# Fused K-operators and the *q*-Onsager algebra

Guillaume Lemarthe, Pascal Baseilhac, Azat M. Gainutdinov

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Guillaume Lemarthe, Pascal Baseilhac, Azat M. Gainutdinov. Fused K-operators and the *q*-Onsager algebra. 2025. hal-03946559

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# 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 introduced 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 $B \otimes H$ satisfying certain axioms, and we consider the case $H = \mathcal{L}U_q sl_2$, the quantum loop algebra for $sl_2$, and $B = A_q$, the alternating central extension of the $q$-Onsager algebra. Considering tensor products of evaluation representations of $\mathcal{L}U_q sl_2$ 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 $j \in \frac{1}{2}\mathbb{N}$ they satisfy the spectral-parameter dependent reflection equation. We provide their explicit expression in terms of elements of the algebra $A_q$ 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 $A_q$ is conjectured based on supporting evidences. Independently, we study K-operator solutions of the twisted intertwining relations associated with the comodule algebra $A_q$, and expand them in the Poincaré-Birkhoff-Witt basis of $A_q$. 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; $q$-Onsager algebra; Fusion for K-operators

## CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>1.1. Background</td><td>2</td></tr><tr><td>1.2. Universal K-matrix and comodule algebras</td><td>5</td></tr><tr><td>1.3. Goal and main results</td><td>7</td></tr><tr><td>1.4. K-operators for A<sub>q</sub> and universal TT-relations</td><td>8</td></tr><tr><td>2. Universal R and K matrices</td><td>10</td></tr><tr><td>2.1. Universal R-matrix</td><td>10</td></tr><tr><td>2.2. Twist pairs</td><td>11</td></tr><tr><td>2.3. Universal K-matrix</td><td>12</td></tr><tr><td>2.4. Example of (H, B)</td><td>15</td></tr><tr><td>3. Tensor product representations of LU<sub>q</sub>sl<sub>2</sub> and sub-representations</td><td>18</td></tr><tr><td>3.1. Analysis of the tensor product representation of LU<sub>q</sub>sl<sub>2</sub></td><td>19</td></tr><tr><td>3.2. Intertwining map E<sup>(j+1/2)</sup> and its pseudo-inverse F<sup>(j+1/2)</sup></td><td>22</td></tr><tr><td>3.3. The maps E&#773;<sup>(j-1/2)</sup> and F&#773;<sup>(j-1/2)</sup></td><td>24</td></tr><tr><td>3.4. Additional properties</td><td>26</td></tr><tr><td>3.5. Formal evaluation representations</td><td>28</td></tr><tr><td>4. Spin-j L- and K-operators</td><td>32</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>4.1. Spin-j L-operators</td>
      <td>32</td>
    </tr>
    <tr>
      <td>4.2. Spin-j K-operators</td>
      <td>42</td>
    </tr>
    <tr>
      <td>4.3. Comodule algebra structure using K-operators</td>
      <td>45</td>
    </tr>
    <tr>
      <td>5. Fused K-operators for A<sub>q</sub></td>
      <td>47</td>
    </tr>
    <tr>
      <td>5.1. The fundamental K-operator for A<sub>q</sub></td>
      <td>47</td>
    </tr>
    <tr>
      <td>5.2. Fused K-operators for A<sub>q</sub></td>
      <td>50</td>
    </tr>
    <tr>
      <td>5.3. Unitarity and invertibility properties</td>
      <td>51</td>
    </tr>
    <tr>
      <td>5.4. Examples of fused K-operators</td>
      <td>52</td>
    </tr>
    <tr>
      <td>5.5. Evaluated coaction of fused K-operators</td>
      <td>55</td>
    </tr>
    <tr>
      <td>5.6. Twisted intertwining relations for fused K-operators</td>
      <td>58</td>
    </tr>
    <tr>
      <td>6. Fused K-operators and the universal K-matrix for A<sub>q</sub></td>
      <td>59</td>
    </tr>
    <tr>
      <td>6.1. Supporting evidences</td>
      <td>61</td>
    </tr>
    <tr>
      <td>6.2. Coaction</td>
      <td>64</td>
    </tr>
    <tr>
      <td>6.3. Comments</td>
      <td>66</td>
    </tr>
    <tr>
      <td>7. K-operators and the PBW basis</td>
      <td>68</td>
    </tr>
    <tr>
      <td>7.1. Spin-1/2 K-operator</td>
      <td>69</td>
    </tr>
    <tr>
      <td>7.2. Spin-1 fused K-operator</td>
      <td>69</td>
    </tr>
    <tr>
      <td>8. Summary and Outlook</td>
      <td>72</td>
    </tr>
    <tr>
      <td>Appendix A. Quantum algebras</td>
      <td>74</td>
    </tr>
    <tr>
      <td>Appendix B. The universal R-matrix</td>
      <td>75</td>
    </tr>
    <tr>
      <td>B.1. Root vectors</td>
      <td>75</td>
    </tr>
    <tr>
      <td>B.2. Khoroshkin-Tolstoy construction</td>
      <td>75</td>
    </tr>
    <tr>
      <td>B.3. Evaluation of the universal R-matrix.</td>
      <td>76</td>
    </tr>
    <tr>
      <td>B.4. The spin-1/2 L-operator L<sup>(1/2)</sup>(u)</td>
      <td>79</td>
    </tr>
    <tr>
      <td>Appendix C. Ordering relations for A<sub>q</sub></td>
      <td>80</td>
    </tr>
    <tr>
      <td>Appendix D. Proof of Theorem 5.7</td>
      <td>81</td>
    </tr>
    <tr>
      <td>References</td>
      <td>84</td>
    </tr>
  </tbody>
</table>

# 1. INTRODUCTION

1.1. **Background.** In the context of quantum integrable systems, the R- and K-matrices are the basic ingredients 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 equation, also known under the name boundary Yang-Baxter equation. For a triple of finite-dimensional vector spaces $V^{(j_k)}$, for $k = 1, 2, 3$, the corresponding Yang-Baxter equation in $End(V^{(j_1)} \otimes V^{(j_2)} \otimes V^{(j_3)})$ takes the form [Ya67, Ba72]:

$$ (1.1) \quad R_{12}^{(j_1,j_2)}(u_1/u_2) R_{13}^{(j_1,j_3)}(u_1/u_3) R_{23}^{(j_2,j_3)}(u_2/u_3) = R_{23}^{(j_2,j_3)}(u_2/u_3) R_{13}^{(j_1,j_3)}(u_1/u_3) R_{12}^{(j_1,j_2)}(u_1/u_2), $$
where $u_k$ are the spectral parameters, and $R_{nm}^{(j_n, j_m)}(u)$ are the R-matrices on corresponding products of spaces: $R_{12} = R \otimes \mathbb{I}$, $R_{23} = \mathbb{I} \otimes R$, $R_{13} = (\mathcal{P} \otimes \mathbb{I}) R_{23} (\mathcal{P} \otimes \mathbb{I})$, with $\mathcal{P}(a \otimes b) = b \otimes a$. Given an R-matrix $R^{(j_1, j_2)}(u)$ satisfying (1.1), the reflection equation in End($V^{(j_1)} \otimes V^{(j_2)}$) is given by [C84, Sk88]

$$ (1.2) \qquad R_{12}^{(j_1, j_2)}(u_1/u_2) K_1^{(j_1)}(u_1) R_{21}^{(j_2, j_1)}(u_1 u_2) K_2^{(j_2)}(u_2) = K_2^{(j_2)}(u_2) R_{12}^{(j_1, j_2)}(u_1 u_2) K_1^{(j_1)}(u_1) R_{21}^{(j_2, j_1)}(u_1/u_2), $$

where

$$ (1.3) \qquad R_{21}^{(j_2, j_1)}(u) = \mathcal{P}^{(j_2, j_1)} R^{(j_2, j_1)}(u) \mathcal{P}^{(j_1, j_2)}. $$

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 $4 \times 4$ R-matrices can be interpreted as intertwining operators for underlying action of the quantum affine algebra $U_q \hat{sl}_2$ on tensor product of so-called evaluation representations of spin-$\frac{1}{2}$, or briefly these are spin-$\frac{1}{2}$ solutions (and similarly for the 3-dimensional or spin-1 solutions). The spin-$\frac{1}{2}$ expressions of the K-matrix are interpreted analogously as ‘twisted’ intertwiners (with respect to the spectral parameter ‘reflection’ $u \to u^{-1}$) for the action of a coideal subalgebra of the quantum affine algebra.

For higher values of spins $j_k$, 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) **Fusion method:** 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-$\frac{1}{2}$, the fused R- and K-matrices of spin-$j$ are obtained inductively by tensoring (or “fusing”) the fundamental representations of $U_q \hat{sl}_2$ and projecting onto the highest spin sub-representation.

(ii) **Intertwining method:** 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 $V^{(j_1)} \otimes V^{(j_2)}$, 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 in 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 $O_q$ [T99, B04].

For instance, for the R-matrix associated with $U_q \hat{sl}_2$ that enjoys P-symmetry [RSV14]

$$ (1.4) \qquad R_{21}^{(j_2, j_1)}(u) = R^{(j_1, j_2)}(u), $$

the corresponding fused solutions using method (i) have been constructed in [KS82], while Intertwining method (ii) was implemented in [KR83]. For the K-matrix $K^{(j)}(u)$ associated with $O_q$ (a coideal subalgebra of $U_q \hat{sl}_2$), the fusion method (i) for spin-1 was implemented in [IOZ96], while the method (ii) for an arbitrary spin was used in [DN02] to produce recurrence relations for entries of the K-matrix. We also note that potentially related recurrence relations for higher-spin K-matrix were also obtained in [ML19].
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], by specialization to the finite-dimensional representations $V^{(j_k)}$ that depend on the spectral parameter $u$. Namely, R-matrix solutions of (1.1) are obtained by specializing L-operators of the form

$$ (1.5) \qquad \hat{\mathbf{L}}^{(j)}(u) \in H \otimes \mathrm{End}(V^{(j)}) $$

that satisfy an equation in $H \otimes \mathrm{End}(V^{(j_1)} \otimes V^{(j_2)})$:

$$ (1.6) \qquad R^{(j_1, j_2)}(u/v) \hat{\mathbf{L}}_1^{(j_1)}(u) \hat{\mathbf{L}}_2^{(j_2)}(v) = \hat{\mathbf{L}}_2^{(j_2)}(v) \hat{\mathbf{L}}_1^{(j_1)}(u) R^{(j_1, j_2)}(u/v). $$

Here $H$ is assumed to be any quantum affine algebra and $V^{(j)}$'s are also known as evaluation representations. Strictly speaking, the first component of $\hat{\mathbf{L}}^{(j)}(u)$ belongs to $H((u))$, the algebra of formal Laurent series in $u$ with coefficients in $H$. For simplicity, in the introduction we often indicate just an algebra like $H$ instead of $H((u))$.

Furthermore, the L-operators themselves can be obtained by specializing 2nd component of the universal R-matrix $\mathfrak{R} \in H \otimes H$ that is known to exist in an appropriate completion of the tensor product. For instance, for $H = U_q \hat{s} l_2$ the universal R-matrix is expressed as an infinite product of $q$-exponentials involving the root vectors [KT92a, LSS93, Da98]$^1$. The evaluation of $\mathfrak{R}$ on finite-dimensional representations requires to treat $u$ as a formal parameter to avoid convergence problems. This was suggested in [Dr86, Sec. 13] and later studied for $H = U_q(\hat{\mathfrak{g}})$ in [FR92, Sec. 4], see [He17, Sec. 1] for a review.

In the framework of the universal R-matrix, the previously existing methods (i) and (ii) find a natural interpretation: they just follow from the universal R-matrix axioms (R1)-(R3) reviewed in Section 2.1. As expected, expressions for the R-matrices previously derived in the literature using Fusion method (i) and Intertwining method (ii) match, up to a scalar function, with those derived through the specializations of the universal R-matrix.

For the reflection equation, it is well known [Sk88] that (1.2) has a representation-theoretic interpretation within the more general framework of reflection algebras, which are generated by the modes of the matrix entries of so-called K-operators

$$ (1.7) \qquad \mathcal{K}^{(j)}(u) \in B \otimes \mathrm{End}(V^{(j)}), $$

satisfying for all $j$ the following reflection equations in $B \otimes \mathrm{End}(V^{(j_1)} \otimes V^{(j_2)})$:

$$ (1.8) \qquad R_{12}^{(j_1, j_2)}(u_1/u_2) \mathcal{K}_1^{(j_1)}(u_1) R_{21}^{(j_2, j_1)}(u_1 u_2) \mathcal{K}_2^{(j_2)}(u_2) = \mathcal{K}_2^{(j_2)}(u_2) R_{12}^{(j_1, j_2)}(u_1 u_2) \mathcal{K}_1^{(j_1)}(u_1) R_{21}^{(j_2, j_1)}(u_1/u_2). $$

<footnote>$^1$Strictly speaking, the universal R-matrices introduced in these works are for quantised Kac-Moody algebras, extensions of $U_q(\hat{\mathfrak{g}})$ by a derivation. However, in our work we will not make any distinction between quantised affine and Kac-Moody algebras, as all representations we consider are of zero central charge which means that we actually work with their quotients called quantum loop algebras.</footnote>

<footnote>$^2$In the case of P-symmetric R-matrices associated to $H = U_q \hat{s} l_2$, the reflection equation (1.8) for the K-operators takes the more standard form (4.67).</footnote>
Here $B$ is in general assumed to be a comodule algebra over $H$, a notion generalizing coideal subalgebras in $H$.<fnref n="3" /> In other words, unlike K-matrices, which have scalar entries, K-operators are matrices with entries belonging to $B$.

In this paper we consider $H = U_q \hat{s} \hat{l}_2$ with zero central charge, so-called quantum loop algebra, and the comodule algebra of primary interest for us is $B = \mathcal{A}_q$, the alternating central extension [BS09, T21a] of the $q$-Onsager algebra $O_q$. Our main motivation in this comodule algebra comes from applications to open spin-chains through the K-operators, as it will be discussed below and in more detail in [LBG24]. It is important to note that we do not have any explicit form of a K-operator for $O_q$, while it is known for the comodule algebra $B = \mathcal{A}_q$ [BS09] in the fundamental case $j = \frac{1}{2}$. Moreover, with this K-operator $K(\frac{1}{2})(u)$ the reflection equation (1.8) for $j_1 = j_2 = \frac{1}{2}$ is the sole defining relation of $\mathcal{A}_q$. We thus have a very compact presentation of $\mathcal{A}_q$ which is analogous to the famous FRT presentation [DF93] of $H = U_q \hat{s} \hat{l}_2$.

The K-operators can also produce their smaller cousins—the K-matrices—by evaluating one-dimensional representations of $B$ on the 1st tensor component of $K^{(j)}(u)$. Indeed, we show in [LBG24] that one-dimensional representations of $\mathcal{A}_q$ produce out of the spin-$\frac{1}{2}$ K-operator $K(\frac{1}{2})(u)$ the most general K-matrix of the open XXZ spin-$\frac{1}{2}$ chain [dVR93, GZ93]. Furthermore, on tensor-product representations of $\mathcal{A}_q$, the K-operator $K^{(j)}(u)$ images match with ‘dressed’ K-matrices or Sklyanin’s operators, and the important problem of functional relations between transfer matrices of different spins, the so-called *TT-relations*, can be investigated at the level of algebraic relations of $\mathcal{A}_q$, as we discuss in more detail in Section 1.4. The main goal of our program is to construct such a general framework that allows the development of universal TT-relations holding in $\mathcal{A}_q$, which would then specialize to the known (conjectured) TT-relations for transfer matrices in the literature such as [FNR07, CYSW14]. This paper provides the first step in this direction by developing a formalism of K-operators with the spectral parameter, as we discuss it now.

**1.2. Universal K-matrix and comodule algebras.** As it was with the L-operators, it is expected that the K-operators $K^{(j)}(u)$ from (1.7) can arise from a universal K-matrix satisfying a universal reflection equation. The concept of a universal K-matrix is not new, it has been studied originally in [CG92, KSS92, tDHO98, DKM02]. More recently, significant progress has been made in the works of [BW13, BKo15, Ko17, AV20]. Each of the definitions of universal K-matrices introduced in these recent references has different axioms that were chosen to serve different purposes. We can highlight three directions corresponding to where a universal K-matrix belongs to:

(1) $H = U_q(\mathfrak{g})$, in [BKo15];

(2) $B \otimes H$, where $B$ is a coideal subalgebra of $H = U_q(\mathfrak{g})$, in [Ko17];

(3) $H = U_q(\hat{\mathfrak{g}})$, with the data of a certain automorphism of $H$, called ‘twist’, in [AV20].

Let us clarify that (1) and (3) imply a choice of $B$ as a coideal subalgebra over $H$ to form a quantum symmetric pair $(H, B)$, and the universal K-matrix satisfies intertwining relations, twisted in (3), with respect to $B$ (and not $H$ as the R-matrix).

<footnote n="3">³There 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, see also [BF11] for quotients of a few higher rank generalizations of $O_q$. However, we are not aware of any comodule algebra structure for these algebras.</footnote>
Let us start with direction (2). Kolb introduced a universal K-matrix $\mathcal{K} \in B \otimes H$ that satisfies certain axioms [Ko17, Def. 2.7]:

$$ (1.9) \qquad \begin{aligned} \mathcal{K} \Delta_B(b) &= \Delta_B(b)\mathcal{K}, && \text{for all } b \in B, \\ (\Delta_B \otimes \text{id})(\mathcal{K}) &= \mathfrak{R}_{32}\mathfrak{K}_{13}\mathfrak{R}_{23}, \\ (\text{id} \otimes \Delta)(\mathcal{K}) &= \mathfrak{R}_{32}\mathfrak{K}_{13}\mathfrak{R}_{23}\mathfrak{K}_{12}. \end{aligned} $$

It satisfies an algebraic (non-matrix) version of the reflection equation that belongs to $B \otimes H \otimes H$

$$ \mathfrak{R}_{32}\mathfrak{K}_{13}\mathfrak{R}_{23}\mathfrak{K}_{12} = \mathfrak{K}_{12}\mathfrak{R}_{32}\mathfrak{K}_{13}\mathfrak{R}_{23}. $$

The specialization of its second tensor component in $H$, to a finite-dimensional representation leads to K-operators that do not depend on a spectral parameter, denoted as $\mathcal{K} \in B \otimes \text{End}(V^{(j)})$, satisfying a reflection equation of the form

$$ (1.10) \qquad R_{12}\mathcal{K}_1R_{21}\mathcal{K}_2 = \mathcal{K}_2R_{12}\mathcal{K}_1R_{21}, $$

which belongs to $B \otimes \text{End}(V^{(j_1)}) \otimes \text{End}(V^{(j_2)})$. Moreover, the evaluation of the second relation in eq. (1.9) takes the form

$$ (1.11) \qquad (\Delta_B \otimes \text{id})(\mathcal{K}) = (\mathcal{L})_{[2]}(\mathcal{K})_{[1]}(\mathcal{L})_{[2]}. $$

Next, let us clarify that direction (1) is a special case of (2). Indeed, applying the counit to the first component of $\mathcal{K}$, we obtain an object $k \in H$ that was introduced by Balagović and Kolb in [BKO15]. Its specialization to a finite-dimensional representation leads to K-matrices (with scalar entries).

However, axiomatics of directions (1) and (2) is not suitable for the reflection equation with spectral parameter (1.2). To change this, Appel and Vlaar proposed in [AV20] a twisted version of the approach (1) in [BKO15], this is direction (3). Their definition of universal K-matrix is associated with a coideal subalgebra of $H = U_q(\hat{\mathfrak{g}})$ and includes a pair of consistent twists. It was shown in [AV20] that such a universal K-matrix exists for a large class of coideals known as quantum symmetric pairs. Furthermore, it was shown in [AV22] that these universal K-matrices are well-defined on finite-dimensional evaluation representations of $H = U_q(\hat{\mathfrak{g}})$ and provide K-matrix solutions of (1.2). Unfortunately, no explicit expression of these universal K-matrices is known even in simplest cases, and the existence result requires a specific form of twists.

Furthermore, whereas the axiomatics and results of [AV20, AV22] are well-suited for coideal subalgebras like $O_q$, they cannot be used for the algebra $A_q$. Indeed, $A_q$ is an extension of $O_q$ by an infinite number of algebraically independent central elements and, to the best of our knowledge, it cannot be realized as a coideal subalgebra of $H = U_q \hat{\mathfrak{sl}}_2$ or any other quasi-triangular Hopf algebra. We therefore cannot use the axiomatics and results of [AV20, AV22], and it requires an adjustment in order to treat general comodule algebras like $A_q$.

In this paper, to produce K-operator solutions of (1.8), we combine the approaches (2) and (3), and introduce a twisted version $\mathfrak{K} \in B \otimes H$ of the universal K-matrix from [Ko17] via a new set of axioms (K1)-(K3) in Section 2.3 that generalize those in (1.9) and at the same time recover the approach (3) via the counit application. The notion of a twist $\psi$ (certain automorphism of $H$) is important here because, as it will be discussed in Section 4, a natural choice of $\psi$ allows the derivation of the K-operator reflection equation (1.8) from the $\psi$-twisted universal reflection equation satisfied by $\mathfrak{K}$ and given in Proposition 2.11. We also note that here $B$ is not assumed to be a coideal in $H$, in particular we do not assume $(H, B)$ to be a quantum symmetric pair, but $B$ is a right comodule algebra over $H$.
We conjecture that such a universal K-matrix $\mathfrak{K}$ for our choice of $\psi = \eta$ defined in (2.34) exists,<fnref n="4" /> so that its 2nd component evaluation on $U_q \hat{sl}_2$ representations leads to the known examples of K-operators (that can be further evaluated to get known examples of K-matrices [LBG24]). However, the examples of K-operators are only known for the fundamental or ‘spin’-$\frac{1}{2}$ representation, the $\mathcal{K}(\frac{1}{2})(u)$ for $B = \mathcal{A}_q$ mentioned above. It is thus natural to investigate further, along the directions (i) or (ii), the spectral parameter dependent K-operators for arbitrary spin representations. The universal K-matrix axioms (K1)-(K3) provide the main guiding directions for such investigation, as it was the case for the interpretation of these methods for L-operators and R-matrices.

**1.3. Goal and main results.** The purpose of this paper is to construct K-operators with a spectral parameter for arbitrary spin representations and show how they relate to specializations of a universal K-matrix $\mathfrak{K} \in B \otimes H$. For this, 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 the comodule algebra $B = \mathcal{A}_q$. The main results are the following:

(I) Universal fusion method: Using the axiom (K2) and the detailed analysis of tensor product representation of $U_q \hat{sl}_2$, we derive fused K-operators $\mathcal{K}^{(j)}(u)$ for arbitrary values of spin-$j$, starting from the spin-$\frac{1}{2}$ K-operator, see Definition 5.6 based on the result in Proposition 4.16. One of our main results (Theorem 5.7) is that they satisfy the reflection equations (1.8) for all $j_1, j_2$ and with no assumption made on existence of $\mathfrak{K}$. For $j = 1, \frac{3}{2}$, we give explicit expressions of the fused K-operators in terms of generating functions of $\mathcal{A}_q$.

(II) Universal intertwining method: K-operators that solve the twisted intertwining relations for $B = \mathcal{A}_q$ are considered in Section 7. These relations, see (5.55), are just specialization of the axiom (K1). For $j = \frac{1}{2}, 1$ and with a reasonably general ansatz of $\mathcal{K}^{(j)}(u)$, solutions of the twisted intertwining relations are obtained using a Poincaré-Birkhoff-Witt (PBW) basis of $\mathcal{A}_q$. 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).

Let us comment more on constructions behind the Universal fusion method (I). Drawing inspiration from the evaluation of one of the axioms of the universal K-matrix, and without making any assumptions about the existence of $\mathfrak{K}$, we define fused spin-$j$ K-operators $\mathcal{K}^{(j)}(u) \in \mathcal{A}_q((u^{-1})) \otimes \text{End}(\mathbb{C}^{2j+1})$ through the following recursion based on the spin-1/2 K-operator from the reflection algebra presentation of $\mathcal{A}_q$ in Theorem 5.1:

$$ (1.12) \qquad \mathcal{K}^{(j)}(u) = \mathcal{F}_{\langle 12 \rangle}^{(j)} \mathcal{K}_1^{(\frac{1}{2})}(uq^{-j+\frac{1}{2}}) \mathcal{R}^{(\frac{1}{2},j-\frac{1}{2})}(u^2q^{-j+1}) \mathcal{K}_2^{(j-\frac{1}{2})}(uq^{\frac{1}{2}}) \mathcal{E}_{\langle 12 \rangle}^{(j)}, $$

and that we call universal fusion procedure. Here, we used $U_q \hat{sl}_2$-intertwiners

$$ \mathcal{E}^{(j)}: \mathbb{C}_u^{2j+1} \to \mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j}, \quad u_1 = uq^{-j+\frac{1}{2}}, \quad u_2 = uq^{\frac{1}{2}}, $$

for evaluation representations, and we denote their pseudo-inverse maps $\mathcal{F}^{(j)}: \mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j} \to \mathbb{C}_u^{2j+1}$. The conditions imposed on $u_1$ and $u_2$ ensure that the tensor product of the evaluation representations

<footnote n="4">⁴In the context of 1-component universal K-matrix $k \in H$, there are existence results for other choices of twists in [AV22]. We discuss this properly at the end of Section 2.</footnote>
of $U_q \hat{s} l_2$ admits a spin-$j$ sub-representation (which is not a direct summand). This reducibility condition is expressed in terms of ratio of the evaluation parameters [CP91, Sect. 4.9]. Using this criterion, we explicitly construct $\mathcal{E}^{(j)}$ and $\mathcal{F}^{(j)}$ by determining their matrix expressions, as described in Section 3.2.

Inspiration for the above universal fusion (1.12) was taken from fusion relations in Proposition 4.16 satisfied by the spin-$j$ K-operator

$$ (1.13) \qquad \mathbf{K}^{(j)}(u) = (\mathrm{id} \otimes \pi_{u^{-1}}^{j})(\mathfrak{k}) \in \mathcal{B}((u^{-1})) \otimes \mathrm{End}(\mathcal{V}^{(j)}). $$

obtained from the 1-component evaluation of $\mathfrak{k}$. Here, the universal K-matrix $\mathfrak{k}$ is considered for the pair of twists $(\psi, J) = (\eta, 1 \otimes 1)$ with $\eta$ defined in (2.34). Moreover, a careful analysis requires treating $u$ as a formal variable, and the representation $\pi_u^j: H \to \mathrm{End}(\mathcal{V}^{(j)}[u^{\pm 1}])$ used in (1.13) is what we call formal evaluation representation of $H = U_q \hat{s} l_2$ defined in Section 3.5. It is an infinite-dimensional analogue of the finite-dimensional evaluation representations of $H$. These representations of quantum affine algebras are not of highest weight type, and also known under the name ‘quantum loop modules’ [CG03]. In Section 3.5, we also analyze tensor products of the formal evaluation representations $\pi_u^j$ and obtain the intertwining operators $\mathcal{E}^{(j)}$ and their pseudo-inverses $\mathcal{F}^{(j)}$ in this framework with $u$ being a formal variable.

It is important to note that, in contrast to the situation with L-operators, there is no single example in the literature of an evaluation (1.13), even for $j = \frac{1}{2}$. On the other side, the formula (1.12) gives explicit expressions for K-operators. Furthermore, we propose a precise relation that links the K-operators $\mathbf{K}^{(j)}(u)$ in (1.13) and the fused spin-$j$ K-operators $\mathcal{K}^{(j)}(u)$ in (1.12) associated with $\mathcal{A}_q$. This is stated in Conjecture 1: the two K-operators equal up to a central and invertible element in $\mathcal{A}_q((u^{-1}))$, which is defined by a functional relation and has a specific coaction. This conjecture is supported by showing independently that the fused K-operators (1.12) satisfy a set of relations arising from the evaluation of the axioms of the universal K-matrix.

### 1.4. K-operators for $\mathcal{A}_q$ and universal TT-relations.
A construction of arbitrary spin K-operators for $\mathcal{A}_q$ is not only important for a better understanding of the universal K-matrix formalism for comodule algebras, but first of all because of applications in quantum integrable systems. Namely, as will be reported in [LBG24], a generating function of a commutative subalgebra of $\mathcal{A}_q$ can be constructed using the spin-$j$ K-operator $\mathcal{K}^{(j)}(u)$, that can be viewed as a spin-$j$ universal transfer matrix $\mathcal{T}^{(j)}(u) \in \mathcal{A}_q((u^{-1}))$. A family of universal TT-relations between the universal transfer matrices can be calculated directly in the algebra $\mathcal{A}_q$:

$$ (1.14) \qquad \mathcal{T}^{(j)}(u) = \mathcal{T}^{(j-\frac{1}{2})}(uq^{-\frac{1}{2}})\mathcal{T}^{(\frac{1}{2})}(uq^{j-\frac{1}{2}}) + f^{(j)}(u)\mathcal{T}^{(j-1)}(uq^{-1}), $$

with $\mathcal{T}^{(0)}(u) \equiv 1$, $\mathcal{T}^{(-\frac{1}{2})}(u) \equiv 0$ and $f^{(j)}(u)$ is a generating function of central elements in $\mathcal{A}_q$ that is explicitly calculated [LBG24]. Considering tensor product (or spin-chain) representations of the algebra $B = \mathcal{A}_q$, one can show that the K-operators $\mathcal{K}^{(j)}(u)$ and the universal transfer matrices $\mathcal{T}^{(j)}(u)$ map, respectively, to the ‘dressed’ K-matrices and transfer matrices associated with various examples of integrable spin chains of XXZ type with generic boundary conditions. Importantly, whereas $f^{(j)}(u)$ in (1.14) is universal, its images with respect to the various $q$-Onsager quotients associated with the spin chains are not: the boundary conditions of the open integrable spin chains dictate what value the image of $f^{(j)}(u)$ takes. In other words, the algebra $\mathcal{A}_q$ in a certain sense governs all known integrable boundary conditions through its quotients and degenerate cases [BB16]. Thus,
many properties of the integrable models can be studied even before specializations to the spin-chain representations of $\mathcal{A}_q$, and so the K-operators of $\mathcal{A}_q$ can be used in the representation-independent analysis of the related integrable models. The applications of our results to integrable models are discussed more in Section 8 and in our forthcoming paper [LBG24].

The text is organized as follows. In Section 2, the formalism of universal R-matrix is reviewed following [Dr86]. Furthermore, we upgrade the universal K-matrix axioms of [Ko17, AV20] in Definition 2.8. In our framework, the K-operators are obtained as evaluations of the universal K-matrix $\mathfrak{K} \in B \otimes H$ (if it exists) satisfying a $\psi$-twisted universal reflection equation, see Proposition 2.11. In this section, we also review basic definitions of the quantum loop algebra $H = \mathcal{L}U_qsl_2$, the $q$-Onsager algebra $O_q$ and its alternating central extension $B = \mathcal{A}_q$.

In Sections 3-5, we develop Universal fusion method (I). Namely, Section 3 is devoted to a detailed analysis of the tensor product of evaluation representations of $\mathcal{L}U_qsl_2$ and their formal-parameter versions. In Section 4, we assume existence of a universal K-matrix $\mathfrak{K}$ for a comodule algebra $B$ and a certain twist pair. The L-operators and K-operators of spin-$j$ are defined as formal evaluations of $\mathfrak{K}$ and $\mathfrak{K}$, respectively; see Definitions 4.1 and 4.15. The main results of this section are the so-called ‘fusion’ relations satisfied by the L- and K-operators of spin-$j$, see Propositions 4.5 and 4.16, respectively. In Section 5, we do not assume existence of a universal K-matrix for $\mathcal{A}_q$, and instead introduce fused K-operators of spin-$j$ in Definition 5.6 by a recursion based on the fundamental K-operator giving the reflection algebra presentation of $\mathcal{A}_q$. Theorem 5.7 shows that they satisfy the reflection equation (1.8) for all choices of $j_1, j_2$. Explicit expressions of the fused K-operators for $j = 1, \frac{3}{2}$ are derived in Section 5.4. We also give compact expressions for the fused R-matrices and the fused K-operators in (5.43), (5.44), solely in terms of the fundamental R-matrix and K-operator.

