$$ (5.48) \qquad + (q^2 - q^{-2}) \left( k_{\mp} q^{\mp \frac{1}{2}} (W_+(u) - U^{-1}W_-(u)) \otimes (w^{\mp 1} S_{\mp} K^{\frac{1}{2}}) \right. \\ \qquad \qquad \qquad \qquad \qquad \qquad \left. + k_{\mp} q^{\pm \frac{1}{2}} (W_-(u) - U^{-1}W_+(u)) \otimes (w^{\pm 1} S_{\mp} K^{-\frac{1}{2}}) \right) \\ \qquad \qquad \qquad + \frac{k_+ k_- (q+q^{-1}) U^{-1}}{(q-q^{-1})} 1 \otimes \left( \frac{k_{\mp}}{k_{\pm}} (q-q^{-1})^2 S_{\mp}^2 - (w^{-2} K^{\pm 1} + w^2 K^{\mp 1}) \right), $$

where we used the shorthand notation $S_+ \equiv E$, $S_- \equiv F$. We note that these expressions were first obtained in [BS09, Prop. 2.2]$^{16}$. Expanding (5.47), (5.48) as power series in $U^{-1}$, it is straightforward to prove Proposition 2.15.

Now, using (5.47), (5.48) we can compute the evaluated coaction of the quantum determinant $\Gamma(u)$ from (5.20):

$$ (5.49) \qquad \delta_w(\Gamma(u)) = \frac{1}{(u^2 q + u^{-2} q^{-1})(u^2 q^3 + u^{-2} q^{-3})} \Gamma(u) \otimes \gamma(u/w)\gamma(uw), $$

where $\gamma(u)$ is given in (4.12). Here we used the ordering relations of $\mathcal{A}_q$ in Lemma C.1 and the PBW basis of $U_q s l_2$ given in Appendix A.

The following result is a natural generalization of Lemma 5.11.

**Proposition 5.12.** The evaluated coaction of $\mathcal{K}^{(j)}(u)$ for $j \in \frac{1}{2}\mathbb{N}_+$ is given by

$$ (5.50) \quad (\delta_w \otimes \text{id})(\mathcal{K}^{(j)}(u)) = \left( \prod_{p=1}^{2j} \frac{U^{-1}}{q+q^{-1}} \bigg|_{u=uq^{j+\frac{1}{2}-p}} \right) \times \left( \mathcal{L}^{(j)}(u/w) \right)_{[2]} \left( \mathcal{K}^{(j)}(u) \right)_{[1]} \left( \mathcal{L}^{(j)}(uw) \right)_{[2]}. $$

*Proof.* An induction argument is used. For $j = \frac{1}{2}$, the relation (5.50) coincides with (5.45). Now, suppose $j \ge 1$. For convenience, we omit the notation [1], [2]. Expand the l.h.s. of (5.50) using the expression of the fused K-operator (5.29) and the evaluated coaction (5.50). It follows:

$$ \begin{align} \delta_w(\mathcal{K}^{(j)}(u)) &= \mathcal{F}_{\langle 12 \rangle}^{(j)}[\delta_w(\mathcal{K}_1^{(\frac{1}{2})}(uq^{-j+\frac{1}{2}}))]R_{12}^{(\frac{1}{2},j-\frac{1}{2})}(u^2q^{-j+1})[\delta_w(\mathcal{K}_2^{(j-\frac{1}{2})}(uq^{\frac{1}{2}}))]\mathcal{E}_{\langle 12 \rangle}^{(j)} \nonumber \\ &= \left( \prod_{p=1}^{2j} \frac{U^{-1}}{q+q^{-1}} \bigg|_{u=uq^{j+\frac{1}{2}-p}} \right) \mathcal{F}_{\langle 12 \rangle}^{(j)} \mathcal{L}_1^{(\frac{1}{2})}(uw^{-1}q^{-j+\frac{1}{2}})\mathcal{K}_1^{(\frac{1}{2})}(uq^{-j+\frac{1}{2}})\mathcal{L}_1^{(\frac{1}{2})}(uwq^{-j+\frac{1}{2}}) \end{align} $$

$$ (5.51) \qquad \times R_{12}^{(\frac{1}{2},j-\frac{1}{2})}(u^2q^{-j+1})\mathcal{L}_2^{(j-\frac{1}{2})}(uw^{-1}q^{\frac{1}{2}})\mathcal{K}_2^{(j-\frac{1}{2})}(uq^{\frac{1}{2}})\mathcal{L}_2^{(j-\frac{1}{2})}(uwq^{\frac{1}{2}})\mathcal{E}_{\langle 12 \rangle}^{(j)}, $$

where we used $\delta_w$ instead of $\delta_w \otimes \text{id}$ for convenience. Multiplying on the left and on the right (4.50) by $(\mathcal{L}_2^{(j)}(v))^{-1}$ and using (4.54), we get

$$ (5.52) \qquad \mathcal{L}_1^{(\frac{1}{2})}(u)R^{(\frac{1}{2},j-\frac{1}{2})}(u/v)\mathcal{L}_2^{(j-\frac{1}{2})}(v^{-1}) = \mathcal{L}_2^{(j-\frac{1}{2})}(v^{-1})R^{(\frac{1}{2},j-\frac{1}{2})}(u/v)\mathcal{L}_1^{(\frac{1}{2})}(u). $$

Using (5.52) for $u \to uwq^{-j+\frac{1}{2}}$, $v \to u^{-1}wq^{-\frac{1}{2}}$ and the commutation $[\mathcal{K}_1^{(j_1)}(u), \mathcal{L}_2^{(j_2)}(v)] = 0$, (5.51) becomes:

$$ (5.53) \qquad = \left( \prod_{p=1}^{2j} \frac{U^{-1}}{q+q^{-1}} \bigg|_{u=uq^{j+\frac{1}{2}-p}} \right) \mathcal{F}_{\langle 12 \rangle}^{(j)} \mathcal{L}_1^{(\frac{1}{2})}(uw^{-1}q^{-j+\frac{1}{2}}) \mathcal{L}_2^{(j-\frac{1}{2})}(uw^{-1}q^{\frac{1}{2}}) $$

<footnote n="16">¹⁶We corrected typos in [BS09], a prefactor was missing.</footnote>
$$ \times \mathcal{K}_1^{(\frac{1}{2})}(uq^{-j+\frac{1}{2}})R_{12}^{(\frac{1}{2},j-\frac{1}{2})}(u^2q^{-j+1})\mathcal{K}_2^{(j-\frac{1}{2})}(uq^{\frac{1}{2}})\mathcal{L}_1^{(\frac{1}{2})}(uwq^{-j+\frac{1}{2}})\mathcal{L}_2^{(j-\frac{1}{2})}(uwq^{\frac{1}{2}})\mathcal{E}_{\langle 12 \rangle}^{(j)}. $$

On the other hand, inserting (4.51), (5.29), in the r.h.s. of (5.50) one gets:

$$ \delta_w(\mathcal{K}^{(j)}(u)) = \left( \prod_{p=1}^{2j} \frac{U^{-1}}{q + q^{-1}} \bigg|_{u=uq^{j+\frac{1}{2}-p}} \right) \mathcal{F}_{\langle 12 \rangle}^{(j)} \mathcal{L}_1^{(\frac{1}{2})}(uw^{-1}q^{-j+\frac{1}{2}}) \mathcal{L}_2^{(j-\frac{1}{2})}(uw^{-1}q^{\frac{1}{2}}) \mathcal{E}_{\langle 12 \rangle}^{(j)} \mathcal{F}_{\langle 12 \rangle}^{(j)} \\ \times \mathcal{K}_1^{(\frac{1}{2})}(uq^{-j+\frac{1}{2}})R_{12}^{(\frac{1}{2},j-\frac{1}{2})}(u^2q^{-j+1})\mathcal{K}_2^{(j-\frac{1}{2})}(uq^{\frac{1}{2}})\mathcal{E}_{\langle 12 \rangle}^{(j)}\mathcal{F}_{\langle 12 \rangle}^{(j)}\mathcal{L}_1^{(\frac{1}{2})}(uwq^{-j+\frac{1}{2}})\mathcal{L}_2^{(j-\frac{1}{2})}(uwq^{\frac{1}{2}})\mathcal{E}_{\langle 12 \rangle}^{(j)}. $$

Now, in order to remove the products $\mathcal{E}_{\langle 12 \rangle}^{(j)}\mathcal{F}_{\langle 12 \rangle}^{(j)}$, multiply first the expression above from the right by $\mathcal{H}_{\langle 12 \rangle}^{(j)}[\mathcal{H}_{\langle 12 \rangle}^{(j)}]^{-1}$. Then, using the relations (3.48)-(3.50), (D.5) and

$$ (5.54) \qquad R^{(\frac{1}{2},j-\frac{1}{2})}(v/u)\mathcal{L}_2^{(j-\frac{1}{2})}(v)\mathcal{L}_1^{(\frac{1}{2})}(u) = \mathcal{L}_1^{(\frac{1}{2})}(u)\mathcal{L}_2^{(j-\frac{1}{2})}(v)R^{(\frac{1}{2},j-\frac{1}{2})}(v/u) $$

which is obtained from the RLL equation, after simplifications the expression matches with (5.53). $\square$

**5.6. Twisted intertwining relations for fused K-operators.** In the next section, we will also need the twisted intertwining relations satisfied by the fused K-operators. For the fundamental K-operator (5.4), the twisted intertwining relations have been given in [BS09, Prop. 4.2]. This result is now extended to higher values of $j$.

**Proposition 5.13.** The following relation holds for any $j \in \frac{1}{2}\mathbb{N}_+$ and all $b \in \mathcal{A}_q$:

$$ (5.55) \qquad \mathcal{K}^{(j)}(v)(\mathrm{id} \otimes \pi^j)[\delta_{v^{-1}}(b)] = (\mathrm{id} \otimes \pi^j)[\delta_v(b)]\mathcal{K}^{(j)}(v). $$

*Proof.* Lemma 5.11 implies that

$$ (5.56) \qquad (\mathrm{id} \otimes \pi^j \otimes \mathrm{id})[(\delta_v \otimes \mathrm{id})(\mathcal{K}^{(\frac{1}{2})}(u))] = \frac{U^{-1}}{q + q^{-1}} R^{(j,\frac{1}{2})}(u/v) \mathcal{K}_2^{(\frac{1}{2})}(u) R^{(j,\frac{1}{2})}(uv). $$

We then notice that $\mathcal{K}^{(j)}(v)$ satisfies the equation

$$ (5.57) \qquad \mathcal{K}_1^{(j)}(v)R^{(j,\frac{1}{2})}(uv)\mathcal{K}_2^{(\frac{1}{2})}(u)R^{(j,\frac{1}{2})}(u/v) = R^{(j,\frac{1}{2})}(u/v)\mathcal{K}_2^{(\frac{1}{2})}(u)R^{(j,\frac{1}{2})}(uv)\mathcal{K}_1^{(j)}(v). $$

This version of reflection equation follows from the standard reflection equation (5.30) for $j_1 = j, j_2 = \frac{1}{2}$ and $u_1 = v, u_2 = u$. Indeed, we multiply (5.30) on the left and the right by $R^{(j_1, j_2)}(u_2/u_1)$ and using (4.55) we obtain (5.57).

Now using (5.56), the equation (5.57) can be rewritten as

$$ (5.58) \qquad \mathcal{K}_1^{(j)}(v)(\mathrm{id} \otimes \pi^j \otimes \mathrm{id})[(\delta_{v^{-1}} \otimes \mathrm{id})(\mathcal{K}^{(\frac{1}{2})}(u))] = (\mathrm{id} \otimes \pi^j \otimes \mathrm{id})[(\delta_v \otimes \mathrm{id})(\mathcal{K}^{(\frac{1}{2})}(u))]\mathcal{K}_1^{(j)}(v). $$

This equation can be thought as an equation in $\mathcal{A}_q[[u^{-1}]]((v^{-1}))\otimes\mathrm{End}(\mathbb{C}^{2j+1})\otimes\mathrm{End}(\mathbb{C}^2)$. Denote the entries of the fundamental K-operator by $\mathcal{K}_{mn}^{(\frac{1}{2})}(u) \in \mathcal{A}_q[[u^{-1}]]$, $m, n = 1, 2$. Now, considering (5.58) as four equations in $\mathcal{A}_q[[u^{-1}]]((v^{-1})) \otimes \mathrm{End}(\mathbb{C}^{2j+1})$, i.e. taking the matrix elements of $\mathrm{End}(\mathbb{C}^2)$, it yields

$$ \mathcal{K}^{(j)}(v) \left( (\mathrm{id} \otimes \pi^j)[\delta_{v^{-1}}(\mathcal{K}_{mn}^{(\frac{1}{2})}(u))] \right) = \left( (\mathrm{id} \otimes \pi^j)[\delta_v(\mathcal{K}_{mn}^{(\frac{1}{2})}(u))] \right) \mathcal{K}^{(j)}(v). $$
Inserting the entries according to (5.4), extracting the independent relations and using (5.1), (5.2), this implies (5.55). □

## 6. FUSED K-OPERATORS AND THE UNIVERSAL K-MATRIX FOR $\mathcal{A}_q$

In this section, we assume there exists a universal K-matrix $\mathcal{K}$ for $\mathcal{A}_q$. We are interested in the precise relationship between the fused K-operators $\mathcal{K}^{(j)}(u)$ constructed in the previous section using (5.29) and the spin-$j$ K-operators defined in (4.56) through the evaluation of the universal K-matrix. Motivated by Lemma 4.9 relating spin-$j$ L-operators (4.1) and fused L-operators (4.27), and by supporting evidences discussed below, we propose the following conjecture.

**Conjecture 1.** For $j \in \frac{1}{2}\mathbb{N}$, we have

$$ (6.1) \qquad \mathbf{K}^{(j)}(u) = \nu^{(j)}(u)\mathcal{K}^{(j)}(u) \in \mathcal{A}_q[[u^{-1}]] \otimes \mathrm{End}(\mathbb{C}^{2j+1}), $$

where $\mathcal{K}^{(j)}(u)$ is defined in (5.29) with

$$ (6.2) \qquad \nu^{(j)}(u) = \left( \prod_{m=0}^{2j-1} \nu(uq^{j-\frac{1}{2}-m}) \right) \left( \prod_{k=0}^{2j-2} \prod_{\ell=0}^{2j-k-2} \pi^{\frac{1}{2}}(\mu(u^2q^{2j-2-2k-\ell})) \right). $$

Here $\pi^{\frac{1}{2}}(\mu(u))$ is given by (4.9) and $\nu(u) \equiv \nu^{(\frac{1}{2})}(u)$ is an invertible central element in $\mathcal{A}_q[[u^{-1}]]$, defined by the functional relation

$$ (6.3) \qquad \pi^{\frac{1}{2}}(\mu(u^2q))\nu(u)\nu(uq)\Gamma(u) = 1, $$

where $\Gamma(u)$ is given in (5.20), and has the evaluated coaction

$$ (6.4) \qquad \delta_w(\nu(u)) = (u^2q + u^{-2}q^{-1})\nu(u) \otimes \mu(u/w)\mu(uw). $$

We note that a solution $\nu(u)$ to (6.3) can be uniquely (up to a sign of its constant term) determined by recursion provided in the next lemma. Here, we assume that $\nu(u)$ is invertible and generates a commutative subalgebra in $\mathcal{A}_q((u^{-1}))$. In what follows we use the following notation

$$ (6.5) \qquad F(u) := \pi^{\frac{1}{2}}(\mu(u^2q))\Gamma(u). $$

We observe that only even modes of $F(u)$ are non-zero, i.e.

$$ (6.6) \qquad F(u) = \sum_{k=0}^{\infty} F_k u^{-2k}. $$

Recall the central elements from Example 5.3, in terms of which the first modes of $F(u)$ read:

$$ (6.7) \qquad F_0 = -\frac{\rho q^{\frac{1}{2}}}{(q - q^{-1})^2}, \quad F_1 = \frac{q^{\frac{1}{2}} c_1}{2(q - q^{-1})} \Delta_1, \quad F_2 = \frac{q^{\frac{1}{2}} c_2}{2(q - q^{-1})} \Delta_2 - \frac{\rho q^{\frac{1}{2}}}{(q^4 - 1)}, $$

with $c_k$ defined in (5.23).

