+�
+[2] .
+Proof. From the fundamental axiom (K2), the l.h.s. of (4.75) can be written as
+(id ⊗ id ⊗ π
+1
+2
+u−1) ◦ (δ ⊗ id)(K) = (id ⊗ id ⊗ π
+1
+2
+u−1) ◦ ((Rη)32K13R23) ,
+where (Rη)32 = (id ⊗ id ⊗ η)(R32). Then, using πj
+u−1 ◦ η = πj
+u together with the definition of the K-operator
+and L±(u) given respectively in (4.47), (4.68), the claim follows.
+□
+In the case when B is generated by the matrix entries of the K-operator K( 1
+2 )(u), eq. (4.75) expresses the
+coaction map for B solely in terms of L- and K-operators. This is the case when B = Aq and it will be
+discussed in Section 6.
+We finally consider the evaluated coaction δw = (id ⊗ evw) ◦ δ: B → B ⊗ Uqsl2, for w ∈ C∗, applied to the
+matrix elements of the spin-j K-operator. The evaluated coaction is obtained by taking the image of (K2)
+under the evaluation map (id ⊗ evw ⊗ πj
+u−1) using (4.1), (4.21), (4.48), and is given by
+(4.76)
+(δw ⊗ id)(K(j)(u)) =
+�
+L(j)(u/w)
+�
+[2]
+�
+K(j)(u)
+�
+[1]
+�
+L(j)(uw)
+�
+[2] .
+Whereas the action of (id ⊗ πj
+v−1) on (K1) gives
+(4.77)
+K(j)(v)(id ⊗ πj)[δv−1(b)] = (id ⊗ πj)[δv(b)]K(j)(v) .
+We call it the twisted intertwining relation for K(j)(u).
+21The index [j] characterizes the ‘quantum space’ V[j] on which the entries of L±(u) act.
+With respect to the ordering
+V[1] ⊗ V[2], one has:
+(4.71)
+((T)[1](T ′)[2])ij =
+2
+�
+k=1
+(T)ik ⊗ (T ′)kj .
+
+32
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+5. Fused K-operators for Aq
+In this section, we consider the comodule algebra B = Aq and related ‘fused’ K-operators. Contrary to
+the previous section, here we do not assume the existence of a universal K-matrix. Instead, we introduce the
+fundamental K-operator for B = Aq and recall the corresponding Faddeev-Reshetikhin-Taktadjan type pre-
+sentation following [BS09, BB17]. Then, in Section 5.2, fused K-operators K(j)(u) built from the fundamental
+K-operator by analogy with (4.63) are shown to satisfy the reflection equation (4.58) for all j ∈ 1
+2N+ where
+K(j)(u) is replaced by K(j)(u). This is the main result in this section. We also establish the unitarity and
+invertiblity properties of K(j)(u) in Section 5.3, and examples of the fused K-operators are derived explicitly
+for small values of j in Section 5.4. In preparation to the discussion in the next section, in Section 5.5 we
+calculate the evaluated coaction for Aq and also establish the twisted intertwining relations for the fused
+K-operators which are similar to (4.77).
+5.1. The fundamental K-operator for Aq. An alternative presentation for Aq besides Definition 2.14 is
+known, which takes the form of a reflection algebra [BS09] that is recalled below. Note that part of the material
+in this subsection is taken from [BS09, BB17, T21a]. Let R( 1
+2 , 1
+2 )(u) be the symmetric R-matrix (4.10) which
+satisfies the quantum Yang-Baxter equation (4.4) with the substitution R(jk,jℓ)(u) → R( 1
+2 , 1
+2 )(u). We now
+introduce the K-operator that provides the reflection algebra presentation of Aq, with the parametrization
+from (2.43).
+Theorem 5.1 ([BS09]). Aq admits a presentation in the form of a reflection algebra. Introduce the generating
+functions:
+W+(u) =
+�
+k∈N
+W−kU −k−1 ,
+W−(u) =
+�
+k∈N
+Wk+1U −k−1 ,
+(5.1)
+G+(u) =
+�
+k∈N
+Gk+1U −k−1 ,
+G−(u) =
+�
+k∈N
+˜Gk+1U −k−1 ,
+(5.2)
+where the shorthand notation U = (qu2 + q−1u−2)/(q + q−1) is used. The defining relations are given by:
+R( 1
+2 , 1
+2 )(u/v) K
+( 1
+2 )
+1
+(u) R( 1
+2 , 1
+2 )(uv) K
+( 1
+2 )
+2
+(v) = K
+( 1
+2 )
+2
+(v) R( 1
+2 , 1
+2 )(uv) K
+( 1
+2 )
+1
+(u) R( 1
+2 , 1
+2 )(u/v)
+(5.3)
+with the R-matrix from (4.10) and
+(5.4)
+K( 1
+2 )(u) =
+�
+uqW+(u) − u−1q−1W−(u)
+1
+k−(q+q−1)G+(u) + k+(q+q−1)
+q−q−1
+1
+k+(q+q−1)G−(u) + k−(q+q−1)
+q−q−1
+uqW−(u) − u−1q−1W+(u)
+�
+.
+Notice that U −1 can be written as a power series in u−2, and (5.1) and (5.2) have no constant terms, so
+K( 1
+2 )(u) is in Aq[[u−1]] ⊗ End(C2). We call K( 1
+2 )(u) the fundamental K-operator for Aq. Explicitly, in terms
+of the generating functions (5.1), (5.2) the defining relations (5.3) read:
+[W±(u), W±(v)] = 0 ,
+(5.5)
+[W+(u), W−(v)] + [W−(u), W+(v)] = 0 ,
+(5.6)
+[Gǫ(u), W±(v)] + [W±(u), Gǫ(v)] = 0,
+ǫ = ± ,
+(5.7)
+[G±(u), G±(v)] = 0 ,
+(5.8)
+[G+(u), G−(v)] + [G−(u), G+(v)] = 0 ,
+(5.9)
+(U − V ) [W±(u), W∓(v)] = (q − q−1)
+ρ(q + q−1) (G±(u)G∓(v) − G±(v)G∓(u))
+(5.10)
++
+1
+(q + q−1) (G±(u) − G∓(u) + G∓(v) − G±(v)) ,
+
+FUSED K-OPERATORS FOR Aq
+33
+W±(u)W±(v) − W∓(u)W∓(v) +
+1
+ρ(q2 − q−2) [G±(u), G∓(v)]
+(5.11)
++ 1 − UV
+U − V (W±(u)W∓(v) − W±(v)W∓(u)) = 0 ,
+U [G∓(v), W±(u)]q − V [G∓(u), W±(v)]q − (q − q−1) (W∓(u)G∓(v) − W∓(v)G∓(u))
+(5.12)
++ ρ (UW±(u) − V W±(v) − W∓(u) + W∓(v)) = 0 ,
+U [W∓(u), G∓(v)]q − V [W∓(v), G∓(u)]q − (q − q−1) (W±(u)G∓(v) − W±(v)G∓(u))
+(5.13)
++ ρ (UW∓(u) − V W∓(v) − W±(u) + W±(v)) = 0 .
+There exists an automorphism of Aq:
+(5.14)
+σ:
+W±(u) → W∓(u) ,
+G±(u) → G∓(u) ,
+k± → k∓ .
+It is obtained from the reflection equation (5.3). Indeed, note that the R-matrix is such that R( 1
+2 , 1
+2 )(u) =
+MR( 1
+2 , 1
+2 )(u)M, with M = σx ⊗ σx. Consider the conjugation of the K-operator by σx. Its entries read:
+(σxK( 1
+2 )(u)σx)i,j = (K( 1
+2 )(u))3−i,3−j for 1 ≤ i, j ≤ 2. Then, multiplying (5.3) on both sides by M ⊗ M, the
+automorphism σ follows.
+In the following, we need the so-called quantum determinant associated with the K-operator.
+It is a
+generating function for central elements of Aq, given by [Sk88]:
+(5.15)
+Γ(u) = tr12
+�
+P−
+12K
+( 1
+2 )
+1
+(u)R( 1
+2 , 1
+2 )(qu2)K
+( 1
+2 )
+2
+(uq)
+�
+.
+Proposition 5.2 ([BB17, T21a]). The quantum determinant of the fundamental K-operator
+(5.16)
+Γ(u) = (u2q2 − u−2q−2)
+2(q − q−1)
+�
+∆( 1
+2 )(u) −
+2ρ
+q − q−1
+�
+,
+with
+∆( 1
+2 )(u) = −(q − q−1)(q2 + q−2)
+�
+W+(u)W+(uq) + W−(u)W−(uq)
+�
++ (q − q−1)(u2q2 + u−2q−2)
+�
+W+(u)W−(uq) + W−(u)W+(uq)
+�
+− (q − q−1)
+ρ
+�
+G+(u)G−(uq) + G−(u)G+(uq)
+�
+− G+(u) − G+(uq) − G−(u) − G−(uq) ,
+(5.17)
+is such that
+�
+Γ(u), K
+( 1
+2 )
+mn(v)
+�
+= 0. It generates the center of Aq.
+We can expand ∆( 1
+2 )(u) as a formal power series in u−1
+∆( 1
+2 )(u) =
+∞
+�
+k=0
+u−2k−2ck+1∆k+1 .
+Explicit expressions for the coefficients in terms of the generators of Aq can be found in [BB17]. We see that
+the constant term of ∆( 1
+2 )(u) − 2ρ/(q − q−1) is −2ρ/(q − q−1) and so it is invertible. Therefore, by [T21d,
+Lem. 4.1] it follows that Γ(u) is invertible too.
+To conclude this subsection, let us point out the invertiblity property of the fundamental K-operator
+K( 1
+2 )(u) given in (5.4), that will be used in Section 6.
+Lemma 5.3.
+(5.18)
+K( 1
+2 )(u−1)K( 1
+2 )(u) = K( 1
+2 )(u)K( 1
+2 )(u−1) = Γ(uq−1)
+c(u−2) I2 ,
+where c(u) and Γ(u) are respectively given in (3.39), (5.16).
+
+34
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Proof. Note that by definition of the generating functions in (5.1), (5.2), one has W±(u−1) = W±(uq−1) and
+G±(u−1) = G±(uq−1). Then, using the ordering relations of Aq given in Appendix B, by straightforward
+calculation one gets (5.18).
+□
+We will call the above property (5.18) the unitarity property of the fundamental K-operator by analogy
+with the L-operator, recall (4.13).
+Remark 5.4. Since Γ(u) is invertible, it follows that K( 1
+2 )(u) is invertible too and its inverse is given by:
+(5.19)
+�
+K( 1
+2 )(u)
+�−1
+= c(u−2)
+Γ(uq−1) K( 1
+2 )(u−1) .
+Remark 5.5. A central element of Aq denoted Z(t) has been proposed [T21a, Def. 8.4]. It is easily checked
+that adapting its expression to our conventions, one has
+(5.20)
+Γ(uq− 1
+2 )
+c(u2q)
+→ Z(t) ,
+with the identification u2q → t, W∓(uq− 1
+2 ) → SW±(S), W∓(uq
+1
+2 ) → T W±(T ), G+(uq− 1
+2 ) + ρ/(q − q−1) →
+G(S), G−(uq
+1
+2 ) + ρ/(q − q−1) → ˜G(T ) , ρ → −(q2 − q−2)2.
+5.2. Fused K-operators for Aq. Recall R( 1
+2 , 1
+2 )(u) and the fundamental K-operator K( 1
+2 )(u), given respec-
+tively by (4.10) and by (5.4), satisfy the reflection equation (5.3). By analogy with (4.63), we now construct
+fused K-operators K(j)(u) ∈ Aq ⊗ End(C2j+1).
+Definition 5.6. For j ∈ 1
+2N+, the fused K-operators for Aq are
+(5.21)
+K(j+ 1
+2 )(u) = F
+(j+ 1
+2 )
+⟨12⟩
+K
+( 1
+2 )
+1
+(uq−j)R( 1
+2 ,j)(u2q−j+ 1
+2 )K(j)
+2 (uq
+1
+2 )E
+(j+ 1
+2 )
+⟨12⟩
+,
+with K( 1
+2 )(u) defined in (5.4).
+The following Theorem is our second main result.
+Theorem 5.7. The fused K-operators K(j)(u) satisfy the reflection equation for any j1, j2 ∈ 1
+2N+:
+(5.22)
+R(j1,j2)(u1/u2)K(j1)
+1
+(u1)R(j1,j2)(u1u2)K(j2)
+2
+(u2) = K(j2)
+2
+(u2)R(j1,j2)(u1u2)K(j1)
+1
+(u1)R(j1,j2)(u1/u2) .
+The proof is by induction on j1, j2. For (j1, j2) = ( 1
+2, 1
+2), the reflection equation (5.22) holds for K( 1
+2 )(u)
+due to [BS09]. The proof is divided into three parts and consists of three lemmas. We first show the case
+(j1, j2) = ( 1
+2, j+ 1
+2), assuming the equation (5.22) holds for (j1, j2) = ( 1
+2, j). Then, we prove the case (j+ 1
+2, 1
+2),
+assuming (5.22) holds for (j1, j2) = (j, 1
+2). Finally, the generic case (j1, j2) follows.
+First of all, the following Yang-Baxter equation holds for any j1, j2 ∈ 1
+2N:
+(5.23)
+R(j1,j2)
+12
+(u1/u2)R(j1,j3)
+13
+(u1/u3)R(j2,j3)
+23
+(u2/u3) = R(j2,j3)
+23
+(u2/u3)R(j1,j3)
+13
+(u1/u3)R(j1,j2)
+12
+(u1/u2) .
+This is due to the fact that R(j1,j2)(u) is proportional to R(j1,j2)(u), see (4.34), and it satisfies the Yang-Baxter
+equation in (4.4). In what follows we use the shorthand notation
+(5.24)
+Rkℓ = R(jk,jℓ)
+kℓ
+(uk/uℓ) ,
+¯Rkℓ = R(jk,jℓ)
+kℓ
+(uℓ/uk) ,
+ˆRkℓ = R(jk,jℓ)
+kℓ
+(ukuℓ) ,
+Kℓ = K(jℓ)
+ℓ
+(uℓ) ,
+and denote by R
+(j+ 1
+2 ,jk)
+⟨ℓ ℓ+1⟩k (u), R
+(jk,j+ 1
+2 )
+k⟨ℓ ℓ+1⟩ (u), K
+(j+ 1
+2 )
+⟨ℓ ℓ+1⟩(u) the resulting R-matrices and K-operators obtained by
+fusing the space at position ℓ with the one at position ℓ + 1, for k, ℓ ∈ N+.
+According to the notation (5.24), the Yang-Baxter equation (5.23) reads
+R12R13R23 = R23R13R12 ,
+or
+R12 ˆR13 ˆR23 = ˆR23 ˆR13R12 ,
+or
+ˆR12 ˆR13 ¯R23 = ¯R23 ˆR13 ˆR12 ,
+(5.25)
+
+FUSED K-OPERATORS FOR Aq
+35
+where the first two relations in (5.25) are related by u3 → u−1
+3 , and the last two relations in (5.25) are related
+by u2 → u−1
+2 . In the following, we will need relations derived from the Yang-Baxter equation:
+(5.26)
+ˆR13 ˆR12R23 = R23 ˆR12 ˆR13 ,
+R13 ˆR12 ˆR23 = ˆR23 ˆR12R13 .
+The first equality is obtained from the first relation in (5.25) by multiplying both sides on the left and on
+the right by ¯R23, using the unitarity property of the R-matrix in (4.46), that is R23 ¯R23 ∝ I, and substituting
+u2 → u−1
+2 , u3 → u−1
+3 . The two equations in (5.26) are related by u3 → u−1
+3 .
+First, we give a relation derived from the reflection equation that will be used later.
+Lemma 5.8. Assuming the reflection equation holds for any j1, j2, j3, then
+(5.27)
+K1 ˆR12K2 ¯R12 = ¯R12K2 ˆR12K1 .
+Proof. According to the notation (5.24), the reflection equation (5.22) reads
+(5.28)
+R12K1 ˆR12K2 = K2 ˆR12K1R12 .
+The equation (5.27) is obtained by multiplying on both sides of (5.28) on the left and on the right by ¯R12
+and using the unitarity property (4.46).
+□
+In the following, we will need various expressions of the fused R-matrices.
+Lemma 5.9. The fused R-matrices defined in (4.32) satisfy the following relations:
+R(j1,j2)(u) = F(j1)
+⟨12⟩R
+(j1− 1
+2 ,j2)
+23
+(uq− 1
+2 )R
+( 1
+2 ,j2)
+13
+(uqj1− 1
+2 )E(j1)
+⟨12⟩ ,
+(5.29)
+= F(j2)
+⟨23⟩R
+(j1, 1
+2 )
+12
+(uq−j2+ 1
+2 )R
+(j1,j2− 1
+2 )
+13
+(uq
+1
+2 )E(j2)
+⟨23⟩ ,
+(5.30)
+= F(j2)
+⟨23⟩R
+(j1,j2− 1
+2 )
+13
+(uq− 1
+2 )R
+(j1, 1
+2 )
+12
+(uqj2− 1
+2 )E(j2)
+⟨23⟩ .