In Section 6, the precise relation between the spin-$j$ K-operators from Section 4 and the fused K-operators from Section 5 leads to Conjecture 1, with a few supporting evidences discussed.

In Section 7, Universal intertwining method (II) is considered, and K-operator solutions of the twisted intertwining relations in Proposition 5.13 are investigated without any additional assumption. Finally, in the concluding 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 $U_qsl_2$ and $U_q\hat{sl}_2$. In Appendix B, we adapt the universal R-matrix constructed by Khoroshkin-Tolstoy in [KT92a] to our conventions. The corresponding Ding-Frenkel type L-operators $\hat{L}^{(\frac{1}{2})}(u) = L^{\pm}(u)$ and the spin-$\frac{1}{2}$ L-operator $L^{(\frac{1}{2})}(u)$ are computed by evaluation of the universal R-matrix. In Appendix C, we give ordering relations for the generating functions of $\mathcal{A}_q$. Finally, in Appendix D, we give a proof of one of our main results, Theorem 5.7, that the fused K-operators satisfy the reflection equations for any pair of spins.

**Notations.** We denote the set of natural numbers by $\mathbb{N} = \{0, 1, 2, \dots\}$ and the positive integers by $\mathbb{N}_+ = \{1, 2, \dots\}$.

All algebras are considered over the field of complex numbers $\mathbb{C}$, 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 $\mathbb{C}$.

Let $q \in \mathbb{C}^*$, and we assume in this paper that $q$ is not a root of unity. The $q$-commutator is

$$[X, Y]_q = qXY - q^{-1}YX$$
and $[X, Y] = [X, Y]_1 = XY - YX$. We denote the $q$-numbers by $[n]_q = (q^n - q^{-n})/(q - q^{-1})$.

We denote by $\mathbb{I}_{2j}$ the $2j \times 2j$ identity matrix. We also use Pauli matrices:

$$ (1.15) \quad \sigma_+ = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad \sigma_- = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \quad \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}. $$

All generating functions considered in this paper like quantum determinants $\gamma(u)$ or $\Gamma(u)$ are formal Laurent series in $u^{-1}$ with coefficients in the corresponding algebra like $\mathcal{A}_q$. In other words, they are of the form

$$ \sum_{n \in \mathbb{Z}} f_n u^{-n}, $$

where $f_n \in \mathcal{A}_q$ and all but finitely many $f_n$ for $n < 0$ vanish. In this case, we use the notation $\mathcal{A}_q((u^{-1}))$, and the convention that every rational function of the form $1/p(u)$, where $p(u)$ is a Laurent polynomial, is expanded in $u^{-1}$. The space of formal power series with coefficients $f_n$ in $\mathcal{A}_q$, i.e. when all $f_n$ are zero for $n < 0$, is denoted by $\mathcal{A}_q[[u^{-1}]]$.

## 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 $(\psi, J)$, and its comodule algebra $B$ – this is an element in $B \otimes H$ that satisfies a universal reflection equation (also called $\psi$-twisted reflection equation), see Proposition 2.11. We also show a precise relation of our notion of the universal K-matrix to those introduced in [Ko17] and [AV20]. In Section 2.4, we introduce the main example of $H$ and $B$ considered in this paper: the quantum loop algebra $H = \mathcal{L}U_q s l_2$ and its comodule algebra $B = \mathcal{A}_q$, the alternating central extension of the $q$-Onsager algebra.

### 2.1. Universal R-matrix.
Let $H$ be a Hopf algebra with coproduct $\Delta: H \to H \otimes H$, the counit $\epsilon: H \to \mathbb{C}$ and the antipode $S: H \to H$, which are subject to consistency conditions, for which we refer to [CP95, Chap. 4]. We denote the opposite coproduct $\Delta^{op} = \mathfrak{p} \circ \Delta$, where $\mathfrak{p}$ is the permutation operator, i.e. $\mathfrak{p}(x \otimes y) = y \otimes x$, for $x, y \in H$. For $\mathfrak{R} \in H \otimes H$, we use the notation $\mathfrak{R}_{12} = \mathfrak{R} \otimes 1$, $\mathfrak{R}_{23} = 1 \otimes \mathfrak{R}$, $\mathfrak{R}_{13} = \mathfrak{p}_{23} \circ \mathfrak{R}_{12}$.

**Definition 2.1 ([Dr86]).** For a Hopf algebra $H$, an invertible element $\mathfrak{R} \in H \otimes H$ is called universal R-matrix if it satisfies

$$ (R1) \qquad \mathfrak{R}\Delta(x) = \Delta^{op}(x)\mathfrak{R}, \quad \forall x \in H, $$

$$ (R2) \qquad (\Delta \otimes \text{id})(\mathfrak{R}) = \mathfrak{R}_{13}\mathfrak{R}_{23}, $$

$$ (R3) \qquad (\text{id} \otimes \Delta)(\mathfrak{R}) = \mathfrak{R}_{13}\mathfrak{R}_{12}. $$

If such $\mathfrak{R}$ exists, then the pair $(H, \mathfrak{R})$ is called a quasi-triangular Hopf algebra.

We note that the universal R-matrix necessarily satisfies

$$ (2.1) \qquad (S \otimes \text{id})(\mathfrak{R}) = \mathfrak{R}^{-1} = (\text{id} \otimes S)(\mathfrak{R}), $$

$$ (2.2) \qquad (\epsilon \otimes \text{id})(\mathfrak{R}) = 1 = (\text{id} \otimes \epsilon)(\mathfrak{R}). $$
Using the relations (R1)–(R3) one can show that the universal R-matrix satisfies the universal Yang-Baxter equation:

$$ (2.3) \qquad \mathfrak{R}_{12}\mathfrak{R}_{13}\mathfrak{R}_{23} = \mathfrak{R}_{23}\mathfrak{R}_{13}\mathfrak{R}_{12}. $$

It is well-known [Dr86] 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.

## 2.2. Twist pairs.
The following is an extension of [AV20, Def. 2.2] from bialgebras to Hopf algebras.

**Definition 2.2.** Let $(H, \mathfrak{R})$ be a quasi-triangular Hopf algebra and $\psi: H \to H$ an algebra automorphism. The $\psi$-twisting of $(H, \mathfrak{R})$ is the quasi-triangular Hopf algebra $(H^{\psi}, \mathfrak{R}^{\psi\psi})$ obtained from $(H, \mathfrak{R})$ by pullback through $\psi$, i.e. $H^{\psi}$ is the Hopf algebra with same multiplication, new coproduct, counit and antipode$^5$:

$$ (2.4) \qquad \Delta^{\psi} = (\psi \otimes \psi) \circ \Delta \circ \psi^{-1}, \quad \epsilon^{\psi} = \epsilon \circ \psi^{-1}, \quad S^{\psi} = \psi \circ S \circ \psi^{-1}, $$

and the universal R-matrix is given by

$$ (2.5) \qquad \mathfrak{R}^{\psi\psi} = (\psi \otimes \psi)(\mathfrak{R}). $$

In what follows, we also use the opposite version of the $\psi$-twisting of $(H, \mathfrak{R})$. Let $H^{\text{cop}}$ be the Hopf algebra with the coproduct $\Delta^{\text{op}}$, the antipode $S^{-1}$ and the other structure maps are the same as $H$. Let $H^{\text{cop},\psi}$ be the $\psi$-twisting of $H^{\text{cop}}$.

**Lemma 2.3.** The pair $(H^{\text{cop},\psi}, \mathfrak{R}_{21}^{\psi\psi})$ is a quasi-triangular Hopf algebra.

*Proof.* We need to show that the following relations hold:

$$ (2.6) \qquad \mathfrak{R}_{21}^{\psi\psi} \Delta^{\text{op},\psi}(x) = \Delta^{\psi}(x)\mathfrak{R}_{21}^{\psi\psi}, \quad \forall x \in H, $$

$$ (2.7) \qquad (\Delta^{\text{op},\psi} \otimes \text{id})(\mathfrak{R}_{21}^{\psi\psi}) = \mathfrak{R}_{31}^{\psi\psi} \mathfrak{R}_{32}^{\psi\psi}, $$

$$ (2.8) \qquad (\text{id} \otimes \Delta^{\text{op},\psi})(\mathfrak{R}_{21}^{\psi\psi}) = \mathfrak{R}_{31}^{\psi\psi} \mathfrak{R}_{21}^{\psi\psi}. $$

Note that as corollaries of (R1)–(R3) we have:

$$ (2.9) \qquad \Delta(x)\mathfrak{R}_{21} = \mathfrak{R}_{21}\Delta^{\text{op}}(x), \quad \forall x \in H, $$

$$ (2.10) \qquad (\Delta^{\text{op}} \otimes \text{id})(\mathfrak{R}_{21}) = \mathfrak{R}_{31}\mathfrak{R}_{32}, $$

$$ (2.11) \qquad (\text{id} \otimes \Delta^{\text{op}})(\mathfrak{R}_{21}) = \mathfrak{R}_{31}\mathfrak{R}_{21}. $$

The relation (2.6) is obtained by applying $(\psi \otimes \psi)$ to (2.9)

$$ \begin{align*} (\psi \otimes \psi)[\Delta(x)\mathfrak{R}_{21}] &= (\psi \otimes \psi)[\mathfrak{R}_{21}\Delta^{\text{op}}(x)] \\ [(\psi \otimes \psi)\Delta(\psi^{-1}\psi(x))]\mathfrak{R}_{21}^{\psi\psi} &= \mathfrak{R}_{21}^{\psi\psi}[(\psi \otimes \psi)\Delta^{\text{op}}(\psi^{-1}\psi(x))] \\ \Delta^{\psi}(\psi(x))\mathfrak{R}_{21}^{\psi\psi} &= \mathfrak{R}_{21}^{\psi\psi}\Delta^{\text{op},\psi}(\psi(x)). \end{align*} $$

Next, rewrite (2.10) as

$$ \begin{align*} (\Delta^{\text{op}} \otimes \text{id})((\psi^{-1} \otimes \psi^{-1})\mathfrak{R}_{21}^{\psi\psi}) &= \mathfrak{R}_{31}\mathfrak{R}_{32}, \\ (\Delta^{\text{op}} \circ \psi^{-1} \otimes \psi^{-1})(\mathfrak{R}_{21}^{\psi\psi}) &= \mathfrak{R}_{31}\mathfrak{R}_{32}, \end{align*} $$

<footnote n="5">⁵It is enough to verify that $\epsilon^{\psi} \circ S^{\psi} = \epsilon^{\psi}$.</footnote>
and apply $(\psi \otimes \psi \otimes \psi)$ to get

$$(\Delta^{op,\psi} \otimes \mathrm{id})(\mathfrak{R}_{21}^{\psi\psi}) = \mathfrak{R}_{31}^{\psi\psi}\mathfrak{R}_{32}^{\psi\psi},$$

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 $J \in H \otimes H$ satisfying the property

$$(2.12) \qquad (\epsilon \otimes \mathrm{id})(J) = 1 = (\mathrm{id} \otimes \epsilon)(J)$$

and the cocycle identity

$$(2.13) \qquad (J \otimes 1)(\Delta \otimes \mathrm{id})(J) = (1 \otimes J)(\mathrm{id} \otimes \Delta)(J).$$

**Example 2.5.** In the literature, there are two natural choices of Drinfeld twists: $J = 1 \otimes 1$ and $J = \mathfrak{R}_{21}\mathfrak{R}$. 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 $(H_J, \mathfrak{R}_J)$ with the coproduct [Dr89b]

$$(2.14) \qquad \Delta_J(x) = J\Delta(x)J^{-1}, \quad \forall x \in H,$$

and the universal R-matrix

$$(2.15) \qquad \mathfrak{R}_J = J_{21}\mathfrak{R}J^{-1}.$$

**Definition 2.6** ([AV20]). Let $(H, \mathfrak{R})$ be a quasi-triangular algebra. A twist pair $(\psi, J)$ is the datum of an algebra automorphism $\psi: H \to H$ and a Drinfeld twist $J \in H \otimes H$ such that $(H^{\mathrm{cop},\psi}, \mathfrak{R}_{21}^{\psi\psi}) = (H_J, \mathfrak{R}_J)$, i.e. such that $\epsilon^\psi = \epsilon$,

$$(2.16) \qquad \begin{aligned} \Delta^{\mathrm{op},\psi}(x) &= J\Delta(x)J^{-1}, & \forall x \in H, \\ \mathfrak{R}_{21}^{\psi\psi} &= J_{21}\mathfrak{R}J^{-1}, \end{aligned}$$

where

$$(2.17) \qquad \Delta^{\mathrm{op},\psi} = (\psi \otimes \psi) \circ \Delta^{\mathrm{op}} \circ \psi^{-1}.$$

**2.3. Universal K-matrix.** Appel and Vlaar introduced the notion of a cylindrical bialgebra [AV20, Def. 2.3] which is a quasi-triangular bialgebra with a twist pair and a universal solution of a twisted reflection equation in $H \otimes H$. This approach is convenient for scalar K-matrix solutions of the parameter-dependent reflection equation (1.2). It is however not a convenient framework for K-operator solutions of more general reflection equations (1.8) associated to general $H$-comodule algebras $B$. To change this, we define a universal K-matrix via axioms (K1)-(K3) and show that it satisfies a universal reflection equation in $B \otimes H \otimes H$ that specializes on representations to the reflection equation (1.8), as will be further demonstrated in Section 4. We begin with the following standard definition.
**Definition 2.7.** An algebra $B$ is called a right comodule algebra over a Hopf algebra $H$ if there exists an algebra map $\delta: B \to B \otimes H$, which we call right coaction, such that the coassociativity and counital conditions hold

$$ (2.18) \qquad (\mathrm{id} \otimes \Delta) \circ \delta = (\delta \otimes \mathrm{id}) \circ \delta, \qquad (\mathrm{id} \otimes \epsilon) \circ \delta = \mathrm{id}. $$

Let $(\psi, J)$ be a twist pair for a Hopf algebra $H$ and $B$ is a right comodule algebra over $H$. We define a universal K-matrix $\mathfrak{K} \in B \otimes H$. Here we use the notation $\mathfrak{K}_{12} = \mathfrak{K} \otimes 1$, $\mathfrak{K}_{13} = \mathfrak{p}_{23} \circ \mathfrak{K}_{12}$.

**Definition 2.8.** We say that $\mathfrak{K} \in B \otimes H$ is universal K-matrix if the following relations hold for all $b \in B$:

$$ (K1) \qquad \mathfrak{K}\delta(b) = \delta^{\psi}(b)\mathfrak{K}, \qquad \text{with } \delta^{\psi} = (\mathrm{id} \otimes \psi) \circ \delta, $$

$$ (K2) \qquad (\delta \otimes \mathrm{id})(\mathfrak{K}) = (\mathfrak{R}^{\psi})_{32}\mathfrak{K}_{13}\mathfrak{R}_{23}, $$

$$ (K3) \qquad (\mathrm{id} \otimes \Delta)(\mathfrak{K}) = J_{23}^{-1}\mathfrak{K}_{13}\mathfrak{R}_{23}^{\psi}\mathfrak{K}_{12}, $$

where

$$ (2.19) \qquad \mathfrak{R}^{\psi} = (\psi \otimes \mathrm{id})(\mathfrak{R}). $$

We note that $\delta^{\psi}$ defines a coalgebra structure on $B$ over $H^{\psi}$. Therefore, by (K1) $\mathfrak{K}$ intertwines two actions of $B$ on $B \otimes H$, given by $\delta$ and $\delta^{\psi}$ 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:

$$ (2.20) \qquad (\mathrm{id} \otimes \Delta \otimes \mathrm{id}) \circ (\delta \otimes \mathrm{id})(\mathfrak{K}) = (\delta \otimes \mathrm{id} \otimes \mathrm{id}) \circ (\delta \otimes \mathrm{id})(\mathfrak{K}). $$

This relation is checked using (R2) and (K2). Indeed, the l.h.s. equals $(\mathfrak{R}^{\psi})_{43}(\mathfrak{R}^{\psi})_{42}\mathfrak{K}_{14}\mathfrak{R}_{24}\mathfrak{R}_{34}$ where we used $(\Delta \otimes \mathrm{id})((\mathfrak{R}^{\psi})_{21}) = (\mathfrak{R}^{\psi})_{32}(\mathfrak{R}^{\psi})_{31}$, 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:

$$ (2.21) \qquad (\mathrm{id} \otimes \epsilon \otimes \mathrm{id}) \circ (\delta \otimes \mathrm{id})(\mathfrak{K}) = (\mathrm{id} \otimes \epsilon \otimes \mathrm{id})((\mathfrak{R}^{\psi})_{32}\mathfrak{K}_{13}\mathfrak{R}_{23}) = \mathfrak{K}. $$

(ii) Recall that the coproduct and the counit of a Hopf algebra satisfy

$$ (2.22) \qquad (\epsilon \otimes \mathrm{id}) \circ \Delta = \mathrm{id} = (\mathrm{id} \otimes \epsilon) \circ \Delta. $$

Then, using (K3), (2.2), (2.12), and the fact that $\epsilon$ is an algebra homomorphism we obtain:

$$ (2.23) \qquad \mathfrak{K} = [(\mathrm{id} \otimes \mathrm{id} \otimes \epsilon) \circ (\mathrm{id} \otimes \Delta)](\mathfrak{K}) = (\mathrm{id} \otimes \mathrm{id} \otimes \epsilon)(J_{23}^{-1}\mathfrak{K}_{13}\mathfrak{R}_{23}^{\psi}\mathfrak{K}_{12}) = (([\mathrm{id} \otimes \epsilon](\mathfrak{K})] \otimes 1)\mathfrak{K}. $$

Applying again $(\mathrm{id} \otimes \epsilon)$ on (2.23), we find that $(\mathrm{id} \otimes \epsilon)(\mathfrak{K})$ is an idempotent. If in addition, $\mathfrak{K}$ is invertible, it follows:

$$ (2.24) \qquad (\mathrm{id} \otimes \epsilon)(\mathfrak{K}) = 1, $$

which is the analogue of (2.2) for the universal R-matrix.
**Remark 2.10.** The previous definitions of universal K-matrices from the works [Ko17, AV20] can be obtained as special cases of Definition 2.8.

(i) It is easy to check that $(\mathrm{id}, \mathfrak{R}_{21}^{-1})$ is a twist pair. In particular, one finds that (K1)-(K3), for $\psi = \mathrm{id}$ and $J = \mathfrak{R}_{21}^{-1}$, correspond to the axioms for the universal K-matrix defined in [Ko17, Def. 2.7].

(ii) Assume that $B$ is a right coideal subalgebra of $H$, that is $\delta = \Delta|_B$, and define

$$ \mathcal{K} = (\epsilon \otimes \mathrm{id})(\mathfrak{R}), \quad \mathcal{K} \in H. $$

Then, applying the counit on the first tensor factor of the universal K-matrix in (K1)-(K3) yields:

$$ (2.25) \qquad \mathcal{K}b = \psi(b)\mathcal{K}, \quad \forall b \in B, $$

$$ (2.26) \qquad \mathfrak{R} = (\mathfrak{R}^{\psi})_{21}\mathcal{K}_{2}\mathfrak{R}, $$

$$ (2.27) \qquad \Delta|_{B}(\mathcal{K}) = J^{-1}\mathcal{K}_{2}\mathfrak{R}^{\psi}\mathcal{K}_{1}. $$

We recall that in this setting of $B$ a coideal subalgebra of $H$, Appel and Vlaar introduced in [AV20] one-component universal K-matrices $k \in H$. The formulas (2.25) and (2.27), with the identification $\mathcal{K} = k$, correspond respectively to the defining relations of the so-called cylindrically invariant subalgebra $B$ and the cylindrical pair $(H, \mathfrak{R})$, see [AV20, Def. 2.3]. We thus get the one-component universal K-matrix of [AV20] from our 2-component $\mathfrak{R}$. The opposite is also true: starting from a solution $\mathcal{K}$ of (2.25) and (2.27), and assuming that $\mathfrak{R}$ defined by (2.26) lies in $B \otimes H$, one can check that $\mathfrak{R}$ satisfies the axioms (K1)-(K3). Indeed, checking (K1) and (K2) is straightforward, while for (K3) we use the equalities

$$ (\mathrm{id} \otimes \Delta)[(\mathfrak{R}^{\psi})_{21}] = J_{23}^{-1}(\mathfrak{R}^{\psi})_{31}(\mathfrak{R}^{\psi})_{21}J_{23}, $$

$$ (\mathfrak{R}^{\psi})_{21}\mathfrak{R}_{23}^{\psi}\mathfrak{R}_{13} = \mathfrak{R}_{13}\mathfrak{R}_{23}^{\psi}(\mathfrak{R}^{\psi})_{21}. $$

The first one is obtained using

$$ \Delta(\psi(x)) = J^{-1}[(\psi \otimes \psi) \circ \Delta^{\mathrm{op}}(x)]J $$

with

$$ (\mathrm{id} \otimes \Delta^{\mathrm{op}})(\mathfrak{R}_{21}) = \mathfrak{R}_{31}\mathfrak{R}_{21}, $$

while we applied $(\mathrm{id} \otimes \psi \otimes \mathrm{id}) \circ \mathfrak{p}_{12}$ to the universal Yang-Baxter equation (2.3) to get the second equality. Then, we get

$$ \begin{aligned} (\mathrm{id} \otimes \Delta)(\mathfrak{R}) &= J_{23}^{-1}(\mathfrak{R}^{\psi})_{31}\mathcal{K}_{3}(\mathfrak{R}^{\psi})_{21}\mathfrak{R}_{23}^{\psi}\mathfrak{R}_{13}\mathcal{K}_{2}\mathfrak{R}_{12} \\ &= J_{23}^{-1}(\mathfrak{R}^{\psi})_{31}\mathcal{K}_{3}\mathfrak{R}_{13}\mathfrak{R}_{23}^{\psi}(\mathfrak{R}^{\psi})_{21}\mathcal{K}_{2}\mathfrak{R}_{12} \\ &= J_{23}^{-1}\mathfrak{R}_{13}\mathfrak{R}_{23}^{\psi}\mathfrak{R}_{12} \end{aligned} $$

which is indeed (K3).

Let us comment on the point (ii) of the above remark. If $B = O_q$, the $q$-Onsager algebra, then an explicit formula for our universal K-matrix in $O_q \otimes \mathcal{L}U_q sl_2$ could be deduced from (2.26) provided $\mathcal{K}$ is known. Note that there are existence results for $\mathcal{K}$ for certain choice of twists [AV20]. However, no expressions for $\mathcal{K}$ in $O_q$, for example in terms of root vectors, are known. In the following, we make a choice of twist pair $(\psi, J)$ that differs from [AV20, Sec. 9.5]. This allows to
obtain the standard reflection equation from the $\psi$-twisted reflection equation (2.28) for our choice of principal gradation. Let us also mention that in the next sections, we define K-operators out of $\mathcal{K} \in \mathcal{A}_q \otimes \mathcal{L}U_q sl_2$, and derive fusion formulas satisfied by the K-operators for our choice of twist pair. As it will be studied in [LBG24], these K-operators are involved in the construction of spin-chain transfer matrices. In view of these applications, it is not clear whether the choice of the twist pair of Appel-Vlaar from [AV20, Sec. 9.5] produces the spin-chain transfer matrices. Indeed, following the approach in (ii) of Remark 2.10 for $B = O_q$ and using their twist pair, the fusion formula is affected by non-trivial $J$ and it provides a priori different spin-$j$ K-operators.

Furthermore, if $B$ is a comodule algebra over $H$ and not a coideal subalgebra of $H$, as in our main case of interest $B = \mathcal{A}_q$, then the formalism from [AV20, AV22] is not suitable. That is why in this section we developed a slight generalization for the two-component universal K-matrices, and we study its consequences in the next sections.

We now derive a universal reflection equation based on the axioms (K1)–(K3) of the universal K-matrix.

**Proposition 2.11.** Let $(\psi, J)$ be a twist pair. The universal K-matrix satisfies the $\psi$-twisted reflection equation

$$ (2.28) \qquad \mathfrak{K}_{12}(\mathfrak{R}^{\psi})_{32}\mathfrak{K}_{13}\mathfrak{R}_{23} = \mathfrak{R}_{32}^{\psi\psi}\mathfrak{K}_{13}\mathfrak{R}_{23}^{\psi}\mathfrak{K}_{12}. $$

*Proof.* Multiply (K2) on the left by $\mathfrak{K}_{12}$ to get

$$ (2.29) \qquad \mathfrak{K}_{12}[(\delta \otimes \mathrm{id})(\mathfrak{K})] = \mathfrak{K}_{12}(\mathfrak{R}^{\psi})_{32}\mathfrak{K}_{13}\mathfrak{R}_{23}. $$

Then, using (K1), the l.h.s. of this equation equals

$$ (2.30) \qquad [(\mathrm{id} \otimes \psi \otimes \mathrm{id}) \circ (\delta \otimes \mathrm{id})(\mathfrak{K})]\mathfrak{K}_{12} = \mathfrak{R}_{32}^{\psi\psi}\mathfrak{K}_{13}\mathfrak{R}_{23}^{\psi}\mathfrak{K}_{12}, $$

where we used again (K2). Equating (2.29) with (2.30), the equation (2.28) follows. $\square$

Note that the $\psi$-twisted reflection equation can also be derived from (K3). Multiply (K3) on the left by $\mathfrak{R}_{23}$ then use (R1) to get

$$ (2.31) \qquad \begin{gathered} \mathfrak{R}_{23}[(\mathrm{id} \otimes \Delta)(\mathfrak{K})] = [(\mathrm{id} \otimes \Delta^{\mathrm{op}})(\mathfrak{K})]\mathfrak{R}_{23} \\ \mathfrak{R}_{23}J_{23}^{-1}\mathfrak{K}_{13}\mathfrak{R}_{23}^{\psi}\mathfrak{K}_{12} = J_{32}^{-1}\mathfrak{K}_{12}(\mathfrak{R}^{\psi})_{32}\mathfrak{K}_{13}\mathfrak{R}_{23}. \end{gathered} $$

Multiply (2.31) on the left by $J_{32}$. Since $(\psi, J)$ is a twist pair, by (2.15) we have $\mathfrak{R}_{32}^{\psi\psi} = J_{32}\mathfrak{R}_{23}J_{23}^{-1}$ and (2.28) follows.

2.4. **Example of $(H, B)$**. We now introduce our main example of the pair $(H, B)$ where the Hopf algebra $H = \mathcal{L}U_q sl_2$ is the quantum loop algebra and its right comodule algebra $B = \mathcal{A}_q$ is a central extension of the $q$-Onsager algebra.

2.4.1. $\mathcal{L}U_q sl_2$. We first recall the definition of the quantum loop algebra $\mathcal{L}U_q sl_2$ corresponding to the presentation of Chevalley type. We refer the reader to Appendix A for definitions of the quantum algebra $U_q sl_2$ and the quantum affine algebra $U_q sl_2$.
**Definition 2.12.** The quantum loop algebra $\mathcal{L}U_q sl_2$ is the unital associative $\mathbb{C}$-algebra generated by the elements $E_i, F_i, K_i^{\pm \frac{1}{2}}$; $i \in \{0, 1\}$ which satisfy the defining relations of the $U_q \hat{sl}_2$ algebra (A.6)-(A.8) with the extra relation

$$ (2.32) \qquad K_0^{\frac{1}{2}} K_1^{\frac{1}{2}} = 1 = K_1^{\frac{1}{2}} K_0^{\frac{1}{2}}. $$

The quantum loop algebra $\mathcal{L}U_q sl_2$ is a Hopf algebra with the coproduct as in (A.9), counit in (A.10) and antipode (A.11) with the substitution $K_0^{\frac{1}{2}} = K_1^{-\frac{1}{2}}$.

Note that a presentation of $\mathcal{L}U_q sl_2$ 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 4.3.

The expression of the universal R-matrix associated to quantum affine algebras is well known. Khoroshkin and Tolstoy first proposed the universal R-matrix associated with the quantized Kac-Moody algebras in [KT92a, Thm. 2] and then the arguments for its explicit form in terms of root vectors for $U_q \hat{sl}_2$ were provided in [KT92b, eq. (58)]$^6$, see also [LSS93, Da98]. In our work, we consider the universal R-matrix $\mathfrak{R}$ associated with the Hopf algebra quotient $\mathcal{L}U_q sl_2$ and review its explicit form in our conventions in Appendix B. It is important to note that the coproduct convention we use is different to the one used in [KT92a]. The two coproducts and corresponding universal R-matrices are related via the automorphism $\nu$ from (A.12):

$$ (2.33) \qquad \Delta^{\mathrm{KT}} = (\nu \otimes \nu) \circ \Delta \circ \nu^{-1}, \quad (\nu^{-1} \otimes \nu^{-1})(\mathfrak{R}^{\mathrm{KT}}) = \mathfrak{R}, $$

where $\mathfrak{R}^{\mathrm{KT}}$ denotes the image of the universal R-matrix given in [KT92b, eq. (58)] under the quotient map to $\mathcal{L}U_q sl_2$, i.e. after setting the central charge to zero.

Let us introduce an automorphism $\eta$ of $\mathcal{L}U_q sl_2$:

$$ (2.34) \qquad \begin{aligned} \eta(E_0) &= F_1, & \eta(E_1) &= F_0, & \eta(K_0^{\frac{1}{2}}) &= K_1^{-\frac{1}{2}}, \\ \eta(F_1) &= E_0, & \eta(F_0) &= E_1, & \eta(K_1^{\frac{1}{2}}) &= K_0^{-\frac{1}{2}}. \end{aligned} $$

**Example 2.13.** For $H = \mathcal{L}U_q sl_2$, both the pairs $(\psi, J) = (\eta, 1 \otimes 1)$ and $(\psi, J) = (\eta, \mathfrak{R}_{21}\mathfrak{R})$, with the automorphism $\eta$ from (2.34), are examples of twist pairs from Definition 2.6. The first choice is the main example considered in this paper.

Let us prove the above statement for $(\psi, J) = (\eta, 1 \otimes 1)$, that is to verify (2.16). Using the definition of $\Delta$ given in (A.9), we get $\Delta^{\mathrm{op},\eta} = \Delta$ and therefore the first equation in (2.16) indeed holds. It remains to show that $\mathfrak{R}_{21}^{\eta\eta} = \mathfrak{R}$. From Lemma 2.3, the pair $(H^{\mathrm{cop},\eta}, \mathfrak{R}_{21}^{\eta\eta})$ satisfies (2.6)–(2.8) with $\psi = \eta$. Recall that an invertible 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 $\mathfrak{R}_{21}^{\eta\eta}$ is of the same form [KT92b, eq. (42)]. Because we have $\Delta^{\mathrm{op},\eta} = \Delta$, then the equations (2.6)–(2.8) for $\mathfrak{R}_{21}^{\eta\eta}$ become simply (R1)–(R3) with the substitution $\mathfrak{R} \to \mathfrak{R}_{21}^{\eta\eta}$. Therefore, as $\mathfrak{R}_{21}^{\eta\eta}$ satisfies the same equations as $\mathfrak{R}$, it follows from the uniqueness that they are equal and $(\eta, 1 \otimes 1)$ is thus a twist pair. Similarly, for the choice $J = \mathfrak{R}_{21}\mathfrak{R}$ using $\Delta_J = \Delta$ and $\mathfrak{R}_J = \mathfrak{R}$, recall (2.14) and (2.15), one shows that $(\eta, \mathfrak{R}_{21}\mathfrak{R})$ is also a twist pair.