**Lemma 6.1.** The generating function $\nu(u) = \nu^{(\frac{1}{2})}(u)$ from (6.1) is central in $\mathcal{A}_q[[u^{-1}]]$ and takes the following form

$$ (6.8) \qquad \nu(u) = \sum_{k=0}^{\infty} \nu_k u^{-2k} $$
where $\nu_0$ is fixed up to a sign by

$$ (6.9) \qquad \nu_0^2 = -\rho^{-1}q^{-\frac{1}{2}}(q-q^{-1})^2, $$

and all $\nu_k$ with $k \ge 1$ are determined recursively by

$$ (6.10) \qquad \nu_k = -\frac{1}{F_0} \sum_{\ell=0}^{k-1} \frac{1-q^{-2(k+\ell)}}{1-q^{-4k}} F_{k-\ell} \nu_\ell. $$

*Proof.* With the notation (6.5), the functional relation (6.3) takes the form

$$ (6.11) \qquad F(u)\nu(u)\nu(uq) = 1. $$

Replacing $u \to uq^{-1}$, we get from it another equation

$$ (6.12) \qquad F(uq^{-1})\nu(uq^{-1})\nu(u) = 1. $$

Comparing both the equations (6.11) and (6.12), and using that by assumption $[\nu(u), \nu(uq^{\pm 1})] = 0$ and $\nu(u)$ is invertible, we get

$$ (6.13) \qquad F(uq)\nu(uq^2) = F(u)\nu(u). $$

Assuming $\nu(u) \in A_q((u^{-1}))$ with an expansion $\nu(u) = \sum_{k=-M}^{\infty} \nu'_k u^{-k}$ for some positive $M$ and using (6.13) with (6.6) we conclude by induction that $M$ has to be zero. We can then assume that $\nu(u) = \sum_{k=0}^{\infty} \nu'_k u^{-k}$. Inserting (6.6) in the above equation (6.13), it is equivalent to

$$ \sum_{p=0}^{\infty} u^{-p} \sum_{k=0}^{\lfloor \frac{p}{2} \rfloor} (q^{-2p+2k}-1) F_k \nu'_{p-2k} = 0 \quad \Leftrightarrow \quad \sum_{k=0}^{\lfloor \frac{p}{2} \rfloor} (q^{-2p+2k}-1) F_k \nu'_{p-2k} = 0, \quad p=0,1,2,... $$

Let $n \in \mathbb{N}$. For $p = 2n + 1$, one gets

$$ (6.14) \qquad \nu'_{2n+1} = -\frac{1}{F_0} \sum_{k=1}^{n} \left( \frac{1-q^{-2(2n+1)+2k}}{1-q^{-2(2n+1)}} \right) F_k \nu'_{2n+1-2k}. $$

For $n = 0$, it gives $\nu'_1 = 0$ which implies by induction that all modes $\nu'_{2n+1} = 0$. Thus, the power series $\nu(u)$ is of the form (6.8). From (6.11) we readily see that $\nu_0$ is fixed as in (6.9). Then, the recurrence relation that follows from (6.13) with (6.8) and (6.6) is now given by

$$ \sum_{l=0}^{p} (q^{-2(p+l)} - 1) F_{p-l} \nu_l = 0, $$

which determines $\nu_l$ as in (6.10).

<figure><img src="image_6.png" /></figure>

**Example 6.2.** Together with (6.7) and (5.24) and (5.25), we get explicit expression of the first few modes of $\nu(u)$ developed in (6.8) in terms of the alternating generators of $A_q$:

$$ (6.15) \qquad \nu_1 = \frac{(q-q^{-1})^2 q^{\frac{1}{2}} \nu_0}{\rho(q+q^{-1})} F_1, \qquad \nu_2 = \frac{(q-q^{-1})^2 q^{3/2} \nu_0}{\rho(q^2+q^{-2})} \left( F_2 + q^{-\frac{1}{2}} \frac{(q-q^{-1})(q^3-q^{-3})}{\rho(q+q^{-1})^2} F_1^2 \right), $$

where $\nu_0$ is fixed as a square root of (6.9).
Let us now show that $\mathbf{K}^{(j)}(u) \in \mathcal{A}_q[[u^{-1}]] \otimes \text{End}(\mathbb{C}^{2j+1})$. First, we note that the leading term of $\nu^{(j)}(u)$ given in (6.2) is at $u^{-4j^2+2j}$, using (6.8) and (4.9). Next, from the definition of $\mathcal{K}^{(j)}(u)$ in (5.29), we find that its leading term is at $u^{4j^2-2j}$, using the fact that the leading terms of $\mathcal{K}^{(\frac{1}{2})}(u)$ and $R^{(\frac{1}{2},j)}(u)$ from (3.41) are respectively at $u^0$ and $u^{2j}$. Finally, we conclude that $\nu^{(j)}(u)\mathcal{K}^{(j)}(u)$ is in $\mathcal{A}_q[[u^{-1}]] \otimes \text{End}(\mathbb{C}^{2j+1})$ with its leading term at $u^0$.

Below, we present supporting evidences for Conjecture 1, and then we derive from this conjecture certain properties of the fused K-operators for $j \ge 0$.

**6.1. Supporting evidences.** For the clarity of the presentation, let us define:

$$ (6.16) \qquad \tilde{\mathbf{K}}^{(j)}(u) = \nu^{(j)}(u)\mathcal{K}^{(j)}(u) \quad \text{for } j \in \frac{1}{2}\mathbb{N}, $$

where we assume $\nu^{(\frac{1}{2})}(u)$ is an invertible central element in $\mathcal{A}_q[[u^{-1}]]$. Importantly, it is not assumed that $\tilde{\mathbf{K}}^{(j)}(u)$ is obtained from the evaluation of a universal K-matrix.

We provide supporting evidences for Conjecture 1. We show that $\tilde{\mathbf{K}}^{(j)}(u)$ for all $j$ satisfy the following systems of equations:

$$ (K1') \qquad \tilde{\mathbf{K}}^{(j)}(v)(\text{id} \otimes \pi^j)[\delta_{v^{-1}}(b)] = (\text{id} \otimes \pi^j)[\delta_v(b)]\tilde{\mathbf{K}}^{(j)}(v), $$

$$ (K2') \qquad (\delta_w \otimes \text{id})(\tilde{\mathbf{K}}^{(j)}(u)) = (\mathbf{L}^{(j)}(u/w))_{[2]} (\tilde{\mathbf{K}}^{(j)}(u))_{[1]} (\mathbf{L}^{(j)}(uw))_{[2]}, $$

$$ (K3') \qquad \tilde{\mathbf{K}}^{(j)}(u) = \mathcal{F}_{(12)}^{(j)} \tilde{\mathbf{K}}_{1}^{(\frac{1}{2})} (uq^{-j+\frac{1}{2}}) \mathcal{R}^{(\frac{1}{2},j-\frac{1}{2})} (u^2q^{-j+1}) \tilde{\mathbf{K}}_{2}^{(j-\frac{1}{2})} (uq^{\frac{1}{2}}) \mathcal{E}_{(12)}^{(j)}, $$

where $\mathcal{R}^{(\frac{1}{2},j)}(u)$ is given in (4.18), if and only if (6.2) and (6.4) hold. Here, (K1'), (K2'), (K3') are direct analogs of (4.84), (4.83), (4.72), respectively. We will also show that $\tilde{\mathbf{K}}^{(j)}(u)$ satisfies the reflection equation (4.67) where $\mathbf{K}^{(j)}(u)$ is replaced by $\tilde{\mathbf{K}}^{(j)}(u)$.

**6.1.1. Fusion relation (K3'), twisted intertwining relations (K1') and evaluated coaction (K2').** We assume that $\nu(u)$ is an invertible central element in $\mathcal{A}_q[[u^{-1}]]$.

**Lemma 6.3.** The K-operators $\tilde{\mathbf{K}}^{(j)}(u)$ for all $j \in \frac{1}{2}\mathbb{N}$ satisfy the fusion relation (K3') if and only if $\nu^{(j)}(u)$ takes the form (6.2).

*Proof.* Note first that (5.29) contains $R^{(\frac{1}{2},j)}(u)$ while (K3') contains $\mathcal{R}^{(\frac{1}{2},j)}(u)$, and they are related by

$$ (6.17) \qquad \mathcal{R}^{(\frac{1}{2},j)}(u) = \left( \prod_{k=0}^{2j-1} \pi^{\frac{1}{2}}(\mu(uq^{-j+\frac{1}{2}+k})) \right) \mathcal{R}^{(\frac{1}{2},j)}(u) $$

due to (4.41) and (4.29). Then, inserting (6.16) in (K3') and using (5.29) and (6.17), the assumption that $\nu(u)$ is an invertible central element in $\mathcal{A}_q[[u^{-1}]]$, and that $\mathcal{K}^{(j)}(u)$ is invertible, see Remark 5.9, the resulting recursion relation on $\nu^{(j)}(u)$ is equivalent to (6.2). $\square$

In what follows, we will assume that (K3') holds, so in particular all $\nu^{(j)}(u)$ are central. Then, using Proposition 5.13, the twisted intertwining relation for $\tilde{\mathbf{K}}^{(j)}(u)$ is immediate:

**Lemma 6.4.** The K-operators $\tilde{\mathbf{K}}^{(j)}(u)$ for all $j \in \frac{1}{2}\mathbb{N}$ satisfy (K1').

We now show that the evaluated coaction (K2') holds for $\tilde{\mathbf{K}}^{(j)}(u)$.
**Lemma 6.5.** The K-operators $\tilde{\mathbf{K}}^{(j)}(u)$ for all $j \in \frac{1}{2}\mathbb{N}$ satisfy (K2') if and only if $\delta_w(\nu(u))$ is given by (6.4).

*Proof.* We first consider the case $j = \frac{1}{2}$. Inserting (6.16) in (K2') for $j = \frac{1}{2}$, using (5.45), and the invertibility of $\mathcal{L}^{(\frac{1}{2})}(u)$, $\mathcal{K}^{(\frac{1}{2})}(u)$, the resulting equation is equivalent to (6.4). It remains to check that given the evaluated coaction (6.4), (K2') holds for higher $j$. Now, consider $j \ge 1$. On one hand, inserting (6.16) in (K2'), the l.h.s reads

$$ (6.18) \quad \delta_w(\nu^{(j)}(u))\delta_w(\mathcal{K}^{(j)}(u)) = \left(\prod_{m=0}^{2j-1} \delta_w(\nu(uq^{j-\frac{1}{2}-m}))\right) \left(\prod_{k=0}^{2j-2} \prod_{\ell=0}^{2j-k-2} \pi^{\frac{1}{2}}(\mu(u^2q^{2j-2-2k-\ell}))\right) \\ \times \left(\prod_{p=1}^{2j} \frac{U^{-1}}{q+q^{-1}}|_{u=uq^{j+\frac{1}{2}-p}}\right) (\mathcal{L}^{(j)}(u/w))_{[2]} (\mathcal{K}^{(j)}(u))_{[1]} (\mathcal{L}^{(j)}(uw))_{[2]}, $$

where we used (5.50), (6.2) and the fact that $\delta_w$ in an algebra homomorphism. On the other hand, the r.h.s. of (K2') is

$$ (6.19) \quad \nu^{(j)}(u) \otimes \mu^{(j)}(u/w)\mu^{(j)}(uw) (\mathcal{L}^{(j)}(u/w))_{[2]} (\mathcal{K}^{(j)}(u))_{[1]} (\mathcal{L}^{(j)}(uw))_{[2]}. $$

Then, replacing $\mu^{(j)}(u)$ and $\nu^{(j)}(u)$ in (6.19) by (4.29), (6.2) respectively, and using (6.4) in (6.18), we find that (6.18) and (6.19) are equal. $\square$

Finally, assuming (K3') holds, so that $\nu^{(j)}(u)$ are central by Lemma 6.3, we show that the fused K-operators $\tilde{\mathbf{K}}^{(j)}(u)$ satisfy the reflection equation.

**Lemma 6.6.** The K-operators $\tilde{\mathbf{K}}^{(j)}(u)$ satisfy the reflection equation (4.67) where $\mathbf{K}^{(j)}(u)$ is replaced by $\tilde{\mathbf{K}}^{(j)}(u)$ for any $j_1, j_2 \in \frac{1}{2}\mathbb{N}_+$.

*Proof.* By Theorem 5.7, the fused K-operators $\mathcal{K}^{(j)}(u)$ satisfy the equation (5.30). Then, multiplying this equation by $\nu^{(j_1)}(u)\nu^{(j_2)}(v)$ and using the fact that they are central, we obtain (4.67) where $\mathbf{K}^{(j)}(u)$ is replaced by $\tilde{\mathbf{K}}^{(j)}(u)$. $\square$

6.1.2. Functional relation on $\nu(u)$. We have seen in Lemma 6.3 that the relation (K3') fixes the normalization factor $\nu^{(j)}(u)$ as (6.2). Here we show that the analog of the reduction relation (4.73) for $\tilde{\mathbf{K}}^{(j)}(u)$ leads to the functional relation (6.3). Recall the functional relation on $\mu(u)$ in (4.30) was obtained by comparing the fusion relation with the reduction relation satisfied by the spin-$j$ L-operators, see Proposition 4.10. We proceed similarly for $\tilde{\mathbf{K}}^{(j)}(u)$.

**Proposition 6.7.** The K-operators $\tilde{\mathbf{K}}^{(j)}(u)$ satisfy (K3') and

$$ (K3') \quad \tilde{\mathbf{K}}^{(j-\frac{1}{2})}(u) = \bar{\mathcal{F}}_{\langle 12 \rangle}^{(j-\frac{1}{2})} \tilde{\mathbf{K}}_1^{(\frac{1}{2})}(uq^{j+1}) \mathcal{R}^{(\frac{1}{2},j)}(u^2q^{j+\frac{3}{2}}) \tilde{\mathbf{K}}_2^{(j)}(uq^{\frac{1}{2}}) \bar{\mathcal{E}}_{\langle 12 \rangle}^{(j-\frac{1}{2})}, $$

for $j=1$ if and only if $\nu(u)$ satisfies the functional relation (6.3).