+(5.31)
+Proof. First, note that the r.h.s. of (5.29)-(5.31) are respectively the analogs22 of the R-matrices in (4.42),
+(4.41), (4.18). Recall that R(j1,j2)(u) is proportional to R(j1,j2)(u), see (4.34) with (4.39), and using (4.28)
+one finds that the coefficient of proportionality reads
+(5.32)
+f (j1,j2)(u) =
+2j1−1
+�
+k=0
+2j2−1
+�
+ℓ=0
+π
+1
+2 (µ(uqj1+j2−1−k−ℓ)) .
+Thus, we have R(j1,j2)(u) = R(j1,j2)(u)/f (j1,j2)(u). Then, inserting in the r.h.s. of the latter relation the
+corresponding R-matrices, the equations (5.29)-(5.31) are obtained provided the following equalities hold
+f (j1− 1
+2 ,j2)(uq− 1
+2 )f ( 1
+2 ,j2)(uqj1− 1
+2 )
+f (j1,j2)(u)
+= f (j1, 1
+2 )(uq−j2+ 1
+2 )f (j1,j2− 1
+2 )(uq
+1
+2 )
+f (j1,j2)(u)
+= f (j1,j2− 1
+2 )(uq− 1
+2 )f (j1, 1
+2 )(uqj2− 1
+2 )
+f (j1,j2)(u)
+= 1 .
+Finally, it is easily checked that they indeed hold using (5.32).
+□
+We now show by induction that the reflection equation (5.22) holds for (j1, j2) = ( 1
+2, j + 1
+2) using the
+R-matrix decomposition in (3.43) and Corollary 3.6 together with the second result in Lemma 4.11.
+Lemma 5.10. The following relation holds:
+R
+( 1
+2 ,j+ 1
+2 )
+1⟨23⟩
+(u/v)K
+( 1
+2 )
+1
+(u)R
+( 1
+2 ,j+ 1
+2 )
+1⟨23⟩
+(uv)K
+(j+ 1
+2 )
+⟨23⟩
+(v) = K
+(j+ 1
+2 )
+⟨23⟩
+(v)R
+( 1
+2 ,j+ 1
+2 )
+1⟨23⟩
+(uv)K
+( 1
+2 )
+1
+(u)R
+( 1
+2 ,j+ 1
+2 )
+1⟨23⟩
+(u/v) .
+(5.33)
+22The relation (4.18) is given for j1 = 1
+2 but it can be generalized to any j1 by computing R(j1,j2)(u) = (πj1 ⊗ id)(L(j2)(u))
+with L(j2)(u) in (4.14).
+
+36
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Proof. We proceed by induction. Recall the reflection equation (5.22) holds for (j1, j2) = ( 1
+2, 1
+2) due to [BS09].
+Assume that (5.22) holds for ( 1
+2, j) with a fixed value of j. Consider the l.h.s. of (5.33) and multiply on the
+right by I2(2j+2) = H
+(j+ 1
+2 )
+⟨23⟩
+[H
+(j+ 1
+2 )
+⟨23⟩
+]−1. Then, using the relations (5.31), (5.30) and the fused K-operator given
+in (5.21), we get (with the necessary steps of the calculation underlined):
+R
+( 1
+2 ,j+ 1
+2 )
+1⟨23⟩
+(u/v)K
+( 1
+2 )
+1
+(u)R
+( 1
+2 ,j+ 1
+2 )
+1⟨23⟩
+(uv)K
+(j+ 1
+2 )
+⟨23⟩
+(v)H
+(j+ 1
+2 )
+⟨23⟩
+[H
+(j+ 1
+2 )
+⟨23⟩
+]−1
+= F⟨23⟩R13R12E⟨23⟩K1F⟨23⟩ ˆR12 ˆR13E⟨23⟩F⟨23⟩K2 ˆR23K3E⟨23⟩H⟨23⟩[H⟨23⟩]−1 ,
+(5.34)
+where we fix in the notations from (5.24)
+(5.35)
+j1 = j2 = 1
+2 ,
+j3 = j ,
+u1 = u ,
+u2 = vq−j ,
+u3 = vq
+1
+2 .
+First, we remove the products E⟨23⟩F⟨23⟩. Recall that R( 1
+2 ,j)(u) from (4.33) agrees with (3.38), see Lemma 4.11,
+then using (3.45) we find that (5.34) equals
+F⟨23⟩R13R12K1E⟨23⟩F⟨23⟩ ˆR12 ˆR13E⟨23⟩F⟨23⟩K2 ˆR23K3 ¯R23E⟨23⟩[H⟨23⟩]−1
+(5.27)
+=
+F⟨23⟩R13R12K1E⟨23⟩F⟨23⟩ ˆR12 ˆR13E⟨23⟩F⟨23⟩ ¯R23K3 ˆR23K2E⟨23⟩[H⟨23⟩]−1
+(3.47)
+=
+F⟨23⟩R13R12K1E⟨23⟩F⟨23⟩ ˆR12 ˆR13 ¯R23K3 ˆR23K2E⟨23⟩[H⟨23⟩]−1
+(5.25)
+=
+F⟨23⟩R13R12K1E⟨23⟩F⟨23⟩ ¯R23 ˆR13 ˆR12K3 ˆR23K2E⟨23⟩[H⟨23⟩]−1
+(3.47)
+=
+F⟨23⟩R13R12K1 ¯R23 ˆR13 ˆR12K3 ˆR23K2E⟨23⟩[H⟨23⟩]−1
+=
+F⟨23⟩R13R12K1 ˆR12 ˆR13K2 ˆR23K3E⟨23⟩ ,
+where we used (5.25), (5.27) and (3.45) to get the last line. Secondly, rearranging the different R-matrices
+and K-operators using [R23, K1] = [R13, K2] = [ ˆR13, K2] = [ ˆR12, K3] = 0 and (5.26), (5.25), (5.28), we get:
+=
+F⟨23⟩R13R12K1 ˆR12K2 ˆR13 ˆR23K3E⟨23⟩
+(5.28)
+=
+F⟨23⟩R13K2 ˆR12K1R12 ˆR13 ˆR23K3E⟨23⟩
+(5.25)
+=
+F⟨23⟩K2R13 ˆR12K1 ˆR23 ˆR13R12K3E⟨23⟩
+(5.26)
+=
+F⟨23⟩K2 ˆR23 ˆR12R13K1 ˆR13K3R12E⟨23⟩
+(5.28)
+=
+F⟨23⟩K2 ˆR23 ˆR12K3 ˆR13K1R13R12E⟨23⟩
+=
+F⟨23⟩K2 ˆR23K3 ˆR12 ˆR13K1R13R12E⟨23⟩
+=
+F⟨23⟩K2 ˆR23K3E⟨23⟩F⟨23⟩ ˆR12 ˆR13E⟨23⟩K1F⟨23⟩R13R12E⟨23⟩ ,
+where we put back the products E⟨23⟩F⟨23⟩ by inserting I2(2j+2) = H
+(j+ 1
+2 )
+⟨23⟩
+[H
+(j+ 1
+2 )
+⟨23⟩
+]−1 and using (5.26),
+(5.27), (3.45)-(3.47). Finally, identifying the fused R-matrices and the fused K-operator, the equation (5.33)
+follows.
+□
+Then, we show by induction that the reflection equation (5.22) holds for (j1, j2) = (j + 1
+2, 1
+2).
+Lemma 5.11. The following relation holds:
+(5.36)
+R
+(j+ 1
+2 , 1
+2 )
+⟨12⟩3
+(u/v)K
+(j+ 1
+2 )
+⟨12⟩
+(u)R
+(j+ 1
+2 , 1
+2 )
+⟨12⟩3
+(uv)K
+( 1
+2 )
+3
+(v) = K
+( 1
+2 )
+3
+(v)R
+(j+ 1
+2 , 1
+2 )
+⟨12⟩3
+(uv)K
+(j+ 1
+2 )
+⟨12⟩
+(u)R
+(j+ 1
+2 , 1
+2 )
+⟨12⟩3
+(u/v) .
+Proof. We proceed by induction similarly to the proof of Lemma 5.10. Recall the reflection equation (5.22)
+holds for (j1, j2) = ( 1
+2, 1
+2) due to [BS09]. Assume that (5.22) holds for (j, 1
+2) with a fixed value of j. Consider
+the l.h.s. of (5.36) and multiply on the right by I2(2j+2) = H
+(j+ 1
+2 )
+⟨12⟩
+[H
+(j+ 1
+2 )
+⟨12⟩
+]−1. Then, using the fused R-
+matrix (4.32) and the fused K-operator given in (5.21), we get (with the necessary steps of the calculation
+
+FUSED K-OPERATORS FOR Aq
+37
+underlined):
+R
+(j+ 1
+2 , 1
+2 )
+⟨12⟩3
+(u/v)K
+(j+ 1
+2 )
+⟨12⟩
+(u)R
+(j+ 1
+2 , 1
+2 )
+⟨12⟩3
+(uv)K
+( 1
+2 )
+3
+(v)H
+(j+ 1
+2 )
+⟨12⟩
+[H
+(j+ 1
+2 )
+⟨12⟩
+]−1
+= F⟨12⟩R13R23E⟨12⟩F⟨12⟩K1 ˆR12K2E⟨12⟩F⟨12⟩ ˆR13 ˆR23E⟨12⟩H⟨12⟩K3[H⟨12⟩]−1 ,
+(5.37)
+where we fix in the notations from (5.24)
+(5.38)
+j1 = j3 = 1
+2 ,
+j2 = j ,
+u1 = uq−j ,
+u2 = uq
+1
+2 ,
+u3 = v .
+Firstly, similarly to Lemma 5.10, we remove the products E⟨12⟩F⟨12⟩ using (3.45)-(3.47), (5.25) and (5.28).
+Then, we find that (5.37) equals
+F⟨12⟩R13R23K1 ˆR12K2 ˆR13 ˆR23K3E⟨12⟩ .
+(5.39)
+Secondly, we rearrange the R-matrices and K-operators in (5.39) as follows:
+F⟨12⟩R13K1R23 ˆR12 ˆR13K2 ˆR23K3E⟨12⟩
+(5.26)
+=
+F⟨12⟩R13K1 ˆR13 ˆR12R23K2 ˆR23K3E⟨12⟩
+(5.28)
+=
+F⟨12⟩R13K1 ˆR13 ˆR12K3 ˆR23K2R23E⟨12⟩
+(5.28)
+=
+F⟨12⟩K3 ˆR13K1R13 ˆR12 ˆR23K2R23E⟨12⟩
+(5.26)
+=
+K3F⟨12⟩ ˆR13K1 ˆR23 ˆR12R13K2R23E⟨12⟩
+=
+K3F⟨12⟩ ˆR13 ˆR23K1 ˆR12K2R13R23E⟨12⟩
+=
+K3F⟨12⟩ ˆR13 ˆR23E⟨12⟩F⟨12⟩K1 ˆR12K2E⟨12⟩F⟨12⟩R13R23E⟨12⟩ ,
+where we put back the products E⟨12⟩F⟨12⟩ inserting I2(2j+2) =H
+(j+ 1
+2 )
+⟨12⟩ [H
+(j+ 1
+2 )
+⟨12⟩
+]−1 and using (3.45)-(3.47), (5.26)
+and (5.27). Finally, identifying the fused R-matrices and the fused K-operator, the equation (5.36) follows.
+□
+Proof of Theorem 5.7. We are now ready to show that the reflection equation (5.22) is satisfied by the fused
+K-operator (5.21) for all j1, j2 ∈
+1
+2N+. We consider separately (5.22) for two distinct cases j1 ≥ j2 and
+j1 ≤ j2:
+(i) (j1, j2) = (j + 1
+2, k + 1
+2) with 0 ≤ k ≤ j .
+(ii) (j1, j2) = (ℓ + 1
+2, j + 1
+2) with 0 ≤ ℓ ≤ j .
+The first case is shown by induction on k. For k = 0, the equation (5.22) holds for (j1, j2) = (j + 1
+2, 1
+2) due
+to Lemma 5.11. Now, assume (5.22) holds for (j1, j2) = (j + 1
+2, k) with a fixed value of k ≤ j. The case (i)
+is shown similarly to the proof of Lemma 5.10. Indeed, fix in (5.34) j1 = j + 1
+2, j2 = 1
+2, j3 = k and u1 = u,
+u2 = vqk, u3 = vq− 1
+2 (instead of (5.35)). Then, the rest of the proof is the same as for Lemma 5.10, using
+now (5.33), (5.36), (4.34) and our assumption.
+The second case is also shown by induction on ℓ. For ℓ = 0, the equation (5.22) holds for (j1, j2) = ( 1
+2, j+ 1
+2)
+due to Lemma 5.10. Assuming that (5.22) holds for (j1, j2) = (ℓ, j + 1
+2) with a fixed value of ℓ ≤ j, then the
+proof of the case (ii) is similar to Lemma 5.11. Fix in (5.37) j1 = 1
+2, j2 = ℓ, j3 = j+ 1
+2 and u1 = uq−ℓ, u2 = uq
+1
+2 ,
+u3 = v (instead of (5.38)). The rest of the proof is the same as Lemma 5.11, using now (5.33), (5.36), (4.34)
+and our assumption.
+This concludes the proof of Theorem 5.7.
+□
+5.3. Unitarity and invertibility properties. We now discuss the unitarity and invertibility properties of
+the fused K-operators K(j)(u) given in (5.21). Recall that K( 1
+2 )(u) satisfies the unitarity property and is
+invertible, see Lemma 5.3 and Remark 5.4, respectively. We generalize these properties for any spin-j.
+Proposition 5.12. Let
+(5.40)
+�K(j+ 1
+2 )(u) = F
+(j+ 1
+2 )
+⟨12⟩
+�K(j)
+2 (uq− 1
+2 )R( 1
+2 ,j)(u2qj− 1
+2 ) �K
+( 1
+2 )
+1
+(uqj)E
+(j+ 1
+2 )
+⟨12⟩
+,
+
+38
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+for j ∈ 1
+2N+ and with �K( 1
+2 )(u) ≡ K( 1
+2 )(u). Then
+K(j)(u) �K(j)(u−1) =
+�2j−1
+�
+k=0
+Γ(uq−j− 1
+2 +k)
+c(u−2q2j−1−2k)
+� �2j−2
+�
+k=0
+2j−k−2
+�
+ℓ=0
+c(u2q2j−1−2k−ℓ)c(u−2q1−k+ℓ)
+�
+I2j+1
+(5.41)
+= �K(j)(u−1)K(j)(u) ,
+where K(j)(u), Γ(u) and c(u) are respectively given in (5.21), (5.16), (3.39).
+Proof. Recall that (5.41) for j = 1
+2 was proven in Lemma 5.3. First, we show that (5.41) holds for j = 1.
+Using (5.21) and (5.40), we find that the product K(1)(u) �K(1)(u−1) equals
+F(1)
+⟨12⟩K
+( 1
+2 )
+1
+(uq− 1
+2 )R( 1
+2 , 1
+2 )(u2)K
+( 1
+2 )
+2
+(uq
+1
+2 )E(1)
+⟨12⟩F(1)
+⟨12⟩ �K
+( 1
+2 )
+2
+(u−1q− 1
+2 )R( 1
+2 , 1
+2 )(u−2) �K
+( 1
+2 )
+1
+(u−1q
+1
+2 )E(1)
+⟨12⟩
+= F(1)
+⟨12⟩K
+( 1
+2 )
+1
+(uq− 1
+2 )R( 1
+2 , 1
+2 )(u2)K
+( 1
+2 )
+2
+(uq
+1
+2 ) �K
+( 1
+2 )
+2
+(u−1q− 1
+2 )R( 1
+2 , 1
+2 )(u−2) �K
+( 1
+2 )
+1
+(u−1q
+1
+2 )E(1)
+⟨12⟩ ,
+where we removed the product E(1)F(1) on the second line, similarly to the derivation of (4.44).
+Then,
+using (3.40) and (5.18) we have
+K(1)(u) �K(1)(u−1) = −Γ(uq− 3
+2 )Γ(uq− 1
+2 ) I3 .
+Similarly, we get
+�K(1)(u−1)K(1)(u) = −Γ(uq− 3
+2 )Γ(uq− 1
+2 ) I3 .
+More generally, by induction we get (5.41). The second line of (5.41) is not obtained using the invariance
+u → u−1 because we do not assume K(j)(u) = �K(j)(u). However, the second line can be shown as the first
+line.
+□
+Remark 5.13. The spin-j fused K-operator K(j)(u) is invertible and its inverse is given by:
+(5.42)
+�
+K(j)(u)
+�−1
+=
+�2j−1
+�
+k=0
+Γ(uq−j− 1
+2 +k)
+c(u−2q2j−1−2k)
+�−1 �2j−2
+�
+k=0
+2j−k−2
+�
+ℓ=0
+c(u2q2j−1−2k−ℓ)c(u−2q1−k+ℓ)
+�−1
+�K(j)(u−1) .
+Remark 5.14. By direct calculations we have checked for j = 1, 3
+2, 2 that �K(j)(u) is equal to K(j)(u) defined
+in (5.21) and we expect this equality holds for any j. Note that K(j)(u) and �K(j)(u), are direct analogs of the
+spin-j K-operators K(j)(u) defined in (4.63) and (4.59), respectively.