<footnote>$^6$As the universal R-matrix $\mathfrak{R}$ of $U_q \hat{sl}_2$ has the form of a product over an infinite set of root vectors, and so strictly speaking $\mathfrak{R}$ is not an element of the tensor product $U_q \hat{sl}_2 \otimes U_q \hat{sl}_2$ but belongs to its appropriate completion, see e.g. [AV22, Sec. 2.5]. This fact however does not affect the use of the R- or K-matrix axioms.</footnote>
### 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 $\mathcal{L}U_q sl_2$. Recall the algebra $U_q sl_2$ defined in Appendix A. First, consider the algebra map

$$ (2.35) \qquad \varphi: \mathcal{L}U_q sl_2 \to U_q sl_2 $$

defined by the equations:

$$ (2.36) \qquad \begin{aligned} \varphi(E_0) &= F, & \varphi(F_0) &= E, & \varphi(K_0^{\frac{1}{2}}) &= K^{-\frac{1}{2}}, \\ \varphi(E_1) &= E, & \varphi(F_1) &= F, & \varphi(K_1^{\frac{1}{2}}) &= K^{\frac{1}{2}}. \end{aligned} $$

Let $\phi_u$ be the $\mathbb{Z}$-gradation automorphism of $\mathcal{L}U_q sl_2$

$$ (2.37) \qquad \phi_u: \mathcal{L}U_q sl_2 \to \mathcal{L}U_q sl_2, $$

with $u \in \mathbb{C}^*$, defined by

$$ (2.38) \qquad \begin{aligned} \phi_u(E_0) &= u^{-1}E_0, & \phi_u(F_0) &= uF_0, & \phi_u(K_0^{\frac{1}{2}}) &= K_0^{\frac{1}{2}}, \\ \phi_u(E_1) &= u^{-1}E_1, & \phi_u(F_1) &= uF_1, & \phi_u(K_1^{\frac{1}{2}}) &= K_1^{\frac{1}{2}}. \end{aligned} $$

Then the evaluation map $\mathrm{ev}_u^7$

$$ (2.39) \qquad \mathrm{ev}_u: \mathcal{L}U_q sl_2 \to U_q sl_2 $$

is defined by the composition

$$ (2.40) \qquad \mathrm{ev}_u = \varphi \circ \phi_u. $$

### 2.4.3. Comodule algebra $\mathcal{A}_q$.
Before giving our example of the comodule algebra $B = \mathcal{A}_q$ over $\mathcal{L}U_q sl_2$, which is a central extension of the $q$-Onsager algebra, we recall the definition of the latter. The $q$-Onsager algebra $O_q$ definition [T99, B04] is given in terms of two generators $W_0, W_1$ satisfying the so-called $q$-Dolan-Grady relations:

$$ (2.41) \qquad [W_0, [W_0, [W_0, W_1]]_{q^{-1}}] = \rho[W_0, W_1], $$

$$ (2.42) \qquad [W_1, [W_1, [W_1, W_0]]_{q^{-1}}] = \rho[W_1, W_0], $$

where we fix

$$ (2.43) \qquad \rho = k_+ k_- (q + q^{-1})^2, \quad \text{with } k_\pm \in \mathbb{C}^*. $$

The algebra $B = \mathcal{A}_q$ has first appeared in [BK05a, BS09] in the form of a reflection algebra that we review in Section 5. It was understood later on that $\mathcal{A}_q$ is isomorphic to the central extension of the $q$-Onsager algebra [BB17, T21a], namely to $O_q \otimes \mathbb{C}[Z]$ where $\mathbb{C}[Z] = \mathbb{C}[z_1, z_2, \dots]$ is a polynomial algebra with infinitely many indeterminates $z_i$, for $i \in \mathbb{N}$. Here, we recall a so-called *compact* presentation of this algebra.

**Definition 2.14** (see [T21b]). $\mathcal{A}_q$ is an associative algebra over $\mathbb{C}$ generated by $W_0, W_1, \{G_{k+1}\}_{k \in \mathbb{N}}$ subject to the following defining relations:

$$ [W_0, [W_0, [W_0, W_1]]_{q^{-1}}] = \rho[W_0, W_1], $$

$$ [W_1, [W_1, [W_1, W_0]]_{q^{-1}}] = \rho[W_1, W_0], $$

$$ [W_1, G_1] = [W_1, [W_1, W_0]]_q, $$

<footnote n="7">⁷For a more general evaluation map, see for instance [BGKNR12, eq. (4.32)]. In this paper we set $s_0 = s_1 = -1$.</footnote>
$$
\begin{align*}
[\mathbf{G}_1, \mathbf{W}_0] &= [[\mathbf{W}_1, \mathbf{W}_0]_q, \mathbf{W}_0], \\
[\mathbf{W}_1, \mathbf{G}_{k+1}] &= \rho^{-1}[\mathbf{W}_1, [\mathbf{W}_1, [\mathbf{W}_0, \mathbf{G}_k]_q]]_q, \quad k \ge 1, \\
[\mathbf{G}_{k+1}, \mathbf{W}_0] &= \rho^{-1}[[[\mathbf{G}_k, \mathbf{W}_1]_q, \mathbf{W}_0]]_q, \quad k \ge 1, \\
[\mathbf{G}_{k+1}, \mathbf{G}_{\ell+1}] &= 0, \quad k, \ell \in \mathbb{N},
\end{align*}
$$

where $\rho$ is given by (2.43).

We discuss the coaction map $\delta: \mathcal{A}_q \to \mathcal{A}_q \otimes \mathcal{L}U_q sl_2$ using the reflection algebra presentation of $\mathcal{A}_q$ in Section 6.2. The corresponding coaction of the fundamental generators $W_0, W_1$ takes the form:

$$
(2.44) \qquad \delta(W_0) = 1 \otimes \left( k_+ q^{\frac{1}{2}} E_1 K_1^{\frac{1}{2}} + k_- q^{-\frac{1}{2}} F_1 K_1^{\frac{1}{2}} \right) + W_0 \otimes K_1,
$$

$$
(2.45) \qquad \delta(W_1) = 1 \otimes \left( k_+ q^{-\frac{1}{2}} F_0 K_0^{\frac{1}{2}} + k_- q^{\frac{1}{2}} E_0 K_0^{\frac{1}{2}} \right) + W_1 \otimes K_0.
$$

In this paper, it will be sufficient to consider the evaluation of the coaction map $\delta$, denoted by $\delta_w$:

$$
(2.46) \qquad \delta_w = (\mathrm{id} \otimes \mathrm{ev}_w) \circ \delta: \mathcal{A}_q \to \mathcal{A}_q \otimes U_q sl_2.
$$

The proof of the following proposition is postponed to the end of Section 5.5.

**Proposition 2.15.** The evaluated coaction map $\delta_w: \mathcal{A}_q \to \mathcal{A}_q \otimes U_q sl_2$ is such that:

$$
(2.47) \qquad \delta_w(\mathbf{W}_0) = 1 \otimes (k_+ q^{\frac{1}{2}} w^{-1} E K^{\frac{1}{2}} + k_- q^{-\frac{1}{2}} w F K^{\frac{1}{2}}) + \mathbf{W}_0 \otimes K,
$$

$$
(2.48) \qquad \delta_w(\mathbf{W}_1) = 1 \otimes (k_+ q^{-\frac{1}{2}} w E K^{-\frac{1}{2}} + k_- q^{\frac{1}{2}} w^{-1} F K^{-\frac{1}{2}}) + \mathbf{W}_1 \otimes K^{-1},
$$

$$
(2.49) \quad \delta_w(\mathbf{G}_{k+1}) = & \frac{k_- (q-q^{-1})^2}{k_+ (q+q^{-1})} (\mathbf{G}_k - \frac{(q+q^{-1})}{\rho} [\mathbf{W}_0, [\mathbf{W}_0, \mathbf{G}_k]_q]) \otimes F^2 \\
& + \frac{(q-q^{-1})}{k_+(q+q^{-1})} (q^{-\frac{1}{2}} w^{-1} [\mathbf{W}_0, \mathbf{G}_{k+1}]_q \otimes F K^{\frac{1}{2}} + q^{\frac{1}{2}} w [\mathbf{G}_{k+1}, \mathbf{W}_1]_q \otimes F K^{-\frac{1}{2}}) \\
& + \mathbf{G}_{k+1} \otimes 1 - \frac{\mathbf{G}_k}{q+q^{-1}} \otimes (w^2 K^{-1} + w^{-2} K),
$$

with initial condition:

$$
\mathbf{G}_k|_{k=0} = k_+ k_- \frac{(q+q^{-1})^2}{q-q^{-1}}.
$$

3. TENSOR PRODUCT REPRESENTATIONS OF $\mathcal{L}U_q sl_2$ AND SUB-REPRESENTATIONS

Our fusion procedure for K-operators requires first analysis of the tensor product of evaluation
representations of $\mathcal{L}U_q sl_2$. 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 in-
tertwining operators. We do it first in Section 3.2 for finite-dimensional evaluation representations,
and then in Section 3.5 for their infinite-dimensional analogues with formal evaluation parameters.
These operators are used in further sections to built fused L- and K-operators for any spin-$j$ starting
from the fundamental L- and K-operators.
First, recall that finite-dimensional irreducible representations of $U_q sl_2$ are labeled 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 $|j, m\rangle$ with $m \in \{-j, -j + 1, \dots, j - 1, j\}$, then

$$ (3.1) \qquad E|j, m\rangle = A_{j,m}|j, m + 1\rangle, \quad F|j, m\rangle = B_{j,m}|j, m - 1\rangle, \quad K^{\pm \frac{1}{2}}|j, m\rangle = q^{\pm m}|j, m\rangle, $$

with

$$ (3.2) \qquad A_{j,m} = \sqrt{[j-m]_q [j+m+1]_q}, \quad B_{j,m} = \sqrt{[j+m]_q [j-m+1]_q}. $$

Let $\pi^j$ be the representation map of $U_q sl_2$:

$$ (3.3) \qquad \pi^j: U_q sl_2 \to \mathrm{End}(\mathbb{C}^{2j+1}). $$

Now, given $u \in \mathbb{C}^*$, we then define the evaluation representations $\pi_u^j: \mathcal{L}U_q sl_2 \to \mathrm{End}(\mathbb{C}^{2j+1})$ by

$$ (3.4) \qquad \pi_u^j = \pi^j \circ \mathrm{ev}_u, $$

where $\mathrm{ev}_u$ is defined in (2.40), and the corresponding $\mathcal{L}U_q sl_2$-module by $\mathbb{C}_u^{2j+1}$. We study now tensor products of these representations

$$ (3.5) \qquad (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^{j}) \circ \Delta: \mathcal{L}U_q sl_2 \to \mathrm{End}(\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1}), $$

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 $\{w_k\}$, $\{v_\ell\}$ for the decomposition with respect to the subalgebra generated by $\{E_1, F_1, K_1\}$. Then, we study the action of $\{E_0, F_0, K_0\}$ 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 $\mathcal{L}U_q sl_2$.** Consider first the subalgebra generated by $\{E_1, F_1, K_1\}$ and construct basis vectors $\{w_k\}$, $\{v_\ell\}$, where $k = 0, 1, \dots, 2j+1$, $\ell = 0, 1, \dots, 2j-1$ and $j \in \frac{1}{2}\mathbb{N}_+$, corresponding to the tensor product decomposition $\mathbb{C}^2 \otimes \mathbb{C}^{2j+1} = \mathbb{C}^{2j+2} \oplus \mathbb{C}^{2j}$. We denote by $w_0$ and $v_0$ the highest weight vectors of the corresponding spins-$(j + \frac{1}{2})$ and $(j - \frac{1}{2})$. These are defined by the relations

$$ (3.6) \qquad [ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^{j}) \Delta(E_1) ] w_0 = 0, \qquad [ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^{j}) \Delta(K_1) ] w_0 = q^{2j+1} w_0, $$

$$ (3.7) \qquad [ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^{j}) \Delta(E_1) ] v_0 = 0, \qquad [ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^{j}) \Delta(K_1) ] v_0 = q^{2j-1} v_0. $$

Solutions to these equations are uniquely determined, up to a scalar, by

$$ (3.8) \qquad w_0 = |\uparrow\rangle \otimes |j, j\rangle, \quad v_0 = |\uparrow\rangle \otimes |j, j-1\rangle - \frac{u_1}{u_2} q^{-j-\frac{1}{2}} A_{j,j-1} |\downarrow\rangle \otimes |j, j\rangle, $$

with $|\uparrow\rangle = |\frac{1}{2}, \frac{1}{2}\rangle$, $|\downarrow\rangle = |\frac{1}{2}, -\frac{1}{2}\rangle$, and where $A_{j,m}$ is given in (3.2). The other basis vectors are constructed via the action of $F_1$:

$$ (3.9) \qquad w_k = \left[ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^{j}) \Delta(F_1) \right]^k w_0, \quad v_\ell = \left[ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^{j}) \Delta(F_1) \right]^\ell v_0, $$

where $0 \le k \le 2j + 1$ and $0 \le \ell \le 2j - 1$, and we set these vectors to zero if their index is outside of the indicated range.
Now, we study the action of the generators $E_0$ and $F_0$ on these basis vectors, and we begin with the action on $w_k$'s:

$$ (3.10) \qquad \begin{bmatrix} (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j) \Delta(E_0) \\ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j) \Delta(F_0) \end{bmatrix} w_k = e_{1,k}(u_1, u_2)w_{k+1} + e_{2,k}(u_1, u_2)v_k, \\ w_k = f_{1,k}(u_1, u_2)w_{k-1} + f_{2,k}(u_1, u_2)v_{k-2}, $$

where the coefficients are

$$ (3.11) \qquad e_{1,k}(u_1, u_2) = u_1^{-2} + [2j]_q u_2^{-2}, \qquad e_{2,k}(u_1, u_2) = \frac{q^{2j+\frac{1}{2}}(u_1^2 - u_2^2 q^{-2j-1})\sqrt{[2j]_q}}{u_1^2 u_2 [2j+1]_q}, $$

$$ (3.12) \qquad f_{1,k}(u_1, u_2) = (u_1^2 + u_2^2 [2j]_q) \frac{[2j-k+2]_q [k]_q}{[2j+1]_q}, \qquad f_{2,k}(u_1, u_2) = \frac{(u_1^2 - u_2^2 q^{-2j-1}) \sqrt{[2j]_q} [k]_q [k-1]_q}{u_2^{-1} q^{-2j-\frac{1}{2}} [2j+1]_q}. $$

We next look at the action on $v_\ell$'s:

$$ (3.13) \qquad \begin{bmatrix} (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j) \Delta(E_0) \\ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j) \Delta(F_0) \end{bmatrix} v_\ell = \bar{e}_{1,\ell}(u_1, u_2)v_{\ell+1} + \bar{e}_{2,\ell}(u_1, u_2)w_{\ell+2}, \\ v_\ell = \bar{f}_{1,\ell}(u_1, u_2)v_{\ell-1} + \bar{f}_{2,\ell}(u_1, u_2)w_\ell, $$

where the coefficients are

$$ (3.14) \qquad \bar{e}_{1,\ell}(u_1, u_2) = \frac{u_2^{-2}[2j+2]_q - u_1^{-2}}{[2j+1]_q}, \qquad \bar{e}_{2,\ell}(u_1, u_2) = \frac{u_1^2 - u_2^2 q^{2j+1}}{q^{2j+\frac{1}{2}} u_1^2 u_2^3 \sqrt{[2j]_q [2j+1]_q}}, $$

$$ (3.15) \qquad \bar{f}_{1,\ell}(u_1, u_2) = (u_1^2 - u_2^2[2j+2]_q) \frac{[\ell-2j]_q[\ell]_q}{[2j+1]_q}, \qquad \bar{f}_{2,\ell}(u_1, u_2) = \frac{(u_1^2 - u_2^2 q^{2j+1})[\ell-2j]_q[2j+1-\ell]_q}{u_2 q^{2j+\frac{1}{2}} \sqrt{[2j]_q [2j+1]_q}}. $$

We first note that in (3.10) and (3.13) the coefficients with indices 2,k and 2,ℓ, respectively, correspond to contribution in the action that mixes the two spin components. In order to find proper sub-representations, we thus need to analyze the roots of these functions $e_{2,k}$, $f_{2,k}$, $\bar{e}_{2,\ell}$ and $\bar{f}_{2,\ell}$. First, for generic values of $u_1/u_2$ these functions do not vanish which corresponds to the well known fact that this tensor product is irreducible. For later convenience, we present this action diagrammatically below in Figure 1 where the red (resp. blue) arrows correspond to the action of $F_0$ (resp. $E_0$), and the branching points correspond to the linear combinations from (3.10) and (3.13).
<figure><img src="image_2.png" /><figcaption>FIGURE 1. Action of $E_i$, $F_i$ on $w_k$, $v_\ell$ for generic values of $u_1$ and $u_2$.</figcaption></figure>

From the expression of the coefficients (3.11) and (3.12), one finds that

$$e_{2,k}(u_1, u_2) = f_{2,k}(u_1, u_2) = 0 \quad \text{iff} \quad u_1/u_2 = \pm q^{-j-\frac{1}{2}}$$

and

$$\bar{e}_{2,\ell}(u_1, u_2) = \bar{f}_{2,\ell}(u_1, u_2) = 0 \quad \text{iff} \quad u_1/u_2 = \pm q^{j+\frac{1}{2}}.$$

Bold arrows in Figure 1 are used to emphasize that they disappear after fixing the ratio $u_1/u_2 = \pm q^{-j-\frac{1}{2}}$. 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 $u_1/u_2$ to the special values we have a sub-representation as depicted below in Figure 2 by the dotted rectangles.

<figure><img src="image_3.png" /><figcaption>FIGURE 2. Action of $E_i$, $F_i$ on $w_k$, $v_\ell$ for fixed values of $u_1/u_2$.</figcaption></figure>

We thus find that fixing $u_1/u_2 = \pm q^{-j-\frac{1}{2}}$ (resp. $u_1/u_2 = \pm q^{j+\frac{1}{2}}$) gives a spin-$(-j+\frac{1}{2})$ (resp. spin-$(-j-\frac{1}{2})$) sub-representation.
3.2. **Intertwining map $\mathcal{E}^{(j+\frac{1}{2})}$ and its pseudo-inverse $\mathcal{F}^{(j+\frac{1}{2})}$.** We now study the spin-(j + 1/2) sub-representation when $u_1/u_2 = q^{-j-1/2}$, and construct the corresponding intertwining operator explicitly. Introduce two linear operators $\mathcal{E}^{(j+\frac{1}{2})}$ and $\mathcal{F}^{(j+\frac{1}{2})}$ for any $j \in \frac{1}{2}\mathbb{N}$ as

$$ (3.16) \qquad \mathcal{E}^{(j+\frac{1}{2})} : \mathbb{C}_u^{2j+2} \to \mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1}, $$

$$ (3.17) \qquad \mathcal{F}^{(j+\frac{1}{2})} : \mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1} \to \mathbb{C}_u^{2j+2}, $$

given by:

$$ (3.18) \qquad \mathcal{E}^{(j+\frac{1}{2})} = \sum_{a=1}^{4j+2} \sum_{b=1}^{2j+2} \mathcal{E}_{a,b}^{(j+\frac{1}{2})} E_{a,b}^{(2j,j)}, \qquad \mathcal{F}^{(j+\frac{1}{2})} \mathcal{E}^{(j+\frac{1}{2})} = \mathbb{I}_{2j+2}, $$

where $\mathcal{E}_{a,b}^{(j+\frac{1}{2})}$ are certain scalars, $E_{a,b}^{(j_1,j_2)}$ denotes the matrix of dimension $(2j_1+2) \times (2j_2+2)$ with 1 at position $(a,b)$ and 0 otherwise. Here, we choose the bases of the source $\{|j+\frac{1}{2}, m\rangle\}$ with $m = j+\frac{1}{2}, j-\frac{1}{2}, \dots, -j+\frac{1}{2}, -j-\frac{1}{2}$ and the target $\{|\uparrow\rangle \otimes |j, j\rangle, \dots, |\uparrow\rangle \otimes |j, -j\rangle, |\downarrow\rangle \otimes |j, j\rangle, \dots, |\downarrow\rangle \otimes |j, -j\rangle\}$.

We now calculate the coefficients $\mathcal{E}_{a,b}^{(j+\frac{1}{2})}$ from (3.18) provided $\mathcal{E}^{(j+\frac{1}{2})}$ is a $\mathcal{L}U_q sl_2$-intertwiner for the condition $u_1/u_2 = q^{-j-1/2}$ and $u_2 = uq^{1/2}$, that was found in the previous subsection for $j \in \frac{1}{2}\mathbb{N}_+$. First of all for $j=0$, we have for any $u$ that $\pi_u^0 = \epsilon$, where the counit is defined in (A.10), i.e. the trivial representation of $\mathcal{L}U_q sl_2$. Then identifying $\mathbb{C}^2 \otimes \mathbb{C}$ with $\mathbb{C}^2$, it follows from (3.16), (3.17) and (3.18) that $\mathcal{E}^{(\frac{1}{2})} = \mathcal{F}^{(\frac{1}{2})} = \mathbb{I}_2$.

**Lemma 3.1.** Let $u_1/u_2 = q^{-j-1/2}$ and $u_2 = uq^{1/2}$, then the map $\mathcal{E}^{(j+\frac{1}{2})}$ in (3.18) is a $\mathcal{L}U_q sl_2$-intertwiner

$$ (3.19) \qquad \mathcal{E}^{(j+\frac{1}{2})}(\pi_u^{j+\frac{1}{2}})(x) = (\pi_{uq^{-j}}^{\frac{1}{2}} \otimes \pi_{uq^{\frac{1}{2}}}^{j})(\Delta(x))\mathcal{E}^{(j+\frac{1}{2})}, \quad \forall x \in \mathcal{L}U_q sl_2, $$

if and only if its entries are given for any $j \in \frac{1}{2}\mathbb{N}_+$ by

$$ (3.20) \qquad \mathcal{E}_{1,1}^{(j+\frac{1}{2})} = 1, \qquad \mathcal{E}_{1+n,1+n}^{(j+\frac{1}{2})} = \prod_{p=0}^{n-1} \frac{B_{j,j-p}}{B_{j+\frac{1}{2},j+\frac{1}{2}-p}}, \qquad \mathcal{E}_{2j+1+m,1+m}^{(j+\frac{1}{2})} = [m]_q \frac{\mathcal{E}_{m,m}^{(j+\frac{1}{2})}}{B_{j+\frac{1}{2},j+\frac{3}{2}-m}}, $$

where $n = 1, 2, \dots, 2j, m = 1, 2, \dots, 2j+1$ and $B_{j,m}$ is given in (3.2), and all the other entries are zero.

**Proof.** Recall that for $u_1 = u_2 q^{-j-1/2}$ we have a spin-(j + 1/2) sub-representation as depicted in left-top of Figure 2, and therefore we also have a corresponding intertwining operator that we are going to construct. In this case, the sub-representation has the basis $\{w_k | 0 \le k \le 2j+1\}$ given in (3.9). Let us introduce a basis $\{w'_k | 0 \le k \le 2j+1\}$ in the source of the map $\mathcal{E}^{(j+\frac{1}{2})}$: we define $w'_0 = |j+\frac{1}{2}, j+\frac{1}{2}\rangle$ and $w'_k = [\pi_u^{j+\frac{1}{2}}(F_1)]^k w'_0$. The intertwining property reads

$$ (3.21) \qquad \mathcal{E}^{(j+\frac{1}{2})} \left[ \pi_u^{j+\frac{1}{2}}(x) \right] w'_k = \left[ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j)(\Delta(x)) \right] \mathcal{E}^{(j+\frac{1}{2})}(w'_k), \quad \forall x \in \mathcal{L}U_q sl_2. $$
This equation for $x = K_0, K_1$ and $k = 0$ gives $\mathcal{E}^{(j+\frac{1}{2})}(w'_0) = w_0$. Then for $k = 0$ and $x = (F_1)^n$, with $n = 1, 2, \dots, 2j+1$, one shows

$$ (3.22) \qquad \mathcal{E}^{(j+\frac{1}{2})}(w'_n) = w_n, \quad \text{for all } n. $$

Using this, we then check directly that (3.21) indeed holds now for all $k$ and $x = F_1$ and $x = E_1$. It remains to check that (3.21) is satisfied for the action of $E_0$ and $F_0$. By straightforward calculations using (2.38), (3.4) and (A.9), one gets:

$$ (3.23) \qquad \left[ \pi_u^{j+\frac{1}{2}}(E_0) \right] w'_k = u^{-2} w'_{k+1}, \qquad \left[ \pi_u^{j+\frac{1}{2}}(F_0) \right] w'_k = u^2 w'_{k-1}, $$

$$ (3.24) \qquad \left[ (\pi_{\tilde{u}_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j) \Delta(E_0) \right] w_k = q u_2^{-2} w_{k+1}, \qquad \left[ (\pi_{\tilde{u}_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j) \Delta(F_0) \right] w_k = q^{-1} u_2^2 w_{k-1}. $$

Using (3.23) and (3.24), one finds that (3.21) holds for $u_2 = uq^{\frac{1}{2}}$. Finally, we get the matrix elements of $\mathcal{E}^{(j+\frac{1}{2})}$. The basis vectors read explicitly:

$$ w_k = u^k \left( \prod_{p=0}^{k-1} B_{j,j-p} |\uparrow\rangle \otimes |j, j-k\rangle + [k]_q \prod_{p=0}^{k-2} B_{j,j-p} |\downarrow\rangle \otimes |j, j-k+1\rangle \right), $$

$$ w'_k = u^k \prod_{p=0}^{k-1} B_{j+\frac{1}{2}, j+\frac{1}{2}-p} |j+\frac{1}{2}, j+\frac{1}{2}-k\rangle, $$

and we set $\prod_{p=0}^n B_{j,j-p} = 1$ if $n$ is negative. Then solving (3.22) for $\mathcal{E}^{(j+\frac{1}{2})}$ in (3.18), one gets (3.20)

(recall that the basis used in (3.18) is $|j+\frac{1}{2}, m\rangle$ with $m = j+\frac{1}{2}, j-\frac{1}{2}, \dots, -j+\frac{1}{2}, -j-\frac{1}{2}$, and not $w'_k$). $\square$

We now give an expression of $\mathcal{F}^{(j+\frac{1}{2})}$ which is a pseudo-inverse of $\mathcal{E}^{(j+\frac{1}{2})}$. It takes the form:

$$ (3.25) \qquad \mathcal{F}^{(j+\frac{1}{2})} = \sum_{a=1}^{2j+2} \sum_{b=1}^{4j+2} \mathcal{F}_{a,b}^{(j+\frac{1}{2})} E_{a,b}^{(j,2j)}, $$

where $\mathcal{F}_{a,b}^{(j+\frac{1}{2})}$ are scalars. The solution of $\mathcal{F}^{(j+\frac{1}{2})}\mathcal{E}^{(j+\frac{1}{2})} = \mathbb{I}_{2j+2}$ is not unique. For instance, we fix the entries of $\mathcal{F}^{(j+\frac{1}{2})}$ for $n = 2, 3, \dots, 2j+1$ as follows:

$$ (3.26) \qquad \mathcal{F}_{1,1}^{(j+\frac{1}{2})} = 1, \qquad \mathcal{F}_{n,n+2j}^{(j+\frac{1}{2})} = \frac{\mathcal{E}_{n+2j,n}^{(j+\frac{1}{2})}}{(\mathcal{E}_{n,n}^{(j+\frac{1}{2})})^2 + (\mathcal{E}_{n+2j,n}^{(j+\frac{1}{2})})^2}, $$

$$ (3.27) \qquad \mathcal{F}_{2j+2,4j+2}^{(j+\frac{1}{2})} = (\mathcal{E}_{4j+2,2j+2}^{(j+\frac{1}{2})})^{-1}, \qquad \mathcal{F}_{n,n}^{(j+\frac{1}{2})} = \frac{1 - \mathcal{F}_{n,n+2j}^{(j+\frac{1}{2})} \mathcal{E}_{n+2j,n}^{(j+\frac{1}{2})}}{\mathcal{E}_{n,n}^{(j+\frac{1}{2})}}, $$

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 $\mathcal{E}^{(j+\frac{1}{2})}$, in particular the one given above, is not a $\mathcal{L}U_q sl_2$-intertwiner because the sub-representation involved is not a direct summand, recall the structure in Figure 2.
3.3. The maps $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$ and $\bar{\mathcal{F}}^{(j-\frac{1}{2})}$. We now study the spin-$\left(j-\frac{1}{2}\right)$ sub-representation when $u_1/u_2 = q^{j+\frac{1}{2}}$. Introduce the two maps $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$ and $\bar{\mathcal{F}}^{(j-\frac{1}{2})}$ for any $j \in \frac{1}{2}\mathbb{N}_+$:

$$ (3.28) \qquad \bar{\mathcal{E}}^{(j-\frac{1}{2})} : \mathbb{C}_u^{2j} \to \mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1}, $$

$$ (3.29) \qquad \bar{\mathcal{F}}^{(j-\frac{1}{2})} : \mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1} \to \mathbb{C}_u^{2j}, $$

given by:

$$ (3.30) \qquad \bar{\mathcal{E}}^{(j-\frac{1}{2})} = \sum_{a=1}^{4j+2} \sum_{b=1}^{2j} \bar{\mathcal{E}}_{a,b}^{(j-\frac{1}{2})} E_{a,b}^{(2j,j-1)}, \qquad \bar{\mathcal{F}}^{(j-\frac{1}{2})} \bar{\mathcal{E}}^{(j-\frac{1}{2})} = \mathbb{I}_{2j}, $$

where $\bar{\mathcal{E}}_{a,b}^{(j-\frac{1}{2})}$ are certain scalars. The bases of the source and the target of $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$ are respectively $\{|j-\frac{1}{2}, m\rangle\}$ with $m = j-\frac{1}{2}, j-\frac{3}{2}, \dots, -j+\frac{3}{2}, -j+\frac{1}{2}$ and $\{|\uparrow\rangle \otimes |j, j\rangle, \dots, |\uparrow\rangle \otimes |j, -j\rangle, |\downarrow\rangle \otimes |j, j\rangle, \dots, |\downarrow\rangle \otimes |j, -j\rangle\}$.

**Lemma 3.2.** Let $u_1/u_2 = q^{j+\frac{1}{2}}$ and $u_2 = uq^{\frac{1}{2}}$, then the map $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$ in (3.30) is a $\mathcal{L}U_q sl_2$-intertwiner

$$ (3.31) \qquad \bar{\mathcal{E}}^{(j-\frac{1}{2})}(\pi_u^{j-\frac{1}{2}})(x) = (\pi_{uq^{j+1}}^{\frac{1}{2}} \otimes \pi_{uq^{\frac{1}{2}}}^{j})(\Delta(x))\bar{\mathcal{E}}^{(j-\frac{1}{2})}, \quad \forall x \in \mathcal{L}U_q sl_2, $$

if and only if its entries are given for any $j \in \frac{1}{2}\mathbb{N}_+$ by

$$ (3.32) \qquad \bar{\mathcal{E}}_{2,1}^{(j-\frac{1}{2})} = 1, \qquad \bar{\mathcal{E}}_{2+n,1+n}^{(j-\frac{1}{2})} = \prod_{p=0}^{n-1} \frac{B_{j,j-p-1}}{B_{j-\frac{1}{2},j-\frac{1}{2}-p}}, \qquad \bar{\mathcal{E}}_{2j+2+m,1+m}^{(j-\frac{1}{2})} = \frac{[m-2j]_q}{B_{j,j-m}} \bar{\mathcal{E}}_{2+m,1+m}^{(j-\frac{1}{2})}, $$

where $n = 1, 2, \dots, 2j-1$, $m = 0, 1, \dots, 2j-1$ and $B_{j,m}$ is given in (3.2), and all the other entries are zero.