*Proof.* The equation (K3') for $j=1$ in terms of fused K-operators reads as

$$ (6.20) \quad \mathcal{K}^{(\frac{1}{2})}(u) = \nu(uq)\nu(uq^2)\pi^{\frac{1}{2}}(\mu(u^2q)\mu(u^2q^2)\mu(u^2q^3))\bar{\mathcal{F}}_{\langle 12 \rangle}^{(\frac{1}{2})} \mathcal{K}_1^{(\frac{1}{2})}(uq^2)\mathcal{R}^{(\frac{1}{2},1)}(u^2q^{\frac{5}{2}})\mathcal{K}_2^{(1)}(uq^{\frac{1}{2}})\bar{\mathcal{E}}_{\langle 12 \rangle}^{(\frac{1}{2})}, $$
where we used $\tilde{\mathbf{K}}^{(\frac{1}{2})}(u) = \nu(u)\mathcal{K}^{(\frac{1}{2})}(u)$, the factorized form (6.2) for $\nu^{(1)}(u)$ due to Lemma 6.3 and we used $\mathcal{R}^{(\frac{1}{2},1)}(u) = \pi^{\frac{1}{2}}(\mu(uq^{\frac{1}{2}})\mu(uq^{-\frac{1}{2}}))R^{(\frac{1}{2},1)}(u)$, recall (6.17). From the relation satisfied by $\mu(u)$ given in (4.30) and using $\pi^j(C) = q^{2j+1} + q^{-2j-1}$, one gets:

$$ (6.21) \qquad \pi^{\frac{1}{2}}(\mu(u)\mu(uq)) = \frac{1}{c(u)c(uq^2)}, $$

where $c(u)$ is given in (3.42) and by our standard convention the right hand side is developed as power series in $u^{-1}$. Then, the equation (6.20) becomes

$$ (6.22) \qquad \mathcal{K}^{(\frac{1}{2})}(u) = \frac{\nu(uq)\nu(uq^2)\pi^{\frac{1}{2}}(\mu(u^2q^3))}{c(u^2q)c(u^2q^3)} \bar{\mathcal{F}}_{(12)}^{(\frac{1}{2})} \mathcal{K}_1^{(\frac{1}{2})}(uq^2) R^{(\frac{1}{2},1)}(u^2q^{\frac{5}{2}}) \mathcal{K}_2^{(1)}(uq^{\frac{1}{2}}) \bar{\mathcal{E}}_{(12)}^{(\frac{1}{2})}. $$

The r.h.s. of (6.22) is now computed using the expressions for $\mathcal{K}^{(\frac{1}{2})}(u)$, $\mathcal{K}^{(1)}(u)$ given respectively in (5.4), (5.37), the fused R-matrix (5.35) and $\bar{\mathcal{E}}_{(12)}^{(\frac{1}{2})}$, $\bar{\mathcal{F}}_{(12)}^{(\frac{1}{2})}$ given by:

$$ \bar{\mathcal{E}}_{(12)}^{(\frac{1}{2})} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \\ 0 & \sqrt{[2]_q} \\ -\sqrt{[2]_q} & 0 \\ 0 & -1 \\ 0 & 0 \end{pmatrix}, \qquad \bar{\mathcal{F}}_{(12)}^{(\frac{1}{2})} = \begin{pmatrix} 0 & \frac{1}{1+[2]_q} & 0 & -\frac{\sqrt{[2]_q}}{1+[2]_q} & 0 & 0 \\ 0 & 0 & \frac{\sqrt{[2]_q}}{1+[2]_q} & 0 & -\frac{1}{1+[2]_q} & 0 \end{pmatrix}. $$

In terms of the quantum determinant (5.20), one finds:

$$ (6.23) \qquad \bar{\mathcal{F}}_{(12)}^{(\frac{1}{2})} \mathcal{K}_1^{(\frac{1}{2})}(uq^2) R^{(\frac{1}{2},1)}(u^2q^{\frac{5}{2}}) \mathcal{K}_2^{(1)}(uq^{\frac{1}{2}}) \bar{\mathcal{E}}_{(12)}^{(\frac{1}{2})} = c(u^2q)c(u^2q^3)\Gamma(uq)\mathcal{K}^{(\frac{1}{2})}(u). $$

This relation is obtained by applying the ordering relations for $\mathcal{A}_q$ given in Appendix C. Inserting this expression in (6.22) and multiplying by $[\mathcal{K}^{(\frac{1}{2})}(u)]^{-1}$ given in (5.27) the relation (6.3) follows. $\square$

**Remark 6.8.** As a consistency check, we observe the following:

(1) The evaluated coaction in (6.4) respects the functional relation (6.3) on $\nu(u)$. Indeed, using (5.49) and the functional relations (4.30) and (6.3), we obtain

$$ \delta_w(\nu(u))\delta_w(\nu(uq))\delta_w(\Gamma(u))\pi^{\frac{1}{2}}(\mu(u^2q)) = 1 \otimes 1. $$

(2) The expression for $\delta_w(\nu(u))$ in (6.4) agrees with the direct calculation of $\delta_w(\nu_1)$. To see it, we first recall the expression of $\nu_1$ in (6.15) together with (6.7) and (5.24). With the expressions from Proposition 2.15, we need to calculate only the evaluated coaction on $\tilde{\mathcal{G}}_1$. This is the 1st mode of the current $\mathcal{G}_-(u)$, recall (5.2), and thus $\delta_w(\tilde{\mathcal{G}}_1)$ is extracted from the 1st mode of (5.48):

$$ \delta_w(\tilde{\mathcal{G}}_1) = \frac{k_+k_-(q+q^{-1})}{q-q^{-1}} \otimes \left( \begin{aligned} & \frac{k_+}{k_-}(q-q^{-1})^2 E^2 - (w^{-2}K^{-1} + w^2K)) \\ & + \tilde{\mathcal{G}}_1 \otimes 1 + (q^2-q^{-2})k_+(q^{\frac{1}{2}}W_0 \otimes (wEK^{\frac{1}{2}}) + q^{-\frac{1}{2}}W_1 \otimes (w^{-1}EK^{-\frac{1}{2}}) \end{aligned} \right). $$
Combining all these expressions of $\delta_w$ together as in (5.24), using that $\delta_w$ is an algebra map for any $w$, and simplifying we obtain

$$ \delta_w(\Delta_1) = \Delta_1 \otimes 1 - \frac{2k_+k_-(w^2 + w^{-2})}{q - q^{-1}} 1 \otimes C, $$

where $C$ is the Casimir element (A.5). Using (6.15) together with the expression of $F_1$ in (6.7), we get the final result

$$ \delta_w(\nu_1) = \nu_1 \otimes 1 + \frac{q^{-1}\nu_0}{q + q^{-1}}(w^2 + w^{-2}) \otimes C $$

which indeed agrees with the term at $u^{-2}$ in the expansion of (6.4).

In summary, the evidence supporting Conjecture 1 can be summarized in Figure 4.

<figure><img src="image_7.png" /><figcaption>FIGURE 4. Supporting evidences for Conjecture 1, where the double-sided arrows connecting two equations signify their equality under the assumption that Conjecture 1 is true.</figcaption></figure>

## 6.2. Coaction.
Motivated by the discussion in Section 4.3 and Proposition 4.20, we propose a right coaction for the components of $\tilde{\mathbf{K}}^{(\frac{1}{2})}(u)$:

$$ (6.24) \qquad (\delta \otimes \mathrm{id})(\tilde{\mathbf{K}}^{(\frac{1}{2})}(u)) = ([\mathbf{L}^{-}(u^{-1})]^{-1})_{[2]} (\tilde{\mathbf{K}}^{(\frac{1}{2})}(u))_{[1]} (\mathbf{L}^{+}(u))_{[2]}, $$

where $\mathbf{L}^{\pm}(u)$ are defined in (4.74) and are calculated in (B.26) and (B.33), respectively. First of all, we show that the proposed coaction (6.24) indeed respects the relations satisfied by the components of $\tilde{\mathbf{K}}^{(\frac{1}{2})}(u)$. Recall that, due to Lemmas 6.6 and 4.13, these relations are

$$ (6.25) \qquad R^{(\frac{1}{2}, \frac{1}{2})}(u/v) \tilde{\mathbf{K}}_{1}^{(\frac{1}{2})}(u) R^{(\frac{1}{2}, \frac{1}{2})}(uv) \tilde{\mathbf{K}}_{2}^{(\frac{1}{2})}(v) = \tilde{\mathbf{K}}_{2}^{(\frac{1}{2})}(v) R^{(\frac{1}{2}, \frac{1}{2})}(uv) \tilde{\mathbf{K}}_{1}^{(\frac{1}{2})}(u) R^{(\frac{1}{2}, \frac{1}{2})}(u/v). $$

**Lemma 6.9.** The following element belonging to $(A_q \otimes \mathcal{L}U_q sl_2)[[u^{-1}]] \otimes \mathrm{End}(\mathbb{C}^2)$ and defined by

$$ (6.26) \qquad \tilde{\mathbf{K}}^{(-,+)}(u) = ([\mathbf{L}^{-}(u^{-1})]^{-1})_{[2]} (\tilde{\mathbf{K}}^{(\frac{1}{2})}(u))_{[1]} (\mathbf{L}^{+}(u))_{[2]} $$

satisfies the reflection equation (6.25) where $\tilde{\mathbf{K}}^{(\frac{1}{2})}(u)$ is replaced by $\tilde{\mathbf{K}}^{(-,+)}(u)$.
*Proof.* We first substitute the K-operators in (6.26) into the reflection equation (6.25). Then, we multiply both sides from the left by $\mathbf{L}_1^-(u^{-1})\mathbf{L}_2^-(v^{-1})$ and from the right by $[\mathbf{L}_2^+(v)]^{-1}[\mathbf{L}_1^+(u)]^{-1}$. One has for the l.h.s. of the resulting equation:

$$ (6.27) \qquad \begin{aligned} & \frac{\mathbf{L}_1^-(u^{-1})\mathbf{L}_2^-(v^{-1})R^{(\frac{1}{2}, \frac{1}{2})}(u/v)[\mathbf{L}_1^-(u^{-1})]^{-1}\tilde{\mathbf{K}}_1^{(\frac{1}{2})}(u)\mathbf{L}_1^+(u)R^{(\frac{1}{2}, \frac{1}{2})}(uv)[\mathbf{L}_2^-(v^{-1})]^{-1}\tilde{\mathbf{K}}_2^{(\frac{1}{2})}(v)[\mathbf{L}_1^+(u)]^{-1}\tilde{\mathbf{K}}_2^{(\frac{1}{2})}(v)}{\mathbf{L}_2^-(v^{-1})\tilde{\mathbf{K}}_1^{(\frac{1}{2})}(u)\mathbf{L}_1^+(u)R^{(\frac{1}{2}, \frac{1}{2})}(uv)[\mathbf{L}_2^-(v^{-1})]^{-1}[\mathbf{L}_1^+(u)]^{-1}\tilde{\mathbf{K}}_2^{(\frac{1}{2})}(v)} \\ & = R^{(\frac{1}{2}, \frac{1}{2})}(u/v)\tilde{\mathbf{K}}_1^{(\frac{1}{2})}(u)R^{(\frac{1}{2}, \frac{1}{2})}(uv)\tilde{\mathbf{K}}_2^{(\frac{1}{2})}(v), \end{aligned} $$

where we underlined the steps of calculation that correspond either to the commutation relations between L- and K-operators associated with different auxiliary spaces or to the use of variations of (4.75), (4.76). For instance, in the first line we use

$$ (6.28) \qquad \mathbf{L}_2^-(v^{-1})R^{(\frac{1}{2}, \frac{1}{2})}(u/v)[\mathbf{L}_1^-(u^{-1})]^{-1} = [\mathbf{L}_1^-(u^{-1})]^{-1}R^{(\frac{1}{2}, \frac{1}{2})}(u/v)\mathbf{L}_2^-(v^{-1}), $$

which is obtained by multiplying the equation (4.75), with the all-minus choice, from the left by $[\mathbf{L}_1^-(u)]^{-1}R^{(\frac{1}{2}, \frac{1}{2})}(v/u)$ and from the right by $R^{(\frac{1}{2}, \frac{1}{2})}(v/u)[\mathbf{L}_1^-(u)]^{-1}$, then using the unitarity property of the R-matrix given in (4.55), and finally, substituting $u \to u^{-1}$, $v \to v^{-1}$.

On the other hand, the r.h.s. of the equation obtained in the first paragraph of this proof takes the following form:

$$ (6.29) \qquad \begin{aligned} & \frac{\mathbf{L}_1(u^{-1})\tilde{\mathbf{K}}_2^{(\frac{1}{2})}(v)\mathbf{L}_2^+(v)R^{(\frac{1}{2}, \frac{1}{2})}(uv)[\mathbf{L}_1(u^{-1})]^{-1}\tilde{\mathbf{K}}_1^{(\frac{1}{2})}(u)\mathbf{L}_1^+(u)R^{(\frac{1}{2}, \frac{1}{2})}(u/v)[\mathbf{L}_2^+(v)]^{-1}[\mathbf{L}_1^+(u)]^{-1}}{\mathbf{L}_2^-(v)\tilde{\mathbf{K}}_1^{(\frac{1}{2})}(v)\mathbf{L}_1^-(u^{-1})\mathbf{L}_2^+(v)R^{(\frac{1}{2}, \frac{1}{2})}(uv)[\mathbf{L}_1^-(u^{-1})]^{-1}\mathbf{L}_1^+(u)[\mathbf{L}_2^+(v)]^{-1}R^{(\frac{1}{2}, \frac{1}{2})}(u/v)} \\ & = \tilde{\mathbf{K}}_2^{(\frac{1}{2})}(v)\mathbf{L}_1^-(u^{-1})\mathbf{L}_2^+(v)R^{(\frac{1}{2}, \frac{1}{2})}(uv)[\mathbf{L}_1^-(u^{-1})]^{-1}\mathbf{L}_1^+(u)[\mathbf{L}_2^+(v)]^{-1}R^{(\frac{1}{2}, \frac{1}{2})}(u/v). \end{aligned} $$

Finally, comparing (6.27) with (6.29), one gets the reflection equation (6.25) that was proven in Lemma 6.6. $\square$