+5.4. Examples of fused K-operators. In this subsection we give examples of spin-1 and spin- 3
+2 fused
+K-operators K(j)(u) for Aq, defined by (5.21). Recall the function c(u) given in (3.39).
+5.4.1. Spin-1 fused K-operator. The expressions of E(j+ 1
+2 ), F(j+ 1
+2 ) in (3.13), (3.14), for j = 1
+2 read:
+E(1) =
+
+
+
+
+
+1
+0
+0
+0
+1
+√
+[2]q 0
+0
+1
+√
+[2]q 0
+0
+0
+1
+
+
+
+
+ ,
+F(1) =
+
+
+
+1
+0
+0
+0
+0
+√
+[2]q
+2
+√
+[2]q
+2
+0
+0
+0
+0
+1
+
+
+ .
+From (4.33), the fused R-matrix reads
+R( 1
+2 ,1)(u) = F(1)
+⟨23⟩R
+( 1
+2 , 1
+2 )
+13
+(uq− 1
+2 )R
+( 1
+2 , 1
+2 )
+12
+(uq
+1
+2 )E(1)
+⟨23⟩
+(5.43)
+
+FUSED K-OPERATORS FOR Aq
+39
+and is given explicitly by
+(5.44)
+R( 1
+2 ,1)(u) = c(uq
+1
+2 )
+
+
+
+
+
+
+
+
+
+c(uq
+3
+2 )
+0
+0
+0
+0
+0
+0
+c(uq
+1
+2 )
+0
+c(q)
+�
+[2]q
+0
+0
+0
+0
+c(uq− 1
+2 )
+0
+c(q)
+�
+[2]q
+0
+0
+c(q)
+�
+[2]q
+0
+c(uq− 1
+2 )
+0
+0
+0
+0
+c(q)
+�
+[2]q
+0
+c(uq
+1
+2 )
+0
+0
+0
+0
+0
+0
+c(uq
+3
+2 )
+
+
+
+
+
+
+
+
+
+.
+From (5.21), the fused K-operator is given by:
+(5.45)
+K(1)(u) = F(1)
+⟨12⟩K
+( 1
+2 )
+1
+(uq− 1
+2 )R( 1
+2 , 1
+2 )(u2)K
+( 1
+2 )
+2
+(uq
+1
+2 )E(1)
+⟨12⟩ .
+Using the above expressions, one finds that the entries K(1)
+mn(u) are explicitly given by:
+K(1)
+11 (u) =
+�
+c(q)−1 + ρ−1G+(uq
+1
+2 )
+��
+ρ + c(q)G−(uq− 1
+2 )
+�
++ (u2q − u−2q−1)
+×
+�
+uq
+3
+2 W+(uq
+1
+2 ) − u−1q− 3
+2 W−(uq
+1
+2 )
+��
+uq
+1
+2 W+(uq− 1
+2 ) − u−1q− 1
+2 W−(uq− 1
+2 )
+�
+,
+K(1)
+12 (u) =
+1
+2k−
+�
+q + q−1
+��
+ρ + c(q)G+(uq
+1
+2 )
+��
+uq
+1
+2 W−(uq− 1
+2 ) − u−1q− 1
+2 W+(uq− 1
+2 )
+�
++ (u2 − u−2)
+�
+ρ(q + q−1)−1 + G+(uq
+1
+2 )
+��
+uq
+1
+2 W+(uq− 1
+2 ) − u−1q− 1
+2 W−(uq− 1
+2 )
+�
++ (u2q − u−2q−1)
+�
+uq
+3
+2 W+(uq
+1
+2 ) − u−1q− 3
+2 W−(uq
+1
+2 )
+��
+ρ(q + q−1)−1 + G+(uq− 1
+2 )
+��
+,
+K(1)
+13 (u) = (u2 − u−2)
+k2
+−c(q2)2
+�
+ρ + c(q)G+(uq
+1
+2 )
+��
+ρ + c(q)G+(uq− 1
+2 )
+�
+,
+K(1)
+21 (u) = k−
+�
+q + q−1
+�
+c(u2q)
+�
+c(q)−1 + ρ−1G−(uq
+1
+2 )
+��
+uq
+1
+2 W+(uq− 1
+2 ) − u−1q− 1
+2 W−(uq− 1
+2 )
+�
+(5.46)
++
+�
+uq
+1
+2 (u2q + u−2q−1(c(q−2) − 1))W+(uq
+1
+2 ) + u−1q− 1
+2 (u2q(c(q2) − 1) + u−2q−1)W−(uq
+1
+2 )
+�
+×
+�
+c(q)−1 + ρ−1G−(uq− 1
+2 )
+�
+,
+K(1)
+22 (u) = (u2q − u−2q−1)
+2
+��
+c(q)−1 + ρ−1G−(uq
+1
+2 )
+��
+ρc(q)−1 + G+(uq− 1
+2 )
+�
++
+�
+c(q)−1 + ρ−1G+(uq
+1
+2 )
+��
+ρc(q)−1 + G−(uq− 1
+2 )
+��
++ 1
+2
+��
+uq− 1
+2 (u2q2 + u−2(c(q−2) − 1))W−(uq
+1
+2 ) + u−1q
+1
+2 (u2(c(q2) − 1) + u−2q−2)W+(uq
+1
+2 )
+�
+×
+�
+uq
+1
+2 W+(uq− 1
+2 ) − u−1q− 1
+2 W−(uq− 1
+2 )
+�
++
+�
+uq− 1
+2 (u2q2 + u−2(c(q−2) − 1))W+(uq
+1
+2 ) + u−1q
+1
+2 (u2(c(q2) − 1) + u−2q−2)W−(uq
+1
+2 )
+�
+×
+�
+uq
+1
+2 W−(uq− 1
+2 ) − u−1q− 1
+2 W+(uq− 1
+2 )
+��
+,
+K(1)
+23 (u) = σ(K(1)
+21 (u)) ,
+K(1)
+31 (u) = σ(K(1)
+13 (u)) ,
+K(1)
+32 (u) = σ(K(1)
+12 (u)) ,
+K(1)
+33 (u) = σ(K(1)
+11 (u)) ,
+
+40
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+where σ is defined in (5.14). The last two lines describe the exchange of the entries of the fused K-operator
+due to the automorphism σ and can be seen graphically:
+(5.47)
+K(1)(u) =
+.
+K(1)
+11 (u)
+K(1)
+12 (u)
+K(1)
+13 (u)
+K(1)
+21 (u)
+K(1)
+22 (u)
+K(1)
+23 (u)
+K(1)
+31 (u)
+K(1)
+32 (u)
+K(1)
+33 (u)
+�
+�
+As shown in Lemma 5.10, the fused K-operator (6.1) for j = 1 satisfies the reflection equation:
+(5.48)
+R( 1
+2 ,1)(u/v)K
+( 1
+2 )
+1
+(u)R( 1
+2 ,1)(uv)K(1)
+2 (v) = K(1)
+2 (v)R( 1
+2 ,1)(uv)K
+( 1
+2 )
+1
+(u)R( 1
+2 ,1)(u/v) .
+Note that the latter equation can be independently checked using the ordering relations given in Lemma B.1.
+5.4.2. Spin- 3
+2 fused K-operator. The elements E(j+ 1
+2 ), F(j+ 1
+2 ) in (3.13), (3.14) for j = 1 read:
+E( 3
+2 ) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+1
+0
+0
+0
+0
+√
+[2]q
+√
+[3]q
+0
+0
+0
+0
+[2]q
+√
+([2]q)2√
+[3]q
+0
+0
+1
+√
+[3]q
+0
+0
+0
+0
+([2]q)
+3
+2
+√
+([2]q)2√
+[3]q
+0
+0
+0
+0
+[2]q
+√
+([2]q)2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+, F( 3
+2 ) =
+
+
+
+
+
+���
+
+
+1
+0
+0
+0
+0
+0
+0
+√
+[2]q√
+[3]q
+1+[2]q
+0
+√
+[3]q
+1+[2]q
+0
+0
+0
+0
+√
+[3]q√
+([2]q)2
+[2]q(1+[2]q)
+0
+√
+[3]q√
+([2]q)2
+√
+[2]q(1+[2]q)
+0
+0
+0
+0
+0
+0
+√
+([2]q)2
+[2]q
+
+
+
+
+
+
+
+
+.
+The fused R-matrix from (4.33) reads
+R( 1
+2 , 3
+2 )(u) = F
+( 3
+2 )
+⟨23⟩R( 1
+2 ,1)
+13
+(uq− 1
+2 )R
+( 1
+2 , 1
+2 )
+12
+(uq)E
+( 3
+2 )
+⟨23⟩ ,
+given explicitly by
+(5.49)
+R( 1
+2 , 3
+2 )(u) = c(u)c(uq)c(q)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+c(uq2)
+c(q)
+0
+0
+0
+0
+0
+0
+0
+0
+c(uq)
+c(q)
+0
+0
+�
+[3]q
+0
+0
+0
+0
+0
+c(u)
+c(q)
+0
+0
+�
+([2]q)2
+0
+0
+0
+0
+0
+c(uq−1)
+c(q)
+0
+0
+�
+[3]q
+0
+0
+�
+[3]q
+0
+0
+c(uq−1)
+c(q)
+0
+0
+0
+0
+0
+�
+([2]q)2
+0
+0
+c(u)
+c(q)
+0
+0
+0
+0
+0
+�
+[3]q
+0
+0
+c(uq)
+c(q)
+0
+0
+0
+0
+0
+0
+0
+0
+c(uq2)
+c(q)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+.
+From (5.21), the fused K-operator reads:
+(5.50)
+K( 3
+2 )(u) = F
+( 3
+2 )
+⟨12⟩K
+( 1
+2 )
+1
+(uq−1)R( 1
+2 ,1)(u2q− 1
+2 )K(1)
+2 (uq
+1
+2 )E
+( 3
+2 )
+⟨12⟩ .
+For instance, the first entry reads:
+K
+( 3
+2 )
+11 (u) = c(u2)(q + q−1)
+2ρc(q)
+�
+ρ + c(q)G+(uq−1)
+��
+c(u2q2)(ρ + c(q)G−(u)
+��
+uq2W+(uq) − u−1q−2W−(uq)
+�
++ u−1�
+(u2(q2 − 2) + u−2q−2)W−(u) + (u4q2 + q−2 − 2)W+(u)
+��
+ρ + c(q)G−(uq)
+��
+
+FUSED K-OPERATORS FOR Aq
+41
++ c(u2)c(u2q)
+�
+uW+(uq−1) − u−1W−(uq−1)
+���
+ρ−1G+(u) + c(q)−1��
+ρ + c(q)G−(uq)
+�
++ c(u2q2)
+�
+uqW+(u) − u−1q−1W−(u)
+��
+uq2W+(uq) − u−1q−2W−(uq)
+��
+.
+The other explicit expressions of the entries in terms of the generating functions for Aq are not reported here
+for simplicity. Under the action of σ from (5.14), the entries exchange according to:
+(5.51)
+K( 3
+2 )(u) =
+,
+K
+( 3
+2 )
+11 (u)
+K
+( 3
+2 )
+12 (u)
+K
+( 3
+2 )
+13 (u)
+K
+( 3
+2 )
+14 (u)
+K
+( 3
+2 )
+21 (u)
+K
+( 3
+2 )
+22 (u)
+K
+( 3
+2 )
+23 (u)
+K
+( 3
+2 )
+24 (u)
+K
+( 3
+2 )
+31 (u)
+K
+( 3
+2 )
+32 (u)
+K
+( 3
+2 )
+33 (u)
+K
+( 3
+2 )
+34 (u)
+K
+( 3
+2 )
+41 (u)
+K
+( 3
+2 )
+42 (u)
+K
+( 3
+2 )
+43 (u)
+K
+( 3
+2 )
+44 (u)
+�
+�
+.
+By Theorem 5.7, the fused K-operator satisfies the reflection equation:
+R( 1
+2 , 3
+2 )(u/v)K
+( 1
+2 )
+1
+(u)R( 1
+2 , 3
+2 )(uv)K
+( 3
+2 )
+2
+(v) = K
+( 3
+2 )
+2
+(v)R( 1
+2 , 3
+2 )(uv)K
+( 1
+2 )
+1
+(u)R( 1
+2 , 3
+2 )(u/v) .
+5.4.3. Spin-j fused K-operator. Specializing the formula (5.21), one gets the fused K-operator K(j)(u) ∈
+Aq ⊗ End(C2j+1) for any value of j starting from (5.4). By analogy with the previous cases, note that one
+has the invariance of the R-matrix (3.38)
+R( 1
+2 ,j)(u) = M (j)R( 1
+2 ,j)(u)M (j)
+with
+M (j) = σx ⊗
+2j+1
+�
+n=1
+E(j,j)
+n,2j+2−n .
+So, due to the automorphism σ in (5.14), the entries of the K-operator of spin-j exchange as K(j)
+m,n(u) =
+σ(K(j)
+2j+2−m,2j+2−n(u)) with 1 ≤ m, n ≤ 2j + 1. This is analogous to the property in (5.47), (5.51).
+From the fusion formulas (4.33) and (5.21) it is clear that the fused R-matrices and K-operators can be
+expressed only in terms of the fundamental K-operator and R-matrix, and the maps E(j) and F(j). They are
+given by:
+(5.52)
+R( 1
+2 ,j)(u) =
+�2j−2
+�
+m=0
+I2m+2 ⊗ F(j− m
+2 )
+� �2j−1
+�
+k=0
+R
+( 1
+2 , 1
+2 )
+1 2j+1−k(uq−j+ 1
+2 +k)
+� �2j−2
+�
+m=0
+I2(2j−1−m) ⊗ E(1+ m
+2 )
+�
+,
+and
+(5.53)
+K(j)(u) =
+�2j−2
+�
+m=0
+I2m ⊗ F(j− m
+2 )
+� 2j
+�
+k=1
+�
+K
+( 1
+2 )
+k
+(uqk−j− 1
+2 )
+�2j−k−1
+�
+ℓ=0
+R
+( 1
+2 , 1
+2 )
+k 2j−ℓ(u2q−2j+2k+ℓ)
+��
+×
+�2j−2
+�
+m=0
+I2(2j−2−m) ⊗ E(1+ m
+2 )
+�
+,
+where the product stands for the usual matrix product and the products are ordered from left to right in an
+increasing way in the indices. The proof of (5.52) is straightforward by induction on j using (4.33), whereas
+the proof of (5.53) is more tedious. We proceed by induction checking (5.21) using (5.52) and (5.53). Then,
+one obtains a formula similar to (5.53) for j → j + 1
+2 but with unwanted products of E(j)F(j). They can
+be removed using the same trick as in the proof of Lemma 5.10. Firstly, multiply (5.21) from the right by
+H(j+ 1
+2 )[H(j+ 1
+2 )]−1 = I2j+2 and use (3.45) to move H(j+ 1
+2 ) to the left. Then, using successively the Yang-
+Baxter equation and the reflection equation, the products E(j)F(j) are removed using the property (3.47).
+
+42
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Note that in the literature, another fusion procedure was developed for the R-matrix, see [Ka79, KRS81],
+and for the K-matrix in [MN92]. In this case, the analogue of the formulas (5.53), (5.52) can be found for
+instance in [FNR07, eq. (2.1), (2.7)].
+5.5. Evaluated coaction of fused K-operators. The fused K-operators K(j)(u) are expected to have a
+simple relation with the spin-j K-operators K(j)(u) as will be discussed in Section 6, similarly to the relations
+between L(j)(u) and L(j)(u). Therefore, the evaluated coaction for K( 1
+2 )(u) is expected to be of the form (4.76)
+up to appropriate normalization.
+Lemma 5.15. The evaluated coaction δw : Aq → Aq ⊗ Uqsl2 is such that23
+(5.54)
+(δw ⊗ id)(K( 1
+2 )(u)) =
+U −1
+q + q−1
+�
+L( 1
+2 )(u/w)
+�
+[2]
+�
+K( 1
+2 )(u)
+�
+[1]
+�
+L( 1
+2 )(uw)
+�
+[2] .
+Proof. Assume the evaluated coaction takes the form
+(5.55)
+(δw ⊗ id)(K( 1
+2 )(u)) = f(u)
+�
+L( 1
+2 )(u/w)
+�
+[2]
+�
+K( 1
+2 )(u)
+�
+[1]
+�
+L( 1
+2 )(uw)
+�
+[2]
+where f(u) is to be central in Aq ⊗ Uqsl2. We show that δw is indeed an algebra homomorphism for a certain
+choice of f(u). It is easily checked using the Yang-Baxter algebra (4.3) satisfied by L( 1
+2 )(u) that (5.55) solves
+the reflection equation (5.3) with the substitution K( 1
+2 )(u) → (δw ⊗ id)(K( 1
+2 )(u)). Then, we fix the function
+f(u) as follows. We compare the l.h.s. of (5.55) using (5.4) to the r.h.s. of (5.55) that is computed using
+L( 1
+2 )(u) given by (4.5). From the matrix entries (1, 1) and (2, 2) of (5.55) one finds
+δw(W+(u)) = f(u)
+�
+W−(u) ⊗
+�
+(q − q−1)2EF − q(K − K−1)
+�
+− (w2 + w−2)W+(u) ⊗ 1
++
+(q − q−1)
+k+k−(q + q−1)
+�
+k+q
+1
+2 G+(u) ⊗ (w−1EK
+1
+2 ) + k−q− 1
+2 G−(u) ⊗ (wFK
+1
+2 )
+�
++ (q + q−1)
+�
+1 ⊗ (k+q
+1
+2 w−1EK
+1
+2 ) + 1 ⊗ (k−q− 1
+2 wFK
+1
+2 )
+�
++ U(q + q−1)W+(u) ⊗ K
+�
+.