**Proof.** Recall that for $u_1 = u_2 q^{j+\frac{1}{2}}$ we have a spin-$\left(j-\frac{1}{2}\right)$ sub-representation as depicted in right-bottom of Figure 2. We also have a corresponding intertwining operator $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$ that we now construct. In this case, the sub-representation has the basis $\{v_\ell | 0 \le \ell \le 2j-1\}$ given in (3.9). Let us introduce a basis $\{v'_k | 0 \le k \le 2j-1\}$ in the source of the map $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$: we define $v'_0 = |j-\frac{1}{2}, j-\frac{1}{2}\rangle$ and $v'_k = [\pi_u^{j-\frac{1}{2}}(F_1)]^k v'_0$. Analogously to the proof of Lemma 3.1, one shows $\bar{\mathcal{E}}^{(j-\frac{1}{2})}(v'_\ell) = v_\ell$. Then, one finds the constraint on the ratio $u/u_2$ and the coefficients (3.32) are obtained by solving the latter equation for $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$ in (3.30). $\square$

We now give expression of $\bar{\mathcal{F}}^{(j-\frac{1}{2})}$ which is a pseudo-inverse of $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$. It takes the form:

$$ (3.33) \qquad \bar{\mathcal{F}}^{(j-\frac{1}{2})} = \sum_{a=1}^{2j} \sum_{b=1}^{4j+2} \bar{\mathcal{F}}_{a,b}^{(j-\frac{1}{2})} E_{a,b}^{(j-1,2j)}, $$

where $\bar{\mathcal{F}}_{a,b}^{(j-\frac{1}{2})}$ are scalars. The solution of $\bar{\mathcal{F}}^{(j-\frac{1}{2})}\bar{\mathcal{E}}^{(j-\frac{1}{2})} = \mathbb{I}_{2j}$ is not unique. Similarly to the fusion case above, we fix the entries of $\bar{\mathcal{F}}^{(j-\frac{1}{2})}$ for $n=1, 2, \dots, 2j$ this way:

$$ (3.34) \qquad \bar{\mathcal{F}}_{n,n+2j+1}^{(j-\frac{1}{2})} = \frac{\bar{\mathcal{E}}_{n+2j+1,n}^{(j-\frac{1}{2})}}{(\bar{\mathcal{E}}_{n+1,n}^{(j-\frac{1}{2})})^2 + (\bar{\mathcal{E}}_{n+2j+1,n}^{(j-\frac{1}{2})})^2}, \qquad \bar{\mathcal{F}}_{n,n+1}^{(j-\frac{1}{2})} = \frac{1 - \bar{\mathcal{F}}_{n,n+2j+1}^{(j-\frac{1}{2})} \bar{\mathcal{E}}_{n+2j+1,n}^{(j-\frac{1}{2})}}{\bar{\mathcal{E}}_{n+1,n}^{(j-\frac{1}{2})}}, $$
and all other entries are zero. We note that, similarly to the previous case, $\bar{\mathcal{F}}^{(j-\frac{1}{2})}$ 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 $\mathcal{L}U_q sl_2$ admits a non-trivial sub-representation either of spin-$\left(j+\frac{1}{2}\right)$ or of spin-$\left(j-\frac{1}{2}\right)$. And we have constructed intertwining operators $\mathcal{E}^{(j+\frac{1}{2})}: \mathbb{C}^2 \otimes \mathbb{C}^{2j+1} \to \mathbb{C}^{2j+2}$ and $\bar{\mathcal{E}}^{(j-\frac{1}{2})}: \mathbb{C}^2 \otimes \mathbb{C}^{2j+1} \to \mathbb{C}^{2j}$, and their pseudo-inverses $\mathcal{F}^{(j+\frac{1}{2})}$ and $\bar{\mathcal{F}}^{(j-\frac{1}{2})}$, respectively.

In what follows, we will need action on tensor product using opposite coproduct. For this new
action$^8$, the corresponding intertwining operators appear at different evaluation parameters.

**Remark 3.3.** Consider the opposite coproduct $\Delta^{op} = \mathfrak{p} \circ \Delta$ with the definition for $\Delta$ in (A.9). Using now the action of $\mathcal{L}U_q sl_2$ on the tensor product given by $\Delta^{op}$, we repeat the sub-representation analysis from Section 3.1 using the corresponding basis $\{\tilde{w}_k | 0 \le k \le 2j+1\}$ and $\{\tilde{v}_\ell | 0 \le \ell \le 2j-1\}$ defined by

$$ \tilde{w}_k = \left[ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j) \Delta^{op}(F_1) \right]^k \tilde{w}_0, \quad \tilde{v}_\ell = \left[ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j) \Delta^{op}(F_1) \right]^\ell \tilde{v}_0, $$

with $\tilde{w}_0 = w_0$ in (3.8), and $\tilde{v}_0$ is the solution of (3.7) with the substitution $\Delta \to \Delta^{op}$

$$ \tilde{v}_0 = |\uparrow\rangle \otimes |j, j-1\rangle - \frac{u_1}{u_2} q^{j+\frac{1}{2}} A_{j, j-1} |\downarrow\rangle \otimes |j, j\rangle . $$

Then, we find that the conditions on the evaluations parameter are different. Indeed, we have for
the spin-$\left(j+\frac{1}{2}\right)$ sub-representation:

$$ (3.35) \qquad u_1/u_2 = \pm q^{j+\frac{1}{2}}, \qquad u = u_2 q^{\frac{1}{2}}, $$

and for the spin-$\left(j-\frac{1}{2}\right)$ sub-representation:

$$ (3.36) \qquad u_1/u_2 = \pm q^{-j-\frac{1}{2}}, \qquad u = u_2 q^{\frac{1}{2}}. $$

Nevertheless, it leads to the intertwining operator $\mathcal{E}^{(j+\frac{1}{2})}$ (resp. $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$) with the same matrix elements
as in (3.20) (resp. in (3.32)). Indeed, the matrix elements are invariant under the replacement of
$q$ by $q^{-1}$. Then, the corresponding intertwining properties for the conditions (3.35), (3.36) and the
choice of the positive sign read respectively for all $x \in \mathcal{L}U_q sl_2$:

$$ (3.37) \qquad \mathcal{E}^{(j+\frac{1}{2})}(\pi_u^{j+\frac{1}{2}})(x) = (\pi_{uq^j}^{\frac{1}{2}} \otimes \pi_{uq^{-\frac{1}{2}}}^j)(\Delta^{op}(x))\mathcal{E}^{(j+\frac{1}{2})}, $$

$$ (3.38) \qquad \bar{\mathcal{E}}^{(j-\frac{1}{2})}(\pi_u^{j-\frac{1}{2}})(x) = (\pi_{uq^{-j-1}}^{\frac{1}{2}} \otimes \pi_{uq^{-\frac{1}{2}}}^j)(\Delta^{op}(x))\bar{\mathcal{E}}^{(j-\frac{1}{2})}. $$

**Remark 3.4.** Let us mention that in the literature there are different conventions for the coproduct of $\mathcal{L}U_q sl_2$. For example, consider the coproduct in [BGKNR10]

$$ (3.39) \qquad \Delta^{KT}(E_i) = E_i \otimes 1 + K_i^{-1} \otimes E_i, \qquad \Delta^{KT}(F_i) = F_i \otimes K_i + 1 \otimes F_i, $$

<footnote>$^8$Recall that for a bialgebra $H$, we can define another bialgebra $H^{cop}$ with the coproduct $\Delta^{op}$. Therefore, $\Delta^{op}$ also defines an action of the algebra $H$ on the tensor product of $H$-modules.</footnote>
with the representation of $U_q s l_2$

$$ (3.40) \qquad \begin{aligned} E |j, m\rangle &= |j, m+1\rangle, \\ F |j, m\rangle &= [j+m]_q [j-m+1]_q |j, m-1\rangle, \\ K^{\pm \frac{1}{2}} |j, m\rangle &= q^{\pm m} |j, m\rangle. \end{aligned} $$

For the coproduct (3.39), the values of $u, u_1$ and $u_2$ coincide with Lemma 3.1. However, the expressions of $\mathcal{E}^{(j+\frac{1}{2})}$, $\mathcal{F}^{(j+\frac{1}{2})}$ are different.

3.4. **Additional properties.** We conclude this section with a few observations on relations between the intertwining operator $\mathcal{E}^{(j+\frac{1}{2})}$, its pseudo-inverse $\mathcal{F}^{(j+\frac{1}{2})}$ and the R-matrix. In the literature the expression of the R-matrix $R^{(\frac{1}{2},j)}(u) \in \text{End}(\mathbb{C}^2 \otimes \mathbb{C}^{2j+1})$ is known [KR83, DN02]. It reads:

$$ (3.41) \qquad R^{(\frac{1}{2},j)}(u) = \prod_{k=0}^{2j-2} c(uq^{j-\frac{1}{2}-k}) \times \left( (q-q^{-1}) (\sigma_+ \otimes \pi^j(F) + \sigma_- \otimes \pi^j(E)) + uq^{\frac{1}{2}(\mathbb{I}_{4j+2} + \sigma_z \otimes \pi^j(H))} - u^{-1}q^{-\frac{1}{2}(\mathbb{I}_{4j+2} + \sigma_z \otimes \pi^j(H))} \right), $$

where $\pi^j(E), \pi^j(F)$ are given in (3.1), $(\pi^j(H))_{mn} = 2(j+1-n)\delta_{m,n}$, with $m, n = 1, 2, \dots, 2j+1$, and where we use the scalar function

$$ (3.42) \qquad c(u) = u - u^{-1}. $$

Note that this R-matrix satisfies the unitarity property

$$ (3.43) \qquad R^{(\frac{1}{2},j)}(u)R^{(\frac{1}{2},j)}(u^{-1}) = \left( \prod_{k=0}^{2j-1} c(uq^{j+\frac{1}{2}-k})c(u^{-1}q^{j+\frac{1}{2}-k}) \right) \mathbb{I}_{4j+2}. $$

Let $\mathcal{H}^{(j+\frac{1}{2})}$ and $\bar{\mathcal{H}}^{(j-\frac{1}{2})}$ be invertible diagonal matrices given by:

$$ (3.44) \qquad \mathcal{H}^{(j+\frac{1}{2})} = \text{Diag}(\mathcal{H}_1^{(j+\frac{1}{2})}, \mathcal{H}_2^{(j+\frac{1}{2})}, \dots, \mathcal{H}_{2j+2}^{(j+\frac{1}{2})}), $$

$$ (3.45) \qquad \bar{\mathcal{H}}^{(j-\frac{1}{2})} = \text{Diag}(\bar{\mathcal{H}}_1^{(j-\frac{1}{2})}, \bar{\mathcal{H}}_2^{(j-\frac{1}{2})}, \dots, \bar{\mathcal{H}}_{2j}^{(j-\frac{1}{2})}), $$

where $\mathcal{H}_m^{(j+\frac{1}{2})}$ and $\bar{\mathcal{H}}_n^{(j-\frac{1}{2})}$ are scalars.

Inspired by [BLN15], the R-matrix (3.41) 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 $\mathcal{E}^{(j+\frac{1}{2})}$ and the operator $\mathcal{F}^{(j+\frac{1}{2})}$, defined above, as follows:

**Lemma 3.5.** The R-matrix (3.41) at the point $u = q^{j+\frac{1}{2}}$ has rank $2j+2$ and decomposes as:

$$ (3.46) \qquad R^{(\frac{1}{2},j)}(q^{j+\frac{1}{2}}) = \mathcal{E}^{(j+\frac{1}{2})} \mathcal{H}^{(j+\frac{1}{2})} \mathcal{F}^{(j+\frac{1}{2})}, $$
where $\mathcal{E}^{(j+\frac{1}{2})}$ is fixed by Lemma 3.1, $\mathcal{F}^{(j+\frac{1}{2})}$ is given in (3.25) with (3.26), (3.27) and the entries of $\mathcal{H}^{(j+\frac{1}{2})}$ are

$$ (3.47) \qquad \begin{aligned} \mathcal{H}_1^{(j+\frac{1}{2})} &= \mathcal{H}_{2j+2}^{(j+\frac{1}{2})} = \left( \prod_{k=0}^{2j-2} c(q^{2j-k}) \right) (q^{2j+1} - q^{-2j-1}), \\ \mathcal{H}_n^{(j+\frac{1}{2})} &= \left( \prod_{k=0}^{2j-2} c(q^{2j-k}) \right) \frac{(q-q^{-1})B_{j,-j-1+n}}{\mathcal{E}_{n,n}^{(j+\frac{1}{2})} \mathcal{F}_{n,n+2j}^{(j+\frac{1}{2})}}, \end{aligned} $$

for $n = 2, 3, \dots, 2j+1$.

*Proof.* Recall that $\mathcal{E}^{(j+\frac{1}{2})}$ is given in Lemma 3.1 and its pseudo-inverse $\mathcal{F}^{(j+\frac{1}{2})}$ is not unique. However, imposing (3.46), with $\mathcal{H}^{(j+\frac{1}{2})}$ defined in (3.44), fixes both $\mathcal{F}^{(j+\frac{1}{2})}$ and $\mathcal{H}^{(j+\frac{1}{2})}$ uniquely as we now show. Indeed, solving $\mathcal{F}^{(j+\frac{1}{2})}\mathcal{E}^{(j+\frac{1}{2})} = \mathbb{I}_{2j+2}$ imposes (3.27) and $\mathcal{F}_{1,1}^{(j+\frac{1}{2})} = 1$. There are still $2j$ unfixed coefficients $\mathcal{F}_{n,n+2j}^{(j+\frac{1}{2})}$, with $n = 2, 3, \dots, 2j+1$. They are fixed as in (3.26), as well as the entries of $\mathcal{H}^{(j+\frac{1}{2})}$, by solving (3.46). $\square$

Then, with the decomposition (3.46) and using the pseudo-inverse property $\mathcal{F}^{(j+\frac{1}{2})}\mathcal{E}^{(j+\frac{1}{2})} = \mathbb{I}_{2j+2}$, we have:

**Corollary 3.6.** The following relations hold:

$$ (3.48) \qquad \mathcal{E}^{(j+\frac{1}{2})}\mathcal{H}^{(j+\frac{1}{2})} = R^{(\frac{1}{2},j)}(q^{j+\frac{1}{2}})\mathcal{E}^{(j+\frac{1}{2})}, $$

$$ (3.49) \qquad \mathcal{H}^{(j+\frac{1}{2})}\mathcal{F}^{(j+\frac{1}{2})} = \mathcal{F}^{(j+\frac{1}{2})}R^{(\frac{1}{2},j)}(q^{j+\frac{1}{2}}), $$

$$ (3.50) \qquad R^{(\frac{1}{2},j)}(q^{j+\frac{1}{2}}) = \mathcal{E}^{(j+\frac{1}{2})}\mathcal{F}^{(j+\frac{1}{2})}R^{(\frac{1}{2},j)}(q^{j+\frac{1}{2}}). $$

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 (3.41) at the point $u = q^{-j-\frac{1}{2}}$ has rank $2j$ and is decomposed as:

$$ (3.51) \qquad R^{(\frac{1}{2},j)}(q^{-j-\frac{1}{2}}) = \bar{\mathcal{E}}^{(j-\frac{1}{2})}\bar{\mathcal{H}}^{(j-\frac{1}{2})}\bar{\mathcal{F}}^{(j-\frac{1}{2})}, $$

where $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$ is fixed by Lemma 3.2, $\bar{\mathcal{F}}^{(j-\frac{1}{2})}$ is given in (3.33) with (3.34) and the entries of $\bar{\mathcal{H}}^{(j-\frac{1}{2})}$ are

$$ (3.52) \qquad \bar{\mathcal{H}}_n^{(j-\frac{1}{2})} = \left( \prod_{k=0}^{2j-2} c(q^{-k-1}) \right) \frac{(q-q^{-1})B_{j,-j+n}}{\bar{\mathcal{E}}_{n+2j+1,n}^{(j-\frac{1}{2})} \bar{\mathcal{F}}_{n,n+1}^{(j-\frac{1}{2})}}, $$

for $n = 1, 2, \dots, 2j$.
**3.5. Formal evaluation representations.** In the next sections, when we discuss L- and K-operators, and in calculations of 1-component evaluation of the universal R-matrix, we actually work with formal parameter $u$, not a complex number, because the evaluated expressions would not be well-defined if $u$ was a number as in the evaluation representations considered above. Actually, in the evaluation of the universal R-matrix, such considerations were already accounted for in [Dr86, Sec. 13] and [FR92, Sec. 4], see [He17, Sec. 1] for a review. For an evaluation of the universal K-matrix, similar precautions should be taken. One way to proceed is to introduce a formal-parameter analogue of the finite-dimensional evaluation representations from (3.4) and to establish the inter-twining properties analogous to Lemmas 3.1 and 3.2 on this formal level. In the literature, these representations of quantum affine algebras are also known [CG03] under the name *quantum loop modules*.

From now on, we assume that $u^{\pm 1}$ is formal, and keeping for brevity the same notations, we define the formal evaluation representations $\pi_u^j$ as

$$ (3.53) \qquad \pi_u^j = \pi^j \circ \mathrm{ev}_u : \mathcal{L}U_q sl_2 \to \mathrm{End}(\mathbb{C}^{2j+1})[u^{\pm 1}] . $$

where the formal evaluation map

$$ (3.54) \qquad \mathrm{ev}_u : \mathcal{L}U_q sl_2 \to U_q sl_2[u^{\pm 1}], $$

is defined by same formulas as in (2.39) but with $u^{\pm 1}$ formal variables. We thus have explicitly on generators:

$$ (3.55) \qquad \begin{aligned} \pi_u^j(E_0) &= u^{-1}\pi^j(F), & \pi_u^j(F_0) &= u\pi^j(E), & \pi_u^j(K_0^{\frac{1}{2}}) &= \pi^j(K^{-\frac{1}{2}}), \\ \pi_u^j(E_1) &= u^{-1}\pi^j(E), & \pi_u^j(F_1) &= u\pi^j(F), & \pi_u^j(K_1^{\frac{1}{2}}) &= \pi^j(K^{\frac{1}{2}}). \end{aligned} $$

In the case $u$ is a complex number, this action is also known as the principal gradation [BDGZ93]. In our formal $u$ case, we will also call it the action of *principal gradation*.

The map (3.53) and formulas (3.55) define the action of $\mathcal{L}U_q sl_2$ on the space $\mathbb{C}^{2j+1}[u^{\pm 1}]$ of Laurent polynomials in $u$ with coefficients being vectors in $\mathbb{C}^{2j+1}$. For example, the action of $E_0$ is given by

$$ \pi_u^j(E_0)(xu^n) = \pi^j(F)(x)u^{n-1}, \quad x \in \mathbb{C}^{2j+1}, n \in \mathbb{Z}, $$

and extended linearly to all elements in $\mathbb{C}^{2j+1}[u^{\pm 1}]$, and similarly for the other generators. Here, we use the same notation for the corresponding representation $\pi_u^j: \mathcal{L}U_q sl_2 \to \mathrm{End}(\mathbb{C}^{2j+1}[u^{\pm 1}])$, and call it *formal evaluation representation*. We denote the corresponding infinite-dimensional module by $\mathbb{C}_u^{2j+1}$ and sometimes call it also *quantum loop module of spin-j*. In what follows, we will often use the notation $\mathbb{C}_{zu}^{2j+1}$, for $z \in \mathbb{C}^*$, which means that the $\mathcal{L}U_q sl_2$ action is on the same underlying vector space $\mathbb{C}^{2j+1}[u^{\pm 1}]$ but one needs to replace $u$ in the action (3.55) by $zu$.

Let $\{x_k\}_{0 \le k \le 2j}$ be a basis in the spin-$j$ module $\mathbb{C}^{2j+1}$ over $U_q sl_2$ such that $x_0$ is the highest weight vector, or $|j, j\rangle$ in the notations of (3.1), and $x_k := \pi^j(F)^k x_0$. Then the vectors

$$ (3.56) \qquad x_{k,n} := x_k u^n, \quad 0 \le k \le 2j, n \in \mathbb{Z}, $$

form a basis in $\mathbb{C}_u^{2j+1}$ with the action of $\mathcal{L}U_q sl_2$ up to scalars described as follows: $E_0$ changes the index $(k, n)$ by $(+1, -1)$, while $F_0$ changes it in the opposite way $(k, n) \to (k-1, n+1)$, and similarly $E_1$ changes the index $(k, n)$ by $(-1, -1)$ while $F_1$ changes it by $(+1, +1)$, and finally $K_0$ and $K_1$ don't change the index $(k, n)$, they act as $K^{-1}$ and $K$, respectively, on $x_k$, i.e. by the corresponding $U_q sl_2$ weight.
For the $\mathcal{L}U_q sl_2$ action in the basis (3.56), let us consider an example of the $j = 1$ case in more details. The action of $\mathcal{L}U_q sl_2$ is better to present diagrammatically as in Figure 3 where each node corresponds to some $x_{k,n}$ while edges are the actions of $E_i$ and $F_i$ described up to scalars above.

<figure><img src="image_4.png" /><figcaption>FIGURE 3. Quantum loop module $\mathbb{C}_u^3$ of spin-1 where the red and blue diagrams correspond to irreducible components of the $\mathcal{L}U_q sl_2$ action.</figcaption></figure>

As we can see from this example, the formal-evaluation representation $\mathbb{C}_u^3$ is fully reducible due to the presence of two different sub-diagrams, one is blue and the other is red. Each of these sub-diagrams describes an irreducible action. It is not hard to see that this action arises from the other version of formal-evaluation representation in the homogeneous gradation. Recall [BDGZ93] that the homogeneous gradation action is as in (3.55) for $E_0$ and $F_0$, while $E_1$ and $F_1$ act just as $E$ and $F$, respectively. Let us denote the corresponding $\mathcal{L}U_q sl_2$-module on the vector space $\mathbb{C}^{2j+1}[u^{\pm 1}]$ by $\tilde{\mathbb{C}}_u^{2j+1}$ and call it *quantum loop module of spin-j in the homogeneous gradation*. It is clear from the action in the basis of $\tilde{\mathbb{C}}_u^{2j+1}$ analogous to $x_k u^n$ that this module is irreducible, while in our example $\mathbb{C}_u^3$ is decomposed as $\mathbb{C}_u^3 \cong \tilde{\mathbb{C}}_u^3 \oplus \tilde{\mathbb{C}}_u^3$.

We have a similar result for arbitrary spin $j$: indeed, the diagram for $j = \frac{1}{2}$ is obtained by simply deleting the last row of nodes labeled by $x_{2,n}$ and all arrows attached to them, so we get a disjoint union of two zig-zag's, while for $j > 1$ one just repeats the obvious pattern of arrows which gives again two independent sub-diagrams of different color. Let us reformulate this observation as the following lemma.

**Lemma 3.8.** $\mathbb{C}_u^{2j+1}$ is a direct sum of two copies of $\tilde{\mathbb{C}}_u^{2j+1}$.

This effect of decomposition of the action in one gradation via actions in the other gradation is in contrast to the finite-dimensional evaluation representations where the choice of gradation is not really important.

We now want to study tensor products of these infinite-dimensional representations

$$ (3.57) \qquad (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j) \circ \Delta: \mathcal{L}U_q sl_2 \to \text{End}(\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1}), $$

where we used that the homomorphism $(\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j) \circ \Delta$ sends $\mathcal{L}U_q sl_2$ to $\text{End}(\mathbb{C}_{u_1}^2) \otimes \text{End}(\mathbb{C}_{u_2}^{2j+1})$ which is a subalgebra in $\text{End}(\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1})$, and this algebra embedding is implicitly used in (3.57).

We then note that as a vector space $\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1}$ is isomorphic to $\mathbb{C}^2 \otimes \mathbb{C}^{2j+1}[u_1^{\pm 1}, u_2^{\pm 1}]$ – the space of Laurent polynomials in two variables $u_1$ and $u_2$ with coefficients in the $2(2j+1)$-dimensional
vector space – or equivalently, to the product of $\mathbb{C}^2 \otimes \mathbb{C}^{2j+1}$ and $\mathbb{C}[u_1^{\pm 1}, u_2^{\pm 1}]$. Below, we will use these identifications implicitly. Then, the action (3.57) can be written as

$$ (3.58) \qquad a(vf(u_1, u_2)) = a(v)f(u_1, u_2) = \left[ (\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{u_2}^j) \circ \Delta(a) \right] (v)f(u_1, u_2) $$

for any $a \in \mathcal{L}U_q sl_2$, $v \in \mathbb{C}^2 \otimes \mathbb{C}^{2j+1}$ and $f(u_1, u_2) \in \mathbb{C}[u_1^{\pm 1}, u_2^{\pm 1}]$. In what follows, we will shortly write the action as on the left-hand side instead of lengthy but more precise expression on the right-hand side.

The difference from the finite-dimensional case where $u_i$'s were generic complex numbers is that these infinite-dimensional tensor products are not irreducible anymore. Indeed, for every non-zero complex number $z$ the Laurent polynomial $u_1 - z^{-1}u_2$ generates a proper ideal in the algebra of Laurent polynomials because its inverse lives in a bigger ring of rational functions in $u_1$ and $u_2$. Similarly, for the module $\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1}$ consider for a fixed $z \in \mathbb{C}^*$ the linear span of vectors of the following form:

$$ (3.59) \qquad I_z := \langle v(u_1 - z^{-1}u_2)f(u_1, u_2) \mid v \in \mathbb{C}^2 \otimes \mathbb{C}^{2j+1}, f(u_1, u_2) \in \mathbb{C}[u_1^{\pm 1}, u_2^{\pm 1}] \rangle, $$

which can be thought of as ‘ideal’ generated by the Laurent polynomials $u_1 - z^{-1}u_2$. Analyzing the action of $F_i$ and $E_i$ on these vectors, recall (3.58), it is clear that $I_z$ is a proper $\mathcal{L}U_q sl_2$-submodule of $\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1}$.

We can then define a so-called partial specialization “identifying $u_2$ with $zu_1$” of the module $\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1}$ as the quotient by the submodule $I_z$ from (3.59). We will use the following short-hand notations for this quotient:

$$ (3.60) \qquad \mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{zu_1}^{2j+1} := (\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1})/I_z, $$

when the same formal parameter appears in both tensor factors. And let us denote the canonical surjection

$$ (3.61) \qquad p_z^j : \mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1} \to \mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{zu_1}^{2j+1}, $$

which is a $\mathcal{L}U_q sl_2$-intertwining operator. We also note that the quotient module $\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{zu_1}^{2j+1}$ can be identified with $\mathbb{C}^2 \otimes \mathbb{C}^{2j+1}[u_1^{\pm 1}]$ as a vector space. Indeed, first note that the images $p_z^j(x \otimes y u_1^n u_2^m)$, for any $x \in \mathbb{C}^2$ and $y \in \mathbb{C}^{2j+1}$ are all proportional to $(x \otimes y)u_1^{n+m}$, therefore every element in the quotient space $\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{zu_1}^{2j+1}$ is a (finite) linear combination of the monomials $(x \otimes y)u_1^n$, for $n \in \mathbb{Z}$, which makes it identical to $\mathbb{C}^2 \otimes \mathbb{C}^{2j+1}[u_1^{\pm 1}]$. With this identification, the $\mathcal{L}U_q sl_2$ representation corresponding to the quotient module (3.60) will be denoted by

$$ (3.62) \qquad \pi_{u_1}^{\frac{1}{2}} \otimes \pi_{zu_1}^j : \mathcal{L}U_q sl_2 \to \text{End}(\mathbb{C}^2 \otimes \mathbb{C}^{2j+1}[u_1^{\pm 1}]). $$

Also, we will often use the notation $\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{zu_1}^j(\Delta(x))$, for $x \in \mathcal{L}U_q sl_2$, to stress that the action in (3.62) uses the coproduct by construction. In other words, $\pi_{u_1}^{\frac{1}{2}} \otimes \pi_{zu_1}^j(\Delta(x))$ stands for the action of $\Delta(x)$ on $\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1}$ reduced to the quotient space by the surjective map $p_z^j$ from (3.62). We note that the intertwining property of $p_z^j$ guarantees that this reduction is well defined, because the kernel of $p_z^j$ is invariant under the action of $\Delta(x)$ for all $x \in \mathcal{L}U_q sl_2$.

In the quotient modules (3.60), we can now identify for appropriate values of $z$ infinite-dimensional submodules isomorphic to certain quantum loop modules. First, denote by $W^j$ the subspace in
$\mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1}$ spanned by vectors $w_{k,n,m}$ obtained via the iterated action of $F_1$ on $w_0 u_1^n u_2^m$ where $w_0$ is the product (3.8) of highest-weight vectors in $U_q sl_2$-modules of spin-$\frac{1}{2}$ and $j$:

$$ (3.63) \qquad w_{k,n,m} := F_1^k(w_0) u_1^n u_2^m, \quad 0 \le k \le 2j+1, n, m \in \mathbb{Z}. $$

Note that as in the finite-dimensional situation if $k > 2j + 1$ then the corresponding vectors $w_{k,n,m}$ are just zero. We will also denote $w_k := w_{k,0,0}$.

**Proposition 3.9.** The image of $W^j = \langle w_{k,n,m} \rangle_{0 \le k \le 2j+1, n,m \in \mathbb{Z}} \subset \mathbb{C}_{u_1}^2 \otimes \mathbb{C}_{u_2}^{2j+1}$ under the projection $p_z^j$ from (3.61) at $z = q^{j+\frac{1}{2}}$ is a $\mathcal{L}U_q sl_2$-submodule isomorphic to the quantum loop module $\mathbb{C}_{u_1 q^j}^{2j+2}$ of spin-$j + \frac{1}{2}$).