We finally show that $\delta$ defined in (6.24) is coassociative and counital, see (2.18). Firstly, we check the coassociativity:

$$ \begin{aligned} (\delta \otimes \mathrm{id} \otimes \mathrm{id}) \circ (\delta \otimes \mathrm{id})(\tilde{\mathbf{K}}^{(\frac{1}{2})}(u)) &= (\delta \otimes \mathrm{id} \otimes \mathrm{id}) \left( ([\mathbf{L}^-(u^{-1})]^{-1})_{[2]} (\tilde{\mathbf{K}}^{(\frac{1}{2})}(u))_{[1]} (\mathbf{L}^+(u))_{[2]} \right) \\ &= ([\mathbf{L}^-(u^{-1})]^{-1})_{[3]} ([\mathbf{L}^-(u^{-1})]^{-1})_{[2]} (\tilde{\mathbf{K}}^{(\frac{1}{2})}(u))_{[1]} (\mathbf{L}^+(u))_{[2]} (\mathbf{L}^+(u))_{[3]} \\ &= (\mathrm{id} \otimes \Delta \otimes \mathrm{id}) \circ (\delta \otimes \mathrm{id})(\tilde{\mathbf{K}}^{(\frac{1}{2})}(u)), \end{aligned} $$

where the coproduct is given in (4.78) and we used $(\Delta \otimes \mathrm{id})([\mathbf{L}^{\mp}(u)]^{-1}) = (\mathbf{L}^{\mp}(u))_{[2]}^{-1}(\mathbf{L}^{\mp}(u))_{[1]}^{-1}$. Secondly, the condition with the counit is checked:

$$ (\mathrm{id} \otimes \epsilon \otimes \mathrm{id}) \circ (\delta \otimes \mathrm{id})(\tilde{\mathbf{K}}^{(\frac{1}{2})}(u)) = (\mathrm{id} \otimes \epsilon \otimes \mathrm{id}) \circ \left( ([\mathbf{L}^{\mp}(u^{-1})]^{-1})_{[2]} (\tilde{\mathbf{K}}^{(\frac{1}{2})}(u))_{[1]} (\mathbf{L}^{\pm}(u))_{[2]} \right) \\ = \tilde{\mathbf{K}}^{(\frac{1}{2})}(u), $$

where we used (4.80).
Let us finally check that the coaction proposed in (6.24) reproduces the ‘standard’ $q$-Onsager coaction (2.44)-(2.45) for the first two generators $W_0$ and $W_1$. Recall from (6.16) that $\tilde{\mathbf{K}}^{(\frac{1}{2})}(u) = \nu(u)\mathcal{K}^{(\frac{1}{2})}(u)$, with $\nu(u)$ calculated in Lemma 6.1 and $\mathcal{K}^{(\frac{1}{2})}(u)$ defined in (5.4). With the Ding-Frenkel L-operators computed in (B.26) and (B.33), we compare leading terms at $u^{-1}$ of the matrix entries (1,1) and (2,2) of both sides of (6.24) which gives

$$ (6.30) \qquad \delta(W_0) = 1 \otimes \left( k_+ q^{\frac{1}{2}} E_1 K_1^{\frac{1}{2}} + k_- q^{-\frac{1}{2}} F_1 K_1^{\frac{1}{2}} \right) + W_0 \otimes K_1, $$

$$ (6.31) \qquad \delta(W_1) = 1 \otimes \left( k_+ q^{-\frac{1}{2}} F_0 K_0^{\frac{1}{2}} + k_- q^{\frac{1}{2}} E_0 K_0^{\frac{1}{2}} \right) + W_1 \otimes K_0. $$

These formulas indeed agree with (2.44) and (2.45), respectively.

**6.3. Comments.** Based on the supporting evidences given in the previous subsection, we believe Conjecture 1 is correct. Some straightforward consequences are now pointed out. Firstly, some relations among the fused K-operators (5.29) are derived. They generalize the relation (6.23).

**Proposition 6.10.** Assume Conjecture 1. Then, the following relations hold for any $j \in \frac{1}{2}\mathbb{N}_+$:

$$ (6.32) \qquad \bar{\mathcal{F}}_{\langle 12 \rangle}^{(j-\frac{1}{2})} \mathcal{K}_1^{(\frac{1}{2})}(uq^{j+1}) R^{(\frac{1}{2},j)}(u^2 q^{j+\frac{3}{2}}) \mathcal{K}_2^{(j)}(uq^{\frac{1}{2}}) \bar{\mathcal{E}}_{\langle 12 \rangle}^{(j-\frac{1}{2})} = \\ \qquad \qquad \left( \prod_{k=0}^{2j-2} c(u^2 q^{2j-1-k}) c(u^2 q^{2j+1-k}) \right) \Gamma(uq^j) \mathcal{K}^{(j-\frac{1}{2})}(u), $$

$$ (6.33) \qquad \bar{\mathcal{F}}_{\langle 12 \rangle}^{(j-\frac{1}{2})} \mathcal{K}_2^{(j)}(uq^{-\frac{1}{2}}) R^{(\frac{1}{2},j)}(u^2 q^{-j-\frac{3}{2}}) \mathcal{K}_1^{(\frac{1}{2})}(uq^{-j-1}) \bar{\mathcal{E}}_{\langle 12 \rangle}^{(j-\frac{1}{2})} = \\ \qquad \qquad \left( \prod_{k=0}^{2j-2} c(u^2 q^{-2j+2+k}) c(u^2 q^{-2j+k}) \right) \Gamma(uq^{-j-1}) \mathcal{K}^{(j-\frac{1}{2})}(u), $$

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

*Proof.* From Remark 3.4, the intertwining property with $\Delta^{op}$ reads:

$$ (6.34) \qquad \bar{\mathcal{E}}^{(j-\frac{1}{2})}(\pi_u^{j-\frac{1}{2}})(x) = (\pi_u^{\frac{1}{2}} \otimes \pi_u^{j-\frac{1}{2}})(\Delta^{op}(x))\bar{\mathcal{E}}^{(j-\frac{1}{2})}, \quad \forall x \in \mathcal{L}U_q s l_2. $$

Now, express the l.h.s. of (6.32) in terms of K-operators and R-matrices. It reads:

$$ (6.35) \qquad \frac{\bar{\mathcal{F}}_{\langle 12 \rangle}^{(j-\frac{1}{2})} \mathbf{K}_1^{(\frac{1}{2})}(uq^{j+1}) \mathcal{R}^{(\frac{1}{2},j)}(u^2 q^{j+\frac{3}{2}}) \mathbf{K}_2^{(j)}(uq^{\frac{1}{2}}) \bar{\mathcal{E}}_{\langle 12 \rangle}^{(j-\frac{1}{2})}}{\nu(uq^{j+1})\nu^{(j)}(uq^{\frac{1}{2}})\pi^{\frac{1}{2}}(\mu^{(j)}(u^2 q^{j+\frac{3}{2}}))} \\ = \frac{(\mathrm{id} \otimes \bar{\mathcal{F}}^{(j-\frac{1}{2})}) [(\mathrm{id} \otimes \pi^{\frac{1}{2}}_{u^{-1}q^{-j-1}} \otimes \pi^{j}_{u^{-1}q^{-\frac{1}{2}}})(\mathrm{id} \otimes \Delta^{op})(\mathfrak{K})] (\mathrm{id} \otimes \bar{\mathcal{E}}^{(j-\frac{1}{2})})}{\nu(uq^{j+1})\nu^{(j)}(uq^{\frac{1}{2}})\pi^{\frac{1}{2}}(\mu^{(j)}(u^2 q^{j+\frac{3}{2}}))} \\ = \frac{\nu^{(j-\frac{1}{2})}(u)}{\nu(uq^{j+1})\nu^{(j)}(uq^{\frac{1}{2}})\pi^{\frac{1}{2}}(\mu^{(j)}(u^2 q^{j+\frac{3}{2}}))} \mathcal{K}^{(j-\frac{1}{2})}(u). $$
Then, simplifying the normalization factors and using (6.3), we get

$$ 
\begin{aligned}
& \left[ \nu(uq^j)\nu(uq^{j-1})\pi^{\frac{1}{2}}(\mu^{(j-\frac{1}{2})}(u^2q^j)\mu^{(j)}(u^2q^{j+\frac{3}{2}})) \right]^{-1} \mathcal{K}^{(j-\frac{1}{2})}(u) \\
& = \left[ \prod_{k=0}^{2j-2} \pi^{\frac{1}{2}}(\mu(u^2q^{2j-k-1})\mu(u^2q^{2j-k})) \right]^{-1} \Gamma(uq^j)\mathcal{K}^{(j-\frac{1}{2})}(u) .
\end{aligned}
$$

Finally, using (6.21), the equation (6.32) follows. The relation (6.33) is obtained similarly. $\square$

Secondly, we analyze the spin-0 K-operator $\mathbf{K}^{(0)}(u)$ and the analog of the quantum determinant (5.19) for the spin-$\frac{1}{2}$ K-operator $\mathbf{K}^{(\frac{1}{2})}(u)$.

**Proposition 6.11.** Assume Conjecture 1, then $\mathbf{K}^{(0)}(u) = 1$. Furthermore, the normalized quantum determinant of the K-operator $\mathbf{K}^{(\frac{1}{2})}(u)$ is equally 1:

$$ (6.36) \qquad \mathrm{tr}_{12}(\mathcal{P}_{12}^{-} \mathbf{K}_{1}^{(\frac{1}{2})}(u) \mathcal{R}^{(\frac{1}{2},\frac{1}{2})}(qu^2) \mathbf{K}_{2}^{(\frac{1}{2})}(uq)) = 1. $$

*Proof.* Specializing (K3") to $j = \frac{1}{2}$ we get

$$ 
\begin{aligned}
(6.37) \qquad & \mathbf{K}^{(0)}(u) = \bar{\mathcal{F}}_{\langle 12 \rangle}^{(0)} \mathbf{K}_{1}^{(\frac{1}{2})}(uq^{\frac{3}{2}}) \mathcal{R}^{(\frac{1}{2},\frac{1}{2})}(u^2q^2) \mathbf{K}_{2}^{(\frac{1}{2})}(uq^{\frac{1}{2}}) \bar{\mathcal{E}}_{\langle 12 \rangle}^{(0)} \\
& = \nu(uq^{\frac{3}{2}})\nu(uq^{\frac{1}{2}})\pi^{\frac{1}{2}}(\mu(u^2q^2)) \bar{\mathcal{F}}_{\langle 12 \rangle}^{(0)} \mathbf{K}_{1}^{(\frac{1}{2})}(uq^{\frac{3}{2}}) \mathcal{R}^{(\frac{1}{2},\frac{1}{2})}(u^2q^2) \mathbf{K}_{2}^{(\frac{1}{2})}(uq^{\frac{1}{2}}) \bar{\mathcal{E}}_{\langle 12 \rangle}^{(0)},
\end{aligned}
$$

where $\bar{\mathcal{E}}^{(0)}$, $\bar{\mathcal{F}}^{(0)}$ are given by

$$ (6.38) \qquad \bar{\mathcal{E}}^{(0)} = \begin{pmatrix} 0 \\ 1 \\ -1 \\ 0 \end{pmatrix}, \qquad \bar{\mathcal{F}}^{(0)} = (0 \quad \frac{1}{2} \quad -\frac{1}{2} \quad 0). $$

Then, noticing that for any two-by-two matrix $A$, one has the property:

$$ (6.39) \qquad \bar{\mathcal{F}}_{\langle 12 \rangle}^{(0)} A \bar{\mathcal{E}}_{\langle 12 \rangle}^{(0)} = \mathrm{tr}_{12}(\mathcal{P}_{12}^{-} A), $$

it follows from (5.19) that

$$ (6.40) \qquad \Gamma(u) = \bar{\mathcal{F}}_{\langle 12 \rangle}^{(0)} \mathcal{K}_{1}^{(\frac{1}{2})}(uq) \mathcal{R}^{(\frac{1}{2},\frac{1}{2})}(u^2q) \mathcal{K}_{2}^{(\frac{1}{2})}(u) \bar{\mathcal{E}}_{\langle 12 \rangle}^{(0)}. $$

Therefore, the r.h.s. of (6.37) becomes

$$ \mathbf{K}^{(0)}(u) = \nu(uq^{\frac{3}{2}})\nu(uq^{\frac{1}{2}})\pi^{\frac{1}{2}}(\mu(u^2q^2))\Gamma(uq^{\frac{1}{2}}) = 1, $$

where we used the functional relation (6.3). We finally note that the quantum determinant in the l.h.s. of (6.36) is $\mathbf{K}^{(0)}(u)$, due to (6.37) and (6.39), and so it equals 1. $\square$

**Proposition 6.12.** Assume Conjecture 1, then $\hat{\mathcal{K}}^{(j)}(u)$ from (5.31) is equal to the fused K-operator $\mathcal{K}^{(j)}(u)$ defined in (5.29).

*Proof.* Recall the K-operators $\mathbf{K}^{(j+\frac{1}{2})}(u)$ can be written either as (4.68) or as (4.72). Using (6.1) with (6.2) and the invertibility of $\nu^{(j)}(u)$, we show by induction (recall that $\hat{\mathcal{K}}^{(\frac{1}{2})}(u) = \mathcal{K}^{(\frac{1}{2})}(u)$) that $\mathcal{K}^{(j)}(u)$ equals $\hat{\mathcal{K}}^{(j)}(u)$. $\square$
## 7. K-OPERATORS AND THE PBW BASIS

As mentioned in the introduction, in the literature the method (ii) has been successfully applied to the derivation of K-matrices with scalar entries solely using twisted intertwining relations in various representations of certain comodule algebras $B$. In this section, solutions $\mathcal{K}^{(j)}(u)$ of the twisted intertwining relation (5.55) are investigated using a PBW basis of $\mathcal{A}_q$ for $j = \frac{1}{2}$, 1. We show that for a reasonably general ansatz, the solutions are uniquely determined (up to an overall factor), and match with the expressions of $\mathcal{K}^{(j)}(u)$'s for $j = \frac{1}{2}$ and $j = 1$ constructed in Section 5.

Different types of PBW bases for $\mathcal{A}_q$ are known [T21a]. For instance, a PBW basis for $\mathcal{A}_q$ can be constructed in terms of the PBW generators

$$ \{\mathcal{W}_{-k}\}_{k \in \mathbb{N}}, \quad \{\mathcal{G}_{\ell+1}\}_{\ell \in \mathbb{N}}, \quad \{\tilde{\mathcal{G}}_{m+1}\}_{m \in \mathbb{N}}, \quad \{\mathcal{W}_{n+1}\}_{n \in \mathbb{N}} $$

in the linear order < that satisfies<fnref n="17" />

$$ (7.1) \qquad \mathcal{W}_{-k} < \mathcal{G}_{\ell+1} < \tilde{\mathcal{G}}_{m+1} < \mathcal{W}_{n+1}, \quad k, \ell, m, n \in \mathbb{N}. $$

We recall that the corresponding generating functions $\mathcal{W}_{\pm}(u)$ and $\mathcal{G}_{\pm}(u)$ were introduced in (5.1) and (5.2), respectively.