+Then, inserting the power series (5.1) and (5.2), one gets
+δw(W0) = f(u)U(q + q−1)
+�
+1 ⊗ (k+q
+1
+2 w−1EK
+1
+2 + k−q− 1
+2 wFK
+1
+2 ) + W0 ⊗ K
+�
+.
+By evaluation of the coaction δ(W0) given in (2.44) it implies f(u) = U −1/(q + q−1). From the analysis of
+δw(W−(u)), we obtain the same result for f(u).
+Now, consider the matrix entry (2, 1) of (5.55). It reads:
+1
+k+(q + q−1)δw(G−(u)) + k−(q + q−1)
+q − q−1
+δw(1) =
+U −1
+k+(q + q−1)2
+�
+k+
+k−
+(q − q−1)2G+(u) ⊗ E2 − G−(u) ⊗ (w−2K−1 + w2K) + U(q + q−1)G−(u) ⊗ 1
++ (q + q−1)(q2 − q−2)
+�
+k+q
+1
+2 (UW+(u)−W−(u)) ⊗ (wEK
+1
+2 ) + k+q− 1
+2 (UW−(u)−W+(u)) ⊗ (w−1EK− 1
+2 )
+�
++ k+k−(q + q−1)2
+(q − q−1)
+1 ⊗
+�k+
+k−
+(q − q−1)2E2 − (w−2K−1 + w2K)
+� �
++ k−(q + q−1)
+q − q−1
+(1 ⊗ 1) .
+By definition δw(1) = 1 ⊗ 1, so the above equation fixes δw(G−(u)).
+□
+23Here, the index [1] is associated with the space for Aq, and [2] for Uqsl2, and we use the convention ((T)[2](T ′)[1](T ′′)[2])ij =
+�2
+k,ℓ=1(T ′)kℓ ⊗ (T)ik(T ′′)ℓj.
+
+FUSED K-OPERATORS FOR Aq
+43
+The evaluated coaction of the generating functions (5.1), (5.2), is readily extracted from (5.54):
+δw(W±(u))=
+U −1
+q + q−1 W∓(u) ⊗
+�
+(q − q−1)2S±S∓ − q(K±1 − K∓1)
+�
+−
+U −1
+q + q−1 (w2 + w−2)W±(u) ⊗ 1
++
+(q − q−1)U −1
+k+k−(q + q−1)2
+�
+k+q± 1
+2 G+(u) ⊗ (w∓1S+K± 1
+2 ) + k−q∓ 1
+2 G−(u) ⊗ (w±1S−K± 1
+2 )
+�
+(5.56)
++ U −1 �
+1 ⊗ (k+q± 1
+2 w∓1S+K± 1
+2 ) + 1 ⊗ (k−q∓ 1
+2 w±1S−K± 1
+2 )
+�
++ W±(u) ⊗ K±1,
+δw(G±(u)) = k∓
+k±
+(q − q−1)2
+(q + q−1) U −1G∓(u) ⊗ S2
+∓ −
+U −1
+q + q−1 G±(u) ⊗ (w−2K±1 + w2K∓1) + G±(u) ⊗ 1
++ (q2 − q−2)
+�
+k∓q∓ 1
+2 (W+(u) − U −1W−(u)) ⊗ (w∓1S∓K
+1
+2 )
+(5.57)
++ k∓q± 1
+2 (W−(u) − U −1W+(u)) ⊗ (w±1S∓K− 1
+2 )
+�
++ k+k−(q + q−1)U −1
+(q − q−1)
+1 ⊗
+�k∓
+k±
+(q − q−1)2S2
+∓ − (w−2K±1 + w2K∓1)
+�
+,
+where we used the shorthand notation S+ ≡ E, S− ≡ F. We note that these expressions were first obtained in
+[BS09, Prop.2.2]24. Expanding (5.56), (5.57) as power series in U −1, it is straightforward to prove Proposition
+2.16.
+Now, using (5.56), (5.57) we can compute the evaluated coaction of the quantum determinant Γ(u)
+from (5.16):
+(5.58)
+δw(Γ(u)) =
+1
+(u2q + u−2q−1)(u2q3 + u−2q−3)Γ(u) ⊗ γ(u/w)γ(uw) ,
+where γ(u) is given in (4.12). Here we used the ordering relations of Aq in Lemma B.1 and the PBW basis of
+Uqsl2 given in Appendix A.
+The following result is a natural generalization of Lemma 5.15.
+Proposition 5.16. The evaluated coaction of K(j)(u) for j ∈ 1
+2N+ is given by
+(5.59)
+(δw ⊗ id)(K(j)(u)) =
+� 2j
+�
+p=1
+U −1
+q + q−1
+��
+u=uqj+ 1
+2 −p
+�
+×
+�
+L(j)(u/w)
+�
+[2]
+�
+K(j)(u)
+�
+[1]
+�
+L(j)(uw)
+�
+[2] .
+Proof. An induction argument is used. For j = 1
+2, the relation (5.59) coincides with (5.54). Now, suppose
+j ≥ 1. For convenience, we omit the notation [1], [2]. Expand the l.h.s. of (5.59) using the expression of the
+fused K-operator (5.21) and the evaluated coaction (5.59). It follows:
+δw(K(j)(u)) = F(j)
+⟨12⟩[δw(K
+( 1
+2 )
+1
+(uq−j+ 1
+2 ))]R
+( 1
+2 ,j− 1
+2 )
+12
+(u2q−j+1)[δw(K
+(j− 1
+2 )
+2
+(uq
+1
+2 ))]E(j)
+⟨12⟩
+=
+� 2j
+�
+p=1
+U −1
+q + q−1
+��
+u=uqj+ 1
+2 −p
+�
+F(j)
+⟨12⟩L
+( 1
+2 )
+1
+(uw−1q−j+ 1
+2 )K
+( 1
+2 )
+1
+(uq−j+ 1
+2 )L
+( 1
+2 )
+1
+(uwq−j+ 1
+2 )
+× R
+( 1
+2 ,j− 1
+2 )
+12
+(u2q−j+1)L
+(j− 1
+2 )
+2
+(uw−1q
+1
+2 )K
+(j− 1
+2 )
+2
+(uq
+1
+2 )L
+(j− 1
+2 )
+2
+(uwq
+1
+2 )E(j)
+⟨12⟩ ,
+(5.60)
+where we used δw instead of δw ⊗ id for convenience.
+Multiplying on the left and on the right (4.3) by
+(L(j)
+2 (v))−1 and using (4.45), we get
+L
+( 1
+2 )
+1
+(u)R( 1
+2 ,j− 1
+2 )(u/v)L
+(j− 1
+2 )
+2
+(v−1) = L
+(j− 1
+2 )
+2
+(v−1)R( 1
+2 ,j− 1
+2 )(u/v)L
+( 1
+2 )
+1
+(u) .
+(5.61)
+24Typos in [BS09] corrected (a prefactor was missing).
+
+44
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Using (5.61) for u → uwq−j+ 1
+2 , v → u−1wq− 1
+2 and the commutation [K(j1)
+1
+(u), L(j2)
+2
+(v)] = 0, (5.60) becomes:
+=
+� 2j
+�
+p=1
+U −1
+q + q−1
+��
+u=uqj+ 1
+2 −p
+�
+F(j)
+⟨12⟩L
+( 1
+2 )
+1
+(uw−1q−j+ 1
+2 )L
+(j− 1
+2 )
+2
+(uw−1q
+1
+2 )
+(5.62)
+× K
+( 1
+2 )
+1
+(uq−j+ 1
+2 )R
+( 1
+2 ,j− 1
+2 )
+12
+(u2q−j+1)K
+(j− 1
+2 )
+2
+(uq
+1
+2 )L
+( 1
+2 )
+1
+(uwq−j+ 1
+2 )L
+(j− 1
+2 )
+2
+(uwq
+1
+2 )E(j)
+⟨12⟩ .
+On the other hand, inserting (4.40), (4.63), in the r.h.s. of (5.59) one gets:
+δw(K(j)(u)) =
+� 2j
+�
+p=1
+U −1
+q + q−1
+��
+u=uqj+ 1
+2 −p
+�
+F(j)
+⟨12⟩L
+( 1
+2 )
+1
+(uw−1q−j+ 1
+2 )L
+(j− 1
+2 )
+2
+(uw−1q
+1
+2 )E(j)
+⟨12⟩F(j)
+⟨12⟩
+× K
+( 1
+2 )
+1
+(uq−j+ 1
+2 )R
+( 1
+2 ,j− 1
+2 )
+12
+(u2q−j+1)K
+(j− 1
+2 )
+2
+(uq
+1
+2 )E(j)
+⟨12⟩F(j)
+⟨12⟩L
+( 1
+2 )
+1
+(uwq−j+ 1
+2 )L
+(j− 1
+2 )
+2
+(uwq
+1
+2 )E(j)
+⟨12⟩ .
+Now, in order to remove the products E(j)
+⟨12⟩F(j)
+⟨12⟩, multiply first the expression above from the right by
+H(j)
+⟨12⟩[H(j)
+⟨12⟩]−1. Then, using the relations (3.45)-(3.47), (5.27) and
+(5.63)
+R( 1
+2 ,j− 1
+2 )(v/u)L
+(j− 1
+2 )
+2
+(v)L
+( 1
+2 )
+1
+(u) = L
+( 1
+2 )
+1
+(u)L
+(j− 1
+2 )
+2
+(v)R( 1
+2 ,j− 1
+2 )(v/u)
+which is obtained from the RLL equation, after simplifications the expression matches with (5.62).
+□
+5.6. Twisted intertwining relations for fused K-operators. In the next section, we will also need the
+twisted intertwining relations satisfied by the fused K-operators. For the fundamental K-operator (5.4), the
+twisted intertwining relations have been given in [BS09, Prop.4.2]. This result is now extended to higher
+values of j.
+Proposition 5.17. The following relation holds for any j ∈ 1
+2N+ and all b ∈ Aq:
+(5.64)
+K(j)(v)(id ⊗ πj)[δv−1(b)] = (id ⊗ πj)[δv(b)]K(j)(v) .
+Proof. Lemma 5.15 implies that
+(5.65)
+(id ⊗ πj ⊗ id)[(δv ⊗ id)(K( 1
+2 )(u))] =
+U −1
+q + q−1 R(j, 1
+2 )(u/v)K
+( 1
+2 )
+2
+(u)R(j, 1
+2 )(uv) .
+We then notice that K(j)(v) satisfies the equation
+(5.66)
+K(j)
+1 (v)R(j, 1
+2 )(uv)K
+( 1
+2 )
+2
+(u)R(j, 1
+2 )(u/v) = R(j, 1
+2 )(u/v)K
+( 1
+2 )
+2
+(u)R(j, 1
+2 )(uv)K(j)
+1 (v) .
+This version of reflection equation follows from the standard reflection equation (5.22) for j1 = j, j2 = 1
+2 and
+u1 = v, u2 = u. Indeed, we multiply (5.22) on the left and the right by R(j1,j2)(u2/u1) and using (4.46) we
+obtain (5.66).
+Now using (5.65), the equation (5.66) can be rewritten as
+(5.67)
+K(j)
+1 (v)(id ⊗ πj ⊗ id)[(δv−1 ⊗ id)(K( 1
+2 )(u))] = (id ⊗ πj ⊗ id)[(δv ⊗ id)(K( 1
+2 )(u))]K(j)
+1 (v) .
+This equation can be thought as an equation in Aq[[u−1]] ⊗ End(C2j+1) ⊗ End(C2). Denote the entries of the
+fundamental K-operator by K
+( 1
+2 )
+mn(u) ∈ Aq[[u−1]], m, n = 1, 2. Now, considering (5.67) as four equations in
+Aq[[u−1]] ⊗ End(C2j+1), i.e. taking the matrix elements of End(C2), it yields
+K(j)(v)
+�
+(id ⊗ πj)[δv−1(K
+( 1
+2 )
+mn(u))]
+�
+=
+�
+(id ⊗ πj)[δv(K
+( 1
+2 )
+mn(u))]
+�
+K(j)(v) .
+Inserting the entries according to (5.4), extracting the independent relations and using (5.1), (5.2), this
+implies (5.64).
+□
+
+FUSED K-OPERATORS FOR Aq
+45
+6. Fused K-operators and the universal K-matrix for Aq
+In this section, we assume there exists a universal K-matrix for Aq.
+We are interested in the precise
+relationship between the fused K-operators K(j)(u) constructed in the previous section using (5.21) and the
+spin-j K-operators defined in (4.47) through the evaluation of the universal K-matrix.
+By analogy with
+Lemma 4.8 relating spin-j L-operators (4.1) and fused L-operators (4.14), we propose:
+Conjecture 1. For j ∈ 1
+2N, we have
+(6.1)
+K(j)(u) = ν(j)(u)K(j)(u) ,
+where K(j)(u) is defined in (5.21) with
+ν(j)(u) =
+�2j−1
+�
+m=0
+ν(uqj− 1
+2 −m)
+� �2j−2
+�
+k=0
+2j−k−2
+�
+ℓ=0
+π
+1
+2 (µ(u2q2j−2−2k−ℓ))
+�
+.
+(6.2)
+Here π
+1
+2 (µ(u)) is given by (4.11) and ν(u) ≡ ν( 1
+2 )(u) is an invertible central element in Aq[[u−1]], defined by
+the functional relation
+(6.3)
+π
+1
+2 (µ(u2q))ν(u)ν(uq)Γ(u) = 1 ,
+where Γ(u) is given in (5.16), and has the evaluated coaction
+(6.4)
+δw(ν(u)) = (u2q + u−2q−1)ν(u) ⊗ µ(u/w)µ(uw) .
+Supporting evidences for Conjecture 1 are now presented. Afterwards, we derive from Conjecture 1 certain
+properties of the fused K-operators for j ≥ 0.
+6.1. Supporting evidences. For the clarity of the presentation, let us define:
+(6.5)
+˜K(j)(u) = ν(j)(u)K(j)(u)
+for j ∈ 1
+2N ,
+where we assume ν( 1
+2 )(u) is an invertible central element in Aq[[u−1]]. Importantly, it is not assumed that
+˜K(j)(u) is obtained from the evaluation of a universal K-matrix.
+We provide supporting evidences for Conjecture 1. We show that ˜K(j)(u) for all j satisfy the following
+systems of equations:
+˜K(j)(v)(id ⊗ πj)[δv−1(b)] = (id ⊗ πj)[δv(b)] ˜K(j)(v) ,
+(K1’)
+(δw ⊗ id)( ˜K(j)(u)) =
+�
+L(j)(u/w)
+�
+[2]
+�
+˜K(j)(u)
+�
+[1]
+�
+L(j)(uw)
+�
+[2] ,
+(K2’)
+˜K(j)(u) = F(j)
+⟨12⟩ ˜K
+( 1
+2 )
+1
+(uq−j+ 1
+2 )R( 1
+2 ,j− 1
+2 )(u2q−j+1) ˜K
+(j− 1
+2 )
+2
+(uq
+1
+2 )E(j)
+⟨12⟩ ,
+(K3’)
+where R( 1
+2 ,j)(u) is given in (4.18), if and only if (6.2) and (6.4) hold. Here, (K1’), (K2’), (K3’) are direct analogs
+of (4.77), (4.76), (4.63), respectively. We will also show that ˜K(j)(u) satisfies the reflection equation (4.58)
+where K(j)(u) is replaced by ˜K(j)(u).
+6.1.1. Fusion relation (K3’), twisted intertwining relations (K1’) and evaluated coaction (K2’). We assume
+that ν(u) is an invertible central element in Aq[[u−1]].
+Lemma 6.1. The K-operators ˜K(j)(u) for all j ∈ 1
+2N satisfy the fusion relation (K3’) if and only if ν(j)(u)
+takes the form (6.2), and so they are central.
+
+46
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Proof. Notice first that (5.21) contains R( 1
+2 ,j)(u) while (K3’) contains R( 1
+2 ,j)(u), and they are related by
+(6.6)
+R( 1
+2 ,j)(u) =
+�2j−1
+�
+k=0
+π
+1
+2 �
+µ(uq−j+ 1
+2 +k)
+�
+�
+R( 1
+2 ,j)(u)
+due to (4.36) and (4.28). Then, inserting (6.5) in (K3’) and using (5.21) and (6.6), the assumption that ν(u)
+is an invertible central element in Aq[[u−1]], and that K(j)(u) is invertible, see Remark 5.13, the resulting
+recursion relation on ν(j)(u) is equivalent to (6.2).