*Proof.* First, the action of $E_1$ preserves $W^j$ because using the commutator relation between $E_1$ and $F_1$ and the fact that all vectors $w_{0,n,m}$ are annihilated by $E_1$ we get

$$ E_1(w_{k,n,m}) \sim w_{k-1,n,m}, $$

and so the image of $W^j$ under $p_z^j$ is equally preserved by the action of $E_1$ because $p_z^j$ commutes with all $E_i$ and $F_i$. Let us now analyze the action of $E_0$ and $F_0$ on the vectors $w_{k,n,m}$. Observing that $w_{k,n,m} = w_k u_1^n u_2^m$, we get for the action the expressions similar to (3.10):

$$ (3.64) \qquad \begin{aligned} E_0(w_{k,n,m}) &= E_0(w_k) u_1^n u_2^m = e_{1,k}(u_1, u_2) w_{k+1,n,m} + e_{2,k}(u_1, u_2) v_k u_1^n u_2^m, \\ F_0(w_{k,n,m}) &= F_0(w_k) u_1^n u_2^m = f_{1,k}(u_1, u_2) w_{k-1,n,m} + f_{2,k}(u_1, u_2) v_{k-2} u_1^n u_2^m, \end{aligned} $$

where $v_k$ is defined as in (3.9) and the coefficients are as in (3.11)-(3.12). Note that the images $p_z^j(w_{k,n,m})$ are all proportional to $p_z^j(w_k) u_1^{n+m}$, so the vectors $w_{k,n} := p_z^j(w_k) u_1^n$, for $n \in \mathbb{Z}$ and $0 \le k \le 2j+1$, form a basis in the image of $W^j$ under the projection $p_z^j$. Then, the action on these basis elements $w_{k,n}$ at $z = q^{j+\frac{1}{2}}$ simplifies to:

$$ (3.65) \qquad \begin{aligned} E_0(w_{k,n}) &= q^{-2j}[2j+1]_q w_{k+1,n-2}, \\ F_0(w_{k,n}) &= q^{2j}[2j+2-k]_q w_{k-1,n+2}. \end{aligned} $$

We thus see that $\mathcal{L}U_q sl_2$ acts on $p_z^j(W^j)$. We are just left to identify the quantum loop module $p_z^j(W^j)$ is isomorphic to. First note from (3.65) that the action of $E_0$ and $F_0$ changes the index $n$ by $\pm 2$ while the action of $E_1$ and $F_1$ keeps it unchanged. Therefore $p_z^j(W^j)$ is decomposed onto a direct sum of two $\mathcal{L}U_q sl_2$-modules: one generated by $w_{0,0}$ and the other by $w_{0,1}$. We identify each of these modules with the quantum loop module $\tilde{\mathbb{C}}_{u_1 q^j}^{2j+2}$ in the homogeneous gradation, using essentially the same analysis as in the proof of Lemma 3.1. Then using Lemma 3.8 we conclude that the image $p_z^j(W^j)$ is isomorphic to the quantum loop module $\mathbb{C}_{u_1 q^j}^{2j+2}$ which proves the proposition. $\square$

We can now construct an intertwining operator corresponding to the quantum loop submodule described in Proposition 3.9:

$$ (3.66) \qquad \mathcal{E}^{(j+\frac{1}{2})} : \mathbb{C}_{uq^j}^{2j+2} \to \mathbb{C}_u^2 \otimes \mathbb{C}_{uq^{j+1/2}}^{2j+1} $$

where for simplicity we use the same notation as in the finite-dimensional setting of Section 3.2. First, we identify the quotient space $\mathbb{C}_u^2 \otimes \mathbb{C}_{uq^{j+1/2}}^{2j+1}$ with $\mathbb{C}^2 \otimes \mathbb{C}^{2j+1}[u^{\pm 1}]$ as discussed below (3.61). Using this identification, we define the action of $\mathcal{E}^{(j+\frac{1}{2})}$ on Laurent polynomials from $\mathbb{C}_{uq^j}^{2j+2} = \mathbb{C}^{2j+2}[u^{\pm 1}]$
simply on their coefficients by the same matrix as in (3.18), with the matrix entries from Lemma 3.1 given in the basis fixed below (3.18). This action on coefficients is well defined because the matrix entries do not depend on $u$. We then extend this action linearly to Laurent polynomials in $u$. With the discussion below (3.62), it is straightforward to check the intertwining property (3.19) of such linear map $\mathcal{E}^{(j+\frac{1}{2})}$ along the same lines as in the proof of Lemma 3.1. Similarly, its pseudo inverse map $\mathcal{F}^{(j+\frac{1}{2})}$ is first defined on coefficients of Laurent polynomials as in (3.25) with the same matrix entries (3.26)-(3.27), and then extended by linearity. Note that $\mathcal{F}^{(j+\frac{1}{2})}$ is not a $\mathcal{L}U_q sl_2$-intertwiner.

Finally, we have a construction of intertwining map $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$ analogous to Section 3.3. This map corresponds to the spin-(j - 1/2) quantum loop submodule in the quotient module $\mathbb{C}_u^2 \otimes \mathbb{C}_{uq^{-j-1/2}}^{2j+1}$. To see this, one can follow the proof of Proposition 3.9 but using the vector $v_0$ from (3.8) instead of $w_0$. We thus get a $\mathcal{L}U_q sl_2$-intertwiner

$$ (3.67) \qquad \bar{\mathcal{E}}^{(j-\frac{1}{2})} : \mathbb{C}_{uq^{-j-1}}^{2j} \to \mathbb{C}_u^2 \otimes \mathbb{C}_{uq^{-j-1/2}}^{2j+1} $$

defined on coefficients of Laurent polynomials by the matrix (3.30), with the matrix entries from Lemma 3.2. This map satisfies the same relation (3.31), now in the context of discussion below (3.62). Similarly, its pseudo inverse map $\bar{\mathcal{F}}^{(j-\frac{1}{2})}$ is defined on coefficients of Laurent polynomials via (3.33) and (3.34).

## 4. SPIN-j L- AND K-OPERATORS

In sections 4.1 and 4.2, we define spin-j L- and K-operators, respectively, as evaluations of universal R- and K-matrices for $H = \mathcal{L}U_q sl_2$ 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’. We also establish the so-called P-symmetry and unitarity properties of the fused L-operators. In the final subsection 4.3, the comodule algebra structure on $B$ is characterized within the framework of the spin-j K-operators and the Ding-Frenkel L-operators.

### 4.1. Spin-j L-operators.

4.1.1. Evaluation of the universal R-matrix. As we noted before, the universal R-matrix $\mathfrak{R}$ for $H = \mathcal{L}U_q sl_2$ is an element of a certain completion of $\mathcal{L}U_q sl_2 \otimes \mathcal{L}U_q sl_2$ (see details e.g. in [AV22, Sec. 2.5]), expressed as a product over an infinite set of root vectors [KT92a, Thm. 1], see also [KT92b, LSS93, Da98]. We give the universal R-matrix expression in our conventions in Appendix B. In this section, we consider 1- and 2-component evaluation of $\mathfrak{R}$ on the formal evaluation representations $\pi_u^j$ studied in the previous section 3.5, recall (3.53) and (3.55).

**Definition 4.1.** For $j \in \frac{1}{2}\mathbb{N}$, we define

$$ (4.1) \qquad \mathbf{L}^{(j)}(u_1/u_2) = (\mathrm{ev}_{u_1} \otimes \pi_{u_2}^j)(\mathfrak{R}) \in U_q sl_2[[u_2/u_1]] \otimes \mathrm{End}(\mathbb{C}^{2j+1}). $$

We call $\mathbf{L}^{(j)}(u)$ the spin-j L-operator. In particular, $\mathbf{L}^{(0)}(u) = 1$ by (2.2).

A few comments for the above definition are necessary:
* Strictly speaking, the right hand side of (4.1) does not make sense literally because $\mathfrak{R}$ does not belong to $\mathcal{L}U_q sl_2 \otimes \mathcal{L}U_q sl_2$ but to its completion which is much bigger and a priori is not of a tensor product form, so we cannot apply a map of this form. What makes sense is to use a bi-gradation on the positive part $\mathcal{L}U_q sl_2^+$ generated by $E_i$: the degree of $E_0$ is (1, 0), while $E_1$ has degree (0, 1). In other words, replacing each $E_i$ by $z_i E_i$ in the $q$-exponent expression (B.4) of $\mathfrak{R}$ produces $\mathfrak{R}(z_0, z_1)$ such that $q^{-\frac{1}{2}h_1 \otimes h_1} \mathfrak{R}(z_0, z_1) \in \mathcal{L}U_q sl_2^+ \otimes \mathcal{L}U_q sl_2^-[z_0, z_1]$. Now, we can apply a map of tensor product form coefficient-wise, and set $z_0 = z_1 = 1$ in the end of the calculation. This is what we will always use implicitly in the exposition below.

* First, we evaluate 2nd tensor component of $\mathfrak{R}$ at $j = \frac{1}{2}$ treating $u_2$ as a formal variable. This is done in Appendix B.3.2 using the exponential form of $\mathfrak{R}$ reviewed in Appendix B.2. This calculation essentially amounts to evaluating infinite series of the root vectors with (3.55), that is why one needs to consider $u_2$ as a formal variable and not just a complex number. The result of such evaluation is the L-operator of type $\hat{\mathbf{L}}^{(\frac{1}{2})}(u_2)$ from (1.5) for $H = \mathcal{L}U_q sl_2$, and it takes the form of the Ding-Frenkel L-operator [DF93]. It is a $2 \times 2$ matrix with entries from $\mathcal{L}U_q sl_2[[u_2]]$.

* In the second step, it is straightforward to evaluate the remaining 1st component of $\mathfrak{R}$ with (3.54), see our calculation in Appendix B.4, and with the final result that depends on the ratio $u = u_1/u_2$ only:

$$ (4.2) \quad \mathbf{L}^{(\frac{1}{2})}(u) = \mu(u)\mathcal{L}^{(\frac{1}{2})}(u), \quad \text{with} \quad \mathcal{L}^{(\frac{1}{2})}(u) = \begin{pmatrix} uq^{\frac{1}{2}}K^{\frac{1}{2}} - u^{-1}q^{-\frac{1}{2}}K^{-\frac{1}{2}} & (q-q^{-1})F \\ (q-q^{-1})E & uq^{\frac{1}{2}}K^{-\frac{1}{2}} - u^{-1}q^{-\frac{1}{2}}K^{\frac{1}{2}} \end{pmatrix}, $$

where the 'normalization' $\mu(u)$ is the following formal power series in $u^{-1}$:

$$ (4.3) \qquad \mu(u) = u^{-1}q^{-\frac{1}{2}}e^{\Lambda(u^{-2}q^{-1})}, $$

and $\Lambda(u)$ is a formal power series in $u$ with coefficients from the center $Z(U_q sl_2)$, see its explicit expression in (B.34). Formally, the result of evaluation in (4.1) belongs to $U_q sl_2[[u_1^{-1}, u_2]] \otimes \text{End}(\mathbb{C}^2)$, however we see that it actually belongs to a smaller space which is $U_q sl_2[[u_2/u_1]] \otimes \text{End}(\mathbb{C}^2)$. Furthermore, we show by the fusion construction discussed below that all the L-operators $\mathbf{L}^{(j)}(u_1/u_2)$ are indeed formal power series belonging to $U_q sl_2[[u^{-1}]] \otimes \text{End}(\mathbb{C}^{2j+1})$ with $u = u_1/u_2$, as indicated in Definition 4.1.

**Remark 4.2.** The spin-$\frac{1}{2}$ L-operator $\mathbf{L}^{(\frac{1}{2})}(u)$ was initially calculated in [BGKNR12, eq. (4.62)]$^9$, with different conventions on the coproduct. The two L-operators are related as follows. Recall that the automorphism (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 $j = \frac{1}{2}$ reads

$$ (4.4) \qquad \mathbf{L}^{(\frac{1}{2})}(u_1/u_2) = (\text{id} \otimes \pi^{\frac{1}{2}}) \circ (\text{ev}_{u_1} \otimes \text{ev}_{u_2}) \circ (\nu^{-1} \otimes \nu^{-1})(\mathfrak{R}^{KT}), $$

where $\nu$ is given in (A.12). Now, consider the automorphism $\tau: U_q sl_2 \to U_q sl_2$ defined by

$$ (4.5) \qquad \tau(E) = q^{-\frac{1}{2}}EK^{-\frac{1}{2}}, \quad \tau(F) = q^{-\frac{1}{2}}FK^{\frac{1}{2}}, \quad \tau(K^{\pm\frac{1}{2}}) = K^{\pm\frac{1}{2}}. $$

<footnote>$^9$With the identification $\zeta \to u, s \to -2$, in particular $e^{\Lambda(q^{-1}\zeta^s)} \to \mu(u)uq^{\frac{1}{2}}$, and $s_0 \to -1, s_1 \to -1$.</footnote>
Noting that it satisfies the property $\mathbf{ev}_u \circ \nu^{-1}(x) = \tau \circ \mathbf{ev}_{uq-\frac{1}{2}}(x)$, for all $x \in \mathcal{L}U_q sl_2$, and $\pi^{\frac{1}{2}} \circ \tau = \pi^{\frac{1}{2}}$, then (4.4) becomes

$$ (4.6) \qquad \mathbf{L}^{(\frac{1}{2})}(u_1/u_2) = (\tau \otimes \pi^{\frac{1}{2}}) \circ (\mathbf{ev}_{u_1 q - \frac{1}{2}} \otimes \mathbf{ev}_{u_2 q - \frac{1}{2}})(\mathfrak{R}^{KT}) = (\tau \otimes \mathrm{id})(\mathbf{L}^{(KT)}(u_1/u_2)), $$

where $\mathbf{L}^{(KT)}(u)$ is the L-operator from [BGKNR12, eq. (4.62)].

Evaluating the first component of $\mathbf{L}^{(j_2)}(u)$ on the spin-$j_1$ representation of $U_q sl_2$ we get the R-matrix for any spin $j_1, j_2$:

$$ (4.7) \qquad \begin{aligned} \mathcal{R}^{(j_1, j_2)}(u_1/u_2) &= (\pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2})(\mathfrak{R}) \\ &= (\pi^{j_1} \otimes \mathrm{id})(\mathbf{L}^{(j_2)}(u_1/u_2)), \end{aligned} $$

which is a matrix from $\mathrm{End}(\mathbb{C}^{2j_1+1} \otimes \mathbb{C}^{2j_2+1})$ with entries in formal power series $\mathbb{C}[[u^{-1}]]$.

**Example 4.3.** For $j_1 = j_2 = \frac{1}{2}$ in (4.7), the corresponding R-matrix is given by:

$$ (4.8) \qquad \mathcal{R}^{(\frac{1}{2}, \frac{1}{2})}(u) = \pi^{\frac{1}{2}}(\mu(u)) \mathcal{R}^{(\frac{1}{2}, \frac{1}{2})}(u), \quad \text{with } \mathcal{R}^{(\frac{1}{2}, \frac{1}{2})}(u) = \begin{pmatrix} c(uq) & 0 & 0 & 0 \\ 0 & c(u) & c(q) & 0 \\ 0 & c(q) & c(u) & 0 \\ 0 & 0 & 0 & c(uq) \end{pmatrix}, $$

where $c(u)$ is given in (3.42) and with

$$ (4.9) \qquad \pi^j(\mu(u)) = u^{-1}q^{-\frac{1}{2}} \exp \left( \sum_{k=1}^{\infty} \frac{q^{k(2j+1)} + q^{-k(2j+1)}}{1+q^{2k}} \frac{u^{-2k}}{k} \right), $$

where we used the evaluation of the coefficients $C_k$ of $\Lambda(u)$ given in (B.34), see [BGKNR12, eq. (4.59)]. Note that $\mathcal{R}^{(\frac{1}{2}, \frac{1}{2})}(u)$ coincides with the expression in (3.41) for $j = \frac{1}{2}$.

We recall that the L-operators satisfy the RLL relations. Indeed, applying $(\mathbf{ev}_{u_1} \otimes \pi_{u_2}^{j_2} \otimes \pi_{u_3}^{j_3})$ to (2.3), one finds$^{10}$

$$ (4.10) \qquad \mathbf{L}_1^{(j_2)}(u_1/u_2)\mathbf{L}_2^{(j_3)}(u_1/u_3)\mathcal{R}_{12}^{(j_2, j_3)}(u_2/u_3) = \mathcal{R}_{12}^{(j_2, j_3)}(u_2/u_3)\mathbf{L}_2^{(j_3)}(u_1/u_3)\mathbf{L}_1^{(j_2)}(u_1/u_2). $$

Recall also that the R-matrix satisfies the Yang-Baxter equation. It is found by applying $(\pi^{j_1} \otimes \mathrm{id} \otimes \mathrm{id})$ to the above equation and setting $u_3 = 1$:

$$ (4.11) \qquad \mathcal{R}_{12}^{(j_1, j_2)}(u_1/u_2)\mathcal{R}_{13}^{(j_1, j_3)}(u_1)\mathcal{R}_{23}^{(j_2, j_3)}(u_2) = \mathcal{R}_{23}^{(j_2, j_3)}(u_2)\mathcal{R}_{13}^{(j_1, j_3)}(u_1)\mathcal{R}_{12}^{(j_1, j_2)}(u_1/u_2). $$

We recall that the so-called quantum determinant is given by [Sk88]

$$ (4.12) \qquad \begin{aligned} \gamma(u) &= \mathrm{tr}_{12}(\mathcal{P}_{12}^{-} \mathcal{L}_{1}^{(\frac{1}{2})}(u) \mathcal{L}_{2}^{(\frac{1}{2})}(uq)) \\ &= u^2 q^2 + u^{-2} q^{-2} - C, \end{aligned} $$

where $\mathrm{tr}_{12}$ stands for the trace over $V_1 \otimes V_2$, and $\mathcal{P}_{12}^{-} = (1 - \mathcal{P})/2$ with the permutation matrix $\mathcal{P} = \mathcal{R}^{(\frac{1}{2}, \frac{1}{2})}(1)/(q - q^{-1})$, and the Casimir element $C$ is defined in (A.5).

<footnote>$^{10}$The RLL equation belongs to the triple product $U_q sl_2 \otimes \mathrm{End}(\mathbb{C}^{2j_1+1}) \otimes \mathrm{End}(\mathbb{C}^{2j_2+1})$, and thus the L-operator should be written as $\mathbf{L}_{0i}^{(j)}(u)$ but here we omit the label 0 corresponding to $U_q sl_2$.</footnote>
**Remark 4.4.** The inverse of $\mathcal{L}^{(\frac{1}{2})}(u)$ is given by

$$ (4.13) \qquad [\mathcal{L}^{(\frac{1}{2})}(u)]^{-1} = -\frac{1}{\gamma(uq^{-1})} \mathcal{L}^{(\frac{1}{2})}(u^{-1}), $$

with $\gamma(u)$ from (4.12). Note that $u^{-2}\gamma(u)$ is invertible in $Z(U_q sl_2)[[u^{-1}]]$ due to [T21d, Lem. 4.1], and so $\gamma(u)$ is invertible too.

4.1.2. L-operators and fusion $(\frac{1}{2}, j) \to (j + \frac{1}{2})$. We study a so-called fusion relation for L-operators that expresses $\mathbf{L}^{(j+\frac{1}{2})}(u)$ in terms of product of $\mathbf{L}^{(j)}(u_1)$ and $\mathbf{L}^{(\frac{1}{2})}(u_2)$. For this, we evaluate 2nd and 3rd tensor components of the universal R-matrix equation (R3) on quantum loop modules with formal evaluation parameters fixed as $u_2 = q^{j+\frac{1}{2}}u_1$. This fixing requires taking the quotient (3.60) to relate the two formal variables $u_1$ and $u_2$ on tensor product (3.57) of two quantum loop modules $\mathbb{C}_{u_1}^2$ and $\mathbb{C}_{u_2}^{2j+1}$, respectively. Using Proposition 3.9, on this quotient module we get a $\mathcal{L}U_q sl_2$-submodule isomorphic to the quantum loop module $\mathbb{C}_{u_1 q^j}^{2j+2}$ of spin-$(j + \frac{1}{2})$, with the corresponding intertwining operator $\mathcal{E}^{(j+\frac{1}{2})}$ described below (3.66). As it was explained there, this intertwiner is defined by Lemma 3.1 and its pseudo-inverse $\mathcal{F}^{(j+\frac{1}{2})}$ is fixed by (3.25) with (3.26), (3.27). Using the associated identity $\mathcal{F}^{(j+\frac{1}{2})}\mathcal{E}^{(j+\frac{1}{2})} = \text{id}_{\mathbb{C}^{2j+2}[u^{\pm 1}]}$, fusion relations satisfied by L-operators and R-matrices are exhibited in the following two propositions.

**Proposition 4.5.** The spin-j L-operators (4.1) satisfy for $j \in \frac{1}{2}\mathbb{N}$ the following relations in $U_q sl_2[[u^{-1}]] \otimes \text{End}(\mathbb{C}^{2j+2})$:

$$ (4.14) \qquad \mathbf{L}^{(j+\frac{1}{2})}(u) = \mathcal{F}_{\langle 12 \rangle}^{(j+\frac{1}{2})} \mathbf{L}_2^{(j)}(uq^{-\frac{1}{2}}) \mathbf{L}_1^{(\frac{1}{2})}(uq^j) \mathcal{E}_{\langle 12 \rangle}^{(j+\frac{1}{2})}, $$

where we use the notation $\langle 12 \rangle$ to indicate which spaces are fused, i.e. where the intertwiner $\mathcal{E}^{(j+\frac{1}{2})}$ acts, and here we use its finite-dimensional matrix form given in (3.18) with (3.20).

*Proof.* By definition of the L-operator we have

$$ (4.15) \qquad \mathbf{L}^{(j+\frac{1}{2})}(w/u) = (\mathfrak{e}v_w \otimes \pi_u^{j+\frac{1}{2}})(\mathfrak{R}). $$

Using the pseudo-inverse property $\mathcal{F}^{(j+\frac{1}{2})}\mathcal{E}^{(j+\frac{1}{2})} = \text{id}$, we get

$$ \begin{align*} \mathbf{L}^{(j+\frac{1}{2})}(w/u) &= (1 \otimes \mathcal{F}^{(j+\frac{1}{2})}\mathcal{E}^{(j+\frac{1}{2})}) [(\mathfrak{e}v_w \otimes \pi_u^{j+\frac{1}{2}})(\mathfrak{R})] \\ &= (1 \otimes \mathcal{F}^{(j+\frac{1}{2})}) [(\mathfrak{e}v_w \otimes \pi_{uq^{-j}}^{\frac{1}{2}} \otimes \pi_{uq^{\frac{1}{2}}}^{j})(\text{id} \otimes \Delta)(\mathfrak{R})] (1 \otimes \mathcal{E}^{(j+\frac{1}{2})}) \\ &= (1 \otimes \mathcal{F}^{(j+\frac{1}{2})}) [(\mathfrak{e}v_w \otimes \pi_{uq^{-j}}^{\frac{1}{2}} \otimes \pi_{uq^{\frac{1}{2}}}^{j})(\mathfrak{R}_{13}\mathfrak{R}_{12})] (1 \otimes \mathcal{E}^{(j+\frac{1}{2})}) \\ &= (1 \otimes \mathcal{F}^{(j+\frac{1}{2})}) \mathbf{L}_2^{(j)}(q^{-\frac{1}{2}}w/u) \mathbf{L}_1^{(\frac{1}{2})}(q^j w/u) (1 \otimes \mathcal{E}^{(j+\frac{1}{2})}), \end{align*} $$

where in the second line we use the convention discussed below (3.62). Here, the second equality is obtained using the intertwining property (3.19) which is applicable in the formal u setting too,
following the discussion below (3.66). Applied to elements of 2nd tensor component of $\mathfrak{R}$ this
intertwining property takes the following form:

$$
(4.16) \quad (1 \otimes \mathcal{E}^{(j+\frac{1}{2})}) \left[ (\mathrm{id} \otimes \pi_u^{j+\frac{1}{2}})(\mathfrak{R}) \right] = \left[ (\mathrm{id} \otimes \pi_{uq^{-j}}^{\frac{1}{2}} \otimes \pi_{uq^{\frac{1}{2}}}^j) \circ (\mathrm{id} \otimes \Delta)(\mathfrak{R}) \right] (1 \otimes \mathcal{E}^{(j+\frac{1}{2})}),
$$

where the maps of the tensor product form are applied to $\mathfrak{R}(z_0, z_1)$ coefficient-wise with respect to the bi-grading we always use to treat properly the universal R-matrix expansion, recall the first comment below Definition 4.1. Then, the third equality in the above calculation is due to (R3), and the last one is by definition of the L-operator. $\square$

**Proposition 4.6.** The R-matrices (4.7) satisfy for all $j_1, j_2 \in \frac{1}{2}\mathbb{N}$ the following relations in $\mathbb{C}[[u^{-1}]]\otimes \text{End}(\mathbb{C}^{2j_1+1} \otimes \mathbb{C}^{2j_2+1}):$

$$
(4.17) \qquad \mathcal{R}^{(j_1,j_2)}(u) = \mathcal{F}_{(12)}^{(j_1)} \mathcal{R}_{13}^{(\frac{1}{2},j_2)} (uq^{-j_1+\frac{1}{2}}) \mathcal{R}_{23}^{(j_1-\frac{1}{2},j_2)} (uq^{\frac{1}{2}}) \mathcal{E}_{(12)}^{(j_1)},
$$

where

$$
(4.18) \qquad \mathcal{R}^{(\frac{1}{2},j+\frac{1}{2})}(u) = \mathcal{F}_{(23)}^{(j+\frac{1}{2})} \mathcal{R}_{13}^{(\frac{1}{2},j)} (uq^{-\frac{1}{2}}) \mathcal{R}_{12}^{(\frac{1}{2},\frac{1}{2})} (uq^j) \mathcal{E}_{(23)}^{(j+\frac{1}{2})}.
$$

Proof. First we show (4.18). By definition we have $\mathcal{R}^{(\frac{1}{2},j_2)}(u) = (\pi^{\frac{1}{2}} \otimes \text{id})(\mathbf{L}^{(j_2)}(u))$, therefore the application of $(\pi^{\frac{1}{2}} \otimes \text{id})$ on (4.14) yields<fnref n="11" /> (4.18). The eq. (4.17) is shown similarly to Proposition 4.5 using now (R2) and (3.19). It gives

$$
\begin{align*}
\mathcal{R}^{(j_1, j_2)}(v/w) &= (\mathcal{F}^{(j_1)} \otimes \text{id}) \circ (\pi_v^{j_1} \otimes \pi_w^{j_2}) (\mathfrak{R}) \\
&= \mathcal{F}_{(12)}^{(j_1)} \left[ (\pi^{\frac{1}{2}}_{vq^{-j_1+\frac{1}{2}}} \otimes \pi^{j_1-\frac{1}{2}}_{vq^{\frac{1}{2}}} \otimes \pi^{j_2}_w)(\mathfrak{R}_{13}\mathfrak{R}_{23}) \right] \mathcal{E}_{(12)}^{(j_1)},
\end{align*}
$$

and using the evaluation of the universal R-matrix from (4.7), the result follows after substituting
$v/w \to u$.
$\square$

**Remark 4.7.** Recall that $\mathcal{F}^{(j+\frac{1}{2})}$ is not uniquely determined, see Section 3.2. From the construction in the proof of Proposition 4.5, it is clear that taking different expressions for $\mathcal{F}^{(j+\frac{1}{2})}$ yields the same L-operators and R-matrices.

4.1.3. L-operators and reduction $(\frac{1}{2}, j) \rightarrow (j - \frac{1}{2})$. We now consider quantum loop submodules of spin-$\left(j - \frac{1}{2}\right)$ and the corresponding $\mathcal{L}U_q s l_2$-intertwiner $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$ in (3.67). With the intertwining property (3.31), we can write:

$$
(1 \otimes \bar{\mathcal{E}}^{(j-\frac{1}{2})}) \left[ (\mathrm{id} \otimes \pi_u^{j-\frac{1}{2}})(\mathfrak{R}) \right] = \left[ (\mathrm{id} \otimes \pi_{uq^{j+1}}^{\frac{1}{2}} \otimes \pi_{uq^{\frac{1}{2}}}^j) \circ (\mathrm{id} \otimes \Delta)(\mathfrak{R}) \right] (1 \otimes \bar{\mathcal{E}}^{(j-\frac{1}{2})}),
$$

with the convention discussed below (3.62). Then, the proof of the following proposition essentially
repeats the one of Proposition 4.5.

<footnote n="11">¹¹The shifting of the labels {0, 1, 2} to {1, 2, 3} is due to the convention that the first tensor component of $\mathbf{L}^{(j)}(u)$ is labeled by 0.</footnote>
**Proposition 4.8.** The L-operators (4.1) satisfy for $j \in \frac{1}{2}\mathbb{N}_+$ the following matrix relations with coefficient in $U_q sl_2[[u^{-1}]]$:

$$ (4.19) \qquad \mathbf{L}^{(j-\frac{1}{2})}(u) = \bar{\mathcal{F}}_{(12)}^{(j-\frac{1}{2})} \mathbf{L}_2^{(j)}(uq^{-\frac{1}{2}}) \mathbf{L}_1^{(\frac{1}{2})}(uq^{-j-1}) \bar{\mathcal{E}}_{(12)}^{(j-\frac{1}{2})}. $$

4.1.4. *P-symmetry of spin-j L-operators.* We now give our first application of the fusion formula in Proposition 4.5 to show the so-called P-symmetry property of the spin-j L-operators, in (4.23) below. As a consequence, we show that the P-symmetry

$$ (4.20) \qquad \mathcal{R}_{21}^{(j_2, j_1)}(u) = \mathcal{R}^{(j_1, j_2)}(u), $$

with

$$ (4.21) \qquad \mathcal{R}_{21}^{(j_2, j_1)}(u) = \mathcal{P}^{(j_2, j_1)}\mathcal{R}^{(j_2, j_1)}(u)\mathcal{P}^{(j_1, j_2)}, $$

holds for any $j_1, j_2$, for the case of $H = \mathcal{L}U_q sl_2$. Note that a proof of this equation for R-matrices can be found in [RSV14, Lem. 2.1]. Here, we give a different proof by showing first a more general relation

$$ (4.22) \qquad (\mathrm{ev}_{u_1^{-1}} \otimes \pi_{u_2^{-1}}^j)(\mathfrak{R}) = (\mathrm{ev}_{u_1} \otimes \pi_{u_2}^j)(\mathfrak{R}_{21}), $$

which can be interpreted as the P-symmetry on the level of L-operators. Indeed, identifying the l.h.s. of (4.22) with the spin-j L-operator defined by (4.1), the equation (4.22) reads

$$ (4.23) \qquad \mathbf{L}^{(j)}(u_2/u_1) = (\mathrm{ev}_{u_1} \otimes \pi_{u_2}^j)(\mathfrak{R}_{21}). $$

The proof of this equation is done by induction on $j$. It is straightforward to check that (4.23) holds for $j = \frac{1}{2}$, by a calculation similar to the evaluation of $\mathfrak{R}$ in Appendix B. Now assume (4.23) holds for a fixed value of $j$, we show it holds for $(j + \frac{1}{2})$. It is done by identifying the r.h.s. of (4.23) with (4.14). Indeed, by an analysis similar to the proof of Proposition 4.5, using the pseudo-inverse property $\mathcal{F}^{(j+\frac{1}{2})}\mathcal{E}^{(j+\frac{1}{2})} = \mathrm{id}$, (3.37) and (2.11), one gets:

$$ \begin{align} (\mathrm{ev}_{u_1} \otimes \pi_{u_2}^{j+\frac{1}{2}})(\mathfrak{R}_{21}) &= (1 \otimes \mathcal{F}^{(j+\frac{1}{2})}) \left[ (\mathrm{ev}_{u_1} \otimes \pi_{u_2 q^j}^{\frac{1}{2}} \otimes \pi_{u_2 q^{-\frac{1}{2}}}^j)(\mathfrak{R}_{31}\mathfrak{R}_{21}) \right] (1 \otimes \mathcal{E}^{(j+\frac{1}{2})}) \tag{4.24} \\ &= \mathcal{F}_{(12)}^{(j+\frac{1}{2})} \mathbf{L}_2^{(\frac{1}{2})} (u_2 q^{-\frac{1}{2}}/u_1) \mathbf{L}_1^{(j)} (u_2 q^j/u_1) \mathcal{E}_{(12)}^{(j+\frac{1}{2})}, \notag \end{align} $$

where we used the assumption (4.23) for a fixed $j$ to get the last line. Then, comparing (4.24) with (4.14), one finds that indeed $(\mathrm{ev}_{u_1} \otimes \pi_{u_2}^{j+\frac{1}{2}})(\mathfrak{R}_{21}) = \mathbf{L}^{(j+\frac{1}{2})}(u_2/u_1)$.