In the following analysis, we will encounter various combinations of the generating functions $\mathcal{W}_{\pm}(u)$ and $\mathcal{G}_{\pm}(u)$ that will need reordering. According to the chosen order (7.1), any words in $\{\mathcal{W}_{\pm}(u_i), \mathcal{G}_{\pm}(u_i)\}_{i \in \mathbb{N}}$ can be written in terms of ordered expressions using Lemma C.1. In particular, from the relations (C.1)-(C.6), one extracts exchange relations that will be used later on for solving the twisted intertwining relations:

$$ \begin{align} \mathcal{W}_+(u)\mathcal{W}_0 &= \mathcal{W}_0\mathcal{W}_+(u), & \mathcal{W}_1\mathcal{W}_-(u) &= \mathcal{W}_-(u)\mathcal{W}_1, \nonumber \\ \mathcal{G}_+(u)\mathcal{W}_0 &= q^2\mathcal{W}_0\mathcal{G}_+(u) + \rho q(\mathcal{W}_0 + \mathcal{W}_-(u) - U\mathcal{W}_+(u)), \nonumber \\ \mathcal{G}_-(u)\mathcal{W}_0 &= q^{-2}\mathcal{W}_0\mathcal{G}_-(u) - \rho q^{-1}(\mathcal{W}_0 + \mathcal{W}_-(u) - U\mathcal{W}_+(u)), \tag{7.2} \\ \mathcal{W}_-(u)\mathcal{W}_0 &= \mathcal{W}_0\mathcal{W}_-(u) + \frac{1}{q+q^{-1}}(\mathcal{G}_+(u) - \mathcal{G}_-(u)), \nonumber \\ \mathcal{W}_1\mathcal{G}_+(u) &= q^2\mathcal{G}_+(u)\mathcal{W}_1 + \rho q(\mathcal{W}_1 + \mathcal{W}_+(u) - U\mathcal{W}_-(u)), \nonumber \\ \mathcal{W}_1\mathcal{G}_-(u) &= q^{-2}\mathcal{G}_-(u)\mathcal{W}_1 - \rho q^{-1}(\mathcal{W}_1 + \mathcal{W}_+(u) - U\mathcal{W}_-(u)), \nonumber \\ \mathcal{W}_1\mathcal{W}_+(u) &= \mathcal{W}_+(u)\mathcal{W}_1 + \frac{1}{q+q^{-1}}(\mathcal{G}_+(u) - \mathcal{G}_-(u)). \nonumber \end{align} $$

From the construction of the fused K-operator $\mathcal{K}^{(j)}(u)$ in (5.29), it is a $(2j+1) \times (2j+1)$ matrix with entries in $\mathcal{A}_q[[u^{-1}]]$ that are linear combinations of monomials of the form (recall the ordering (7.1))

$$ (7.3) \qquad f(u) \prod_{r=1}^{R} \mathcal{W}_{+}(uq^{a_r}) \prod_{p=1}^{P} \mathcal{G}_{+}(uq^{b_p}) \prod_{m=1}^{M} \mathcal{G}_{-}(uq^{c_m}) \prod_{t=1}^{T} \mathcal{W}_{-}(uq^{d_t}), \quad R + P + M + T \le 2j $$

and for some choice of $a_r, b_p, c_m, d_t \in \frac{1}{2}\mathbb{Z}$, and $f(u)$ is a Laurent polynomial in $u$ of maximal degree $2(R + P + M + T)$. Below, we successively show for the cases $j = \frac{1}{2}$ and $j = 1$ that the twisted

<footnote n="17">¹⁷For a different choice of ordering, the proof of the PBW basis is given in [T21a].</footnote>
intertwining relations (5.55) admit a unique solution of the above form, and it agrees with $\mathcal{K}^{(j)}(u)$'s constructed in Section 5.

Before proceeding, we first notice that it is enough to solve the relations (5.55) for $b = W_0$ and $b = W_1$. This follows from the fact that $\mathcal{A}_q$ is the central extension of $O_q$ – it is generated by $W_0, W_1$ and its center. We can take the quantum determinant $\Gamma(u)$ as the generating function of the center of $\mathcal{A}_q$, recall its definition in (5.20). This central element has a particularly simple expression (5.49) for the evaluated coaction $\delta_v$ – both the tensor components are central and $\delta_v(\Gamma(u)) = \delta_{v-1}(\Gamma(u))$. From this observation, it is clear that the relations (5.55) hold for $b = \Gamma(u)$ and whatever choice of $\mathcal{K}^{(j)}(u)$, that is, they do not give any constraints on the form of $\mathcal{K}^{(j)}(u)$.

### 7.1. Spin-$\frac{1}{2}$ K-operator.
The above discussed ansatz for $\mathcal{K}^{(\frac{1}{2})}(u)$ takes the form:

$$ (7.4) \qquad \mathcal{K}^{(\frac{1}{2})}(u) = A_{1,1}^{(\frac{1}{2})}(u)W_{+}(u) + A_{1,2}^{(\frac{1}{2})}(u)W_{-}(u) + A_{1,3}^{(\frac{1}{2})}(u)\mathcal{G}_{+}(u) + A_{1,4}^{(\frac{1}{2})}(u)\mathcal{G}_{-}(u) + A_{0}^{(\frac{1}{2})}(u), $$

where $A_{1,i}^{(\frac{1}{2})}(u)$ and $A_{0}^{(\frac{1}{2})}(u)$ are two-by-two matrices. We are now solving the twisted intertwining relation (5.55) for $b = W_0, W_1$ using the ordering relations (7.2). For $b = W_0$ it is straightforward to show using (2.47) that the twisted intertwining relation (5.55) is equivalent to the system of four equations:

$$ \begin{align} \left[ W_0, \mathcal{K}_{11}^{(\frac{1}{2})}(u) \right] &= u^{-1}q^{-1}(k_-\mathcal{K}_{12}^{(\frac{1}{2})}(u) - k_+\mathcal{K}_{21}^{(\frac{1}{2})}(u)), \tag{7.5} \\ \left[ W_0, \mathcal{K}_{22}^{(\frac{1}{2})}(u) \right] &= -uq(k_-\mathcal{K}_{12}^{(\frac{1}{2})}(u) - k_+\mathcal{K}_{21}^{(\frac{1}{2})}(u)), \nonumber \\ \left[ W_0, \mathcal{K}_{12}^{(\frac{1}{2})}(u) \right]_q &= k_+(u\mathcal{K}_{11}^{(\frac{1}{2})}(u) - u^{-1}\mathcal{K}_{22}^{(\frac{1}{2})}(u)), \nonumber \\ \left[ W_0, \mathcal{K}_{21}^{(\frac{1}{2})}(u) \right]_{q^{-1}} &= -k_-(u\mathcal{K}_{11}^{(\frac{1}{2})}(u) - u^{-1}\mathcal{K}_{22}^{(\frac{1}{2})}(u)). \nonumber \end{align} $$

For $b = W_1$, the analogous system of equations is obtained by substituting $W_0 \mapsto W_1, u \mapsto u^{-1}, q \mapsto q^{-1}$ into the above equations. Inserting the K-operator's ansatz (7.4) into these two sets of inter-twining relations and using (7.2), one extracts a set of linearly independent equations that determine uniquely (up to an overall factor) the entries of the matrices $A_{1,i}^{(\frac{1}{2})}(u)$, $A_{0}^{(\frac{1}{2})}(u)$. Adjusting the overall normalization for convenience, they read:

$$ \begin{align*} A_{1,1}^{(\frac{1}{2})}(u) &= \begin{pmatrix} uq & 0 \\ 0 & -u^{-1}q^{-1} \end{pmatrix}, & A_{1,2}^{(\frac{1}{2})}(u) &= \begin{pmatrix} -u^{-1}q^{-1} & 0 \\ 0 & uq \end{pmatrix}, & A_{1,3}^{(\frac{1}{2})}(u) &= \begin{pmatrix} 0 & \frac{1}{k_-(q+q^{-1})} \\ 0 & 0 \end{pmatrix}, \\ A_{1,4}^{(\frac{1}{2})}(u) &= \begin{pmatrix} 0 & 0 \\ \frac{1}{k_+(q+q^{-1})} & 0 \end{pmatrix}, & A_{0}^{(\frac{1}{2})}(u) &= \begin{pmatrix} 0 & \frac{k_+(q+q^{-1})}{q-q^{-1}} \\ \frac{k_-(q+q^{-1})}{q-q^{-1}} & 0 \end{pmatrix}. \end{align*} $$

Clearly, it is seen that the K-operator (7.4) matches with the fundamental K-operator (5.4).

### 7.2. Spin-1 fused K-operator.
For $j = 1$, an ansatz for the K-operator is built along the same line. Quadratic and linear combinations of the generating functions $\{W_{\pm}(u), \mathcal{G}_{\pm}(u)\}$ are now expected.
Let $\alpha, \beta$ be half-integers. The ansatz reads:

$$ (7.6) \qquad \begin{aligned} \mathcal{K}^{(1)}(u) = & A_{2,1}^{(1)}(u)\mathcal{W}_+(uq^\alpha)\mathcal{W}_+(uq^\beta) + A_{2,2}^{(1)}(u)\mathcal{W}_+(uq^\alpha)\mathcal{G}_+(uq^\beta) + A_{2,3}^{(1)}(u)\mathcal{W}_+(uq^\beta)\mathcal{G}_+(uq^\alpha) \\ & + A_{2,4}^{(1)}(u)\mathcal{W}_+(uq^\alpha)\mathcal{G}_-(uq^\beta) + A_{2,5}^{(1)}(u)\mathcal{W}_+(uq^\beta)\mathcal{G}_-(uq^\alpha) + A_{2,6}^{(1)}(u)\mathcal{W}_+(uq^\alpha)\mathcal{W}_-(uq^\beta) \\ & + A_{2,7}^{(1)}(u)\mathcal{W}_+(uq^\beta)\mathcal{W}_-(uq^\alpha) + A_{2,8}^{(1)}(u)\mathcal{G}_+(uq^\alpha)\mathcal{G}_+(uq^\beta) + A_{2,9}^{(1)}(u)\mathcal{G}_+(uq^\alpha)\mathcal{G}_-(uq^\beta) \\ & + A_{2,10}^{(1)}(u)\mathcal{G}_+(uq^\beta)\mathcal{G}_-(uq^\alpha) + A_{2,11}^{(1)}(u)\mathcal{G}_+(uq^\alpha)\mathcal{W}_-(uq^\beta) + A_{2,12}^{(1)}(u)\mathcal{G}_+(uq^\beta)\mathcal{W}_-(uq^\alpha) \\ & + A_{2,13}^{(1)}(u)\mathcal{G}_-(uq^\alpha)\mathcal{G}_-(uq^\beta) + A_{2,14}^{(1)}(u)\mathcal{G}_-(uq^\alpha)\mathcal{W}_-(uq^\beta) + A_{2,15}^{(1)}(u)\mathcal{G}_-(uq^\beta)\mathcal{W}_-(uq^\alpha) \\ & + A_{2,16}^{(1)}(u)\mathcal{W}_-(uq^\alpha)\mathcal{W}_-(uq^\beta) + A_{1,1}^{(1)}(u)\mathcal{W}_+(uq^\alpha) + A_{1,2}^{(1)}(u)\mathcal{W}_+(uq^\beta) + A_{1,3}^{(1)}(u)\mathcal{G}_+(uq^\alpha) \\ & + A_{1,4}^{(1)}(u)\mathcal{G}_+(uq^\beta) + A_{1,5}^{(1)}(u)\mathcal{G}_-(uq^\alpha) + A_{1,6}^{(1)}(u)\mathcal{G}_-(uq^\beta) + A_{1,7}^{(1)}(u)\mathcal{W}_-(uq^\alpha) \\ & + A_{1,8}^{(1)}(u)\mathcal{W}_-(uq^\beta) + A_0^{(1)}, \end{aligned} $$

where $A_{i,j}^{(1)}(u)$ and $A_0^{(1)}$ are three-by-three matrices. Again, assume the twisted intertwining relations (5.55) hold for $b = W_0, W_1$, then for $b = W_0$ we have:

$$ \begin{aligned} [W_0, \mathcal{K}_{11}^{(1)}(u)] &= u^{-1}q^{-\frac{3}{2}}\sqrt{[2]_q} (k_-\mathcal{K}_{12}^{(1)}(u) - k_+\mathcal{K}_{21}^{(1)}(u)), \\ [W_0, \mathcal{K}_{12}^{(1)}(u)]_q &= \sqrt{[2]_q}q^{-\frac{1}{2}} (u^{-1}q^{-1}k_-\mathcal{K}_{13}^{(1)}(u) + k_+(u\mathcal{K}_{11}^{(1)}(u) - u^{-1}\mathcal{K}_{22}^{(1)}(u))), \\ [W_0, \mathcal{K}_{13}^{(1)}(u)]_{q^2} &= k_+\sqrt{[2]_q} (uq^{-\frac{1}{2}}\mathcal{K}_{12}^{(1)}(u) - u^{-1}q^{\frac{1}{2}}\mathcal{K}_{23}^{(1)}(u)), \\ [W_0, \mathcal{K}_{21}^{(1)}(u)]_{q^{-1}} &= \sqrt{[2]_q}q^{-\frac{3}{2}}u^{-1} (k_-q(\mathcal{K}_{22}^{(1)}(u) - u^2\mathcal{K}_{31}^{(1)}(u)) - k_+\mathcal{K}_{31}^{(1)}(u)), \\ [W_0, \mathcal{K}_{22}^{(1)}(u)] &= \sqrt{[2]_q}q^{-\frac{1}{2}} (k_+(uq\mathcal{K}_{21}^{(1)}(u) - u^{-1}\mathcal{K}_{32}^{(1)}(u)) + k_-(u^{-1}\mathcal{K}_{23}^{(1)}(u) - uq\mathcal{K}_{12}^{(1)}(u))), \\ [W_0, \mathcal{K}_{23}^{(1)}(u)]_q &= q^{\frac{1}{2}}\sqrt{[2]_q} (k_+(u\mathcal{K}_{22}^{(1)}(u) - u^{-1}\mathcal{K}_{33}^{(1)}(u)) - k_-uq\mathcal{K}_{13}^{(1)}(u)), \\ [W_0, \mathcal{K}_{31}^{(1)}(u)]_{q^{-2}} &= k_-\sqrt{[2]_q} (u^{-1}q^{\frac{1}{2}}\mathcal{K}_{32}^{(1)}(u) - uq^{-\frac{1}{2}}\mathcal{K}_{21}^{(1)}(u)), \\ [W_0, \mathcal{K}_{32}^{(1)}(u)]_{q^{-1}} &= q^{\frac{1}{2}}\sqrt{[2]_q} (k_+uq\mathcal{K}_{31}^{(1)}(u) - k_-(u\mathcal{K}_{22}^{(1)}(u) - u^{-1}\mathcal{K}_{33}^{(1)}(u))), \\ [W_0, \mathcal{K}_{33}^{(1)}(u)] &= uq^{\frac{3}{2}}\sqrt{[2]_q} (k_+\mathcal{K}_{32}^{(1)}(u) - k_-\mathcal{K}_{23}^{(1)}(u)). \end{aligned} $$