+□
+In what follows, we will assume that (K3’) holds, so in particular all ν(j)(u) are central. Then, using
+Proposition 5.17, the twisted intertwining relation for ˜K(j)(u) is immediate:
+Lemma 6.2. The K-operators ˜K(j)(u) for all j ∈ 1
+2N satisfy (K1’).
+We now show that the evaluated coaction (K2’) holds for ˜K(j)(u).
+Lemma 6.3. The K-operators ˜K(j)(u) for all j ∈ 1
+2N satisfy (K2’) if and only if δw(ν(u)) is given by (6.4).
+Proof. We first consider the case j = 1
+2. Inserting (6.5) in (K2’) for j = 1
+2, using (5.54), and the invertibility
+of L( 1
+2 )(u), K( 1
+2 )(u), the resulting equation is equivalent to (6.4). It remains to check that given the evaluated
+coaction (6.4), (K2’) holds for higher j. Now, consider j ≥ 1. On one hand, inserting (6.5) in (K2’), the l.h.s.
+reads
+δw
+�
+ν(j)(u)
+�
+δw(K(j)(u)) =
+�2j−1
+�
+m=0
+δw(ν(uqj− 1
+2 −m))
+� �2j−2
+�
+k=0
+2j−k−2
+�
+ℓ=0
+π
+1
+2 (µ(u2q2j−2−2k−ℓ))
+�
+(6.7)
+×
+� 2j
+�
+p=1
+U −1
+q + q−1
+��
+u=uqj+ 1
+2 −p
+� �
+L(j)(u/w)
+�
+[2]
+�
+K(j)(u)
+�
+[1]
+�
+L(j)(uw)
+�
+[2] ,
+where we used (5.59), (6.2) and the fact that δw in an algebra homomorphism. On the other hand, the r.h.s.
+of (K2’) is
+(6.8)
+ν(j)(u) ⊗ µ(j)(u/w)µ(j)(uw)
+�
+L(j)(u/w)
+�
+[2]
+�
+K(j)(u)
+�
+[1]
+�
+L(j)(uw)
+�
+[2] .
+Then, replacing µ(j)(u) and ν(j)(u) in (6.8) by (4.28), (6.2) respectively, and using (6.4) in (6.7), we find
+that (6.7) and (6.8) are equal.
+□
+Finally, assuming (K3’) holds so that ν(j)(u) are central, see Lemma 6.1, we show that the fused K-operators
+˜K(j)(u) satisfy the reflection equation.
+Lemma 6.4. The K-operators ˜K(j)(u) satisfy the reflection equation (4.58) where K(j)(u) is replaced by
+˜K(j)(u) for any j1, j2 ∈ 1
+2N+.
+Proof. By Theorem 5.7, the fused K-operators K(j)(u) satisfy the equation (5.22). Then, multiplying this
+equation by ν(j1)(u)ν(j2)(v) and using the fact that they are central, we obtain (4.58) where K(j)(u) is
+replaced by ˜K(j)(u).
+□
+6.1.2. Functional relation on ν(u). We have seen in Lemma 6.1 that the relation (K3’) fixes the normalization
+factor ν(j)(u) as (6.2). Here we show that the analog of the reduction relation (4.64) for ˜K(j)(u) leads to
+the functional relation (6.3). Recall the functional relation on µ(u) in (4.29) was obtained by comparing the
+fusion relation with the reduction relation satisfied by the spin-j L-operators, see Lemma 4.9. We proceed
+similarly for ˜K(j)(u).
+
+FUSED K-OPERATORS FOR Aq
+47
+Proposition 6.5. The K-operators ˜K(j)(u) satisfy (K3’) and
+(K3”)
+˜K(j− 1
+2 )(u) = ¯F
+(j− 1
+2 )
+⟨12⟩
+˜K
+( 1
+2 )
+1
+(uqj+1)R( 1
+2 ,j)(u2qj+ 3
+2 ) ˜K(j)
+2 (uq
+1
+2 ) ¯E
+(j− 1
+2 )
+⟨12⟩
+,
+for j = 1 if and only if ν(u) satisfies the functional relation (6.3).
+Proof. The equation (K3”) for j = 1 in terms of fused K-operators reads as
+K( 1
+2 )(u) = ν(uq)ν(uq2)π
+1
+2 (µ(u2q)µ(u2q2)µ(u2q3)) ¯F
+( 1
+2 )
+⟨12⟩K
+( 1
+2 )
+1
+(uq2)R( 1
+2 ,1)(u2q
+5
+2 )K(1)
+2 (uq
+1
+2 ) ¯E
+( 1
+2 )
+⟨12⟩ ,
+(6.9)
+where we used ˜K( 1
+2 )(u) = ν(u)K( 1
+2 )(u), the factorized form (6.2) for ν(1)(u) due to Lemma 6.1 and we used
+R( 1
+2 ,1)(u) = π
+1
+2 (µ(uq
+1
+2 )µ(uq− 1
+2 ))R( 1
+2 ,1)(u), recall (6.6). From the relation satisfied by µ(u) given in (4.29) and
+using πj(C) = q2j+1 + q−2j−1, one gets:
+π
+1
+2 (µ(u)µ(uq)) =
+1
+c(u)c(uq2) ,
+(6.10)
+where c(u) is given in (3.39). Then, the equation (6.9) becomes
+K( 1
+2 )(u) = ν(uq)ν(uq2)π
+1
+2 (µ(u2q3))
+c(u2q)c(u2q3)
+¯F
+( 1
+2 )
+⟨12⟩K
+( 1
+2 )
+1
+(uq2)R( 1
+2 ,1)(u2q
+5
+2 )K(1)
+2 (uq
+1
+2 ) ¯E
+( 1
+2 )
+⟨12⟩ .
+(6.11)
+The r.h.s. of (6.11) is now computed using the expressions for K( 1
+2 )(u), K(1)(u) given respectively in (5.4),
+(5.46), the fused R-matrix (5.44) and ¯E
+( 1
+2 )
+⟨12⟩, ¯F
+( 1
+2 )
+⟨12⟩ given by:
+¯E
+( 1
+2 )
+⟨12⟩ =
+
+
+
+
+
+
+
+
+0
+0
+1
+0
+0
+�
+[2]q
+−
+�
+[2]q
+0
+0
+−1
+0
+0
+
+
+
+
+
+
+
+
+,
+¯F
+( 1
+2 )
+⟨12⟩ =
+
+0
+1
+1+[2]q
+0
+−
+√
+[2]q
+1+[2]q
+0
+0
+0
+0
+√
+[2]q
+1+[2]q
+0
+−
+1
+1+[2]q 0
+
+ .
+In terms of the quantum determinant (5.16), one finds:
+(6.12)
+¯F
+( 1
+2 )
+⟨12⟩K
+( 1
+2 )
+1
+(uq2)R( 1
+2 ,1)(u2q
+5
+2 )K(1)
+2 (uq
+1
+2 ) ¯E
+( 1
+2 )
+⟨12⟩ = c(u2q)c(u2q3)Γ(uq)K( 1
+2 )(u) .
+This relation is obtained by applying the ordering relations for Aq given in Appendix B. Inserting this
+expression in (6.11) and multiplying by [K( 1
+2 )(u)]−1 — recall Remark 5.4 — the relation (6.3) follows.
+□
+Remark 6.6. As a consistency check, we observe that the evaluated coaction in (6.4) respects the functional
+relation (6.3) on ν(u). Indeed, using (5.58) and the functional relations (4.29) and (6.3), we obtain
+(6.13)
+δw(ν(u))δw(ν(uq))δw(Γ(u))π
+1
+2 (µ(u2q)) = 1 ⊗ 1 .
+6.1.3. Coaction. We propose a right coaction for the components of ˜K( 1
+2 )(u):
+(6.14)
+(δ ⊗ id)( ˜K( 1
+2 )(u)) =
+�
+[L−(u−1)]−1�
+[2]
+�
+˜K( 1
+2 )(u)
+�
+[1]
+�
+L+(u)
+�
+[2] ,
+where L±(u) are defined in (4.68), which is the direct analog of (4.75).
+First of all, we show that the
+coaction as defined by (6.14) respects the relations satisfied by the components of ˜K( 1
+2 )(u). Recall that, due
+to Lemma 6.4, these relations are
+(6.15)
+R( 1
+2 , 1
+2 )(u/v) ˜K
+( 1
+2 )
+1
+(u)R( 1
+2 , 1
+2 )(uv) ˜K
+( 1
+2 )
+2
+(v) = ˜K
+( 1
+2 )
+2
+(v)R( 1
+2 , 1
+2 )(uv) ˜K
+( 1
+2 )
+1
+(u)R( 1
+2 , 1
+2 )(u/v) .
+
+48
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Lemma 6.7. The K-operators
+(6.16)
+˜K(∓,±)(u) =
+�
+[L∓(u−1)]−1�
+[2]
+�
+˜K( 1
+2 )(u)
+�
+[1]
+�
+L±(u)
+�
+[2]
+satisfy the reflection equation (6.15) where ˜K( 1
+2 )(u) is replaced by ˜K(∓,±)(u).
+Proof. We first substitute the K-operators in (6.16) into the l.h.s. of (6.15). Then, we multiply from the
+left by L∓
+1 (u−1)L∓
+2 (v−1) and from the right by [L±
+2 (v)]−1[L±
+1 (u)]−1. One has for the l.h.s. of the resulting
+equation:
+L∓
+1 (u−1)L∓
+2 (v−1)R( 1
+2 , 1
+2 )(u/v)[L∓
+1 (u−1)]−1 ˜K
+( 1
+2 )
+1
+(u)L±
+1 (u)R( 1
+2 , 1
+2 )(uv)[L∓
+2 (v−1)]−1 ˜K
+( 1
+2 )
+2
+(v)[L±
+1 (u)]−1
+= R( 1
+2 , 1
+2 )(u/v)L∓
+2 (v−1) ˜K
+( 1
+2 )
+1
+(u) L±
+1 (u)R( 1
+2 , 1
+2 )(uv)[L∓
+2 (v−1)]−1[L±
+1 (u)]−1 ˜K
+( 1
+2 )
+2
+(v)
+= R( 1
+2 , 1
+2 )(u/v) ˜K
+( 1
+2 )
+1
+(u)R( 1
+2 , 1
+2 )(uv) ˜K
+( 1
+2 )
+2
+(v) ,
+(6.17)
+where we underlined the steps of calculation that correspond either to the commutation relations between L-
+and K-operators associated with different auxiliary spaces or to the use of variations of (4.69), (4.70). For
+instance, on the first line we use
+L∓
+2 (v−1)R( 1
+2 , 1
+2 )(u/v)[L∓
+1 (u−1)]−1 = [L∓
+1 (u−1)]−1R( 1
+2 , 1
+2 )(u/v)L∓
+2 (v−1)
+which is obtained by multiplying (4.69) from the left by [L±(u)]−1R( 1
+2 , 1
+2 )(v/u) and from the right by
+R( 1
+2 , 1
+2 )(v/u)[L±(u)]−1, using the relation R( 1
+2 , 1
+2 )(u)R( 1
+2 , 1
+2 )(u−1) ∝ I4 and substituting u → u−1, v → v−1.
+On the other hand, the r.h.s. reads:
+L∓
+1 (u−1) ˜K
+( 1
+2 )
+2
+(v)L±
+2 (v)R( 1
+2 , 1
+2 )(uv)[L∓
+1 (u−1)]−1 ˜K
+( 1
+2 )
+1
+(u)L±
+1 (u)R( 1
+2 , 1
+2 )(u/v)[L±
+2 (v)]−1[L±
+1 (u)]−1
+= ˜K
+( 1
+2 )
+2
+(v)L∓
+1 (u−1)L±
+2 (v)R( 1
+2 , 1
+2 )(uv)[L∓
+1 (u−1)]−1 ˜K
+( 1
+2 )
+1
+(u)[L±
+2 (v)]−1R( 1
+2 , 1
+2 )(u/v)
+= ˜K
+( 1
+2 )
+2
+(v)R( 1
+2 , 1
+2 )(uv) ˜K
+( 1
+2 )
+1
+(u)R( 1
+2 , 1
+2 )(u/v) .
+(6.18)
+Finally, comparing (6.17) with (6.18), one gets the reflection equation (6.15) that was proven in Lemma 6.4.
+□
+We finally show that δ defined in (6.14) is coassociative and counital, see (2.18). Firstly, we check the
+coassociativity:
+(δ ⊗ id ⊗ id) ◦ (δ ⊗ id)( ˜K( 1
+2 )(u)) = (δ ⊗ id ⊗ id)
+��
+[L−(u−1)]−1�
+[2]
+�
+˜K( 1
+2 )(u)
+�
+[1]
+�
+L+(u)
+�
+[2]
+�
+=
+�
+[L−(u−1)]−1�
+[3]
+�
+[L−(u−1)]−1�
+[2]
+�
+˜K( 1
+2 )(u)
+�
+[1]
+�
+L+(u)
+�
+[2]
+�
+L+(u)
+�
+[3]
+= (id ⊗ ∆ ⊗ id) ◦ (δ ⊗ id)( ˜K( 1
+2 )(u)) ,
+where the coproduct is given in (4.72) and we used (∆ ⊗ id)([L∓(u)]−1) = (L∓(u))−1
+[2] (L∓(u))−1
+[1] . Secondly,
+the condition with the counit is checked:
+(id ⊗ ǫ ⊗ id) ◦ (δ ⊗ id)( ˜K( 1
+2 )(u)) = (id ⊗ ǫ ⊗ id) ◦
+��
+[L∓(u−1)]−1�
+[2]
+�
+˜K( 1
+2 )(u)
+�
+[1]
+�
+L±(u)
+�
+[2]
+�
+= ˜K( 1
+2 )(u) ,
+where we used (4.74).
+To conclude this section, let us comment on the coaction of Aq, that is expected to be closely related with
+the coaction of Oq, see the discussion in Section 2.4.3. By assumption, ν(u) = �∞
+k=0 νku−k where νk ∈ Z(Aq).
+
+FUSED K-OPERATORS FOR Aq
+49
+From (6.3) it is easily checked that ν0 is a scalar satisfying
+(6.19)
+−
+ν2
+0ρq
+1
+2
+(q − q−1)2 = 1 .
+Provided Conjecture 1 holds, the coaction of Aq is given by (6.14) with (6.5), (5.4) and (C.26), (C.33), recall
+the discussion in Section 4.3 and Proposition 4.18. A comparison of the leading term u−1 of the matrix entries
+(1, 1) and (2, 2) of both sides of (6.14) gives:
+δ(W0) = 1 ⊗
+�
+k+q
+1
+2 E1K
+1
+2
+1 + k−q− 1
+2 F1K
+1
+2
+1
+�
++ W0 ⊗ K1 ,
+(6.20)
+δ(W1) = 1 ⊗
+�
+k+q− 1
+2 F0K
+1
+2
+0 + k−q
+1
+2 E0K
+1
+2
+0
+�
++ W1 ⊗ K0 .
+(6.21)
+Note that these equations indeed agree with the coaction of Oq given in (2.44), (2.45), where the embed-
+ding (2.46) has been used.
+To construct the coaction of W−k, Wk+1, Gk+1, ˜Gk+1 for general values of k,
+the properties of the generating function ν(u) need to be investigated further starting from the functional
+relation (6.3).
+6.2. Comments. Based on the supporting evidences given in the previous subsection, we believe Conjecture 1
+is correct. Some straightforward consequences are now pointed out. Firstly, some relations among the fused
+K-operators (5.21) are derived. They generalize the relation (6.12).
+Proposition 6.8. Assume Conjecture 1. Then, the following relations hold for any j ∈ 1
+2N+:
+(6.22)
+¯F
+(j− 1
+2 )
+⟨12⟩
+K
+( 1
+2 )
+1
+(uqj+1)R( 1
+2 ,j)(u2qj+ 3
+2 )K(j)
+2 (uq
+1
+2 ) ¯E
+(j− 1
+2 )
+⟨12⟩
+=
+�2j−2
+�
+k=0
+c(u2q2j−1−k)c(u2q2j+1−k)
+�
+Γ(uqj)K(j− 1
+2 )(u) ,
+(6.23)
+¯F
+(j− 1
+2 )
+⟨12⟩
+K(j)
+2 (uq− 1
+2 )R( 1
+2 ,j)(u2q−j− 3
+2 )K
+( 1
+2 )
+1
+(uq−j−1) ¯E
+(j− 1
+2 )
+⟨12⟩
+=
+�2j−2
+�
+k=0
+c(u2q−2j+2+k)c(u2q−2j+k)
+�
+Γ(uq−j−1)K(j− 1
+2 )(u) ,
+where ¯E(j− 1
+2 ) is fixed by Lemma 3.2 and ¯F(j− 1
+2 ) is given in (3.30) with (3.31).
+Proof. From Remark 3.4, the intertwining property with ∆op reads:
+(6.24)
+¯E(j− 1
+2 )(π
+j− 1
+2
+u
+)(x) = (π
+1
+2
+uq−j−1 ⊗ πj
+uq− 1
+2 )(∆op(x)) ¯E(j− 1
+2 ) ,
+∀x ∈ LUqsl2 .