Now, by specializing the first tensor component of (4.23) on the spin-$j_1$ representation and for $j = j_2$, one obtains

$$ (4.25) \qquad \mathcal{R}^{(j_1, j_2)}(u_2/u_1) = (\pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2})(\mathfrak{R}_{21}). $$

Finally, the immediate corollary of the latter relation is the P-symmetry (4.20). Indeed, the R-matrix defined in (4.21) can be interpreted as the evaluation of the flipped universal R-matrix

$$ (4.26) \qquad \mathcal{R}_{21}^{(j_2, j_1)}(u_2/u_1) = (\pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2})(\mathfrak{R}_{21}), $$

and thus (4.20) holds.
4.1.5. Fused L-operators and fused R-matrices. For further technical needs, let us introduce a higher spin generalization of 2 × 2 matrices $\mathcal{L}^{(\frac{1}{2})}(u)$ from (4.2). For any $j \in \frac{1}{2}\mathbb{N}_+$, we define the fused L-operators $\mathcal{L}^{(j)}(u) \in U_q sl_2[u, u^{-1}] \otimes \text{End}(\mathbb{C}^{2j+1})$ as:

$$ (4.27) \qquad \mathcal{L}^{(j+\frac{1}{2})}(u) := \mathcal{F}_{(12)}^{(j+\frac{1}{2})} \mathcal{L}_2^{(j)}(uq^{-\frac{1}{2}}) \mathcal{L}_1^{(\frac{1}{2})}(uq^j) \mathcal{E}_{(12)}^{(j+\frac{1}{2})}. $$

Although not needed here, it can be proven directly by induction that $\mathcal{L}^{(j)}(u)$'s satisfy the Yang-Baxter equation (4.10), where $\mathbf{L}^{(j)}(u)$ are replaced by $\mathcal{L}^{(j)}(u)$. 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 (4.27).

**Lemma 4.9.** The spin-j L-operators and the fused L-operators are related as follows:

$$ (4.28) \qquad \mathbf{L}^{(j)}(u) = \mu^{(j)}(u)\mathcal{L}^{(j)}(u), $$

where

$$ (4.29) \qquad \mu^{(j)}(u) = \prod_{k=0}^{2j-1} \mu(uq^{j-\frac{1}{2}-k}) $$

is central in $U_q sl_2[[u^{-1}]]$.

*Proof.* The relation (4.28) is shown by induction on $j$ using (4.14) and (4.27) and the fact that $\mu(u)$ is central in $U_q sl_2$. The first step of the induction is given by (4.2). $\square$

As pointed out in [BGKNR12], using the explicit expression (B.34) one can show that the “normalising” generating function $\mu(u)$ in (4.2), given by (4.3), satisfies the following functional relation:

$$ (4.30) \qquad \mu(u)\mu(uq)\gamma(u) = 1, $$

with $\gamma(u)$ introduced in (4.12). Actually, this functional relation can be independently derived from the fusion and reduction relations obtained in Propositions 4.5 and 4.8 without the knowledge of (4.3), as we now show.

**Proposition 4.10.** Assume the spin-$\frac{1}{2}$ L-operator is of the form (4.2) for some formal Laurent series $\mu(u)$ from $Z(U_q sl_2)((u^{-1}))$. Then, the functional relation (4.30) holds.

*Proof.* Comparing $\mathbf{L}^{(\frac{1}{2})}(u) = \mu(u)\mathcal{L}^{(\frac{1}{2})}(u)$ with (4.19) for $j=1$ it follows:

$$ (4.31) \qquad \mathcal{L}^{(\frac{1}{2})}(u) = \mu(uq^{-1})\mu(uq^{-2})\bar{\mathcal{F}}_{(12)}^{(\frac{1}{2})} \mathcal{L}_2^{(1)}(uq^{-\frac{1}{2}}) \mathcal{L}_1^{(\frac{1}{2})}(uq^{-2}) \bar{\mathcal{E}}_{(12)}^{(\frac{1}{2})}, $$

where we used $\mu^{(1)}(u)$ in (4.29). For the r.h.s. of (4.31), after a direct calculation we find that

$$ \bar{\mathcal{F}}_{(12)}^{(\frac{1}{2})} \mathcal{L}_2^{(1)}(uq^{-\frac{1}{2}}) \mathcal{L}_1^{(\frac{1}{2})}(uq^{-2}) \bar{\mathcal{E}}_{(12)}^{(\frac{1}{2})} = \gamma(uq^{-2}) \mathcal{L}^{(\frac{1}{2})}(u). $$

Then, since $\mathcal{L}^{(\frac{1}{2})}(u)$ is invertible by (4.13), the relation (4.30) follows. $\square$

We obtain the following corollary of the functional relation (4.30).

**Corollary 4.11.**

(i) The inverse of $\mathbf{L}^{(\frac{1}{2})}(u)$ is given by

$$ (4.32) \qquad [\mathbf{L}^{(\frac{1}{2})}(u)]^{-1} = -\mu(uq^{-1})\mathcal{L}^{(\frac{1}{2})}(u^{-1}). $$
(ii) The quantum determinant of the L-operator $\mathbf{L}^{(\frac{1}{2})}(u)$ is such that

$$ (4.33) \qquad \mathrm{tr}_{12}(\mathcal{P}_{12}^{-} \mathbf{L}_{1}^{(\frac{1}{2})}(u) \mathbf{L}_{2}^{(\frac{1}{2})}(uq)) = 1, $$

where $\mathcal{P}^{-}$ is given below (4.12).

*Proof.* Recall first from (4.3) and Remark 4.4 that $\mu(u)$ and $\gamma(u)$ are invertible. Then, to prove the first statement we use the first equality in (4.2), (4.13) and (4.30). Equation (4.33) follows directly from the first equality in (4.2), (4.12) and the functional relation (4.30). $\square$

**Remark 4.12.** Assuming that $\mu(u)$ in (4.2) is an invertible Laurent series in $u$ with coefficients in a commutative subalgebra of $U_q sl_2$, its explicit form (4.3) can be derived, up to a sign, from the quadratic functional equation (4.30). Indeed, we can write $\mu(u) = \mu_0 u^{-p} g(u)$ for some scalar $\mu_0$ and positive integer $p$, and $g(u)$ is an invertible power series with coefficients in a commutative subalgebra of $U_q sl_2$ with constant term $g_0 = 1$. Using (4.12), the relation (4.30) fixes $p = 1$ and $\mu_0 = \pm q^{-1/2}$. Therefore, $\mu(u) = \pm u^{-1} q^{-\frac{1}{2}} g(u)$ where by (4.30) $g(u)$ satisfies

$$ (4.34) \qquad g(u)g(uq)\xi(u) = 1 \quad \text{with} \quad \xi(u) = 1 - Cu^{-2}q^{-2} + u^{-4}q^{-4}. $$

Equating the l.h.s. of the first equation in (4.34) with the one for $u$ replaced by $uq^{-1}$ is equivalent to $g(uq)\xi(u) = g(uq^{-1})\xi(uq^{-1})$. Inserting $g(u) = \sum_{k=0}^{\infty} g_k u^{-k}$ in the last equation, we get the following three-term recurrence relation:

$$ (4.35) \qquad g_k = - \frac{1}{(q^k - q^{-k})} \left( -(q^k - q^{-k+2}) C q^{-2} g_{k-2} + (q^k - q^{-k+4}) q^{-4} g_{k-4} \right), $$

with initial conditions $g_0 = 1$ and $g_k = 0$ for all $k < 0$. It gives $g_1 = 0$, which implies $g_{2\ell+1} = 0$ for all $\ell \ge 0$. Thus, $g(u) \in U_q sl_2[[u^{-2}]]$ and by assumption $[g_{2k}, g_{2\ell}] = 0$ for all $\ell, k > 0$. As every formal power series can be written in an exponential form using the Bell polynomials<fnref n="12" />, let us define $g(u) = \exp(\Lambda(u^{-2}q^{-1}))$ with $\Lambda(u) = \sum_{k=1} \Lambda_k u^k$. Now, taking the logarithm of the relation (4.34) with shifted argument $u \to uq^{-1}$, we get the relation:

$$ (4.36) \qquad \Lambda(u^{-2}q) + \Lambda(u^{-2}q^{-1}) = -\log(1 - Cu^{-2} + u^{-4}). $$

Introduce the central elements $C_k \in Z(U_q sl_2)$ defined by (B.13) in order to rewrite the r.h.s as the power series $\sum_{k=1}^{\infty} C_k \frac{u^{-2k}}{k}$. Comparing the coefficients of the power series on both sides of the relation above determines uniquely $\Lambda_k = \frac{C_k}{(q^k+q^{-k})k}$, which concludes the proof.

We now introduce fused R-matrices by analogy with (4.17) and (4.18):

$$ (4.37) \qquad R^{(j_1,j_2)}(u) = \mathcal{F}_{\langle 12 \rangle}^{(j_1)} R_{13}^{(\frac{1}{2},j_2)}(uq^{-j_1+\frac{1}{2}}) R_{23}^{(j_1-\frac{1}{2},j_2)}(uq^{\frac{1}{2}}) \mathcal{E}_{\langle 12 \rangle}^{(j_1)}, $$

for $j_1 \ge 1$ and where

$$ (4.38) \qquad R^{(\frac{1}{2},j+\frac{1}{2})}(u) = \mathcal{F}_{\langle 23 \rangle}^{(j+\frac{1}{2})} R_{13}^{(\frac{1}{2},j)}(uq^{-\frac{1}{2}}) R_{12}^{(\frac{1}{2},\frac{1}{2})}(uq^j) \mathcal{E}_{\langle 23 \rangle}^{(j+\frac{1}{2})}, $$

with $R^{(\frac{1}{2},\frac{1}{2})}(u)$ given in the right part of (4.8), and show that (4.38) indeed agrees with the explicit expression for $R^{(\frac{1}{2},j)}(u)$ given in (3.41).

<footnote n="12">¹²Note that $g_{2k} = \frac{B_k(q^{-1}\Lambda_1, \dots, k!q^{-k}\Lambda_k)}{k!}$, where $B_k(x_1, \dots, x_k)$ are the complete Bell polynomials.</footnote>
**Lemma 4.13.** The R-matrices (4.7) and the fused R-matrices (4.37) are related by

$$ (4.39) \qquad \mathcal{R}^{(j_1,j_2)}(u) = f^{(j_1,j_2)}(u)R^{(j_1,j_2)}(u), $$

where

$$ (4.40) \qquad f^{(j_1,j_2)}(u) = u^{-4j_1j_2} q^{-2j_1j_2} \exp \left( \sum_{k=1}^{\infty} \frac{q^{2k} + q^{-2k}}{1+q^{2k}} [2j_1]_{q^k} [2j_2]_{q^k} \frac{u^{-2k}}{k} \right), $$

and (4.38) agrees with (3.41).

*Proof.* Firstly, we prove (4.39) for $j_1 = \frac{1}{2}$. We show by induction on $j_2$ that

$$ (4.41) \qquad \mathcal{R}^{(\frac{1}{2},j_2)}(u) = \pi^{\frac{1}{2}}(\mu^{(j_2)}(u)) \mathcal{R}^{(\frac{1}{2},j_2)}(u), $$

and we identify $\pi^{\frac{1}{2}}(\mu^{(j_2)})(u)$ with $f^{(\frac{1}{2},j_2)}(u)$. For $j_2 = \frac{1}{2}$, it is given in (4.8). Now, assuming (4.41) holds for a fixed value of $j_2$, we show it holds for $j_2 + \frac{1}{2}$. Inserting (4.18) and (4.38) in (4.41) for $j_2 \to j_2 + \frac{1}{2}$ and using (4.29), one finds that the equality indeed holds. Then, using (4.29) and (4.9) we find that

$$ (4.42) \qquad \pi^{\frac{1}{2}}(\mu^{(j_2)})(u) = u^{-2j_2} q^{-j_2} \exp \left( \sum_{k=1}^{\infty} \frac{q^{2k} + q^{-2k}}{1+q^{2k}} [2j_2]_{q^k} \frac{u^{-2k}}{k} \right), $$

and it coincides with (4.40) for $j_1 = \frac{1}{2}$.

Secondly, we show that (4.38) agrees with (3.41). On one hand, using (4.2) it is straightforward to find that (recall the notation in (4.21))

$$ (4.43) \qquad \mathcal{R}_{21}^{(j, \frac{1}{2})}(u) = \mathcal{P}^{(j, \frac{1}{2})}[(\pi^j \otimes \mathrm{id})(\mathbf{L}^{(\frac{1}{2})}(u))]\mathcal{P}^{(\frac{1}{2}, j)} $$

is proportional to (3.41). On the other hand, from the P-symmetry in (4.20), one has $\mathcal{R}_{21}^{(j, \frac{1}{2})}(u) = \mathcal{R}^{(\frac{1}{2},j)}(u)$. Therefore, $\mathcal{R}^{(\frac{1}{2},j)}(u)$ is equally proportional to (3.41), as well as the expression in (4.38), recall (4.41). Finally, comparing the matrix entry (1,1) of (3.41) and (4.38), one finds that they are equal.

Thirdly, assuming that $\mathcal{R}^{(j_1,j_2)}(u)$ is proportional to $\mathcal{R}^{(j_1,j_2)}(u)$ as in (4.39), we show that $f^{(j_1,j_2)}(u)$ takes the form (4.40). Replacing the R-matrices and the fused R-matrices in (4.39) by (4.17), (4.37), and using (4.41) one gets:

$$ (4.44) \qquad \begin{aligned} f^{(j_1,j_2)}(u) &= \pi^{\frac{1}{2}}(\mu^{(j_2)}(u q^{-j_1+\frac{1}{2}})) f^{(j_1-\frac{1}{2},j_2)}(u q^{\frac{1}{2}}) \\ &= \prod_{k=0}^{2j_1-1} \left[ \pi^{\frac{1}{2}}(\mu^{(j_2)}(u q^{j_1-\frac{1}{2}-k})) \right], \end{aligned} $$

where we set $f^{(0,j_2)}(u) = 1$. Finally, using (4.42) in the latter relation, one finds that $f^{(j_1,j_2)}(u)$ is indeed given by (4.40) and so the claim follows. $\square$

As a consequence of the above Lemma 4.13, we see that the P-symmetry in (1.4) follows from (4.20) with the fact that $f^{(j_1,j_2)}(u) = f^{(j_2,j_1)}(u)$ and that $\mu(u)$ is invertible.
### 4.1.6. Unitarity properties of fused 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 4.14.** The following relations hold:

$$ (4.45) \qquad \mathbf{L}^{(j+\frac{1}{2})}(u) = \mathcal{F}_{(12)}^{(j+\frac{1}{2})} \mathbf{L}_1^{(\frac{1}{2})}(uq^{-j}) \mathbf{L}_2^{(j)}(uq^{\frac{1}{2}}) \mathcal{E}_{(12)}^{(j+\frac{1}{2})}, $$

$$ (4.46) \qquad \mathcal{R}^{(j_1, j+\frac{1}{2})}(u) = \mathcal{F}_{(23)}^{(j+\frac{1}{2})} \mathcal{R}_{12}^{(j_1, \frac{1}{2})}(uq^{-j}) \mathcal{R}_{13}^{(j_1, j)}(uq^{\frac{1}{2}}) \mathcal{E}_{(23)}^{(j+\frac{1}{2})}, $$

$$ (4.47) \qquad \mathcal{R}^{(j+\frac{1}{2}, j_2)}(u) = \mathcal{F}_{(12)}^{(j+\frac{1}{2})} \mathcal{R}_{23}^{(j_1, j_2)}(uq^{-\frac{1}{2}}) \mathcal{R}_{13}^{(\frac{1}{2}, j_2)}(uq^j) \mathcal{E}_{(12)}^{(j+\frac{1}{2})}. $$

**Proof.** Recall the discussion in Remark 3.3 about the action on tensor product of evaluation representations given by the opposite coproduct. The same comments also apply to the tensor product of quantum loop modules studied in Section 3.5. In particular, the intertwining property (3.37) holds also at the level of formal $u$. Now, combining (3.37) with $(\mathrm{id} \otimes \Delta^{op})(\mathfrak{R}) = \mathfrak{R}_{12}\mathfrak{R}_{13}$, which is obtained by applying $\mathfrak{p}_{23}$ to (R3), then (4.45) is proven similarly to Proposition 4.5. Eq. (4.46) follows from (4.45) by specialization of the 1st tensor component to the spin-$j_1$ representation. The last relation (4.47) is proven similarly to (4.17) but using (3.37) and $(\Delta^{op} \otimes \mathrm{id})(\mathfrak{R}) = \mathfrak{R}_{23}\mathfrak{R}_{13}$, coming from the application of $\mathfrak{p}_{12}$ to (R2), instead of (3.19) and (R2). $\square$

Let us note that, using Remark 4.4, the L-operator (4.2) satisfies the unitarity property

$$ (4.48) \qquad \mathcal{L}^{(\frac{1}{2})}(u)\mathcal{L}^{(\frac{1}{2})}(u^{-1}) = -\gamma(uq^{-1})\mathbb{I}_2. $$

We now study spin-$j$ generalization of this property starting with $j=1$. First, we need another form of the RLL equation (4.10):

$$ (4.49) \qquad \mathcal{R}_{12}^{(j_1, j_2)}(u_1/u_2) \mathbf{L}_1^{(j_1)}(u_1) \mathbf{L}_2^{(j_2)}(u_2) = \mathbf{L}_2^{(j_2)}(u_2) \mathbf{L}_1^{(j_1)}(u_1) \mathcal{R}_{12}^{(j_1, j_2)}(u_1/u_2), $$

which is found by applying $(\mathrm{ev}_u \otimes \pi_v^{j_1} \otimes \pi_w^{j_2}) \circ \mathfrak{p}_{23}$ to the universal Yang-Baxter equation (2.3), using (4.26) and substituting $v \to u/u_1, w \to u/u_2$, where we also used the P-symmetry property (4.25). Recall from Lemma 4.9 that $\mathbf{L}^{(j)}(u)$ is proportional to $\mathcal{L}^{(j)}(u)$, and so we also have

$$ (4.50) \qquad \mathcal{R}_{12}^{(j_1, j_2)}(u_1/u_2) \mathcal{L}_1^{(j_1)}(u_1) \mathcal{L}_2^{(j_2)}(u_2) = \mathcal{L}_2^{(j_2)}(u_2) \mathcal{L}_1^{(j_1)}(u_1) \mathcal{R}_{12}^{(j_1, j_2)}(u_1/u_2), $$

$$ (4.51) \qquad \mathcal{L}^{(j+\frac{1}{2})}(u) = \mathcal{F}_{(12)}^{(j+\frac{1}{2})} \mathcal{L}_1^{(\frac{1}{2})}(uq^{-j}) \mathcal{L}_2^{(j)}(uq^{\frac{1}{2}}) \mathcal{E}_{(12)}^{(j+\frac{1}{2})}, $$

where we used (4.45) for 2nd equality and the fact that all $\mu^{(j)}(u)$ are invertible. Recall that the fused L-operators are defined in (4.27). This expression matches the fusion relation (4.51) because, due to Lemma 4.9, both follow from the fusion relations satisfied by the spin-$j$ L-operators in (4.14) and (4.45). Indeed, comparing the two fusion relations written in terms of fused L-operators $\mathcal{L}^{(j)}(u)$, the coefficients $\mu^{(j)}(uq^{-\frac{1}{2}})\mu(uq^j)$ and $\mu^{(j)}(uq^{\frac{1}{2}})\mu(uq^{-j})$ on both sides cancel each other, as they are both equal to $\mu^{(j+\frac{1}{2})}(u)$, see (4.29). Now, using (4.27) and (4.51) for $j = \frac{1}{2}$, we get

$$ (4.52) \qquad \mathcal{L}^{(1)}(u)\mathcal{L}^{(1)}(u^{-1}) = \mathcal{F}_{(12)}^{(1)} \mathcal{L}_1^{(\frac{1}{2})}(uq^{-\frac{1}{2}}) \mathcal{L}_2^{(\frac{1}{2})}(uq^{\frac{1}{2}}) \mathcal{E}_{(12)}^{(1)} \mathcal{F}_{(12)}^{(1)} \mathcal{L}_2^{(\frac{1}{2})}(u^{-1}q^{-\frac{1}{2}}) \mathcal{L}_1^{(\frac{1}{2})}(u^{-1}q^{\frac{1}{2}}) \mathcal{E}_{(12)}^{(1)}. $$

Then, we need to use (4.48) to show that (4.52) is proportional to the identity matrix. However, there is an unwanted product $\mathcal{E}^{(1)}\mathcal{F}^{(1)}$ which is removed as follows. We recall first Corollary 3.6,
then insert $\mathcal{H}_{\langle 12 \rangle}^{(1)} [\mathcal{H}_{\langle 12 \rangle}^{(1)}]^{-1} = \mathbb{I}_3$ in the r.h.s. of (4.52) and use (3.48), which gives:

$$ (4.53) \qquad \begin{aligned} \mathcal{L}^{(1)}(u)\mathcal{L}^{(1)}(u^{-1}) &= \mathcal{F}_{\langle 12 \rangle}^{(1)} \mathcal{L}_{1}^{(\frac{1}{2})}(uq^{-\frac{1}{2}})\mathcal{L}_{2}^{(\frac{1}{2})}(uq^{\frac{1}{2}})\mathcal{E}_{\langle 12 \rangle}^{(1)} \mathcal{F}_{\langle 12 \rangle}^{(1)} \mathcal{L}_{2}^{(\frac{1}{2})}(\frac{1}{u}q^{-\frac{1}{2}})\mathcal{L}_{1}^{(\frac{1}{2})}(\frac{1}{u}q^{\frac{1}{2}})R^{(\frac{1}{2}, \frac{1}{2})}(q)\mathcal{E}_{\langle 12 \rangle}^{(1)}[\mathcal{H}_{\langle 12 \rangle}^{(1)}]^{-1} \\ &= \mathcal{F}_{\langle 12 \rangle}^{(1)} \mathcal{L}_{1}^{(\frac{1}{2})}(uq^{-\frac{1}{2}})\mathcal{L}_{2}^{(\frac{1}{2})}(uq^{\frac{1}{2}})\mathcal{E}_{\langle 12 \rangle}^{(1)} \mathcal{F}_{\langle 12 \rangle}^{(1)} R^{(\frac{1}{2}, \frac{1}{2})}(q)\mathcal{L}_{1}^{(\frac{1}{2})}(\frac{1}{u}q^{\frac{1}{2}})\mathcal{L}_{2}^{(\frac{1}{2})}(\frac{1}{u}q^{-\frac{1}{2}})\mathcal{E}_{\langle 12 \rangle}^{(1)}[\mathcal{H}_{\langle 12 \rangle}^{(1)}]^{-1} \\ &= \mathcal{F}_{\langle 12 \rangle}^{(1)} \mathcal{L}_{1}^{(\frac{1}{2})}(uq^{-\frac{1}{2}})\mathcal{L}_{2}^{(\frac{1}{2})}(uq^{\frac{1}{2}})\mathcal{L}_{2}^{(\frac{1}{2})}(u^{-1}q^{-\frac{1}{2}})\mathcal{L}_{1}^{(\frac{1}{2})}(u^{-1}q^{\frac{1}{2}})\mathcal{E}_{\langle 12 \rangle}^{(1)}, \end{aligned} $$

where we used the RLL equation given in (4.50) to get the second line, and the property (3.50), the RLL equation and (3.48) to get the third line. Finally, using (4.48), we get

$$ \mathcal{L}^{(1)}(u^{-1})\mathcal{L}^{(1)}(u) = \mathcal{L}^{(1)}(u)\mathcal{L}^{(1)}(u^{-1}) = \gamma(uq^{-\frac{1}{2}})\gamma(uq^{-\frac{3}{2}})\mathbb{I}_3. $$

More generally, by induction one gets the unitarity property for any spin-j:

$$ (4.54) \qquad \mathcal{L}^{(j)}(u^{-1})\mathcal{L}^{(j)}(u) = \mathcal{L}^{(j)}(u)\mathcal{L}^{(j)}(u^{-1}) = \left( \prod_{k=0}^{2j-1} - \gamma(uq^{-j-\frac{1}{2}+k}) \right) \mathbb{I}_{2j+1}. $$

Therefore, because the evaluation representation of the fused L-operators gives R-matrices, we also have

$$ (4.55) \qquad R^{(j_1, j_2)}(u)R^{(j_1, j_2)}(u^{-1}) = R^{(j_1, j_2)}(u^{-1})R^{(j_1, j_2)}(u) \propto \mathbb{I}_{(2j_1+1)(2j_2+1)}, $$

where the proportionality coefficient is a Laurent polynomial in $u$.

## 4.2. Spin-j K-operators.
Here, we introduce analogs of spin-j L-operators, that we call spin-j K-operators, and show that they satisfy the reflection equation (1.8) together with fusion and reduction relations.

As always, we consider the case $H = \mathcal{L}U_q sl_2$ as introduced in Section 2.4, however, without specifying its comodule algebra $B$. Assume that a universal K-matrix $\mathcal{K}$ exists for a choice of $B$ and the twist pair $(\psi, J) = (\eta, 1 \otimes 1)$ where $\eta$ is defined in (2.34). Similarly to the situation with the universal R-matrix $\mathfrak{R}$ for $\mathcal{L}U_q sl_2$, by this assumption we actually mean that there exists a “root vector” basis in $B$ such that $\mathfrak{R}$ can be formally written as a possibly infinite sum of products of the root vectors, i.e. each term in this sum is an element in $B \otimes \mathcal{L}U_q sl_2$. In particular, a basis of root vectors for $B = O_q$ was constructed in [BK17] and for $B = A_q$ in [T21c]. With this assumption, we can consider evaluation of $\mathfrak{R}$ on the second tensor component:

**Definition 4.15.** For $j \in \frac{1}{2}\mathbb{N}$, introduce

$$ (4.56) \qquad \mathbf{K}^{(j)}(u) = (\mathrm{id} \otimes \pi_{u^{-1}}^j)(\mathfrak{R}) \quad \in B((u^{-1})) \otimes \mathrm{End}(\mathbb{C}^{2j+1}). $$

We call $\mathbf{K}^{(j)}(u)$ the spin-j K-operator.

Similarly to the case of L-operators, recall Definition 4.1 and comments below it, we consider here $u$ as a formal variable. In contrast to the L-operators, we do not have examples of explicit evaluations of $\mathfrak{R}$, and that is why in the above definition we assumed that the entries of K-operators are in the ring of formal Laurent series $B((u^{-1}))$ instead of power series $B[[u^{-1}]]$. However, in the case of $B = A_q$ we will see by Conjecture 1 in Section 6 that $\mathbf{K}^{(\frac{1}{2})}(u)$ is in $B[[u^{-1}]] \otimes \mathrm{End}(\mathbb{C}^2)$, compare also with the similar Appel-Vlaar's result [AV22] on 1-leg universal K-matrices for $B = O_q$.
By Proposition 2.11, the universal K-matrix $\mathfrak{K}$ satisfies the $\psi$-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 $\psi$-twisted universal R-matrices (2.19). Firstly, note that $\psi = \eta$ from (2.34) is such that (recall the definition in (2.40))

$$ (4.57) \qquad \mathrm{ev}_u \circ \eta = \mathrm{ev}_{u^{-1}}. $$

Then the evaluations of the $\psi$-twisted universal R-matrices read:

$$ (4.58) \qquad (\pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2}) (\mathfrak{R}^\eta) = \mathcal{R}^{(j_1, j_2)}(1/(u_1 u_2)), $$

$$ (4.59) \qquad (\pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2}) ((\mathfrak{R}^\eta)_{21}) = \mathcal{R}_{21}^{(j_2, j_1)}(1/(u_1 u_2)), $$

$$ (4.60) \qquad (\pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2}) ((\mathfrak{R}_{21})^\eta) = \mathcal{R}_{21}^{(j_2, j_1)}(u_1 u_2), $$

$$ (4.61) \qquad (\pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2}) (\mathfrak{R}_{21}^{\eta\eta}) = \mathcal{R}_{21}^{(j_2, j_1)}(u_1/u_2), $$

$$ (4.62) \qquad (\pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2}) (\mathfrak{R}^{\eta\eta}) = \mathcal{R}^{(j_1, j_2)}(u_2/u_1), $$

where $\mathcal{R}_{21}^{(j_2, j_1)}(u)$ is defined in (4.21). Applying $p_{23}$ to (2.28) leads to $\mathfrak{K}_{13}(\mathfrak{R}^\eta)_{23}\mathfrak{K}_{12}\mathfrak{R}_{32} = \mathfrak{R}_{23}^{\eta\eta}\mathfrak{K}_{12}(\mathfrak{R}^\eta)_{32}\mathfrak{K}_{13}$. Finally, applying $(\mathrm{id} \otimes \pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2})$ to the latter equation using (4.58)-(4.62), it follows that the K-operator (4.56) satisfies the reflection equation<fnref n="13" />

$$ (4.63) \qquad \mathcal{R}_{12}^{(j_1, j_2)}(u_1/u_2) \mathbf{K}_1^{(j_1)}(u_1) \mathcal{R}_{21}^{(j_2, j_1)}(u_1 u_2) \mathbf{K}_2^{(j_2)}(u_2) = \\ \qquad \qquad \mathbf{K}_2^{(j_2)}(u_2) \mathcal{R}_{12}^{(j_1, j_2)}(u_1 u_2) \mathbf{K}_1^{(j_1)}(u_1) \mathcal{R}_{21}^{(j_2, j_1)}(u_1/u_2), $$

for any value of $j_1, j_2$.

Due to the P-symmetry (4.20), the relations (4.59)-(4.61) become

$$ (4.64) \qquad (\pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2}) ((\mathfrak{R}^\eta)_{21}) = \mathcal{R}^{(j_1, j_2)}(1/(u_1 u_2)), $$

$$ (4.65) \qquad (\pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2}) ((\mathfrak{R}_{21})^\eta) = \mathcal{R}^{(j_1, j_2)}(u_1 u_2), $$

$$ (4.66) \qquad (\pi_{u_1}^{j_1} \otimes \pi_{u_2}^{j_2}) (\mathfrak{R}_{21}^{\eta\eta}) = \mathcal{R}^{(j_1, j_2)}(u_1/u_2), $$

and the reflection equation (4.63) becomes the standard reflection equation

$$ (4.67) \qquad \mathcal{R}^{(j_1, j_2)}(u_1/u_2) \mathbf{K}_1^{(j_1)}(u_1) \mathcal{R}^{(j_1, j_2)}(u_1 u_2) \mathbf{K}_2^{(j_2)}(u_2) = \\ \qquad \qquad \mathbf{K}_2^{(j_2)}(u_2) \mathcal{R}^{(j_1, j_2)}(u_1 u_2) \mathbf{K}_1^{(j_1)}(u_1) \mathcal{R}^{(j_1, j_2)}(u_1/u_2), $$

where we can also replace $\mathcal{R}^{(j_1, j_2)}(u)$ by $\mathcal{R}^{(j_1, j_2)}(u)$ due to Lemma 4.13.

4.2.1. K-operators and fusion $(\frac{1}{2}, j) \rightarrow (j + \frac{1}{2})$. We follow here the same approach used for L-operators based on sub-representations in the tensor product of formal evaluation representations of $\mathcal{L}U_q sl_2$, now considering (K3) instead of (R3). Recall the intertwining operator $\mathcal{E}^{(j+\frac{1}{2})}$ is fixed by Lemma (3.1) and its pseudo-inverse $\mathcal{F}^{(j+\frac{1}{2})}$ is given in (3.25) with (3.26) and (3.27). We now obtain our first main result.