By substituting $W_0 \mapsto W_1, u \mapsto u^{-1}, q \mapsto q^{-1}$ into the above equations, we obtain the system of equations associated with $b = W_1$. Inserting the K-operator's ansatz (7.6) into these equations and using (7.2) we extract a system of linearly independent equations for the entries of $\mathcal{K}^{(1)}(u)$. Similarly to the case $j = \frac{1}{2}$, one finds that it determines uniquely $\alpha = \frac{1}{2}, \beta = -\frac{1}{2}$ and the matrices $A_{i,j}^{(1)}(u)$
and $A_0^{(1)}(u)$ are given by

$$A_{1,1}^{(1)}(u) = \frac{c(u^2)(q+q^{-1})^2}{c(q)c(u^2q)} \begin{pmatrix} 0 & uq^{\frac{3}{2}}k_{+} \frac{(u^2q^2+u^{-2}q^{-2})}{\sqrt{q+q^{-1}}} & 0 \\ -u^{-1}q^{-\frac{3}{2}}k_{-}\sqrt{q+q^{-1}} & 0 & -uq^{\frac{3}{2}}k_{+}\sqrt{q+q^{-1}} \\ 0 & u^{-1}q^{-\frac{3}{2}}k_{-} \frac{(u^2q^2+u^{-2}q^{-2})}{\sqrt{q+q^{-1}}} & 0 \end{pmatrix},$$

$$A_{1,3}^{(1)}(u) = \frac{1}{c(q)} \begin{pmatrix} q & 0 & \frac{k_{+}c(u^2)}{k_{-}} \\ 0 & \frac{u^{-4}q^{-2}+c(q^2)-1}{c(u^2q)} & 0 \\ 0 & 0 & q \end{pmatrix}, \quad A_{2,8}^{(1)}(u) = \begin{pmatrix} 0 & 0 & \frac{k_{+}c(u^2)}{k_{-}\rho} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix},$$

$$A_{2,2}^{(1)}(u) = \frac{uq^{\frac{3}{2}}c(u^2)}{k_{-}c(u^2q)} \begin{pmatrix} 0 & \frac{(u^2q^2+u^{-2}q^{-2})}{\sqrt{q+q^{-1}}} & 0 \\ 0 & 0 & -\sqrt{q+q^{-1}} \\ 0 & 0 & 0 \end{pmatrix}, \quad A_{2,3}^{(1)}(u) = \frac{q^{\frac{1}{2}}c(u^2)}{u k_{-}c(u^2q)} \begin{pmatrix} 0 & -\sqrt{q+q^{-1}} & 0 \\ 0 & 0 & \frac{(u^2q^2+u^{-2}q^{-2})}{\sqrt{q+q^{-1}}} \\ 0 & 0 & 0 \end{pmatrix},$$

$$A_{2,11}^{(1)}(u) = \frac{q^{\frac{1}{2}}c(u^2)}{u k_{-}c(u^2q)} \begin{pmatrix} 0 & \frac{(u^2q^2+u^{-2}q^{-2})}{\sqrt{q+q^{-1}}} & 0 \\ 0 & 0 & -\sqrt{q+q^{-1}} \\ 0 & 0 & 0 \end{pmatrix}, \quad A_{2,12}^{(1)}(u) = \frac{uq^{\frac{3}{2}}c(u^2)}{k_{-}c(u^2q)} \begin{pmatrix} 0 & -\sqrt{q+q^{-1}} & 0 \\ 0 & 0 & \frac{(u^2q^2+u^{-2}q^{-2})}{\sqrt{q+q^{-1}}} \\ 0 & 0 & 0 \end{pmatrix},$$

$$A_0^{(1)}(u) = \frac{\rho}{c(q)} \begin{pmatrix} 1 & 0 & \frac{k_+ c(u^2)}{k_- c(q)} \\ 0 & \frac{c(u^2 q)}{c(q)} & 0 \\ \frac{k_- c(u^2)}{k_+ c(q)} & 0 & 1 \end{pmatrix},$$

$$A_{2,16}^{(1)}(u) = \text{Diag}(q^{-1} - u^{-4}q^{-3}, -u^2q + u^{-2}(q + q^{-1} - q^3), u^4q^3 - q^3 + q^{-1} - q^{-3}),$$

$$A_{2,6}^{(1)}(u) = \text{Diag}(-c(u^2q)q, -u^{-4}q^{-2} - q^2 + 1 + q^{-2}, -c(u^2q)q),$$

$$A_{1,2}^{(1)}(u) = A_{1,1}^{(1)t}(u)|_{k_{\pm} \to k_{\mp}q^{\mp 2}}, \quad A_{1,7}^{(1)}(u) = A_{1,1}^{(1)t}(u)|_{k_{\pm} \to k_{\mp}u^{\mp 2}q^{\mp 3}}, \quad A_{1,8}^{(1)}(u) = A_{1,1}^{(1)}(u)|_{k_{\pm} \to k_{\pm}u^{\mp 2}q^{\mp 1}},$$

$$A_{1,4}^{(1)}(u) = A_{1,3}^{(1)}(u^{-1})|_{q \to q^{-1}}, \quad A_{1,5}^{(1)}(u) = A_{1,3}^{(1)t}(u^{-1})|_{k_{\pm} \to k_{\mp}, q \to q^{-1}}, \quad A_{1,6}^{(1)}(u) = A_{1,3}^{(1)t}(u)|_{k_{+} \to k_{-}},$$

$$A_{2,1}^{(1)}(u) = A_{2,16}^{(1)}(u^{-1})|_{q \to q^{-1}}, \quad A_{2,13}^{(1)}(u) = A_{2,8}^{(1)t}(u)|_{k_{+} \to k_{-}}, \quad A_{2,4}^{(1)}(u) = A_{2,3}^{(1)t}(u)|_{k_{-} \to k_{+}q^2},$$

$$A_{2,5}^{(1)}(u) = A_{2,2}^{(1)t}(u)|_{k_{-} \to k_{+}q^2}, \quad A_{2,7}^{(1)}(u) = -A_{2,6}^{(1)}(u^{-1})|_{q \to q^{-1}}, \quad A_{2,9}^{(1)}(u) = -(\rho c(u^2q))^{-1} A_{2,6}^{(1)}(u),$$

$$A_{2,10}^{(1)}(u) = -(\rho c(u^2q))^{-1} A_{2,6}^{(1)}(u^{-1})|_{q \to q^{-1}}, \quad A_{2,14}^{(1)}(u) = A_{2,12}^{(1)}(u)|_{k_{\pm} \to k_{\mp}q^2}, \quad A_{2,15}^{(1)}(u) = A_{2,11}^{(1)}(u)|_{k_{\pm} \to k_{\mp}q^2}.$$

As expected, one checks that the K-operator in (7.6) together with the above solution matches with the expression for the fused K-operator (5.37) after applying the ordering relations given in Appendix C.

To summarize, K-operators of the form (7.4), (7.6) for $j = \frac{1}{2}$, 1, respectively, are uniquely determined (up to an overall factor) by the twisted intertwining relations (5.55). Importantly, it is sufficient to consider the relations for $b = W_0, W_1$. Furthermore, the corresponding expressions match with the fused ones derived in Section 5. Based on these evidences, we conjecture that spin-$j$ fused K-operator solutions of (5.55) are unique (up to an overall scalar factor), their matrix entries
are linear combinations of monomials of the form (7.3) and match with the fused expression given by (5.29).

## 8. SUMMARY AND OUTLOOK

To briefly summarize our main results, we provided a new set of K-operator solutions to the spectral parameter dependent reflection equation (1.8) in terms of generating functions of the centrally extended $q$-Onsager algebra $\mathcal{A}_q$. The central formula of this work is the recursion (5.29) for the fused K-operators of arbitrary spin $j \in \frac{1}{2}\mathbb{N}$ as well as Theorem 5.7 on the reflection equation they satisfy. We also gave formulas for the fused R-matrices and the fused K-operators in (5.43), (5.44), whose expressions contain only the fundamental R-matrix and K-operator. These results were established within a general framework of universal K-matrices that we developed in Section 2.3, extending the previously known approaches (discussed in Introduction). In particular, the central formula (5.29) is based on the results in Proposition 4.16 and in Remark 4.19. We also provided in Section 5.4 a few explicit examples of the fused K-operators (for spins $j = 1$ and $j = \frac{3}{2}$) in terms of generating functions of $\mathcal{A}_q$.

As the existence of a universal K-matrix (for our choice of algebras $H = \mathcal{L}U_qsl_2$ and $B = \mathcal{A}_q$ and the compatible twists) is still an open fundamental question, we have investigated whether the central formula (5.29) satisfies the (evaluated version of) universal K-matrix axioms (K1)-(K3) which is resulted in Conjecture 1. One of the key problems here is to understand better the central element $\nu(u) \in \mathcal{A}_q[[u^{-1}]]$, in particular to derive its coaction $\delta(\nu(u))$ so that it reproduces the evaluated coaction in (6.4).

It is also important to mention a few possible applications of our results in integrable models. In the literature on quantum integrable systems, K-operators and their images in the tensor product (or spin-chain) representations of the algebra $\mathcal{A}_q$ – known as Sklyanin's operators – are the basic building elements for the construction of mutually commuting quantities, for instance the Hamiltonian of open spin chains with integrable boundary conditions [Sk88]. For the quotient of $\mathcal{A}_q$ known as the $q$-Onsager algebra, the fundamental K-operator (5.4) is the essential ingredient in the open XXZ spin-$\frac{1}{2}$ chain with generic boundary conditions [BK05b]. For the generic diagonal boundary conditions in this spin chain, the fundamental K-operator (5.4) generates another quotient$^{18}$ of $\mathcal{A}_q$ known as the augmented $q$-Onsager algebra [BB12, Sec. 2]. Furthermore, for the quotient of $\mathcal{A}_q$ known as the Askey-Wilson algebra, the K-operator (5.4) leads to an integrable model with three-spin interaction terms [BP19]. For other cases, the corresponding quotients of $\mathcal{A}_q$ appearing in the finite or half-infinite spin chains are identified in [BB16, Sec. 3].

The K-operators play a crucial role in the spectrum analysis of the corresponding quantum spin chains, for instance, in the diagonalization of the Hamiltonian of the half-infinite XXZ spin-chain with diagonal boundary conditions [JKKKM94] (see also [BB12] in Onsager's approach) and with triangular ones [BB12], or describe hidden non-abelian symmetries [BB16] of the model. The K-operators are also used to construct the Baxter's $Q$-operator for diagonal boundary conditions [BT17, VW20] and triangular boundary conditions [Ts19, Ts20]. For all these models, the transfer matrix is the image in the spin-chain representation of a generating function $T^{(\frac{1}{2})}(u)$ built from the K-operator (5.4) and a dual solution of the reflection equation for a spin-$\frac{1}{2}$ auxiliary space. Importantly,

<footnote>$^{18}$More precisely, it is a degenerate specialization at $\rho \to 0$ of the $q$-Onsager quotient.</footnote>
$\mathcal{T}^{(\frac{1}{2})}(u)$ reads as a linear combination of some fundamental generators $\{\mathcal{I}_{2k+1}|k \in \mathbb{N}\}$ of a maximal commutative subalgebra of $\mathcal{A}_q$. Therefore, in this approach the diagonalization of the transfer matrix reduces to the diagonalization of the image of the commutative subalgebra.

The fused K-operators of spin-$j$ constructed in this paper open a route to the representation-independent analysis of related integrable models beyond the case of the fundamental spin-$\frac{1}{2}$ auxiliary space, for instance, of the open XXZ spin-$j$ chain with generic integrable boundary conditions. And the following problems can be addressed here:

* Firstly, it is natural to ask about relations between any local$^{19}$ or non-local mutually commut- ing quantities of quantum spin chains and the generators $\{\mathcal{I}_{2k+1}|k \in \mathbb{N}\}$ of the commutative subalgebra in $\mathcal{A}_q$. For instance, in the spin-$\frac{1}{2}$ case the differentiation of the transfer ma- trix leads to the expression of the Hamiltonian in terms of the operators $\mathcal{I}_{2k+1}$'s [BK05b, eq. (39)]. We thus also expect that the transfer matrix for the models based on the auxiliary space of arbitrary spin-$j$ admits a unified formulation as the image of a generating function $\mathcal{T}^{(j)}(u)$ in the commutative subalgebra of $\mathcal{A}_q[[u^{-1}]]$. In a forthcoming paper [LBG24], the structure of the universal transfer matrices $\mathcal{T}^{(j)}(u)$ for higher spin-$j$ auxiliary space repre- sentation will be studied in details. In particular, the so called TT-relations – a recursion on $\mathcal{T}^{(j)}(u)$ given in (1.14) – are derived at the algebraic level, independently of a representation chosen. Generalizing the spin-$\frac{1}{2}$ case, it will be shown that $\mathcal{T}^{(j)}(u)$ is a power series in $u^{-1}$ with coefficients being polynomials of degree $2j$ in the generators $\{\mathcal{I}_{2k+1}|k \in \mathbb{N}\}$, and that the universal $\mathcal{T}^{(j)}(u)$'s indeed agree with the ‘physical’ transfer matrices on the spin chains.

* Secondly, given those transfer matrices generated from various images of $\mathcal{A}_q$, the problem of characterizing their spectral properties – leading to the eigenstates and eigenvalues of the Hamiltonian – is consequently reduced to the diagonalization of the images of $\{\mathcal{I}_{2k+1}|k \in \mathbb{N}\}$. For the simplest example of the quotient of $\mathcal{A}_q$ known as the Askey-Wilson algebra, for irreducible finite-dimensional representations the problem is solved in [BP19], combining the theory of Leonard pairs and the so-called modified algebraic Bethe ansatz [BC13, B14, BP14]. A similar analysis for $\mathcal{A}_q$ remains to be done.

* Let us mention that the construction of a universal $Q$-operator for $\mathcal{A}_q$ and corresponding TQ-relations may be also addressed. In that case, it would be desirable to construct the analogue of the fused K-operator for $j \to \infty$ as suggested in [YNZ05], see also [VW20, Ts20].

* Finally, it is very desirable to construct K-operators of arbitrary complex spins, i.e., asso- ciated to $U_q s l_2$ Verma modules of complex weights, as it would give essentially the corre- sponding universal K-matrix. Such integrable quantum spin-chains with integrable bound- ary conditions based on the Verma modules were recently introduced [CGS22], and it was shown [CGJS22] a deep connection to the $q$-Onsager algebra via the common XXZ spin chain spectrum with the integrable non-diagonal boundary conditions.

**Acknowledgments:** We thank A. Appel, P. Terwilliger, and B. Vlaar for stimulating discussions. P.B. and A.M.G. are supported by C.N.R.S. The work of A.M.G. was also partially supported by the ANR grant JCJC ANR-18-CE40-0001.

<footnote>$^{19}$In the case of integrable spin chains, an operator is said to be local whenever it is a product of a finite number of spin matrices $S_\pm, S_3$, or a sum of such products.</footnote>
APPENDIX A. QUANTUM ALGEBRAS

We recall the definitions of the quantum algebra $U_qsl_2$ and the quantum affine algebra $U_q\hat{sl}_2$.