+Now, express the l.h.s. of (6.22) in terms of K-operators and R-matrices. It reads25:
+¯F
+(j− 1
+2 )
+⟨12⟩
+K
+( 1
+2 )
+1
+(uqj+1)R( 1
+2 ,j)(u2qj+ 3
+2 )K(j)
+2 (uq
+1
+2 ) ¯E
+(j− 1
+2 )
+⟨12⟩
+ν(uqj+1)ν(j)(uq
+1
+2 )π
+1
+2 (µ(j)(u2qj+ 3
+2 ))
+=
+(id ⊗ ¯F(j− 1
+2 ))
+�
+(id ⊗ π
+1
+2
+u−1q−j−1 ⊗ πj
+u−1q− 1
+2 )(id ⊗ ∆op)(K)
+�
+(id ⊗ ¯E(j− 1
+2 ))
+ν(uqj+1)ν(j)(uq
+1
+2 )π
+1
+2 (µ(j)(u2qj+ 3
+2 ))
+and using (6.24), it becomes:
+=
+ν(j− 1
+2 )(u)
+ν(uqj+1)ν(j)(uq
+1
+2 )π
+1
+2 (µ(j)(u2qj+ 3
+2 ))
+K(j− 1
+2 )(u) .
+(6.25)
+25Here we assume u takes generic values such that π
+1
+2 (µ(j)(u2qj+ 3
+2 )) ̸= 0.
+
+50
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Then, simplifying the normalization factors and using (6.3), we get
+�
+ν(uqj)ν(uqj−1)π
+1
+2 (µ(j− 1
+2 )(u2qj)µ(j)(u2qj+ 3
+2 ))
+�−1
+K(j− 1
+2 )(u)
+=
+�2j−2
+�
+k=0
+π
+1
+2 (µ(u2q2j−k−1)µ(u2q2j−k))
+�−1
+Γ(uqj)K(j− 1
+2 )(u) .
+Finally, using (6.10), the equation (6.22) follows. The relation (6.23) is obtained similarly.
+□
+Secondly, we analyze the spin-0 K-operator K(0)(u) and the analog of the quantum determinant (5.15) for
+the spin- 1
+2 K-operator K( 1
+2 )(u).
+Proposition 6.9. Assume Conjecture 1, then K(0) = 1.
+Furthermore, the quantum determinant of the
+K-operator K( 1
+2 )(u) is equally 1:
+(6.26)
+tr12
+�
+P−
+12K
+( 1
+2 )
+1
+(u)R( 1
+2 , 1
+2 )(qu2)K
+( 1
+2 )
+2
+(uq)
+�
+= 1 .
+Proof. Specializing (K3”) to j = 1
+2 we get
+K(0)(u) = ¯F(0)
+⟨12⟩K
+( 1
+2 )
+1
+(uq
+3
+2 )R( 1
+2 , 1
+2 )(u2q2)K
+( 1
+2 )
+2
+(uq
+1
+2 ) ¯E(0)
+⟨12⟩
+(6.27)
+= ν(uq
+3
+2 )ν(uq
+1
+2 )π
+1
+2 (µ(u2q2)) ¯F(0)
+⟨12⟩K
+( 1
+2 )
+1
+(uq
+3
+2 )R( 1
+2 , 1
+2 )(u2q2)K
+( 1
+2 )
+2
+(uq
+1
+2 ) ¯E(0)
+⟨12⟩ ,
+where ¯E(0), ¯F(0) are given by
+(6.28)
+¯E(0) =
+
+
+
+
+0
+1
+−1
+0
+
+
+
+ ,
+¯F(0) =
+�0 1
+2 − 1
+2 0�
+.
+Then, noticing that for any two-by-two matrix A, one has the property:
+(6.29)
+¯F(0)
+⟨12⟩A ¯E(0)
+⟨12⟩ = tr12(P−
+12A) ,
+it follows from (5.15) that
+(6.30)
+Γ(u) = ¯F(0)
+⟨12⟩K
+( 1
+2 )
+1
+(uq)R( 1
+2 , 1
+2 )(u2q)K
+( 1
+2 )
+2
+(u) ¯E(0)
+⟨12⟩ .
+Therefore, the r.h.s. of (6.27) becomes
+K(0)(u) = ν(uq
+3
+2 )ν(uq
+1
+2 )π
+1
+2 (µ(u2q2))Γ(uq
+1
+2 )
+= 1 ,
+(6.31)
+where we used the functional relation (6.3).
+We finally note that the quantum determinant in the l.h.s.
+of (6.26) is K(0)(u), due to (6.27) and (6.29), and so it equals 1.
+□
+Proposition 6.10. Assume Conjecture 1, then �K(j)(u) from (5.40) is equal to the fused K-operator K(j)(u)
+defined in (5.21).
+Proof. Recall the K-operators K(j+ 1
+2 )(u) can be written either as (4.59) or as (4.63). Using (6.1) with (6.2)
+and the invertibility of ν(j)(u), we show by induction (recall that �K( 1
+2 )(u) = K( 1
+2 )(u)) that K(j)(u) equals
+�K(j)(u).
+□
+
+FUSED K-OPERATORS FOR Aq
+51
+7. K-operators and the PBW basis
+As mentioned in the introduction, in the literature the method (ii) has been successfully applied to the
+derivation of K-matrices with scalar entries solely using twisted intertwining relations in various represen-
+tations of certain comodule algebras B. In this section, solutions K(j)(u) of the twisted intertwining rela-
+tion (5.64) are investigated using a PBW basis of Aq for j = 1
+2, 1. We show that for a reasonably general
+ansatz, the solutions are uniquely determined (up to an overall factor), and match with the expressions of
+K(j)(u)’s for j = 1
+2 and j = 1 constructed in Section 5.
+Different types of PBW bases for Aq are known [T21a]. For instance, a PBW basis for Aq can be constructed
+in terms of the PBW generators
+{W−k}k∈N ,
+{Gℓ+1}ℓ∈N ,
+{˜Gm+1}m∈N ,
+{Wn+1}n∈N
+in the linear order < that satisfies26
+(7.1)
+W−k < Gℓ+1 < ˜Gm+1 < Wn+1 ,
+k, ℓ, m, n ∈ N .
+We recall that the corresponding generating functions W±(u) and G±(u) were introduced in (5.1) and (5.2),
+respectively.
+In the following analysis, we will encounter various combinations of the generating functions W±(u) and
+G±(u) that will need reordering. According to the chosen order (7.1), any words in {W±(ui), G±(ui)}i∈N can
+be written in terms of ordered expressions using Lemma B.1. In particular, from the relations (B.1)-(B.6),
+one extracts exchange relations that will be used later on for solving the twisted intertwining relations:
+(7.2)
+W+(u)W0 = W0W+(u) ,
+W1W−(u) = W−(u)W1 ,
+G+(u)W0 = q2W0G+(u) + ρq(W0 + W−(u) − UW+(u)) ,
+G−(u)W0 = q−2W0G−(u) − ρq−1(W0 + W−(u) − UW+(u)) ,
+W−(u)W0 = W0W−(u) +
+1
+q + q−1 (G+(u) − G−(u)) ,
+W1G+(u) = q2G+(u)W1 + ρq(W1 + W+(u) − UW−(u)) ,
+W1G−(u) = q−2G−(u)W1 − ρq−1(W1 + W+(u) − UW−(u)) ,
+W1W+(u) = W+(u)W1 +
+1
+q + q−1 (G+(u) − G−(u)) .
+From the construction of the fused K-operator K(j)(u) in (5.21), it is a (2j + 1) × (2j + 1) matrix with
+entries in Aq[[u−1]] that are linear combinations of monomials of the form (recall the ordering (7.1))
+(7.3)
+f(u)
+R
+�
+r=1
+W+(uqar)
+P
+�
+p=1
+G+(uqbp)
+M
+�
+m=1
+G−(uqcm)
+T
+�
+t=1
+W−(uqdt) ,
+R + P + M + T ≤ 2j
+and for some choice of ar, bp, cm, dt ∈ 1
+2Z, and f(u) is a Laurent polynomial in u of maximal degree 2(R+P +
+M+T ). Below, we successively show for the cases j = 1
+2 and j = 1 that the twisted intertwining relations (5.64)
+admit a unique solution of the above form, and it agrees with K(j)(u)’s constructed in Section 5.
+Before proceeding, we first notice that it is enough to solve the relations (5.64) for b = W0 and b = W1.
+This follows from the fact that Aq is the central extension of Oq – it is generated by W0, W1 and its center.
+We can take the quantum determinant Γ(u) as the generating function of the center of Aq, recall its definition
+in (5.16). This central element has a particularly simple expression (5.58) for the evaluated coaction δv –
+both the tensor components are central and δv(Γ(u)) = δv−1(Γ(u)). From this observation, it is clear that the
+26For a different choice of ordering, the proof of the PBW basis is given in [T21a].
+
+52
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+relations (5.64) hold for b = Γ(u) and whatever choice of K(j)(u), that is, they do not give any constraints on
+the form of K(j)(u).
+7.1. Spin- 1
+2 K-operator. The above discussed ansatz for K( 1
+2 )(u) takes the form:
+(7.4)
+K( 1
+2 )(u) = A
+( 1
+2 )
+1,1 (u)W+(u) + A
+( 1
+2 )
+1,2 (u)W−(u) + A
+( 1
+2 )
+1,3 (u)G+(u) + A
+( 1
+2 )
+1,4 (u)G−(u) + A
+( 1
+2 )
+0
+(u) ,
+where A
+( 1
+2 )
+1,i (u) and A
+( 1
+2 )
+0
+(u) are two-by-two matrices. We are now solving the twisted intertwining relation
+(5.64) for b = W0, W1 using the ordering relations (7.2). For b = W0 it is straightforward to show using (2.48)
+that the twisted intertwining relation (5.64) is equivalent to the system of four equations:
+(7.5)
+�
+W0, K
+( 1
+2 )
+11 (u)
+�
+= u−1q−1(k−K
+( 1
+2 )
+12 (u) − k+K
+( 1
+2 )
+21 (u)) ,
+�
+W0, K
+( 1
+2 )
+22 (u)
+�
+= −uq(k−K
+( 1
+2 )
+12 (u) − k+K
+( 1
+2 )
+21 (u)) ,
+�
+W0, K
+( 1
+2 )
+12 (u)
+�
+q = k+(uK
+( 1
+2 )
+11 (u) − u−1K
+( 1
+2 )
+22 (u)) ,
+�
+W0, K
+( 1
+2 )
+21 (u)
+�
+q−1 = −k−(uK
+( 1
+2 )
+11 (u) − u−1K
+( 1
+2 )
+22 (u)) .
+For b = W1, the analogous system of equations is obtained by substituting W0 �→ W1, u �→ u−1, q �→ q−1 into
+the above equations. Inserting the K-operator’s ansatz (7.4) into these two sets of intertwining relations and
+using (7.2), one extracts a set of linearly independent equations that determine uniquely (up to an overall
+factor) the entries of the matrices A
+( 1
+2 )
+1,i (u), A
+( 1
+2 )
+0
+(u). Adjusting the overall normalization for convenience, they
+read:
+A
+( 1
+2 )
+1,1 (u) =
+�
+uq
+0
+0 −u−1q−1
+�
+,
+A
+( 1
+2 )
+1,2 (u) =
+�
+−u−1q−1 0
+0
+uq
+�
+,
+A
+( 1
+2 )
+1,3 (u) =
+�0
+1
+k−(q+q−1)
+0
+0
+�
+,
+A
+( 1
+2 )
+1,4 (u) =
+�
+0
+0
+1
+k+(q+q−1) 0
+�
+,
+A
+( 1
+2 )
+0
+(u) =
+�
+0
+k+(q+q−1)
+q−q−1
+k−(q+q−1)
+q−q−1
+0
+�
+.
+Clearly, it is seen that the K-operator (7.4) matches with the fundamental K-operator (5.4).
+7.2. Spin-1 fused K-operator. For j = 1, an ansatz for the K-operator is built along the same line.
+Quadratic and linear combinations of the generating functions {W±(u), G±(u)} are now expected. Let α, β
+be half-integers. The ansatz reads:
+(7.6)
+K(1)(u) = A(1)
+2,1(u)W+(uqα)W+(uqβ) + A(1)
+2,2(u)W+(uqα)G+(uqβ) + A(1)
+2,3(u)W+(uqβ)G+(uqα)
++ A(1)
+2,4(u)W+(uqα)G−(uqβ) + A(1)
+2,5(u)W+(uqβ)G−(uqα) + A(1)
+2,6(u)W+(uqα)W−(uqβ)
++ A(1)
+2,7(u)W+(uqβ)W−(uqα) + A(1)
+2,8(u)G+(uqα)G+(uqβ) + A(1)
+2,9(u)G+(uqα)G−(uqβ)
++ A(1)
+2,10(u)G+(uqβ)G−(uqα) + A(1)
+2,11(u)G+(uqα)W−(uqβ) + A(1)
+2,12(u)G+(uqβ)W−(uqα)
++ A(1)
+2,13(u)G−(uqα)G−(uqβ) + A(1)
+2,14(u)G−(uqα)W−(uqβ) + A(1)
+2,15(u)G−(uqβ)W−(uqα)
++ A(1)
+2,16(u)W−(uqα)W−(uqβ) + A(1)
+1,1(u)W+(uqα) + A(1)
+1,2(u)W+(uqβ) + A(1)
+1,3(u)G+(uqα)
++ A(1)
+1,4(u)G+(uqβ) + A(1)
+1,5(u)G−(uqα) + A(1)
+1,6(u)G−(uqβ) + A(1)
+1,7(u)W−(uqα)
++ A(1)
+1,8(u)W−(uqβ) + A(1)
+0
+,
+
+FUSED K-OPERATORS FOR Aq
+53
+where A(1)
+i,j (u) and A(1)
+0
+are three-by-three matrices. Again, assume the twisted intertwining relations (5.64)
+hold for b = W0, W1, then for b = W0 we have:
+(7.7)
+�
+W0, K(1)
+11 (u)
+�
+=
+u−1q− 3
+2
+�
+[2]q
+�
+k−K(1)
+12 (u) − k+K(1)
+21 (u)
+�
+,
+�
+W0, K(1)
+12 (u)
+�
+q
+=
+�
+[2]qq− 1
+2
+�
+u−1q−1k−K(1)
+13 (u) + k+(uK(1)
+11 (u) − u−1K(1)
+22 (u))
+�
+,
+�
+W0, K(1)
+13 (u)
+�
+q2
+=
+k+
+�
+[2]q
+�
+uq− 1
+2 K(1)
+12 (u) − u−1q
+1
+2 K(1)
+23 (u)
+�
+,
+�
+W0, K(1)
+21 (u)
+�
+q−1
+=
+�
+[2]qq− 3
+2 u−1 �
+k−q(K(1)
+22 (u) − u2K(1)
+31 (u)) − k+K(1)
+31 (u)
+�
+,
+�
+W0, K(1)
+22 (u)
+�
+=
+�
+[2]qq− 1
+2
+�
+k+(uqK(1)
+21 (u) − u−1K(1)
+32 (u)) + k−(u−1K(1)
+23 (u) − uqK(1)
+12 (u))
+�
+,
+�
+W0, K(1)
+23 (u)
+�
+q
+=
+q
+1
+2
+�
+[2]q
+�
+k+(uK(1)
+22 (u) − u−1K(1)
+33 (u)) − k−uqK(1)
+13 (u)
+�
+,
+�
+W0, K(1)
+31 (u)
+�
+q−2
+=
+k−
+�
+[2]q
+�
+u−1q
+1
+2 K(1)
+32 (u) − uq− 1
+2 K(1)
+21 (u)
+�
+,
+�
+W0, K(1)
+32 (u)
+�
+q−1
+=
+q
+1
+2
+�
+[2]q
+�
+k+uqK(1)
+31 (u) − k−(uK(1)
+22 (u) − u−1K(1)
+33 (u))
+�
+,
+�
+W0, K(1)
+33 (u)
+�
+=
+uq
+3
+2
+�
+[2]q(k+K(1)
+32 (u) − k−K(1)
+23 (u)) .