<footnote n="13">¹³As for the L-operator $\mathbf{L}^{(j)}(u)$, the K-operator should be written as $\mathbf{K}_{0i}^{(j)}(u)$ but here we use standard notation $\mathbf{K}_i^{(j)}(u)$ where we omit the label 0 corresponding to B.</footnote>
**Proposition 4.16.** The K-operators (4.56) satisfy for $j \in \frac{1}{2}\mathbb{N}$:

$$ (4.68) \qquad \mathbf{K}^{(j+\frac{1}{2})}(u) = \mathcal{F}_{\langle 12 \rangle}^{(j+\frac{1}{2})} \mathbf{K}_2^{(j)}(uq^{-\frac{1}{2}}) \mathcal{R}^{(\frac{1}{2},j)}(u^2 q^{j-\frac{1}{2}}) \mathbf{K}_1^{(\frac{1}{2})}(uq^j) \mathcal{E}_{\langle 12 \rangle}^{(j+\frac{1}{2})}. $$

*Proof.* By definition of the K-operator we have

$$ \mathbf{K}^{(j+\frac{1}{2})}(u) = (\mathrm{id} \otimes \pi_{u-1}^{j+\frac{1}{2}})(\mathfrak{K}). $$

Using the pseudo-inverse property $\mathcal{F}^{(j+\frac{1}{2})} \mathcal{E}^{(j+\frac{1}{2})} = \mathbb{I}_{2j+2}$, we get

$$ \begin{align*} (\mathrm{id} \otimes \pi_{u-1}^{j+\frac{1}{2}})(\mathfrak{K}) &= (1 \otimes \mathcal{F}^{(j+\frac{1}{2})} \mathcal{E}^{(j+\frac{1}{2})})(\mathrm{id} \otimes \pi_{u-1}^{j+\frac{1}{2}})(\mathfrak{K}) \\ &= (1 \otimes \mathcal{F}^{(j+\frac{1}{2})})[(\mathrm{id} \otimes \pi_{u-1q^{-j}}^{\frac{1}{2}} \otimes \pi_{u-1q^{\frac{1}{2}}}^{j}) \circ (\mathrm{id} \otimes \Delta)(\mathfrak{K})](1 \otimes \mathcal{E}^{(j+\frac{1}{2})}) \\ &= (1 \otimes \mathcal{F}^{(j+\frac{1}{2})})[(\mathrm{id} \otimes \pi_{u-1q^{-j}}^{\frac{1}{2}} \otimes \pi_{u-1q^{\frac{1}{2}}}^{j})(\mathfrak{K}_{13}\mathfrak{R}_{23}^{\eta}\mathfrak{K}_{12})](1 \otimes \mathcal{E}^{(j+\frac{1}{2})}). \end{align*} $$

The second equality is obtained using the intertwining property (3.19):

$$ (4.69) \qquad (1 \otimes \mathcal{E}^{(j+\frac{1}{2})})(\mathrm{id} \otimes \pi_{u-1}^{j+\frac{1}{2}})(\mathfrak{K}) = [(\mathrm{id} \otimes \pi_{u-1q^{-j}}^{\frac{1}{2}} \otimes \pi_{u-1q^{\frac{1}{2}}}^{j}) \circ (\mathrm{id} \otimes \Delta)(\mathfrak{K})](1 \otimes \mathcal{E}^{(j+\frac{1}{2})}). $$

Then, the third equality is due to (K3) and finally, from the definition of the K-operator (4.56) and the evaluation of the twisted universal R-matrix (4.58), the claim follows. $\square$

We see from the above Proposition 4.16 that $\mathbf{K}^{(j)}(u)$ is a formal power series in $u^{-1}$, i.e. it is in $B[[u^{-1}]] \otimes \mathrm{End}(\mathbb{C}^{2j+1})$, whenever $\mathbf{K}^{(\frac{1}{2})}(u)$ is so.

**Remark 4.17.** Similarly to Remark 4.7, it is clear from the proof of Proposition 4.16 that the K-operator does not depend on the choice of $\mathcal{F}^{(j)}$, as long as it satisfies $\mathcal{F}^{(j)}\mathcal{E}^{(j)} = \mathbb{I}_{2j+1}$.

4.2.2. K-operators and reduction $(\frac{1}{2}, j) \to (j - \frac{1}{2})$. 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 $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$ is fixed by Lemma 3.2 and its pseudo-inverse $\bar{\mathcal{F}}^{(j-\frac{1}{2})}$ is given in (3.33) with (3.34).

**Proposition 4.18.** The K-operators (4.56) satisfy for $j \in \frac{1}{2}\mathbb{N}_+$:

$$ (4.70) \qquad \mathbf{K}^{(j-\frac{1}{2})}(u) = \bar{\mathcal{F}}_{\langle 12 \rangle}^{(j-\frac{1}{2})} \mathbf{K}_2^{(j)}(uq^{-\frac{1}{2}}) \mathcal{R}^{(\frac{1}{2},j)}(u^2 q^{-j-\frac{3}{2}}) \mathbf{K}_1^{(\frac{1}{2})}(uq^{-j-1}) \bar{\mathcal{E}}_{\langle 12 \rangle}^{(j-\frac{1}{2})}. $$

Recall that we assumed that the universal K-matrix exists for a given choice of $B$ and the twist pair $(\eta, 1 \otimes 1)$. 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 $\mathbf{K}^{(j)}(u)$ either using the fusion for $(\frac{1}{2}, j - \frac{1}{2}) \to j$ or using the reduction $(\frac{1}{2}, j + \frac{1}{2}) \to j$.

**Remark 4.19.** Consider the opposite coproduct $\Delta^{op} = \mathfrak{p} \circ \Delta$ with the definition (A.9). It follows from (K3):

$$ (4.71) \qquad (\mathrm{id} \otimes \Delta^{op})(\mathfrak{K}) = \mathfrak{K}_{12}(\mathfrak{R}^{\psi})_{32}\mathfrak{K}_{13}. $$
Recall that we obtained for $\Delta^{op}$ an intertwining relation in (3.37) where $\mathcal{E}^{(j+\frac{1}{2})}$ is fixed as in (3.20), see Remark 3.3. Thus, we also take $\mathcal{F}^{(j+\frac{1}{2})}$ as defined in (3.26), (3.27). Then, using (3.37) and (4.71), we obtain a new fusion relation for any $j \in \frac{1}{2}\mathbb{N}$:

$$ (4.72) \qquad \mathbf{K}^{(j+\frac{1}{2})}(u) = \mathcal{F}_{(12)}^{(j+\frac{1}{2})} \mathbf{K}_{1}^{(\frac{1}{2})}(uq^{-j}) \mathcal{R}^{(\frac{1}{2},j)}(u^2q^{-j+\frac{1}{2}}) \mathbf{K}_{2}^{(j)}(uq^{\frac{1}{2}}) \mathcal{E}_{(12)}^{(j+\frac{1}{2})}. $$

Similarly, we also have the intertwining relation for $\Delta^{op}$ given in (3.36) with $\bar{\mathcal{E}}^{(j-\frac{1}{2})}$ fixed as in (3.32), see Remark 3.3, and we take $\bar{\mathcal{F}}^{(j-\frac{1}{2})}$ as defined in (3.33) with (3.34). Then, using (4.71) and (3.38), we obtain a new reduction relation for any $j \in \frac{1}{2}\mathbb{N}_+$:

$$ (4.73) \qquad \mathbf{K}^{(j-\frac{1}{2})}(u) = \bar{\mathcal{F}}_{(12)}^{(j-\frac{1}{2})} \mathbf{K}_{1}^{(\frac{1}{2})}(uq^{j+1}) \mathcal{R}^{(\frac{1}{2},j)}(u^2q^{j+\frac{3}{2}}) \mathbf{K}_{2}^{(j)}(uq^{\frac{1}{2}}) \bar{\mathcal{E}}_{(12)}^{(j-\frac{1}{2})}. $$

To conclude this subsection, let us comment on the other choice of twist pair $(\psi, J) = (\eta, \mathfrak{R}_{21}\mathfrak{R})$ from Example 2.13. The universal reflection equation (2.28) with $\psi = \eta$ produces the same reflection equation (4.67) as with the first choice of the twist pair. However, the fusion formula based on the axiom (K3) requires evaluating $J$ on tensor product of evaluation representations. Since this evaluation might not converge, due to infinite sums of Casimir eigenvalues, we do not consider it in detail.

### 4.3. Comodule algebra structure using K-operators.
Given the Hopf algebra $H = \mathcal{L}U_q sl_2$, it is known that the coproduct, antipode and counit can be formulated solely in terms of so-called Ding-Frenkel L-operators [DF93]:

$$ (4.74) \qquad \mathbf{L}^{+}(u) = (\mathrm{id} \otimes \pi_{u-1}^{\frac{1}{2}})(\mathfrak{R}), \qquad \mathbf{L}^{-}(u) = [(\mathrm{id} \otimes \pi_{u-1}^{\frac{1}{2}})(\mathfrak{R}_{21})]^{-1}, $$

They are computed in Appendix B, see (B.26) and (B.33). Then, one finds that $\mathbf{L}^{\pm}(u)$ are formal power series in $u^{\mp 1}$, i.e. $\mathbf{L}^{\pm}(u)$ are in $\mathcal{L}U_q sl_2[[u^{\mp 1}]] \otimes \mathrm{End}(\mathbb{C}^2)$. The modes of the entries of $\mathbf{L}^{\pm}(u)$ generate $\mathcal{L}U_q sl_2$ and they satisfy the Yang-Baxter algebra relations:

$$ (4.75) \qquad R^{(\frac{1}{2}, \frac{1}{2})}(u/v) \mathbf{L}_1^{\pm}(u) \mathbf{L}_2^{\pm}(v) = \mathbf{L}_2^{\pm}(v) \mathbf{L}_1^{\pm}(u) R^{(\frac{1}{2}, \frac{1}{2})}(u/v), $$

$$ (4.76) \qquad R^{(\frac{1}{2}, \frac{1}{2})}(u/v) \mathbf{L}_1^{\pm}(u) \mathbf{L}_2^{\mp}(v) = \mathbf{L}_2^{\mp}(v) \mathbf{L}_1^{\pm}(u) R^{(\frac{1}{2}, \frac{1}{2})}(u/v), $$

with $R^{(\frac{1}{2}, \frac{1}{2})}(u)$ defined in (4.8). These four relations are obtained by applying $(\mathrm{id} \otimes \pi_{u-1}^{\frac{1}{2}} \otimes \pi_{v-1}^{\frac{1}{2}})$ to the following four equations:

$$ \begin{aligned} \mathfrak{p}_{23}(2.3), & \qquad \mathfrak{R}_{31}^{-1} \mathfrak{R}_{21}^{-1} [\mathfrak{p}_{13}(2.3)] \mathfrak{R}_{21}^{-1} \mathfrak{R}_{31}^{-1}, \\ \mathfrak{R}_{31}^{-1} [\mathfrak{p}_{23} \circ \mathfrak{p}_{12}(2.3)] \mathfrak{R}_{31}^{-1}, & \qquad \mathfrak{R}_{23}^{-1} \mathfrak{R}_{21}^{-1} [\mathfrak{p}_{12}(2.3)] \mathfrak{R}_{21}^{-1} \mathfrak{R}_{23}^{-1}, \end{aligned} $$
and using (4.74), (4.7) with (4.8) and (4.20), while the unitarity property (4.55) is also used for the last one. The coproduct, antipode and counit of $\mathcal{L}U_qsl_2$ are given, respectively, by<fnref n="14" />:

$$ (4.78) \qquad (\Delta \otimes \mathrm{id})(\mathbf{L}^{\pm}(u)) = (\mathbf{L}^{\pm}(u))_{[1]} (\mathbf{L}^{\pm}(u))_{[2]}, $$

$$ (4.79) \qquad (S \otimes \mathrm{id})(\mathbf{L}^{\pm}(u)) = (\mathbf{L}^{\pm}(u))^{-1}, $$

$$ (4.80) \qquad (\epsilon \otimes \mathrm{id})(\mathbf{L}^{\pm}(u)) = 1. $$

These relations are easily understood using (R2) and (2.1)-(2.2): the relation (4.78) is obtained by applying $(\mathrm{id} \otimes \mathrm{id} \otimes \pi_{u-1}^{\frac{1}{2}})$ on (R2), and similarly for $\mathbf{L}^{-}(u)$ using $(\pi_{u-1}^{\frac{1}{2}} \otimes \mathrm{id} \otimes \mathrm{id}) \circ (\mathrm{id} \otimes \Delta)(\mathfrak{R}^{-1})$. The relations (4.79), (4.80) follow immediately from (2.1), (2.2).

Consider the subalgebra in $B$ generated by the matrix entries of the K-operator $\mathbf{K}^{(\frac{1}{2})}(u)$. They satisfy the reflection equation (4.67) for $j_1 = j_2 = \frac{1}{2}$. Similarly to the coproduct of $\mathcal{L}U_qsl_2$ discussed above, the coaction for this subalgebra can be expressed in terms of L- and K-operators.

**Proposition 4.20.** The coaction map $\delta: B \to B \otimes \mathcal{L}U_qsl_2$ restricted to the subalgebra generated by the matrix entries of $\mathbf{K}^{(\frac{1}{2})}(u)$ is such that

$$ (4.81) \qquad (\delta \otimes \mathrm{id})(\mathbf{K}^{(\frac{1}{2})}(u)) = ([\mathbf{L}^{-}(u^{-1})]^{-1})_{[2]} (\mathbf{K}^{(\frac{1}{2})}(u))_{[1]} (\mathbf{L}^{+}(u))_{[2]}. $$

*Proof.* From the fundamental axiom (K2), the l.h.s. of (4.81) can be written as

$$ (\mathrm{id} \otimes \mathrm{id} \otimes \pi_{u-1}^{\frac{1}{2}}) \circ (\delta \otimes \mathrm{id})(\mathfrak{R}) = (\mathrm{id} \otimes \mathrm{id} \otimes \pi_{u-1}^{\frac{1}{2}}) \circ ((\mathfrak{R}^{\eta})_{32}\mathfrak{R}_{13}\mathfrak{R}_{23}), $$

where $(\mathfrak{R}^{\eta})_{32} = (\mathrm{id} \otimes \mathrm{id} \otimes \eta)(\mathfrak{R}_{32})$. Then, using $\pi_{u-1}^{j} \circ \eta = \pi_{u}^{j}$ together with the definition of the K-operator and $\mathbf{L}^{\pm}(u)$ given respectively in (4.56), (4.74), the claim follows. $\square$

In the case when $B$ is generated by the matrix entries of the K-operator $\mathbf{K}^{(\frac{1}{2})}(u)$, eq. (4.81) expresses the coaction map for $B$ solely in terms of L- and K-operators. This is the case when $B = \mathcal{A}_q$ and it will be discussed in Section 6.

Let us now introduce a formal generalization of the evaluated coaction (2.46)

$$ (4.82) \qquad \delta_w = (\mathrm{id} \otimes \mathrm{ev}_w) \circ \delta: \quad B \to B \otimes U_qsl_2[w^{\pm 1}] $$

with a formal variable $w$. We will nevertheless call $\delta_w$ the evaluated coaction for brevity. We now consider the evaluated coaction applied to the matrix elements of the spin-$j$ K-operator, which is obtained by taking the image of (K2) under the formal evaluation map $(\mathrm{id} \otimes \mathrm{ev}_w \otimes \pi_{u-1}^{j})$ using (4.1), (4.23), (4.57), and is given by

$$ (4.83) \qquad (\delta_w \otimes \mathrm{id})(\mathbf{K}^{(j)}(u)) = (\mathbf{L}^{(j)}(u/w))_{[2]} (\mathbf{K}^{(j)}(u))_{[1]} (\mathbf{L}^{(j)}(uw))_{[2]}. $$

$$ (4.77) \qquad ((T)_{[1]}(T')_{[2]})_{ij} = \sum_{k=1}^{2} (T)_{ik} \otimes (T')_{kj}. $$

<footnote n="14">¹⁴The index $[j]$ characterizes the ‘quantum space’ $V_{[j]}$ on which the entries of $\mathbf{L}^{\pm}(u)$ act. With respect to the ordering $V_{[1]} \otimes V_{[2]}$, one has:</footnote>
Whereas the action of $(\mathrm{id} \otimes \pi_{v-1}^j)$ on (K1) gives

$$ (4.84) \qquad \mathbf{K}^{(j)}(v)(\mathrm{id} \otimes \pi^j)[\delta_{v-1}(b)] = (\mathrm{id} \otimes \pi^j)[\delta_v(b)]\mathbf{K}^{(j)}(v). $$

We call this equation the twisted intertwining relation for $\mathbf{K}^{(j)}(u)$.

## 5. FUSED K-OPERATORS FOR $\mathcal{A}_q$

In this section, we consider the comodule algebra $B = \mathcal{A}_q$ 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 $B = \mathcal{A}_q$ and recall the corresponding Faddeev-Reshetikhin-Taktadjan type presentation following [BS09, BB17]. Then, in Section 5.2, fused K-operators $\mathcal{K}^{(j)}(u)$ built from the fundamental K-operator by analogy with (4.72) are shown to satisfy the reflection equation (4.67) for all $j \in \frac{1}{2}\mathbb{N}_+$ where $\mathbf{K}^{(j)}(u)$ is replaced by $\mathcal{K}^{(j)}(u)$. This is the main result in this section. We also establish the unitarity and invertibility properties of $\mathcal{K}^{(j)}(u)$ 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 $\mathcal{A}_q$ and also establish in Section 5.6 the twisted intertwining relations for the fused K-operators which are similar to (4.84).

**5.1. The fundamental K-operator for $\mathcal{A}_q$.** An alternative presentation for $\mathcal{A}_q$ 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 $R^{(\frac{1}{2}, \frac{1}{2})}(u)$ be the symmetric $R$-matrix (4.8) which satisfies the quantum Yang-Baxter equation (4.11) with the substitution $R^{(j_k, j_l)}(u) \to R^{(\frac{1}{2}, \frac{1}{2})}(u)$. We now introduce the K-operator that provides the reflection algebra presentation of $\mathcal{A}_q$, with the parametrization from (2.43).

**Theorem 5.1 ([BS09]).** $\mathcal{A}_q$ admits a presentation in the form of a reflection algebra. Introduce the generating functions:

$$ (5.1) \qquad \mathcal{W}_+(u) = \sum_{k \in \mathbb{N}} \mathcal{W}_{-k} U^{-k-1}, \qquad \mathcal{W}_-(u) = \sum_{k \in \mathbb{N}} \mathcal{W}_{k+1} U^{-k-1}, $$

$$ (5.2) \qquad \mathcal{G}_+(u) = \sum_{k \in \mathbb{N}} \mathcal{G}_{k+1} U^{-k-1}, \qquad \mathcal{G}_-(u) = \sum_{k \in \mathbb{N}} \tilde{\mathcal{G}}_{k+1} U^{-k-1}, $$

where the shorthand notation $U = (qu^2 + q^{-1}u^{-2})/(q+q^{-1})$ is used. The defining relations are given by:

$$ (5.3) \qquad R^{(\frac{1}{2}, \frac{1}{2})}(u/v) K_1^{(\frac{1}{2})}(u) R^{(\frac{1}{2}, \frac{1}{2})}(uv) K_2^{(\frac{1}{2})}(v) = K_2^{(\frac{1}{2})}(v) R^{(\frac{1}{2}, \frac{1}{2})}(uv) K_1^{(\frac{1}{2})}(u) R^{(\frac{1}{2}, \frac{1}{2})}(u/v) $$

with the $R$-matrix from (4.8) and

$$ (5.4) \qquad \mathcal{K}^{(\frac{1}{2})}(u) = \left( \frac{uq\mathcal{W}_+(u) - u^{-1}q^{-1}\mathcal{W}_-(u)}{\frac{1}{k_+(q+q^{-1})}\mathcal{G}_-(u) + \frac{k_-(q+q^{-1})}{q-q^{-1}}} \frac{\frac{1}{k_-(q+q^{-1})}\mathcal{G}_+(u) + \frac{k_+(q+q^{-1})}{q-q^{-1}}}{uq\mathcal{W}_-(u) - u^{-1}q^{-1}\mathcal{W}_+(u)} \right), $$

and with the identification (2.43).
We call $K^{(\frac{1}{2})}(u)$ the fundamental $K$-operator for $\mathcal{A}_q$. We note that $U^{-1}$ can be written as a power series in $u^{-2}$ with the following expansion

$$ (5.5) \qquad U^{-1} = (1 + q^{-2})u^{-2} \sum_{\ell=0}^{\infty} (-u^{-4}q^{-2})^{\ell}. $$

Thus, the generating functions $W_{\pm}(u)$, $G_{\pm}(u)$ start with $u^{-2}$. Consequently, the leading term of the diagonal entries of the $K$-operator in (5.4) is at $u^{-1}$, while the leading term of the off-diagonal entries is at $u^0$. Therefore, $K^{(\frac{1}{2})}(u)$ is in $\mathcal{A}_q[[u^{-1}]] \otimes \text{End}(\mathbb{C}^2)$.

When we write $W_{\pm}(u^{-1})$ and $G_{\pm}(u^{-1})$ it means replacing $u$ by $u^{-1}$ in $U$ and developing in power series in $u^{-2}$, this way we get

$$ (5.6) \qquad W_{\pm}(u^{-1}) = W_{\pm}(uq^{-1}) \quad \text{and} \quad G_{\pm}(u^{-1}) = G_{\pm}(uq^{-1}). $$

In particular, this shows that $K^{(\frac{1}{2})}(u^{-1})$ is equally in $\mathcal{A}_q[[u^{-1}]] \otimes \text{End}(\mathbb{C}^2)$.

Explicitly, in terms of the generating functions (5.1), (5.2) the defining relations (5.3) read:

$$ (5.7) \qquad [W_{\pm}(u), W_{\pm}(v)] = 0, $$

$$ (5.8) \qquad [W_{+}(u), W_{-}(v)] + [W_{-}(u), W_{+}(v)] = 0, $$

$$ (5.9) \qquad [G_{\epsilon}(u), W_{\pm}(v)] + [W_{\pm}(u), G_{\epsilon}(v)] = 0, \quad \epsilon = \pm, $$

$$ (5.10) \qquad [G_{\pm}(u), G_{\pm}(v)] = 0, $$

$$ (5.11) \qquad [G_{+}(u), G_{-}(v)] + [G_{-}(u), G_{+}(v)] = 0, $$

$$ (5.12) \qquad (U - V)[W_{\pm}(u), W_{\mp}(v)] = \frac{(q - q^{-1})}{\rho(q + q^{-1})} (G_{\pm}(u)G_{\mp}(v) - G_{\pm}(v)G_{\mp}(u)) \\ \qquad \qquad + \frac{1}{(q + q^{-1})} (G_{\pm}(u) - G_{\mp}(u) + G_{\mp}(v) - G_{\pm}(v)), $$

$$ (5.13) \qquad W_{\pm}(u)W_{\pm}(v) - W_{\mp}(u)W_{\mp}(v) + \frac{1}{\rho(q^2 - q^{-2})} [G_{\pm}(u), G_{\mp}(v)] \\ \qquad \qquad + \frac{1 - UV}{U - V} (W_{\pm}(u)W_{\mp}(v) - W_{\pm}(v)W_{\mp}(u)) = 0, $$

$$ (5.14) \qquad U[G_{\mp}(v), W_{\pm}(u)]_q - V[G_{\mp}(u), W_{\pm}(v)]_q - (q - q^{-1})(W_{\mp}(u)G_{\mp}(v) - W_{\mp}(v)G_{\mp}(u)) \\ \qquad \qquad + \rho(UW_{\pm}(u) - VW_{\pm}(v) - W_{\mp}(u) + W_{\mp}(v)) = 0, $$

$$ (5.15) \qquad U[W_{\mp}(u), G_{\mp}(v)]_q - V[G_{\mp}(v), G_{\mp}(u)]_q - (q - q^{-1})(W_{\pm}(u)G_{\mp}(v) - W_{\pm}(v)G_{\mp}(u)) \\ \qquad \qquad + \rho(UW_{\mp}(u) - VW_{\mp}(v) - W_{\pm}(u) + W_{\pm}(v)) = 0. $$

We note that the corresponding relations for the modes $W_0$, $W_1$, $\{G_{k+1}\}_{k \in \mathbb{N}}$ give the compact presentation from Definition 2.14. The modes $\{W_{-k}, W_{k+1}, G_{k+1}, \tilde{G}_{k+1}\}_{k \in \mathbb{N}}$ give actually PBW type generators, called *alternating*. The expressions of the alternating generators $W_{-k}$ and $W_{k+1}$ in terms of the generators of the compact presentation (called *essential* generators in [T21b]) are obtained recursively from two of the defining relations of $\mathcal{A}_q$ [BS09, Def. 3.1 & eqs. (3.2),(3.3)]:

$$ (5.16) \qquad W_{-k-1} = W_{k+1} + \frac{1}{\rho}[W_0, G_{k+1}]_q, $$

$$ (5.17) \qquad W_{k+2} = W_{-k} + \frac{1}{\rho}[G_{k+1}, W_1]_q. $$
We introduce the following automorphism of $\mathcal{A}_q$:

$$ (5.18) \qquad \sigma: \quad W_{\pm}(u) \to W_{\mp}(u), \quad G_{\pm}(u) \to G_{\mp}(u), \quad k_{\pm} \to k_{\mp}. $$

It is obtained from the reflection equation (5.3). Indeed, note that the R-matrix is such that $R^{(\frac{1}{2}, \frac{1}{2})}(u) = MR^{(\frac{1}{2}, \frac{1}{2})}(u)M$, with $M = \sigma_x \otimes \sigma_x$. Consider the conjugation of the K-operator by $\sigma_x$. Its entries read: $(\sigma_x K^{(\frac{1}{2})}(u)\sigma_x)_{i,j} = (K^{(\frac{1}{2})}(u))_{3-i,3-j}$ for $1 \le i, j \le 2$. Then, multiplying (5.3) on both sides by $M \otimes M$, the automorphism $\sigma$ 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 $\mathcal{A}_q$, given by [Sk88]:

$$ (5.19) \qquad \Gamma(u) = \mathrm{tr}_{12} (\mathcal{P}_{12}^{-} K_{1}^{(\frac{1}{2})}(u) R^{(\frac{1}{2}, \frac{1}{2})}(qu^{2}) K_{2}^{(\frac{1}{2})}(uq)), $$

where $\mathcal{P}^{-}$ is defined below (4.12).

**Proposition 5.2** ([BB17, T21a]). *The quantum determinant of the fundamental K-operator*

$$ (5.20) \qquad \Gamma(u) = \frac{(u^2 q^2 - u^{-2} q^{-2})}{2(q - q^{-1})} \left( \Delta^{(\frac{1}{2})}(u) - \frac{2\rho}{q - q^{-1}} \right), $$

with

$$ \begin{align} \Delta^{(\frac{1}{2})}(u) &= -(q - q^{-1})(q^2 + q^{-2}) (W_{+}(u)W_{+}(uq) + W_{-}(u)W_{-}(uq)) \nonumber \\ &\quad + (q - q^{-1})(u^2 q^2 + u^{-2} q^{-2}) (W_{+}(u)W_{-}(uq) + W_{-}(u)W_{+}(uq)) \tag{5.21} \\ &\quad - \frac{(q - q^{-1})}{\rho} (G_{+}(u)G_{-}(uq) + G_{-}(u)G_{+}(uq)) - G_{+}(u) - G_{+}(uq) - G_{-}(u) - G_{-}(uq), \nonumber \end{align} $$

is such that $[\Gamma(u), K_{\min}^{(\frac{1}{2})}(v)] = 0$. It generates the center of $\mathcal{A}_q$, denoted by $Z(\mathcal{A}_q)$.

We can expand $\Delta^{(\frac{1}{2})}(u)$ as a formal power series in $u^{-2}$

$$ (5.22) \qquad \Delta^{(\frac{1}{2})}(u) = \sum_{k=0}^{\infty} u^{-2k-2} c_{k+1} \Delta_{k+1}, $$

where

$$ (5.23) \qquad c_k = -q^{-2k} (q + q^{-1})^k (q^k + q^{-k}). $$

An explicit formula expressing the central elements $\Delta_{k+1}$ in terms of the alternating generators we introduced above is given in [BB17, Lem. 2.1]. Here, we give a few of them, as they are used below.

**Example 5.3.** *First few modes from (5.22), as elements of $Z(\mathcal{A}_q)$, are*

$$ (5.24) \qquad \Delta_1 = G_1 + \tilde{G}_1 - (q - q^{-1})(W_0 W_1 + W_1 W_0), $$

$$ (5.25) \qquad \Delta_2 = G_2 + \tilde{G}_2 - \frac{(q^2 - q^{-2})}{(q^2 + q^{-2})} (q^{-1}W_0 W_2 + qW_2 W_0 + q^{-1}W_1 W_{-1} + qW_{-1} W_1) \\ \qquad \qquad + \frac{(q - q^{-1})}{(q^2 + q^{-2})} \left( (q^2 + q^{-2})(W_0^2 + W_1^2) + \frac{\tilde{G}_1 G_1 + G_1 \tilde{G}_1}{\rho} \right), $$

where $W_{-1}$ and $W_2$ are determined by (5.16) and (5.17) at $k = 0$, respectively.
**Lemma 5.4.**

$$
(5.26) \qquad \mathcal{K}^{(\frac{1}{2})}(u^{-1})\mathcal{K}^{(\frac{1}{2})}(u) = \mathcal{K}^{(\frac{1}{2})}(u)\mathcal{K}^{(\frac{1}{2})}(u^{-1}) = \frac{\Gamma(uq^{-1})}{c(u^{-2})} \mathbb{I}_2,
$$

where $c(u)$ and $\Gamma(u)$ are respectively given in (3.42) and (5.20), and where the right hand side we consider as an element in $Z(\mathcal{A}_q)[[u^{-1}]]$.

Proof. Using our developing in series convention in (5.6) and the ordering relations of $\mathcal{A}_q$ given in Appendix C, it is straightforward to calculate (5.26). $\square$

We will call the above property (5.26) the unitarity property of the fundamental K-operator by
analogy with the L-operator.

Let us now show invertibility property of the fundamental K-operator $\mathcal{K}^{(\frac{1}{2})}(u)$ from (5.4). We first show that $\Gamma(u)$ is invertible in the ring $\mathcal{A}_q((u^{-1}))$. We notice that the constant term of $\Delta^{(\frac{1}{2})}(u)$ from (5.21) is zero, which implies using (5.20) that the constant term of $u^{-2}\Gamma(u)$ is non-zero. Therefore, the latter formal power series is invertible in $Z(\mathcal{A}_q)[[u^{-1}]]$ by [T21d, Lem. 4.1]. We thus obtain that $\Gamma(u)$ is invertible in the ring $\mathcal{A}_q((u^{-1}))$. Explicit expressions of the modes of $\Gamma(u)^{-1}$ can be deduced from [T21d, Lem. 4.1] shifting the mode index by $-2$. Since $\Gamma(u)$ is invertible, it follows by Lemma 5.4 that $\mathcal{K}^{(\frac{1}{2})}(u)$ is invertible too and its inverse is given by:

$$
(5.27) \qquad \left[ \mathcal{K}^{(\frac{1}{2})}(u) \right]^{-1} = \frac{c(u^{-2})}{\Gamma(uq^{-1})} \mathcal{K}^{(\frac{1}{2})}(u^{-1}) .
$$

**Remark 5.5.** A central element of $\mathcal{A}_q$ denoted $\mathcal{Z}(t)$ has been proposed [T21a, Def. 8.4]. It is easily checked that adapting its expression to our conventions, one has

$$
(5.28) \qquad \frac{\Gamma(uq^{-\frac{1}{2}})}{c(u^2q)} \to \mathcal{Z}(t),
$$

with the identification

$$
\begin{align*}
\rho \rightarrow -(q^2 - q^{-2})^2, \quad q^{-1}u^{-2} \rightarrow t, \quad \mathcal{W}_{\mp}(uq^{\frac{1}{2}}) \rightarrow S\mathcal{W}^{\pm}(S), \quad \mathcal{W}_{\mp}(uq^{-\frac{1}{2}}) \rightarrow T\mathcal{W}^{\pm}(T) \\
\mathcal{G}_{+}(uq^{\frac{1}{2}}) + \rho/(q - q^{-1}) \rightarrow \mathcal{G}(S), \quad \mathcal{G}_{-}(uq^{-\frac{1}{2}}) + \rho/(q - q^{-1}) \rightarrow \tilde{\mathcal{G}}(T).
\end{align*}
$$

5.2. Fused K-operators for $\mathcal{A}_q$. Recall $R^{(\frac{1}{2}, \frac{1}{2})}(u)$ and the fundamental K-operator $\mathcal{K}^{(\frac{1}{2})}(u)$, given respectively in (4.8) and (5.4), satisfy the reflection equation (5.3). By analogy with (4.72), we now introduce fused K-operators $\mathcal{K}^{(j)}(u) \in \mathcal{A}_q((u^{-1})) \otimes \text{End}(\mathbb{C}^{2j+1})$:

**Definition 5.6.** For $j \in \frac{1}{2}\mathbb{N}_+$, the fused K-operators for $\mathcal{A}_q$ are

$$
(5.29) \qquad \mathcal{K}^{(j+\frac{1}{2})}(u) = \mathcal{F}_{(12)}^{(j+\frac{1}{2})} \mathcal{K}_{1}^{(\frac{1}{2})}(uq^{-j}) R^{(\frac{1}{2},j)}(u^2 q^{-j+\frac{1}{2}}) \mathcal{K}_{2}^{(j)}(uq^{\frac{1}{2}}) \mathcal{E}_{(12)}^{(j+\frac{1}{2})},
$$

with $\mathcal{K}^{(\frac{1}{2})}(u)$ defined in (5.4). We note that $\mathcal{K}^{(j)}(u) \in u^{4j^2-2j}\mathcal{A}_q[[u^{-1}]] \otimes \text{End}(\mathbb{C}^{2j+1})$.