**Definition A.1.** The algebra $U_qsl_2$ is a Hopf algebra generated by the elements $E, F, K^{\pm\frac{1}{2}}$ satisfying:

$$ (A.1) \qquad K^{\frac{1}{2}}E = qEK^{\frac{1}{2}}, \quad K^{\frac{1}{2}}F = q^{-1}FK^{\frac{1}{2}}, \quad [E, F] = \frac{K-K^{-1}}{q-q^{-1}}, \quad K^{\frac{1}{2}}K^{-\frac{1}{2}} = K^{-\frac{1}{2}}K^{\frac{1}{2}} = 1. $$

Let $\Delta: U_qsl_2 \to U_qsl_2 \otimes U_qsl_2$, $\epsilon: U_qsl_2 \to \mathbb{C}$ and $S: U_qsl_2 \to U_qsl_2$ be respectively the coproduct, the counit and the antipode. They are given by:

$$ (A.2) \qquad \Delta(E) = E \otimes K^{-\frac{1}{2}} + K^{\frac{1}{2}} \otimes E, \quad \Delta(F) = F \otimes K^{-\frac{1}{2}} + K^{\frac{1}{2}} \otimes F, \quad \Delta(K^{\pm\frac{1}{2}}) = K^{\pm\frac{1}{2}} \otimes K^{\pm\frac{1}{2}}, $$

$$ (A.3) \qquad \epsilon(E) = \epsilon(F) = 0, \quad \epsilon(K^{\pm\frac{1}{2}}) = 1, $$

$$ (A.4) \qquad S(E) = -q^{-1}E, \quad S(F) = -qF, \quad S(K^{\pm\frac{1}{2}}) = K^{\mp\frac{1}{2}}. $$

We also recall that the Casimir central element of $U_qsl_2$ is given by

$$ (A.5) \qquad \begin{aligned} C &= (q - q^{-1})^2 FE + qK + q^{-1}K^{-1} \\ &= (q - q^{-1})^2 EF + q^{-1}K + qK^{-1}. \end{aligned} $$

Note that the monomials $\{E^r K^{\pm\frac{s}{2}} F^t \mid r, s, t \in \mathbb{N}\}$ provide a PBW basis for $U_qsl_2$, see for instance [KS12, Chap. 3].

**Definition A.2.** Define the extended Cartan matrix $a_{ij}$ with $a_{ii} = 2$, $a_{ij} = -2$ for $i \neq j$. The quantum affine algebra $U_q\hat{sl}_2$ is a Hopf algebra generated by the elements $E_i, F_i, K_i^{\pm\frac{1}{2}}, i \in \{0, 1\}$ satisfying:

$$ (A.6) \qquad K_i^{\frac{1}{2}}E_j = q^{\frac{a_{ij}}{2}}E_jK_i^{\frac{1}{2}}, \quad K_i^{\frac{1}{2}}F_j = q^{-\frac{a_{ij}}{2}}F_jK_i^{\frac{1}{2}}, \quad [E_i, F_j] = \delta_{i,j} \frac{K_i - K_i^{-1}}{q - q^{-1}}, $$

$$ (A.7) \qquad K_i^{\frac{1}{2}}K_i^{-\frac{1}{2}} = K_i^{-\frac{1}{2}}K_i^{\frac{1}{2}} = 1, \quad K_0^{\frac{1}{2}}K_1^{\frac{1}{2}} = K_1^{\frac{1}{2}}K_0^{\frac{1}{2}}, $$

with the $q$-Serre relations:

$$ (A.8) \qquad [E_i, [E_i, [E_i, E_j]]_{q^{-1}}] = 0, \quad [F_i, [F_i, [F_i, F_j]]_{q^{-1}}] = 0. $$

Let $\Delta: U_q\hat{sl}_2 \to U_q\hat{sl}_2 \otimes U_q\hat{sl}_2$, $\epsilon: U_q\hat{sl}_2 \to \mathbb{C}$ and $S: U_q\hat{sl}_2 \to U_q\hat{sl}_2$ be respectively the coproduct, the counit and the antipode. They are given by:

$$ (A.9) \qquad \Delta(E_i) = E_i \otimes K_i^{\frac{1}{2}} + K_i^{-\frac{1}{2}} \otimes E_i, \quad \Delta(F_i) = F_i \otimes K_i^{\frac{1}{2}} + K_i^{-\frac{1}{2}} \otimes F_i, \quad \Delta(K_i^{\pm\frac{1}{2}}) = K_i^{\pm\frac{1}{2}} \otimes K_i^{\pm\frac{1}{2}}, $$

$$ (A.10) \qquad \epsilon(E_i) = \epsilon(F_i) = 0, \quad \epsilon(K_i^{\pm\frac{1}{2}}) = 1, $$

$$ (A.11) \qquad S(E_i) = -qE_i, \quad S(F_i) = -q^{-1}F_i, \quad S(K_i^{\pm\frac{1}{2}}) = K_i^{\mp\frac{1}{2}}. $$

The element $K_0^{\frac{1}{2}}K_1^{\frac{1}{2}}$ is central. The algebra $U_q\hat{sl}_2$ has an automorphism $\nu$ defined by

$$ (A.12) \qquad \nu(E_i) = E_i K_i^{\frac{1}{2}}, \quad \nu(F_i) = K_i^{-\frac{1}{2}} F_i, \quad \nu(K_i^{\pm\frac{1}{2}}) = K_i^{\mp\frac{1}{2}}. $$
## APPENDIX B. THE UNIVERSAL R-MATRIX

In this appendix, we compute evaluations of the universal R-matrix. Firstly, we recall the construction of the universal R-matrix of Khoroshkin-Tolstoy [KT92a] for the Hopf algebra $H = \mathcal{L}U_q sl_2$ in terms of root vectors. Secondly, evaluations of the universal R-matrix are considered. In particular, we give expressions of the Ding-Frenkel L-operators $\mathbf{L}^+(u)$ and $[\mathbf{L}^-(u^{-1})]^{-1}$, as defined in (4.74). Finally, the spin-$\frac{1}{2}$ L-operator $\mathbf{L}^{(\frac{1}{2})}(u)$ is computed by evaluating $\mathbf{L}^+(u)$.

B.1. **Root vectors.** Let us first recall the definition of the root vectors of $\mathcal{L}U_q sl_2$. We adapt the construction in [BGKNR12] to our choice of coproduct, recall the relation (2.33). Let us set

$$ (B.1) \qquad e_{\alpha} = E_1 K_1^{-\frac{1}{2}}, \quad e_{\delta-\alpha} = E_0 K_0^{-\frac{1}{2}}, \quad f_{\alpha} = K_1^{\frac{1}{2}} F_1, \quad f_{\delta-\alpha} = K_0^{\frac{1}{2}} F_0. $$

The other root vectors are defined by the recursion relations:

$$ \begin{align} e'_{k\delta} &= q^{-1}[e_{\alpha+(k-1)\delta}, e_{\delta-\alpha}]_q, \notag \\ e_{\alpha+k\delta} &= [2]_q^{-1}[e_{\alpha+(k-1)\delta}, e'_{\delta}], \notag \\ e_{\delta-\alpha+k\delta} &= [2]_q^{-1}[e'_{\delta}, e_{\delta-\alpha+(k-1)\delta}], \notag \\ f'_{k\delta} &= q[f_{\delta-\alpha}, f_{\alpha+(k-1)\delta}]_{q^{-1}}, \notag \\ f_{\alpha+k\delta} &= [2]_q^{-1}[f'_{\delta}, f_{\alpha+(k-1)\delta}], \notag \\ f_{\delta-\alpha+k\delta} &= [2]_q^{-1}[f_{\delta-\alpha+(k-1)\delta}, f'_{\delta}], \quad k \in \mathbb{N}_+. \end{align} $$

The root vectors $e_{k\delta}$, $f_{k\delta}$ are defined via the generating functions

$$ \begin{align*} (q - q^{-1}) \sum_{k=1}^{\infty} e_{k\delta} z^{-k} &= \log \left( 1 + (q - q^{-1}) \sum_{k=1}^{\infty} e'_{k\delta} z^{-k} \right), \\ -(q - q^{-1}) \sum_{k=1}^{\infty} f_{k\delta} z^{-k} &= \log \left( 1 - (q - q^{-1}) \sum_{k=1}^{\infty} f'_{k\delta} z^{-k} \right). \end{align*} $$

B.2. **Khoroshkin-Tolstoy construction.** Let $\{\alpha + k\delta\}_{k=0}^{\infty} \cup \{k\delta\}_{k=0}^{\infty} \cup \{\delta - \alpha + k\delta\}_{k=0}^{\infty}$ be the positive root system of $\widehat{sl}_2$. We choose the root ordering as

(B.3) $\alpha, \alpha + \delta, \dots, \alpha + k\delta, \dots, \delta, 2\delta, \dots, \ell\delta, \dots, \dots, (\delta - \alpha) + m\delta, \dots, (\delta - \alpha) + \delta, \delta - \alpha,$ for any $k, \ell, m \in \mathbb{N}$. Then, the universal R-matrix obtained by Khoroshkin and Tolstoy takes the following factorized form

$$ (B.4) \qquad \mathcal{R} = \mathcal{R}^{+} \mathcal{R}^{0} \mathcal{R}^{-} q^{\frac{1}{2}h_1 \otimes h_1}, $$

where$^{20}$

$$ (B.5) \qquad \mathcal{R}^{+} = \prod_{k=0}^{\infty} \exp_{q^{-2}} \left( (q - q^{-1}) e_{\alpha + k\delta} \otimes f_{\alpha + k\delta} \right), $$

<footnote>$^{20}$Here we use the notation $\prod_{k=0}^{\infty} a(k) = a(n)a(n-1)\dots a(0)$ and $\prod_{k=0}^{\infty} a(k) = a(0)a(1)\dots a(n)$, for any function $a(n)$.</footnote>
$$
(B.6) \qquad \mathcal{R}^0 = \exp \left( (q - q^{-1}) \sum_{k=1}^{\infty} \frac{k}{[2k]_q} e_{k\delta} \otimes f_{k\delta} \right),
$$

$$
(B.7) \qquad \mathcal{R}^{-} = \prod_{k=0}^{\infty} \exp_{q^{-2}} ((q - q^{-1}) e_{\delta - \alpha + k\delta} \otimes f_{\delta - \alpha + k\delta}),
$$

with $q^{h_1} \equiv K_1$ and the $q$-exponential is

$$
(B.8) \qquad \exp_q(x) = 1 + \sum_{k=1}^{\infty} \frac{x^k}{(k)_q!}, \quad (k)_q! = (1)_q (2)_q \cdots (k)_q, \quad (k)_q = \frac{q^k - 1}{q - 1}.
$$

We also notice that $[e_{k\delta}, e_{\ell\delta}] = 0 = [f_{k\delta}, f_{\ell\delta}]$ for any $k, \ell$, and so the exponent in (B.6) can be also written in the form of semi-infinite product of exponents involving only one term $\sim e_{k\delta} \otimes f_{k\delta}$.

B.3. Evaluation of the universal R-matrix. In the previous subsection, the explicit form of the universal R-matrix was recalled. It is expressed as a product of q-exponentials with root vectors in the arguments. Now, we evaluate the second tensor product component of the universal R-matrix by taking its image under the formal fundamental evaluation representation as introduced in Section 3.5.

B.3.1. *Evaluation of the root vectors.* The action of the evaluation map defined in (2.40) on the first root vectors gives

$$
(B.9) \qquad \mathrm{ev}_u(e_{\delta-\alpha}) = u^{-1}FK^{\frac{1}{2}}, \qquad \mathrm{ev}_u(e_\alpha) = u^{-1}EK^{-\frac{1}{2}}, \qquad \mathrm{ev}_u(K_0^{\frac{1}{2}}) = K^{-\frac{1}{2}},
$$

$$
(B.10) \qquad \mathrm{ev}_u(f_{\delta-\alpha}) = uq^{-1}EK^{-\frac{1}{2}}, \quad \mathrm{ev}_u(f_\alpha) = uq^{-1}FK^{\frac{1}{2}}, \quad \mathrm{ev}_u(K_1^{\frac{1}{2}}) = K^{\frac{1}{2}}.
$$

The image of the other root vectors of $\mathcal{L}U_q s l_2$ in (B.2) under the evaluation map are obtained by induction similarly to [BGKNR12, Sect. 4.4]. They are given for $k \in \mathbb{N}$ by:

$$
\begin{align*}
\mathrm{ev}_u(e_{\alpha+k\delta}) &= (-1)^k u^{-2k-1} q^{-k} EK^{-k-\frac{1}{2}}, \\
\mathrm{ev}_u(e_{\delta-\alpha+k\delta}) &= (-1)^k u^{-2k-1} q^k FK^{-k+\frac{1}{2}}, \\
\mathrm{ev}_u(f_{\alpha+k\delta}) &= (-1)^k u^{2k+1} q^{-k-1} FK^{k+\frac{1}{2}}, \\
\mathrm{ev}_u(f_{\delta-\alpha+k\delta}) &= (-1)^k u^{2k+1} q^{k-1} EK^{k-\frac{1}{2}},
\end{align*}
$$

and for $k \in \mathbb{N}_+$

$$
\begin{align*}
\mathrm{ev}_u(e'_{k\delta}) &= (-1)^{k-1} u^{-2k} [E, F]_{q^k} K^{-k+1}, \\
\mathrm{ev}_u(e_{k\delta}) &= \frac{(-1)^{k-1} u^{-2k}}{(q-q^{-1})k} (C_k - (q^k + q^{-k})K^{-k}), \\
\mathrm{ev}_u(f'_{k\delta}) &= (-1)^{k-1} u^{2k} [E, F]_{q^{-k}} K^{k-1}, \\
\mathrm{ev}_u(f_{k\delta}) &= \frac{(-1)^k u^{2k}}{(q-q^{-1})k} (C_k - (q^k + q^{-k})K^k),
\end{align*}
$$

where the elements $C_k$ are defined by the generating function

$$
(B.13) \qquad \sum_{k=1}^{\infty} (-1)^{k-1} C_k \frac{z^{-k}}{k} = \log(1 + Cz^{-1} + z^{-2}), \quad z \in \mathbb{C},
$$
and where $C$ is the central element of $U_qsl_2$ given in (A.5). For instance by expanding (B.13) we get the first elements of $C_k$

$$ (B.14) \qquad C_1 = C, \quad C_2 = C^2 - 2, \quad C_3 = C^3 - 3C, \quad C_4 = C^4 - 4C^2 + 2. $$

Recall $E_{ab}$ is the matrix with zero everywhere except 1 in the entry $(a, b)$. The matrix multiplication obeys

$$ (B.15) \qquad E_{ab}E_{cd} = \delta_{b,c}E_{ad}. $$

In this notation, the spin-$\frac{1}{2}$ finite-dimensional representation of $U_qsl_2$ reads

$$ (B.16) \qquad \pi^{\frac{1}{2}}(K^m) = q^m E_{11} + q^{-m} E_{22}, \qquad \pi^{\frac{1}{2}}(E) = E_{12}, \qquad \pi^{\frac{1}{2}}(F) = E_{21}, $$

and the central elements $C_k$ become, for all $k \ge 1$,

$$ (B.17) \qquad \pi^{\frac{1}{2}}(C_k) = (q^{2k} + q^{-2k})\mathbb{I}_2. $$

In order to obtain Ding-Frenkel L-operators and R-matrices from the universal R-matrix, one also needs the image of the root vectors under the representation map $\pi_u^{\frac{1}{2}}: \mathcal{L}U_qsl_2 \to \text{End}(\mathbb{C}^2)$, which is given by:

$$ \begin{align} \pi_u^{\frac{1}{2}}(e_{\alpha+k\delta}) &= (-1)^k u^{-2k-1} q^{\frac{1}{2}} E_{12}, \notag \\ \pi_u^{\frac{1}{2}}(e_{\delta-\alpha+k\delta}) &= (-1)^k u^{-2k-1} q^{\frac{1}{2}} E_{21}, \notag \\ \pi_u^{\frac{1}{2}}(f_{\alpha+k\delta}) &= (-1)^k u^{2k+1} q^{-\frac{1}{2}} E_{21}, \notag \\ \pi_u^{\frac{1}{2}}(f_{\delta-\alpha+k\delta}) &= (-1)^k u^{2k+1} q^{-\frac{1}{2}} E_{12}, \tag{B.18} \\ \pi_u^{\frac{1}{2}}(e'_{k\delta}) &= (-1)^{k-1} u^{-2k} (qE_{11} - q^{-1}E_{22}), \notag \\ \pi_u^{\frac{1}{2}}(e_{k\delta}) &= (-1)^{k-1} u^{-2k} \frac{[k]_q}{k} (q^k E_{11} - q^{-k} E_{22}), \notag \\ \pi_u^{\frac{1}{2}}(f'_{k\delta}) &= (-1)^{k-1} u^{2k} (q^{-1}E_{11} - qE_{22}), \notag \\ \pi_u^{\frac{1}{2}}(f_{k\delta}) &= (-1)^{k-1} u^{2k} \frac{[k]_q}{k} (q^{-k}E_{11} - q^k E_{22}). \notag \end{align} $$

Recall that $\mathbf{L}^{\pm}(u)$ are defined in (4.74). We now compute explicitly $\mathbf{L}^{+}(u)$ and $[\mathbf{L}^{-}(u^{-1})]^{-1}$.