+By substituting W0 �→ W1, u �→ u−1, q �→ q−1 into the above equations, we obtain the system of equations
+associated with b = W1. Inserting the K-operator’s ansatz (7.6) into these equations and using (7.2) we
+extract a system of linearly independent equations for the entries of K(1)(u). Similarly to the case j = 1
+2, one
+finds that it determines uniquely α = 1
+2, β = − 1
+2 and the matrices A(1)
+i,j (u) and A(1)
+0
+are given by
+A(1)
+1,1(u) = c(u2)(q + q−1)2
+c(q)c(u2q)
+
+
+
+
+0
+uq
+3
+2 k+
+(u2q2+u−2q−2)
+√
+q+q−1
+0
+−u−1q− 3
+2 k−
+�
+q + q−1
+0
+−uq
+3
+2 k+
+�
+q + q−1
+0
+u−1q− 3
+2 k−
+(u2q2+u−2q−2)
+√
+q+q−1
+0
+
+
+
+ ,
+A(1)
+1,3(u) =
+1
+c(q)
+
+
+
+q
+0
+k+
+k− c(u2)
+0 u−4q−2+c(q2)−1
+c(u2q)
+0
+0
+0
+q
+
+
+ ,
+A(1)
+2,8(u) =
+
+
+
+0 0 k+c(u2)
+k−ρ
+0 0
+0
+0 0
+0
+
+
+ ,
+A(1)
+2,2(u) = uq
+3
+2 c(u2)
+k−c(u2q)
+
+
+
+0 (u2q2+u−2q−2)
+√
+q+q−1
+0
+0
+0
+−
+�
+q + q−1
+0
+0
+0
+
+
+, A(1)
+2,3(u) =
+q
+1
+2 c(u2)
+uk−c(u2q)
+
+
+
+0 −
+�
+q + q−1
+0
+0
+0
+(u2q2+u−2q−2)
+√
+q+q−1
+0
+0
+0
+
+
+ ,
+A(1)
+2,11(u) =
+q
+1
+2 c(u2)
+uk−c(u2q)
+
+
+
+0 (u2q2+u−2q−2)
+√
+q+q−1
+0
+0
+0
+−
+�
+q + q−1
+0
+0
+0
+
+
+, A(1)
+2,12(u)= uq
+3
+2 c(u2)
+k−c(u2q)
+
+
+
+0 −
+�
+q + q−1
+0
+0
+0
+(u2q2+u−2q−2)
+√
+q+q−1
+0
+0
+0
+
+
+,
+A(1)
+0 (u) =
+ρ
+c(q)
+
+
+
+
+1
+0
+k+c(u2)
+k−c(q)
+0
+c(u2q)
+c(q)
+0
+k−c(u2)
+k+c(q)
+0
+1
+
+
+
+ ,
+A(1)
+2,16(u) = Diag(q−1 − u−4q−3, −u2q + u−2(q + q−1 − q3), u4q3 − q3 + q−1 − q−3) ,
+
+54
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+A(1)
+2,6(u) = Diag(−c(u2q)q, −u−4q−2 − q2 + 1 + q−2, −c(u2q)q) ,
+A(1)
+1,2(u) = A(1)t
+1,1 (u)|k±→k∓q∓2 , A(1)
+1,7(u) = A(1)t
+1,1 (u)|k±→k∓u∓2q∓3 , A(1)
+1,8(u) = A(1)
+1,1(u)|k±→k±u∓2q∓1 ,
+A(1)
+1,4(u) = A(1)
+1,3(u−1)|q→q−1 , A(1)
+1,5(u) = A(1)t
+1,3 (u−1)|k±→k∓,q→q−1 , A(1)
+1,6(u) = A(1)t
+1,3 (u)|k+→k−,
+A(1)
+2,1(u) = A(1)
+2,16(u−1)|q→−q−1 , A(1)
+2,13(u) = A(1)t
+2,8 (u)|k+→k− , A(1)
+2,4(u) = A(1)t
+2,3 (u)|k−→k+q2 ,
+A(1)
+2,5(u) = A(1)t
+2,2 (u)|k−→k+q2 , A(1)
+2,7(u) = −A(1)
+2,6(u−1)|q→q−1 , A(1)
+2,9(u) = −(ρc(u2q))−1A(1)
+2,6(u) ,
+A(1)
+2,10(u) = −(ρc(u2q))−1A(1)
+2,6(u−1)|q→q−1 , A(1)
+2,14(u) = A(1)
+2,12(u)|k±→k∓q2 , A(1)
+2,15(u) = A(1)
+2,11(u)|k±→k∓q2 .
+As expected, one checks that the K-operator in (7.6) together with the above solution matches with the
+expression for the fused K-operator (5.46) after applying the ordering relations given in Appendix B.
+To summarize, K-operators of the form (7.4), (7.6) for j = 1
+2, 1, respectively, are uniquely determined (up
+to an overall factor) by the twisted intertwining relations (5.64). Importantly, it is sufficient to consider the
+relations for b = W0, W1. Furthermore, the corresponding expressions match with the fused ones derived in
+Section 5. Based on these evidences, we conjecture that spin-j fused K-operator solutions of (5.64) are unique
+(up to an overall scalar factor), their matrix entries are linear combinations of monomials of the form (7.3)
+and match with the fused expression given by (5.21).
+8. Summary and Outlook
+To briefly summarize our main results, we provided a new set of K-operator solutions to the spectral
+parameter dependent reflection equation (1.8) in terms of generating functions of the centrally extended
+q-Onsager algebra Aq. The central formula of this work is the recursion (5.21) for the fused K-operators
+of arbitrary spin j ∈
+1
+2N as well as Theorem 5.7 on the reflection equation they satisfy.
+We also gave
+formulas for the fused R-matrices and the fused K-operators in (5.52), (5.53), whose expressions contain only
+the fundamental R-matrix and K-operator. These results were established within a general framework of
+universal K-matrices that we developed in Section 2.3, extending the previously known approaches (discussed
+in Introduction). In particular, the central formula (5.21) is based on the results in Proposition 4.14 and in
+Remark 4.17. We also provided in Section 5.4 a few explicit examples of the fused K-operators (for spins
+j = 1 and j = 3
+2) in terms of generating functions of Aq.
+As the existence of a universal K-matrix (for our choice of algebras H = LUqsl2 and B = Aq and the com-
+patible twists) is still an open fundamental question, we have investigated whether the central formula (5.21)
+satisfies the (evaluated version of) universal K-matrix axioms (K1)-(K3) which is resulted in Conjecture 1.
+One of the key problems here is to understand better the central element ν(u) ∈ Aq[[u−1]], in particular to
+derive its coaction δ(ν(u)) so that it reproduces the evaluated coaction in (6.4).
+It is also important to mention a few possible applications of our results in integrable models. In the
+literature on quantum integrable systems, K-operators and their images in the tensor product (or spin-
+chain) representations of the algebra Aq – known as Sklyanin’s operators – are the basic building elements
+for the construction of mutually commuting quantities, for instance the Hamiltonian of open spin chains
+with integrable boundary conditions [Sk88]. For the quotient of Aq known as the q-Onsager algebra, the
+fundamental K-operator (5.4) is the essential ingredient in the open XXZ spin- 1
+2 chain with generic boundary
+conditions [BK05b]. For the generic diagonal boundary conditions in this spin chain, the fundamental K-
+operator (5.4) generates another quotient27 of Aq known as the augmented q-Onsager algebra [BB12, Sec. 2].
+Furthermore, for the quotient of Aq known as the Askey-Wilson algebra, the K-operator (5.4) leads to an
+27More precisely, it is a degenerate specialisation at ρ → 0 of the q-Onsager quotient.
+
+FUSED K-OPERATORS FOR Aq
+55
+integrable model with three-spin interaction terms [BP19]. For other cases, the corresponding quotients of Aq
+appearing in the finite or half-infinite spin chains are identified in [BB16, Sec. 3].
+The K-operators (and their cousins) also play a crucial role in the spectrum analysis of the corresponding
+quantum spin chains, for instance, in the diagonalization of the Hamiltonian of the half-infinite XXZ spin-chain
+with diagonal boundary conditions [JKKKM94] (see also [BB12] in Onsager’s approach) and with triangular
+ones [BB12], or describe hidden non-abelian symmetries [BB16] of the model. The K-operators are also used to
+construct the Baxter’s Q-operator for diagonal boundary conditions [BT17, VW20] and triangular boundary
+conditions [Ts19, Ts20]. For all these models, the transfer matrix is the image in the spin-chain representation
+of a generating function t( 1
+2 )(u) built from the K-operator (5.4) and a dual solution of the reflection equation
+for a spin- 1
+2 auxiliary space. Importantly, t( 1
+2 )(u) reads as a linear combination of some fundamental generators
+{I2k+1|k ∈ N} of a maximal commutative subalgebra of Aq. Therefore, in this approach the diagonalization
+of the transfer matrix reduces to the diagonalization of the image of the commutative subalgebra.
+The fused K-operators of spin-j constructed in this paper open a route to the representation-independent
+analysis of related integrable models beyond the case of the fundamental auxiliary space, for instance, of
+the open XXZ spin-j chain with generic integrable boundary conditions. And the following problems can be
+addressed here:
+• Firstly, it is natural to ask for the relation between any local28 or non-local mutually commuting
+quantities of quantum spin chains and the generators {I2k+1|k ∈ N} of the commutative subalgebra
+in Aq. For instance, in the spin- 1
+2 case the differentiation of the transfer matrix leads to the expression
+of the Hamiltonian in terms of the operators I2k+1’s [BK05b, eq. (39)]. We thus also expect that the
+transfer matrix for the models based on the auxiliary space of arbitrary spin-j admits a unified
+formulation as the image of a generating function t(j)(u) in the commutative subalgebra of Aq[[u−1]].
+In a forthcoming paper [LBG23], the structure of t(j)(u) for higher spin-j auxiliary space representation
+will be studied in details. In particular, the so called T T -relations – a recursive definition of t(j)(u) –
+will be constructed at the algebraic level, independently of a representation chosen. Generalizing the
+spin- 1
+2 case, it will be shown that t(j)(u) is a power series in u−1 with coefficients being polynomials
+of degree 2j in the generators {I2k+1|k ∈ N}.
+• Secondly, given those transfer matrices generated from various images of Aq, the problem of cha-
+racterizing their spectral properties – leading to the eigenstates and eigenvalues of the Hamiltonian
+– is consequently reduced to the diagonalization of the images of {I2k+1|k ∈ N}. For the simplest
+example of the quotient of Aq known as the Askey-Wilson algebra, for irreducible finite-dimensional
+representations the problem is solved in [BP19], combining the theory of Leonard pairs and the so-
+called modified algebraic Bethe ansatz [BC13, B14, BP14]. A similar analysis for Aq remains to be
+done.
+• Let us mention that the construction of a universal Q-operator for Aq and corresponding T Q-relations
+may be also addressed. In that case, it would be desirable to construct the analogue of the fused K-
+operator for j → ∞ as suggested in [YNZ05], see also [VW20, Ts20].
+• Finally, it is very desirable to construct K-operators of arbitrary complex spins, i.e., associated to
+Uqsl2 Verma modules of complex weights, as it would give essentially the corresponding universal
+K-matrix. Such integrable quantum spin-chains with integrable boundary conditions based on the
+Verma modules were recently introduced [CGS22], and it was shown [CGJS22] a deep connection to
+the q-Onsager algebra via the common XXZ spin chain spectrum with the integrable non-diagonal
+boundary conditions.
+28In the case of integrable spin chains, an operator is said to be local whenever it is a product of a finite number of spin
+matrices S±, S3, or a sum of such products.
+
+56
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+Acknowledgments: We thank B. Vlaar for a discussion at the early stage of this work. P.B. and A.M.G.
+are supported by C.N.R.S. The work of A.M.G. was also partially supported by the ANR grant JCJC ANR-
+18-CE40-0001 and the RSF Grant No. 20-61-46005.
+Appendix A. Quantum algebras
+We recall the definitions of the quantum algebra Uqsl2 and the quantum affine algebra Uq �sl2.
+Definition A.1. The algebra Uqsl2 is a Hopf algebra generated by the elements E, F, K± 1
+2 satisfying:
+(A.1)
+K
+1
+2 E = qEK
+1
+2 ,
+K
+1
+2 F = q−1FK
+1
+2 ,
+[E, F] = K − K−1
+q − q−1
+,
+K
+1
+2 K− 1
+2 = K− 1
+2 K
+1
+2 = 1 .
+Let ∆: Uqsl2 → Uqsl2 ⊗ Uqsl2, ǫ: Uqsl2 → C and S : Uqsl2 → Uqsl2 be respectively the coproduct, the counit
+and the antipode. They are given by:
+∆(E) = E ⊗ K− 1
+2 + K
+1
+2 ⊗ E ,
+∆(F) = F ⊗ K− 1
+2 + K
+1
+2 ⊗ F ,
+∆(K± 1
+2 ) = K± 1
+2 ⊗ K± 1
+2 ,
+(A.2)
+ǫ(E) = ǫ(F) = 0 ,
+ǫ(K± 1
+2 ) = 1 ,
+(A.3)
+S(E) = −q−1E ,
+S(F) = −qF ,
+S(K± 1
+2 ) = K∓ 1
+2 .
+(A.4)
+We also recall that the Casimir central element of Uqsl2 is given by
+(A.5)
+C = (q − q−1)2FE + qK + q−1K−1
+= (q − q−1)2EF + q−1K + qK−1 .
+Note that the monomials {ErK± s
+2 F t | r, s, t �� N} provide a PBW basis for Uqsl2, see for instance [KS12,
+Chap. 3].
+Definition A.2. Define the extended Cartan matrix aij with aii = 2, aij = −2 for i ̸= j. The quantum affine
+algebra Uq �sl2 is a Hopf algebra generated by the elements Ei, Fi, K
+± 1
+2
+i
+, i ∈ {0, 1} satisfying:
+K
+1
+2
+i Ej = q
+aij
+2 EjK
+1
+2
+i ,
+K
+1
+2
+i Fj = q−
+aij
+2 FjK
+1
+2
+i ,
+[Ei, Fj] = δi,j
+Ki − K−1
+i
+q − q−1
+,
+(A.6)
+K
+1
+2
+i K
+− 1
+2
+i
+= K
+− 1
+2
+i
+K
+1
+2
+i = 1 ,
+K
+1
+2
+0 K
+1
+2
+1 = K
+1
+2
+1 K
+1
+2
+0 ,
+(A.7)
+with the q-Serre relations:
+(A.8)
+[Ei, [Ei, [Ei, Ej]q]q−1] = 0 ,
+[Fi, [Fi, [Fi, Fj]q]q−1] = 0 .
+Let ∆: Uq �sl2 → Uq �sl2 ⊗ Uq �sl2, ǫ: Uq �sl2 → C and S : Uq �sl2 → Uq �sl2 be respectively the coproduct, the counit
+and the antipode. They are given by:
+∆(Ei) = Ei ⊗ K
+1
+2
+i + K
+− 1
+2
+i
+⊗ Ei ,
+∆(Fi) = Fi ⊗ K
+1
+2
+i + K
+− 1
+2
+i
+⊗ Fi ,
+∆(K
+± 1
+2
+i
+) = K
+± 1
+2
+i
+⊗ K
+± 1
+2
+i
+,
+(A.9)
+ǫ(Ei) = ǫ(Fi) = 0 ,
+ǫ(K
+± 1
+2
+i
+) = 1 ,
+(A.10)
+S(Ei) = −qEi ,
+S(Fi) = −q−1Fi ,
+S(K
+± 1
+2
+i
+) = K
+∓ 1
+2
+i
+.
+(A.11)
+The element K
+1
+2
+0 K
+1
+2
+1 is central. The algebra Uq �sl2 has an automorphism ν defined by
+(A.12)
+ν(Ei) = EiK
+1
+2
+i ,
+ν(Fi) = K
+− 1
+2
+i
+Fi ,
+ν(K
+± 1
+2
+i
+) = K
+± 1
+2
+i
+.
+
+FUSED K-OPERATORS FOR Aq
+57
+Appendix B. Ordering relations for Aq
+Lemma B.1. The following relations hold in Aq[[u−1]]:
+G−(v)G+(u) = G+(u)G−(v) + ρ(q2 − q−2)
+�
+W+(u)W+(v) − W−(u)W−(v)
+(B.1)
++ 1 − UV
+U − V (W+(u)W−(v) − W+(v)W−(u))
+�
+,
+W−(v)W+(u) = W+(u)W−(v) +
+1
+V − U
+� (q − q−1)
+ρ(q + q−1)
+�
+G+(u)G−(v) − G+(v)G−(u)
+�
+(B.2)
++
+1
+q + q−1
+�
+G+(u) − G−(u) + G−(v) − G+(v)
+��
+,
+W−(v)G+(u) =
+q
+U − V
+��
+Uq−1 − V q
+�
+G+(u)W−(v) − (q − q−1)
+�
+W+(u)G+(v)
+(B.3)
+− W+(v)G+(u) − UG+(v)W−(u)
+�
++ ρ
+�
+UW−(u) − V W−(v) − W+(u) + W+(v)
+��
+,
+W−(v)G−(u) =
+1
+q(V − U)
+��
+V q−1 − Uq
+�
+G−(u)W−(v) − (q − q−1)
+�
+W+(u)G−(v) − W+(v)G−(u)
+(B.4)
+− UG−(v)W−(u)
+�
++ ρ
+�
+UW−(u) − V W−(v) − W+(u) + W+(v)
+��
+,
+G+(v)W+(u) =
+q
+U − V
+��
+Uq − V q−1�
+W+(u)G+(v) − (q − q−1)
+�
+G+(v)W−(u)
+(B.5)
++ ρ
+�
+UW+(u) − V W+(v) − W−(u) + W−(v)
+��
+,
+G−(v)W+(u) =
+1
+q(V − U)
+��
+V q − Uq−1�
+W+(u)G−(v) − (q − q1)
+�
+G−(v)W−(u) − G−(u)W−(v)
+(B.6)
++ V W+(v)G−(u)
+�
++ ρ
+�
+UW+(u) − V W+(v) − W−(u) + W−(v)
+��
+.
+Proof. The first two ordering relations (B.1), (B.2), are obtained directly from (5.7) and (5.8). The third
+relation (B.3) follows from (5.9) and (5.11) by replacing the element which is not in the chosen order. The
+relations (B.4)–(B.6) are derived similarly.