The following Theorem is our second main result.

**Theorem 5.7.** The fused K-operators $\mathcal{K}^{(j)}(u)$ satisfy the reflection equation for any $j_1, j_2 \in \frac{1}{2}\mathbb{N}_+$:

$$
(5.30) \qquad R^{(j_1, j_2)}(u_1/u_2) \mathcal{K}_1^{(j_1)}(u_1) R^{(j_1, j_2)}(u_1 u_2) \mathcal{K}_2^{(j_2)}(u_2) = \\
\qquad \qquad \qquad \qquad \mathcal{K}_2^{(j_2)}(u_2) R^{(j_1, j_2)}(u_1 u_2) \mathcal{K}_1^{(j_1)}(u_1) R^{(j_1, j_2)}(u_1/u_2).
$$
The proof of this theorem is very technical and was delegated to Appendix D. We proceed by induction on $j_1, j_2$, using several expressions for the fused R-matrices $R^{(j_1, j_2)}(u)$ defined in (4.37). Moreover, the decomposition of $R^{(\frac{1}{2}, j)}(q^{j+\frac{1}{2}})$, expressed in terms of $\mathcal{E}^{(j+\frac{1}{2})}$, $\mathcal{F}^{(j+\frac{1}{2})}$, and $\mathcal{H}^{(j+1/2)}$ as given in Lemma 3.5, is crucial for the proof of this theorem, along with the relations from Corollary 3.6.

**5.3. Unitarity and invertibility properties.** We now discuss the unitarity and invertibility properties of the fused K-operators $\mathcal{K}^{(j)}(u)$ defined by (5.29). Recall that $\mathcal{K}^{(\frac{1}{2})}(u)$ satisfies the unitarity property and is invertible, see Lemma 5.4 and expression in (5.27), respectively. We generalize these properties for any spin-$j$.

**Proposition 5.8.** Let

$$ (5.31) \qquad \hat{\mathcal{K}}^{(j+\frac{1}{2})}(u) = \mathcal{F}_{\langle 12 \rangle}^{(j+\frac{1}{2})} \hat{\mathcal{K}}_2^{(j)}(uq^{-\frac{1}{2}}) R^{(\frac{1}{2}, j)}(u^2 q^{j-\frac{1}{2}}) \hat{\mathcal{K}}_1^{(\frac{1}{2})}(uq^j) \mathcal{E}_{\langle 12 \rangle}^{(j+\frac{1}{2})}, $$

for $j \in \frac{1}{2}\mathbb{N}_+$ and with $\hat{\mathcal{K}}^{(\frac{1}{2})}(u) \equiv \mathcal{K}^{(\frac{1}{2})}(u)$. Then

$$ (5.32) \qquad \begin{aligned} \mathcal{K}^{(j)}(u) \hat{\mathcal{K}}^{(j)}(u^{-1}) &= \left( \prod_{k=0}^{2j-1} \frac{\Gamma(uq^{-j-\frac{1}{2}+k})}{c(u^{-2}q^{2j-1-2k})} \right) \left( \prod_{k=0}^{2j-2} \prod_{\ell=0}^{2j-k-2} c(u^2q^{2j-1-2k-\ell})c(u^{-2}q^{1-k+\ell}) \right) \mathbb{I}_{2j+1} \\ &= \hat{\mathcal{K}}^{(j)}(u^{-1}) \mathcal{K}^{(j)}(u), \end{aligned} $$

where $\mathcal{K}^{(j)}(u)$, $\Gamma(u)$ and $c(u)$ are respectively given in (5.29), (5.20), (3.42).

**Proof.** Recall that (5.32) for $j = \frac{1}{2}$ was proven in Lemma 5.4. First, we show that (5.32) holds for $j = 1$. Using (5.29) and (5.31), we find that the product $\mathcal{K}^{(1)}(u)\hat{\mathcal{K}}^{(1)}(u^{-1})$ equals

$$ \begin{aligned} & \mathcal{F}_{\langle 12 \rangle}^{(1)} \mathcal{K}_1^{(\frac{1}{2})}(uq^{-\frac{1}{2}}) R^{(\frac{1}{2}, \frac{1}{2})}(u^2) \mathcal{K}_2^{(\frac{1}{2})}(uq^{\frac{1}{2}}) \mathcal{E}_{\langle 12 \rangle}^{(1)} \mathcal{F}_{\langle 12 \rangle}^{(1)} \hat{\mathcal{K}}_2^{(\frac{1}{2})}(u^{-1}q^{-\frac{1}{2}}) R^{(\frac{1}{2}, \frac{1}{2})}(u^{-2}) \hat{\mathcal{K}}_1^{(\frac{1}{2})}(u^{-1}q^{\frac{1}{2}}) \mathcal{E}_{\langle 12 \rangle}^{(1)} \\ &= \mathcal{F}_{\langle 12 \rangle}^{(1)} \mathcal{K}_1^{(\frac{1}{2})}(uq^{-\frac{1}{2}}) R^{(\frac{1}{2}, \frac{1}{2})}(u^2) \mathcal{K}_2^{(\frac{1}{2})}(uq^{\frac{1}{2}}) \hat{\mathcal{K}}_2^{(\frac{1}{2})}(u^{-1}q^{-\frac{1}{2}}) R^{(\frac{1}{2}, \frac{1}{2})}(u^{-2}) \hat{\mathcal{K}}_1^{(\frac{1}{2})}(u^{-1}q^{\frac{1}{2}}) \mathcal{E}_{\langle 12 \rangle}^{(1)}, \end{aligned} $$

where we removed the product $\mathcal{E}^{(1)}\mathcal{F}^{(1)}$ on the second line, similarly to the derivation of (4.53). Then, using (3.43) and (5.26) we have

$$ \mathcal{K}^{(1)}(u) \hat{\mathcal{K}}^{(1)}(u^{-1}) = -\Gamma(uq^{-\frac{3}{2}})\Gamma(uq^{-\frac{1}{2}}) \mathbb{I}_3. $$

Similarly, we get

$$ \hat{\mathcal{K}}^{(1)}(u^{-1}) \mathcal{K}^{(1)}(u) = -\Gamma(uq^{-\frac{3}{2}})\Gamma(uq^{-\frac{1}{2}}) \mathbb{I}_3. $$

More generally, by induction we get both equalities in (5.32) for all spin $j$.

<figure><img src="image_5.png" /></figure>

**Remark 5.9.** The spin-$j$ fused K-operator $\mathcal{K}^{(j)}(u)$ is invertible and its inverse lies in $\mathcal{A}_q[[u^{-1}]] \otimes \text{End}(\mathbb{C}^{2j+1})$ and is given by:

$$ (5.33) \qquad [\mathcal{K}^{(j)}(u)]^{-1} = \left[ \prod_{k=0}^{2j-1} \frac{\Gamma(uq^{-j-\frac{1}{2}+k})}{c(u^{-2}q^{2j-1-2k})} \right]^{-1} \left[ \prod_{k=0}^{2j-2} \prod_{\ell=0}^{2j-k-2} c(u^2q^{2j-1-2k-\ell})c(u^{-2}q^{1-k+\ell}) \right]^{-1} \hat{\mathcal{K}}^{(j)}(u^{-1}). $$

**Remark 5.10.** By direct calculations we have checked for $j = 1, \frac{3}{2}, 2$ that $\hat{\mathcal{K}}^{(j)}(u)$ is equal to $\mathcal{K}^{(j)}(u)$ defined in (5.29) and we expect this equality holds for any $j$. Note that $\mathcal{K}^{(j)}(u)$ and $\hat{\mathcal{K}}^{(j)}(u)$, are direct analogs of the spin-$j$ K-operators $\mathcal{K}^{(j)}(u)$ defined in (4.72) and (4.68), respectively.
5.4. **Examples of fused K-operators.** In this subsection we give examples of spin-1 and spin-$\frac{3}{2}$ fused K-operators $\mathcal{K}^{(j)}(u)$ for $\mathcal{A}_q$, defined by (5.29). Recall the function $c(u)$ given in (3.42).

5.4.1. Spin-1 fused K-operator. The expressions of $\mathcal{E}^{(j+\frac{1}{2})}$, $\mathcal{F}^{(j+\frac{1}{2})}$ in (3.16), (3.17), for $j = \frac{1}{2}$ read:

$$ \mathcal{E}^{(1)} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{[2]_q}} & 0 \\ 0 & \frac{1}{\sqrt{[2]_q}} & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad \mathcal{F}^{(1)} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}. $$

From (4.38), the fused R-matrix reads

$$ (5.34) \qquad R^{(\frac{1}{2},1)}(u) = \mathcal{F}_{\langle 23 \rangle}^{(1)} R_{13}^{(\frac{1}{2},\frac{1}{2})} (u q^{-\frac{1}{2}}) R_{12}^{(\frac{1}{2},\frac{1}{2})} (u q^{\frac{1}{2}}) \mathcal{E}_{\langle 23 \rangle}^{(1)} $$

and is given explicitly by

$$ (5.35) \qquad R^{(\frac{1}{2},1)}(u) = c(uq^{\frac{1}{2}}) \begin{pmatrix} c(uq^{\frac{3}{2}}) & 0 & 0 & 0 & 0 & 0 \\ 0 & c(uq^{\frac{1}{2}}) & 0 & c(q)\sqrt{[2]_q} & 0 & 0 \\ 0 & 0 & c(uq^{-\frac{1}{2}}) & 0 & c(q)\sqrt{[2]_q} & 0 \\ 0 & c(q)\sqrt{[2]_q} & 0 & c(uq^{-\frac{1}{2}}) & 0 & 0 \\ 0 & 0 & c(q)\sqrt{[2]_q} & 0 & c(uq^{\frac{1}{2}}) & 0 \\ 0 & 0 & 0 & 0 & 0 & c(uq^{\frac{3}{2}}) \end{pmatrix}. $$

From (5.29), the fused K-operator is given by:

$$ (5.36) \qquad \mathcal{K}^{(1)}(u) = \mathcal{F}_{\langle 12 \rangle}^{(1)} \mathcal{K}_{1}^{(\frac{1}{2})}(u q^{-\frac{1}{2}}) \mathcal{R}^{(\frac{1}{2},\frac{1}{2})}(u^2) \mathcal{K}_{2}^{(\frac{1}{2})}(u q^{\frac{1}{2}}) \mathcal{E}_{\langle 12 \rangle}^{(1)}. $$

Using the above expressions, one finds that the entries $\mathcal{K}_{mn}^{(1)}(u)$ are explicitly given by:

$$ \mathcal{K}_{11}^{(1)}(u) = (c(q)^{-1} + \rho^{-1}\mathcal{G}_{+}(uq^{-\frac{1}{2}}))(\rho + c(q)\mathcal{G}_{-}(uq^{\frac{1}{2}})) \\ + c(u^2q)(uq^{\frac{1}{2}}\mathcal{W}_{+}(uq^{-\frac{1}{2}}) - u^{-1}q^{-\frac{1}{2}}\mathcal{W}_{-}(uq^{-\frac{1}{2}}))(uq^{\frac{3}{2}}\mathcal{W}_{+}(uq^{\frac{1}{2}}) - u^{-1}q^{-\frac{3}{2}}\mathcal{W}_{-}(uq^{\frac{1}{2}})) $$

$$ \mathcal{K}_{12}^{(1)}(u) = \frac{(q + q^{-1})^{-\frac{3}{2}}}{k_{-}} \left( c(u^2) (\rho c(q)^{-1} + \mathcal{G}_{+}(uq^{-\frac{1}{2}})) (uq^{\frac{3}{2}}\mathcal{W}_{+}(uq^{\frac{1}{2}}) - u^{-1}q^{-\frac{3}{2}}\mathcal{W}_{-}(uq^{\frac{1}{2}})) \right. \\ + (\rho + c(q)\mathcal{G}_{+}(uq^{-\frac{1}{2}}))(uq^{\frac{3}{2}}\mathcal{W}_{-}(uq^{\frac{1}{2}}) - u^{-1}q^{-\frac{3}{2}}\mathcal{W}_{-}(uq^{\frac{1}{2}})) \\ + c(u^2q)(uq^{\frac{1}{2}}\mathcal{W}_{+}(uq^{-\frac{1}{2}}) - u^{-1}q^{-\frac{1}{2}}\mathcal{W}_{-}(uq^{-\frac{1}{2}}))(\rho c(q)^{-1} + \mathcal{G}_{+}(uq^{\frac{1}{2}})) \left. \right), $$

$$ \mathcal{K}_{13}^{(1)}(u) = \frac{c(u^2)}{k_{-}^2 c(q^2)^2} (\rho + c(q)\mathcal{G}_{+}(uq^{-\frac{1}{2}}))(\rho + c(q)\mathcal{G}_{+}(uq^{\frac{1}{2}})), $$

$$ (5.37) \qquad \mathcal{K}_{21}^{(1)}(u) = \frac{c(q)^{-1}}{2k_{+}\sqrt{q+q^{-1}}} \left( c(u^2q)(\rho + c(q)\mathcal{G}_{-}(uq^{-\frac{1}{2}}))(uq^{\frac{3}{2}}\mathcal{W}_{+}(uq^{\frac{1}{2}}) - u^{-1}q^{-\frac{3}{2}}\mathcal{W}_{-}(uq^{\frac{1}{2}})) \right. \\ + (q^{-\frac{1}{2}}(u^{-3} + u(-2+q^2))\mathcal{W}_{-}(uq^{-\frac{1}{2}}) + q^{\frac{1}{2}}(u^3 + u^{-1}(-2+q^{-2}))\mathcal{W}_{+}(uq^{-\frac{1}{2}})) \left. \right) \\ \times (\rho + c(q)\mathcal{G}_{-}(uq^{\frac{1}{2}})), $$
$$
\begin{align*}
\mathcal{K}_{22}^{(1)}(u) &= \frac{c(u^2q)}{2c(q)^2\rho} \left( (\rho + c(q)\mathcal{G}_+(uq^{-\frac{1}{2}}))(\rho + c(q)\mathcal{G}_-(uq^{\frac{1}{2}})) \right. \\
&\qquad \left. + (\rho + c(q)\mathcal{G}_-(uq^{-\frac{1}{2}}))(\rho + c(q)\mathcal{G}_+(uq^{\frac{1}{2}})) \right) \\
&\qquad + \frac{1}{2}(q^{-\frac{1}{2}}(u^{-3} + u(-2 + q^2))\mathcal{W}_+(uq^{-\frac{1}{2}}) + q^{\frac{1}{2}}(u^{-1}(-2 + q^2) + u^3)\mathcal{W}_-(uq^{-\frac{1}{2}})) \\
&\qquad \times (uq^{\frac{3}{2}}\mathcal{W}_+(uq^{\frac{1}{2}}) - u^{-1}q^{-\frac{3}{2}}\mathcal{W}_-(uq^{\frac{1}{2}})) \\
&\qquad + \frac{1}{2}(q^{-\frac{1}{2}}(u^{-3} + u(-2 + q^2))\mathcal{W}_-(uq^{-\frac{1}{2}}) + q^{\frac{1}{2}}(u^{-1}(-2 + q^2) + u^3)\mathcal{W}_+(uq^{-\frac{1}{2}})) \\
&\qquad \times (uq^{\frac{3}{2}}\mathcal{W}_-(uq^{\frac{1}{2}}) - u^{-1}q^{-\frac{3}{2}}\mathcal{W}_+(uq^{\frac{1}{2}})),
\end{align*}
\begin{align*}
\mathcal{K}_{23}^{(1)}(u) &= \sigma(\mathcal{K}_{21}^{(1)}(u)), & \mathcal{K}_{31}^{(1)}(u) &= \sigma(\mathcal{K}_{13}^{(1)}(u)), \\
\mathcal{K}_{32}^{(1)}(u) &= \sigma(\mathcal{K}_{12}^{(1)}(u)), & \mathcal{K}_{33}^{(1)}(u) &= \sigma(\mathcal{K}_{11}^{(1)}(u)),
\end{align*}
$$

where $\sigma$ is defined in (5.18). The last two lines describe the exchange of the entries of the fused K-operator due to the automorphism $\sigma$ and can be seen graphically:

$$
(5.38) \qquad \mathcal{K}^{(1)}(u) = \left( \begin{array}{c|c} \mathcal{K}_{11}^{(1)}(u) & \mathcal{K}_{12}^{(1)}(u) \mathcal{K}_{13}^{(1)}(u) \\ \hline \mathcal{K}_{21}^{(1)}(u) & \mathcal{K}_{22}^{(1)}(u) \mathcal{K}_{23}^{(1)}(u) \\ \hline \mathcal{K}_{31}^{(1)}(u) & \mathcal{K}_{32}^{(1)}(u) \mathcal{K}_{33}^{(1)}(u) \end{array} \right) .
$$

As shown in Lemma D.3, the fused K-operator (6.1) for $j=1$ satisfies the reflection equation:

$$
(5.39) \quad R^{(\frac{1}{2},1)}(u/v) \mathcal{K}_1^{(\frac{1}{2})}(u) R^{(\frac{1}{2},1)}(uv) \mathcal{K}_2^{(1)}(v) = \mathcal{K}_2^{(1)}(v) R^{(\frac{1}{2},1)}(uv) \mathcal{K}_1^{(\frac{1}{2})}(u) R^{(\frac{1}{2},1)}(u/v) .
$$

Note that the latter equation can be independently checked using the ordering relations given in Lemma C.1.

5.4.2. Spin-$\frac{3}{2}$ fused K-operator. The elements $\mathcal{E}^{(j+\frac{1}{2})}$, $\mathcal{F}^{(j+\frac{1}{2})}$ in (3.16), (3.17) for $j=1$ read:

$$
\mathcal{E}^{(\frac{3}{2})} =
\begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & \frac{\sqrt{[2]_q}}{\sqrt{[3]_q}} & 0 & 0 \\
0 & 0 & \frac{[2]_q}{\sqrt{([2]_q)^2}\sqrt{[3]_q}} & 0 \\
0 & \frac{1}{\sqrt{[3]_q}} & 0 & 0 \\
0 & 0 & \frac{([2]_q)^{\frac{3}{2}}}{\sqrt{([2]_q)^2}\sqrt{[3]_q}} & 0 \\
0 & 0 & 0 & \frac{[2]_q}{\sqrt{([2]_q)^2}}
\end{pmatrix},
\mathcal{F}^{(\frac{3}{2})} =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{\sqrt{[2]_q}\sqrt{[3]_q}}{1+[2]_q} & 0 & \frac{\sqrt{[3]_q}}{1+[2]_q} & 0 & 0 \\
0 & 0 & \frac{\sqrt{[3]_q}\sqrt{([2]_q)^2}}{[2]_q(1+[2]_q)} & 0 & \frac{\sqrt{[3]_q}\sqrt{([2]_q)^2}}{\sqrt{[2]_q(1+[2]_q)}} & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{\sqrt{([2]_q)^2}}{[2]_q}
\end{pmatrix}.
$$

The fused R-matrix from (4.38) reads

$$
R^{(\frac{1}{2}, \frac{3}{2})}(u) = \mathcal{F}_{(23)}^{(\frac{3}{2})} R_{13}^{(\frac{1}{2}, 1)}(uq^{-\frac{1}{2}}) R_{12}^{(\frac{1}{2}, \frac{1}{2})}(uq) \mathcal{E}_{(23)}^{(\frac{3}{2})},
$$
given explicitly by

$$
(5.40) \quad R^{(\frac{1}{2}, \frac{3}{2})}(u) = c(u)c(uq)c(q) \begin{pmatrix} \frac{c(uq^2)}{c(q)} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{c(uq)}{c(q)} & 0 & 0 & \sqrt{[3]_q} & 0 & 0 & 0 \\ 0 & 0 & \frac{c(u)}{c(q)} & 0 & 0 & \sqrt{([2]_q)^2} & 0 & 0 \\ 0 & 0 & 0 & \frac{c(uq^{-1})}{c(q)} & 0 & 0 & \sqrt{[3]_q} & 0 \\ 0 & \sqrt{[3]_q} & 0 & 0 & \frac{c(uq^{-1})}{c(q)} & 0 & 0 & 0 \\ 0 & 0 & \sqrt{([2]_q)^2} & 0 & 0 & \frac{c(u)}{c(q)} & 0 & 0 \\ 0 & 0 & 0 & \sqrt{[3]_q} & 0 & 0 & \frac{c(uq)}{c(q)} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{c(uq^2)}{c(q)} \end{pmatrix} .
$$

From (5.29), the fused K-operator reads:

$$
(5.41) \qquad \mathcal{K}^{(\frac{3}{2})}(u) = \mathcal{F}_{(12)}^{(\frac{3}{2})} \mathcal{K}_{1}^{(\frac{1}{2})}(uq^{-1}) R^{(\frac{1}{2},1)}(u^2 q^{-\frac{1}{2}}) \mathcal{K}_{2}^{(1)}(uq^{\frac{1}{2}}) \mathcal{E}_{(12)}^{(\frac{3}{2})}.
$$

For instance, the first entry reads:

$$
\begin{equation}
\begin{split}
\mathcal{K}_{11}^{(\frac{3}{2})}(u) = \frac{c(u^2)(q+q^{-1})}{2\rho c(q)} & \left(\rho + c(q)\mathcal{G}_+(uq^{-1})\right) \left(c(u^2q^2)(\rho + c(q)\mathcal{G}_-(u)) (uq^2\mathcal{W}_+(uq) - u^{-1}q^{-2}\mathcal{W}_-(uq))\right. \\
& \left. \qquad + u^{-1}\left((u^2(q^2-2) + u^{-2}q^{-2})\mathcal{W}_-(u) + (u^4q^2 + q^{-2} - 2)\mathcal{W}_+(u)\right) (\rho + c(q)\mathcal{G}_-(uq))\right) \\
& + c(u^2)c(u^2q)(u\mathcal{W}_+(uq^{-1}) - u^{-1}\mathcal{W}_-(uq^{-1})) \left( (\rho^{-1}\mathcal{G}_+(u) + c(q)^{-1})(\rho + c(q)\mathcal{G}_-(uq)) \right. \\
& \left. \qquad + c(u^2q^2)(uq\mathcal{W}_+(u) - u^{-1}q^{-1}\mathcal{W}_-(u)) (uq^2\mathcal{W}_+(uq) - u^{-1}q^{-2}\mathcal{W}_-(uq)) \right).
\end{split}
\end{equation}
$$

The other explicit expressions of the entries in terms of the generating functions for $\mathcal{A}_q$ are not
reported here for simplicity. Under the action of $\sigma$ from (5.18), the entries exchange according to:

$$
(5.42) \quad \mathcal{K}^{(\frac{3}{2})}(u) = \begin{pmatrix}
\mathcal{K}_{11}^{(\frac{3}{2})}(u) & \mathcal{K}_{12}^{(\frac{3}{2})}(u) & \mathcal{K}_{13}^{(\frac{3}{2})}(u) & \mathcal{K}_{14}^{(\frac{3}{2})}(u) \\
\mathcal{K}_{21}^{(\frac{3}{2})}(u) & \mathcal{K}_{22}^{(\frac{3}{2})}(u) & \mathcal{K}_{23}^{(\frac{3}{2})}(u) & \mathcal{K}_{24}^{(\frac{3}{2})}(u) \\
\mathcal{K}_{31}^{(\frac{3}{2})}(u) & \mathcal{K}_{32}^{(\frac{3}{2})}(u) & \mathcal{K}_{33}^{(\frac{3}{2})}(u) & \mathcal{K}_{34}^{(\frac{3}{2})}(u) \\
\mathcal{K}_{41}^{(\frac{3}{2})}(u) & \mathcal{K}_{42}^{(\frac{3}{2})}(u) & \mathcal{K}_{43}^{(\frac{3}{2})}(u) & \mathcal{K}_{44}^{(\frac{3}{2})}(u)
\end{pmatrix} .
$$

By Theorem 5.7, the fused K-operator satisfies the reflection equation:

$$
R^{(\frac{1}{2}, \frac{3}{2})}(u/v) \mathcal{K}_{1}^{(\frac{1}{2})}(u) R^{(\frac{1}{2}, \frac{3}{2})}(uv) \mathcal{K}_{2}^{(\frac{3}{2})}(v) = \mathcal{K}_{2}^{(\frac{3}{2})}(v) R^{(\frac{1}{2}, \frac{3}{2})}(uv) \mathcal{K}_{1}^{(\frac{1}{2})}(u) R^{(\frac{1}{2}, \frac{3}{2})}(u/v).
$$

5.4.3. Spin-j fused K-operator. Specializing the formula (5.29), one gets the fused K-operator $\mathcal{K}^{(j)}(u)$ 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 (3.41)

$$
R^{(\frac{1}{2},j)}(u) = M^{(j)} R^{(\frac{1}{2},j)}(u) M^{(j)} \quad \text{with} \quad M^{(j)} = \sigma_x \otimes \sum_{n=1}^{2j+1} E_{n,2j+2-n}^{(j,j)}.
$$
So, due to the automorphism $\sigma$ in (5.18), the entries of the K-operator of spin-$j$ exchange as $K_{m,n}^{(j)}(u) = \sigma(K_{2j+2-m,2j+2-n}^{(j)}(u))$ with $1 \le m, n \le 2j+1$. This is analogous to the property in (5.38), (5.42).

From the fusion formulas (4.38) and (5.29) 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 $\mathcal{E}^{(j)}$ and $\mathcal{F}^{(j)}$. They are given by:

$$ (5.43) \quad R^{(\frac{1}{2},j)}(u) = \left( \prod_{m=0}^{2j-2} \mathbb{I}_{2m+2} \otimes \mathcal{F}^{(j-\frac{m}{2})} \right) \left( \prod_{k=0}^{2j-1} R_{12j+1-k}^{(\frac{1}{2},\frac{1}{2})} (uq^{-j+\frac{1}{2}+k}) \right) \left( \prod_{m=0}^{2j-2} \mathbb{I}_{2(2j-1-m)} \otimes \mathcal{E}^{(1+\frac{m}{2})} \right), $$

and

$$ (5.44) \quad \begin{aligned} \mathcal{K}^{(j)}(u) = & \left( \prod_{m=0}^{2j-2} \mathbb{I}_{2m} \otimes \mathcal{F}^{(j-\frac{m}{2})} \right) \prod_{k=1}^{2j} \left\{ \mathcal{K}_k^{(\frac{1}{2})} (uq^{k-j-\frac{1}{2}}) \left[ \prod_{\ell=0}^{2j-k-1} R_{k2j-\ell}^{(\frac{1}{2},\frac{1}{2})} (u^2 q^{-2j+2k+\ell}) \right] \right\} \\ & \times \left( \prod_{m=0}^{2j-2} \mathbb{I}_{2(2j-2-m)} \otimes \mathcal{E}^{(1+\frac{m}{2})} \right), \end{aligned} $$

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 (5.43) is straightforward by induction on $j$ using (4.38), whereas the proof of (5.44) is more tedious. We proceed by induction checking (5.29) using (5.43) and (5.44). Then, one obtains a formula similar to (5.44) for $j \to j + \frac{1}{2}$ but with unwanted products of $\mathcal{E}^{(j)}\mathcal{F}^{(j)}$. They can be removed using the same trick as in eq. (4.53). Firstly, multiply (5.29) from the right by $\mathcal{H}^{(j+\frac{1}{2})}[\mathcal{H}^{(j+\frac{1}{2})}]^{-1} = \mathbb{I}_{2j+2}$ and use (3.48) to move $\mathcal{H}^{(j+\frac{1}{2})}$ to the left. Then, using successively the Yang-Baxter equation and the reflection equation, the products $\mathcal{E}^{(j)}\mathcal{F}^{(j)}$ are removed using the property (3.50).

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 (5.44), (5.43) can be found for instance in [FNR07, eq. (2.1), (2.7)].

### 5.5. Evaluated coaction of fused K-operators.
The fused K-operators $\mathcal{K}^{(j)}(u)$ are expected to have a simple relation with the spin-$j$ K-operators $\mathbf{K}^{(j)}(u)$ as will be discussed in Section 6, similarly to the relations between $\mathbf{L}^{(j)}(u)$ and $\mathcal{L}^{(j)}(u)$. Therefore, the evaluated coaction $\delta_w$ defined in (4.82) and applied to the entries of $\mathcal{K}^{(\frac{1}{2})}(u)$ is expected to be of the form (4.83) up to appropriate normalization.

**Lemma 5.11.** The evaluated coaction $\delta_w: \mathcal{A}_q \to \mathcal{A}_q \otimes U_q sl_2[w^{\pm 1}]$ is such that$^{15}$

$$ (5.45) \quad (\delta_w \otimes \mathrm{id})(\mathcal{K}^{(\frac{1}{2})}(u)) = \frac{U^{-1}}{q+q^{-1}} \left(\mathcal{L}^{(\frac{1}{2})}(u/w)\right)_{[2]} \left(\mathcal{K}^{(\frac{1}{2})}(u)\right)_{[1]} \left(\mathcal{L}^{(\frac{1}{2})}(uw)\right)_{[2]} $$

*Proof.* Assume the evaluated coaction takes the form

$$ (5.46) \quad (\delta_w \otimes \mathrm{id})(\mathcal{K}^{(\frac{1}{2})}(u)) = f(u) \left(\mathcal{L}^{(\frac{1}{2})}(u/w)\right)_{[2]} \left(\mathcal{K}^{(\frac{1}{2})}(u)\right)_{[1]} \left(\mathcal{L}^{(\frac{1}{2})}(uw)\right)_{[2]} $$

<footnote>$^{15}$Here, the index $[1]$ is associated with the space for $\mathcal{A}_q$, and $[2]$ for $U_q sl_2$, and we use the convention $((T)_{[2]}(T')_{[1]}(T'')_{[2]})_{ij} = \sum_{k,l=1}^2 (T')_{kl} \otimes (T)_{ik}(T'')_{lj}$.</footnote>