+□
+Appendix C. The universal R-matrix
+In this appendix, we compute evaluations of the universal R-matrix. Firstly, we recall the construction
+of the universal R-matrix of Khoroshkin-Tolstoy [KT92a] for the Hopf algebra H = LUqsl2 in terms of root
+vectors. Secondly, evaluations of the universal R-matrix are considered. In particular, we give expressions
+of the Ding-Frenkel L-operators L+(u) and [L−(u−1)]−1, as defined in (4.68). Finally, the spin- 1
+2 L-operator
+L( 1
+2 )(u) is computed by evaluating L+(u).
+C.1. Root vectors. Let us first recall the definition of the root vectors of LUqsl2. We adapt the construction
+in [BGKNR12] to our choice of coproduct, recall the relation (2.33). Let us set
+(C.1)
+eα = E1K
+− 1
+2
+1
+,
+eδ−α = E0K
+− 1
+2
+0
+,
+fα = K
+1
+2
+1 F1 ,
+fδ−α = K
+1
+2
+0 F0 .
+
+58
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+The other root vectors are defined by the recursion relations:
+(C.2)
+e′
+kδ = q−1[eα+(k−1)δ, eδ−α]q ,
+eα+kδ = [2]−1
+q [eα+(k−1)δ, e′
+δ] ,
+eδ−α+kδ = [2]−1
+q [e′
+δ, eδ−α+(k−1)δ] ,
+f ′
+kδ = q[fδ−α, fα+(k−1)δ]q−1 ,
+fα+kδ = [2]−1
+q [f ′
+δ, fα+(k−1)δ] ,
+fδ−α+kδ = [2]−1
+q [fδ−α+(k−1)δ, f ′
+δ] ,
+k ∈ N+ .
+The root vectors ekδ, fkδ are defined via the generating functions
+(q − q−1)
+∞
+�
+k=1
+ekδz−k = log
+�
+1 + (q − q−1)
+∞
+�
+k=1
+e′
+kδz−k
+�
+,
+−(q − q−1)
+∞
+�
+k=1
+fkδz−k = log
+�
+1 − (q − q−1)
+∞
+�
+k=1
+f ′
+kδz−k
+�
+.
+C.2. Khoroshkin-Tolstoy construction. Let {α+ kδ}∞
+k=0 ∪{kδ}∞
+k=0 ∪{δ − α+ kδ}∞
+k=0 be the positive root
+system of �
+sl2. We choose the root ordering as
+(C.3)
+α, α + δ, . . . , α + kδ, . . . , δ, 2δ, . . . , ℓδ, . . . , . . . , (δ − α) + mδ, . . . , (δ − α) + δ, δ − α ,
+for any k, ℓ, m ∈ N. Then, the universal R-matrix obtained by Khoroshkin and Tolstoy takes the following
+factorized form
+(C.4)
+R = R+R0R−q
+1
+2 h1⊗h1 ,
+where29
+R+ =
+−
+→
+∞
+�
+k=0
+expq−2
+�
+(q − q−1)eα+kδ ⊗ fα+kδ
+�
+,
+(C.5)
+R0 = exp
+�
+(q − q−1)
+∞
+�
+k=1
+k
+[2k]q
+ekδ ⊗ fkδ
+�
+,
+(C.6)
+R− =
+←
+−
+∞
+�
+k=0
+expq−2
+�
+(q − q−1)eδ−α+kδ ⊗ fδ−α+kδ
+�
+,
+(C.7)
+with qh1 ≡ K1 and the q-exponential is
+(C.8)
+expq(x) = 1 +
+∞
+�
+k=1
+xk
+(k)q! ,
+(k)q! = (1)q(2)q · · · (k)q, (k)q = qk − 1
+q − 1 .
+C.3. Evaluation of the universal R-matrix. In the previous subsection, the explicit form of the universal
+R-matrix was recalled. It is expressed as a product of q-exponentials with root vectors in the arguments.
+Now, we evaluate the second tensor product component of the universal R-matrix by taking its image under
+the fundamental evaluation representation. As it is well known, evaluations of the universal R-matrix lead to
+L-operators and R-matrices.
+29Here we use the notation
+←
+−
+n
+�
+k=0
+a(k) = a(n)a(n − 1) . . . a(0) and
+−
+→
+n
+�
+k=0
+a(k) = a(0)a(1) . . . a(n), for any function a(n).
+
+FUSED K-OPERATORS FOR Aq
+59
+C.3.1. Evaluation of the root vectors. The action of the evaluation map defined in (2.40) on the first root
+vectors gives
+evu(eδ−α) = u−1FK
+1
+2 ,
+evu(eα) = u−1EK− 1
+2 ,
+evu(K
+1
+2
+0 ) = K− 1
+2 ,
+(C.9)
+evu(fδ−α) = uq−1EK− 1
+2 ,
+evu(fα) = uq−1FK
+1
+2 ,
+evu(K
+1
+2
+1 ) = K
+1
+2 .
+(C.10)
+The image of the other root vectors of LUqsl2 in (C.2) under the evaluation map are obtained by induction
+similarly to [BGKNR12, Sect. 4.4]. They are given for k ∈ N by:
+(C.11)
+evu(eα+kδ) = (−1)ku−2k−1q−kEK−k− 1
+2 ,
+evu(eδ−α+kδ) = (−1)ku−2k−1qkFK−k+ 1
+2 ,
+evu(fα+kδ) = (−1)ku2k+1q−k−1FKk+ 1
+2 ,
+evu(fδ−α+kδ) = (−1)ku2k+1qk−1EKk− 1
+2 ,
+and for k ∈ N+
+(C.12)
+evu(e′
+kδ) = (−1)k−1u−2k[E, F]qkK−k+1 ,
+evu(ekδ) = (−1)k−1u−2k
+(q − q−1)k (Ck − (qk + q−k)K−k) ,
+evu(f ′
+kδ) = (−1)k−1u2k[E, F]q−kKk−1 ,
+evu(fkδ) = −(−1)k−1u2k
+(q − q−1)k (Ck − (qk + q−k)Kk) ,
+where the elements Ck are defined by the generating function
+(C.13)
+∞
+�
+k=1
+(−1)k−1Ck
+z−k
+k
+= log(1 + Cz−1 + z−2) ,
+z ∈ C ,
+and where C is the central element of Uqsl2 given in (A.5). For instance by expanding (C.13) we get the first
+elements of Ck
+(C.14)
+C1 = C ,
+C2 = C2 − 2 ,
+C3 = C3 − 3C ,
+C4 = C4 − 4C2 + 2 .
+Recall Eab is the matrix with zero everywhere except 1 in the entry (a, b). The matrix multiplication obeys
+(C.15)
+EabEcd = δb,cEad .
+In this notation, the spin- 1
+2 finite-dimensional representation of Uqsl2 reads
+(C.16)
+π
+1
+2 (Km) = qmE11 + q−mE22 ,
+π
+1
+2 (E) = E12 ,
+π
+1
+2 (F) = E21 ,
+and the central elements become
+(C.17)
+π
+1
+2 (C) = (q2 + q−2)I2 ,
+π
+1
+2 (Ck) = (q2k + q−2k)I2 .
+In order to obtain Ding-Frenkel L-operators and R-matrices from the universal R-matrix, one also needs
+the image of the root vectors under the representation map π
+1
+2u : LUqsl2 → End(C2), which is given by:
+π
+1
+2u (eα+kδ) = (−1)ku−2k−1q
+1
+2 E12 ,
+π
+1
+2u (eδ−α+kδ) = (−1)ku−2k−1q
+1
+2 E21 ,
+π
+1
+2u (fα+kδ) = (−1)ku2k+1q− 1
+2 E21 ,
+π
+1
+2u (fδ−α+kδ) = (−1)ku2k+1q− 1
+2 E12 ,
+
+60
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+π
+1
+2u (e′
+kδ) = (−1)k−1u−2k(qE11 − q−1E22) ,
+(C.18)
+π
+1
+2u (ekδ) = (−1)k−1u−2k [k]q
+k (qkE11 − q−kE22) ,
+π
+1
+2u (f ′
+kδ) = (−1)k−1u2k(q−1E11 − qE22) ,
+π
+1
+2u (fkδ) = (−1)k−1u2k [k]q
+k (q−kE11 − qkE22) .
+Recall that L±(u) are defined in (4.68). We now compute explicitly L+(u) and [L−(u−1)]−1.
+C.3.2. L+(u). Recall the factorized form of the universal R-matrix (C.4). We now compute the image of R±,
+R0, q
+1
+2 h1⊗h1 under the action of (id ⊗ π
+1
+2
+u−1). First, from (C.5) and with (C.18) we get:
+(id ⊗ π
+1
+2
+u−1)(R+) =
+−
+→
+∞
+�
+k=0
+expq−2
+�
+(−u−2)ku−1(q − q−1)q− 1
+2 eα+kδ ⊗ E21
+�
+= 1 ⊗ (E11 + E22) + e+(u) ⊗ E21 ,
+(C.19)
+where
+(C.20)
+e+(u) = (q − q−1)q− 1
+2 u−1
+� ∞
+�
+k=0
+(−u−2)keα+kδ
+�
+.
+Similarly, from (C.7) and with (C.18) we have:
+(id ⊗ π
+1
+2
+u−1)(R−) =
+←
+−
+∞
+�
+k=0
+expq−2
+�
+(−u−2)ku−1(q − q−1)q− 1
+2 eδ−α+kδ ⊗ E12
+�
+= 1 ⊗ (E11 + E22) + f +(u) ⊗ E12 ,
+(C.21)
+where
+(C.22)
+f +(u) = (q − q−1)q− 1
+2 u−1
+� ∞
+�
+k=0
+(−u−2)keδ−α+kδ
+�
+.
+A straightforward calculation from (C.6) and using (C.18) yields
+(id ⊗ π
+1
+2
+u−1)(R0) = k+(u) ⊗ E11 + ˜k+(u) ⊗ E22 ,
+(C.23)
+where
+(C.24)
+k+(u) = exp
+�
+−(q − q−1)
+∞
+�
+k=1
+(−u−2q−1)k
+qk + q−k
+ekδ
+�
+,
+˜k+(u) = exp
+�
+(q − q−1)
+∞
+�
+k=1
+(−u−2q)k
+qk + q−k ekδ
+�
+.
+Then, using π
+1
+2u (h1) = E11 − E22, we get
+(id ⊗ π
+1
+2
+u−1)(q
+1
+2 h1⊗h1) = K
+1
+2
+1 ⊗ E11 + K
+− 1
+2
+1
+⊗ E22 .
+(C.25)
+Finally, combining (C.19)-(C.25), we get
+(C.26)
+L+(u) =
+�
+k+(u)K
+1
+2
+1
+k+(u)f +(u)K
+��� 1
+2
+1
+e+(u)k+(u)K
+1
+2
+1 ˜k+(u)K
+− 1
+2
+1
++ e+(u)k+(u)f +(u)K
+− 1
+2
+1
+�
+.
+Note that from the definition of e+(u), f +(u), k+(u), ˜k+(u) in (C.20), (C.22), (C.24) it is easy to see that
+L+(u) is a formal power series in u−1, i.e. L+(u) is in LUqsl2[[u−1]] ⊗ End(C2).
+
+FUSED K-OPERATORS FOR Aq
+61
+C.3.3. [L−(u−1)]−1. Consider p ◦ R± = R±
+21, p ◦ R0 = R0
+21. We now compute their image under the action of
+(id ⊗ π
+1
+2
+u−1) to obtain the expression of [L−(u−1)]−1 defined in (4.68). First, it follows from (C.5) and (C.18)
+(id ⊗ π
+1
+2u )(R+
+21) =
+−
+→
+∞
+�
+k=0
+expq−2
+�
+(−u−2)ku−1(q − q−1)q
+1
+2 fα+kδ ⊗ E12
+�
+= 1 ⊗ (E11 + E22) + f −(u) ⊗ E12 ,
+(C.27)
+where
+(C.28)
+f −(u) = (q − q−1)q
+1
+2 u−1
+� ∞
+�
+k=0
+(−u−2)kfα+kδ
+�
+.
+Similarly, from (C.7) and using (C.18)
+(id ⊗ π
+1
+2u )(R−
+21) =
+←
+−
+∞
+�
+k=0
+expq−2
+�
+(−u−2)ku−1(q − q−1)q
+1
+2 fδ−α+kδ ⊗ E21
+�
+= 1 ⊗ (E11 + E22) + e−(u) ⊗ E21 ,
+(C.29)
+where
+(C.30)
+e−(u) = (q − q−1)q
+1
+2 u−1
+� ∞
+�
+k=0
+(−u−2)kfδ−α+kδ
+�
+.
+Then from (C.7) and using (C.18) we get
+(id ⊗ π
+1
+2u )(R0
+21) = k−(u) ⊗ E11 + ˜k−(u) ⊗ E22 ,
+(C.31)
+where
+(C.32)
+k−(u) = exp
+�
+−(q − q−1)
+∞
+�
+k=1
+(−u−2q)k
+qk + q−k fkδ
+�
+,
+˜k−(u) = exp
+�
+(q − q−1)
+∞
+�
+k=1
+(−u−2q−1)k
+qk + q−k
+fkδ
+�
+,
+Finally, combining (C.27)-(C.31) and (C.25), we get
+(C.33)
+[L−(u−1)]−1 =
+�
+k−(u)K
+1
+2
+1 + f −(u)˜k−(u)e−(u)K
+1
+2
+1 f −(u)˜k−(u)K
+− 1
+2
+1
+˜k−(u)e−(u)K
+1
+2
+1
+˜k−(u)K
+− 1
+2
+1
+�
+.
+Note that from the definition of e−(u), f −(u), k−(u), ˜k−(u) in (C.30), (C.28), (C.32) it is easy to see that
+[L−(u−1)]−1 is a formal power series in u−1, i.e. [L−(u−1)]−1 is in LUqsl2[[u−1]] ⊗ End(C2).
+C.4. The spin- 1
+2 L-operator L( 1
+2 )(u). We now compute the spin- 1
+2 L-operator L( 1
+2 )(u) defined in (4.1). It
+is obtained by taking the image of L+(u) under the evaluation with (evv ⊗ id).
+Recall the expression of L+(u) in (C.26). The spin- 1
+2 L-operator is then obtained by evaluating e+(u),
+f +(u), k+(u), ˜k+(u) defined in (C.20), (C.22), (C.24). Let us first introduce the function [BGKNR12]
+(C.34)
+Λ(u) =
+∞
+�
+k=1
+Ck
+(qk + q−k)
+uk
+k ,
+where the central elements Ck are defined by (C.13). Note that it satisfies
+(C.35)
+Λ(uq) + Λ(uq−1) = − log(1 − Cu + u2) .
+
+62
+GUILLAUME LEMARTHE, PASCAL BASEILHAC, AND AZAT M. GAINUTDINOV
+From the evaluated root vectors (C.9) and (C.12), we get:
+(C.36)
+evv(e+(u)) = (q − q−1)q− 1
+2 u−1v−1EK− 1
+2 �
+1 − u−2v−2q−1K−1�−1 ,
+evv(f +(u)) = (q − q−1)q− 1
+2 u−1v−1FK
+1
+2 �
+1 − u−2v−2qK−1�−1 ,
+evv(k+(u)) = eΛ(u−2v−2q−1) �
+1 − u−2v−2q−1K−1�
+,
+evv(˜k+(u)) = e−Λ(u−2v−2q) �
+1 − u−2v−2qK−1�−1 .
+For instance, let us now compute the evaluation of the matrix entry (2, 2) of L+(u) in (C.26), it reads:
+evv((L+(u))22) = evv
+�
+˜k+(u)K
+− 1
+2
+1
++ e+(u)k+(u)f +(u)K
+− 1
+2
+1
+�
+=
+�
+e−Λ(u−2v−2q) + eΛ(u−2v−2q−1)u−2v−2(q − q−1)2EF
+� �
+1 − u−2v−2qK−1�−1 K− 1
+2
+= eΛ(u−2v−2q−1)�
+1 + u−4v−4 − u−2v−2(C − (q − q−1)2EF)
+� �
+1 − u−2v−2qK−1�−1 K− 1
+2 ,
+where we used (C.35) on the third line, and where C is defined in (A.5). Then, we obtain
+evv((L+(u))22) = eΛ(u−2v−2q−1) �
+K− 1
+2 − u−2v−2q−1K
+1
+2
+�
+.
+Computing the other entries, we have
+(C.37)
+L( 1
+2 )(uv) = (evv ⊗ id)(L+(u)) = µ(uv)L( 1
+2 )(uv) ,
+where µ(u) and L( 1
+2 )(u) are given respectively in (4.6) and (4.5).
+Similarly, evaluating the Ding-Frenkel L-operator in (C.33), we have
+(C.38)
+(evv ⊗ id)([L−(u−1)]−1) = µ(u/v)L( 1
+2 )(u/v) = L( 1
+2 )(u/v) .
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+Email address: pascal.baseilhac@idpoisson.fr, guillaume.lemarthe@idpoisson.fr, azat.gainutdinov@idpoisson.fr
+