### B.3.2. $\mathbf{L}^{+}(u)$. Recall the factorized form of the universal R-matrix (B.4). We now compute the image of $\mathfrak{R}^{\pm}$, $\mathfrak{R}^0$, $q^{\frac{1}{2}h_1 \otimes h_1}$ under the action of $(\text{id} \otimes \pi_{u^{-1}}^{\frac{1}{2}})$. First, from (B.5) and with (B.18) we get:

$$ \begin{align} (\text{id} \otimes \pi_{u^{-1}}^{\frac{1}{2}})(\mathfrak{R}^{+}) &= \prod_{k=0}^{\infty} \exp_{q^{-2}} \left( (-u^{-2})^k u^{-1} (q-q^{-1}) q^{-\frac{1}{2}} e_{\alpha+k\delta} \otimes E_{21} \right) \tag{B.19} \\ &= 1 \otimes \mathbb{I}_2 + e^+(u) \otimes E_{21}, \notag \end{align} $$

where

$$ (B.20) \qquad e^+(u) = (q - q^{-1}) q^{-\frac{1}{2}} u^{-1} \left( \sum_{k=0}^{\infty} (-u^{-2})^k e_{\alpha+k\delta} \right). $$
Similarly, from (B.7) and with (B.18) we have:

$$ (\mathrm{id} \otimes \pi_{u-1}^{\frac{1}{2}})(\mathfrak{R}^{-}) = \prod_{k=0}^{\infty} \exp_{q-2} \left( (-u^{-2})^k u^{-1} (q-q^{-1}) q^{-\frac{1}{2}} e_{\delta-\alpha+k\delta} \otimes E_{12} \right) \\ (B.21) \qquad = 1 \otimes \mathbb{I}_2 + f^+(u) \otimes E_{12}, $$

where

$$ (B.22) \qquad f^+(u) = (q - q^{-1}) q^{-\frac{1}{2}} u^{-1} \left( \sum_{k=0}^{\infty} (-u^{-2})^k e_{\delta-\alpha+k\delta} \right). $$

A straightforward calculation from (B.6) and using (B.18) yields

$$ (B.23) \qquad (\mathrm{id} \otimes \pi_{u-1}^{\frac{1}{2}})(\mathfrak{R}^{0}) = k^{+}(u) \otimes E_{11} + \tilde{k}^{+}(u) \otimes E_{22}, $$

where

$$ (B.24) \qquad k^{+}(u) = \exp \left( -(q-q^{-1}) \sum_{k=1}^{\infty} \frac{(-u^{-2}q^{-1})^k}{q^k + q^{-k}} e_{k\delta} \right), \quad \tilde{k}^{+}(u) = \exp \left( (q-q^{-1}) \sum_{k=1}^{\infty} \frac{(-u^{-2}q)^k}{q^k + q^{-k}} e_{k\delta} \right). $$

Then, using $\pi_u^{\frac{1}{2}}(h_1) = E_{11} - E_{22}$, we get

$$ (B.25) \qquad (\mathrm{id} \otimes \pi_{u-1}^{\frac{1}{2}})(q^{\frac{1}{2}h_1 \otimes h_1}) = K_1^{\frac{1}{2}} \otimes E_{11} + K_1^{-\frac{1}{2}} \otimes E_{22}. $$

Finally, combining (B.19)-(B.25), we get

$$ (B.26) \qquad \mathbf{L}^{+}(u) = \begin{pmatrix} k^{+}(u) K_{1}^{\frac{1}{2}} & k^{+}(u) f^{+}(u) K_{1}^{-\frac{1}{2}} \\ e^{+}(u) k^{+}(u) K_{1}^{\frac{1}{2}} & \tilde{k}^{+}(u) K_{1}^{-\frac{1}{2}} + e^{+}(u) k^{+}(u) f^{+}(u) K_{1}^{-\frac{1}{2}} \end{pmatrix}. $$

Note that from the definition of $e^{+}(u)$, $f^{+}(u)$, $k^{+}(u)$, $\tilde{k}^{+}(u)$ in (B.20), (B.22), (B.24) it is easy to see that $\mathbf{L}^{+}(u)$ is a formal power series in $u^{-1}$, i.e. $\mathbf{L}^{+}(u)$ is in $\mathcal{L}U_q s l_2[[u^{-1}]] \otimes \mathrm{End}(\mathbb{C}^2)$.

B.3.3. $[\mathbf{L}^{-}(u^{-1})]^{-1}$. Consider $\mathfrak{p} \circ \mathfrak{R}^{\pm} = \mathfrak{R}_{21}^{\pm}$, $\mathfrak{p} \circ \mathfrak{R}^{0} = \mathfrak{R}_{21}^{0}$. We now compute their image under the action of $(\mathrm{id} \otimes \pi_{u-1}^{\frac{1}{2}})$ to obtain the expression of $[\mathbf{L}^{-}(u^{-1})]^{-1}$ defined in (4.74). First, it follows from (B.5) and (B.18)

$$ (\mathrm{id} \otimes \pi_u^{\frac{1}{2}})(\mathfrak{R}_{21}^{+}) = \prod_{k=0}^{\infty} \exp_{q-2} \left( (-u^{-2})^k u^{-1} (q-q^{-1}) q^{\frac{1}{2}} f_{\alpha+k\delta} \otimes E_{12} \right) \\ (B.27) \qquad = 1 \otimes \mathbb{I}_2 + f^{-}(u) \otimes E_{12}, $$

where

$$ (B.28) \qquad f^{-}(u) = (q - q^{-1}) q^{\frac{1}{2}} u^{-1} \left( \sum_{k=0}^{\infty} (-u^{-2})^{k} f_{\alpha+k\delta} \right). $$

Similarly, from (B.7) and using (B.18)

$$ (\mathrm{id} \otimes \pi_u^{\frac{1}{2}})(\mathfrak{R}_{21}^{-}) = \prod_{k=0}^{\infty} \exp_{q-2} \left( (-u^{-2})^k u^{-1} (q-q^{-1}) q^{\frac{1}{2}} f_{\delta-\alpha+k\delta} \otimes E_{21} \right) $$
$$ (B.29) \qquad = 1 \otimes \mathbb{I}_2 + e^{-}(u) \otimes E_{21}, $$

where

$$ (B.30) \qquad e^{-}(u) = (q - q^{-1}) q^{\frac{1}{2}} u^{-1} \left( \sum_{k=0}^{\infty} (-u^{-2})^{k} f_{\delta-\alpha+k\delta} \right). $$

Then from (B.7) and using (B.18) we get

$$ (B.31) \qquad (\mathrm{id} \otimes \pi_u^{\frac{1}{2}})(\mathfrak{R}_{21}^0) = k^{-}(u) \otimes E_{11} + \tilde{k}^{-}(u) \otimes E_{22}, $$

where

$$ (B.32) \qquad k^{-}(u) = \exp \left( -(q - q^{-1}) \sum_{k=1}^{\infty} \frac{(-u^{-2}q)^{k}}{q^{k} + q^{-k}} f_{k\delta} \right), \quad \tilde{k}^{-}(u) = \exp \left( (q - q^{-1}) \sum_{k=1}^{\infty} \frac{(-u^{-2}q^{-1})^{k}}{q^{k} + q^{-k}} f_{k\delta} \right). $$

Finally, combining (B.27)-(B.31) and (B.25), we get

$$ (B.33) \qquad [\mathbf{L}^{-}(u^{-1})]^{-1} = \begin{pmatrix} k^{-}(u)K_{1}^{\frac{1}{2}} + f^{-}(u)\tilde{k}^{-}(u)e^{-}(u)K_{1}^{\frac{1}{2}} & f^{-}(u)\tilde{k}^{-}(u)K_{1}^{-\frac{1}{2}} \\ \tilde{k}^{-}(u)e^{-}(u)K_{1}^{\frac{1}{2}} & \tilde{k}^{-}(u)K_{1}^{-\frac{1}{2}} \end{pmatrix}. $$

Note that from the definition of $e^{-}(u)$, $f^{-}(u)$, $k^{-}(u)$, $\tilde{k}^{-}(u)$ in (B.30), (B.28), (B.32) it is easy to see that $[\mathbf{L}^{-}(u^{-1})]^{-1}$ is a formal power series in $u^{-1}$, i.e. $[\mathbf{L}^{-}(u^{-1})]^{-1}$ is in $\mathcal{L}U_q sl_2[[u^{-1}]] \otimes \mathrm{End}(\mathbb{C}^2)$.

### B.4. The spin-$\frac{1}{2}$ L-operator $\mathbf{L}^{(\frac{1}{2})}(u)$. We now compute the spin-$\frac{1}{2}$ L-operator $\mathbf{L}^{(\frac{1}{2})}(u)$ defined in (4.1). It is obtained by taking the image of $\mathbf{L}^{+}(u)$ under the evaluation with $(ev_v \otimes \mathrm{id})$.

Recall the expression of $\mathbf{L}^{+}(u)$ in (B.26). The spin-$\frac{1}{2}$ L-operator is then obtained by evaluating $e^{+}(u)$, $f^{+}(u)$, $k^{+}(u)$, $\tilde{k}^{+}(u)$ defined in (B.20), (B.22), (B.24). Let us first introduce the function [BGKNR12]

$$ (B.34) \qquad \Lambda(u) = \sum_{k=1}^{\infty} \frac{C_k}{(q^k + q^{-k})} \frac{u^k}{k}, $$

where the central elements $C_k$ are defined by (B.13). Note that it satisfies

$$ (B.35) \qquad \Lambda(uq) + \Lambda(uq^{-1}) = -\log(1 - Cu + u^2). $$

From the evaluated root vectors (B.9) and (B.12), we get:

$$ \begin{align} \mathrm{ev}_v(e^+(u)) &= (q-q^{-1})q^{-\frac{1}{2}}u^{-1}v^{-1}EK^{-\frac{1}{2}}(1-u^{-2}v^{-2}q^{-1}K^{-1})^{-1}, \nonumber \\ \mathrm{ev}_v(f^+(u)) &= (q-q^{-1})q^{-\frac{1}{2}}u^{-1}v^{-1}FK^{\frac{1}{2}}(1-u^{-2}v^{-2}qK^{-1})^{-1}, \nonumber \\ \mathrm{ev}_v(k^+(u)) &= e^{\Lambda(u^{-2}v^{-2}q^{-1})}(1-u^{-2}v^{-2}q^{-1}K^{-1}), \nonumber \\ \mathrm{ev}_v(\tilde{k}^+(u)) &= e^{-\Lambda(u^{-2}v^{-2}q)}(1-u^{-2}v^{-2}qK^{-1})^{-1}. \end{align} $$

For instance, let us now compute the evaluation of the matrix entry (2,2) of $\mathbf{L}^{+}(u)$ in (B.26), it reads:

$$ \begin{align*} \mathrm{ev}_v((\mathbf{L}^+(u))_{22}) &= \mathrm{ev}_v(\tilde{k}^+(u)K_1^{-\frac{1}{2}} + e^+(u)k^+(u)f^+(u)K_1^{-\frac{1}{2}}) \\ &= (e^{-\Lambda(u^{-2}v^{-2}q)} + e^{\Lambda(u^{-2}v^{-2}q^{-1})}u^{-2}v^{-2}(q-q^{-1})^2EF)(1-u^{-2}v^{-2}qK^{-1})^{-1}K^{-\frac{1}{2}} \end{align*} $$
$$= e^{\Lambda(u^{-2}v^{-2}q^{-1})} (1 + u^{-4}v^{-4} - u^{-2}v^{-2}(C - (q - q^{-1})^2EF)) (1 - u^{-2}v^{-2}qK^{-1})^{-1} K^{-\frac{1}{2}},$$

where we used (B.35) on the third line, and where C is defined in (A.5). Then, we obtain

$$\mathbf{e}\mathbf{v}_v((\mathbf{L}^+(u))_{22}) = e^{\Lambda(u^{-2}v^{-2}q^{-1})} (K^{-\frac{1}{2}} - u^{-2}v^{-2}q^{-1}K^{\frac{1}{2}}).$$

Computing the other entries, we have

$$(B.37) \qquad \mathbf{L}^{(\frac{1}{2})}(uv) = (\mathbf{e}\mathbf{v}_v \otimes \mathrm{id})(\mathbf{L}^+(u)) = \mu(uv)\mathcal{L}^{(\frac{1}{2})}(uv),$$

where $\mu(u)$ and $\mathcal{L}^{(\frac{1}{2})}(u)$ are given respectively in (4.3) and (4.2).

Similarly, evaluating the Ding-Frenkel L-operator in (B.33), we have

$$(B.38) \qquad (\mathbf{e}\mathbf{v}_v \otimes \mathrm{id})([\mathbf{L}^{-}(u^{-1})]^{-1}) = \mu(u/v)\mathcal{L}^{(\frac{1}{2})}(u/v) = \mathbf{L}^{(\frac{1}{2})}(u/v).$$

## APPENDIX C. ORDERING RELATIONS FOR $\mathcal{A}_q$

**Lemma C.1.** The following relations hold in $\mathcal{A}_q[[u^{-1}]]$:

$$(C.1) \qquad \mathcal{G}_-(v)\mathcal{G}_+(u) = \mathcal{G}_+(u)\mathcal{G}_-(v) + \rho(q^2 - q^{-2})\left(\mathcal{W}_+(u)\mathcal{W}_+(v) - \mathcal{W}_-(u)\mathcal{W}_-(v)\right. \\ \qquad \left. + \frac{1 - UV}{U - V}(\mathcal{W}_+(u)\mathcal{W}_-(v) - \mathcal{W}_+(v)\mathcal{W}_-(u))\right),$$

$$(C.2) \qquad \mathcal{W}_-(v)\mathcal{W}_+(u) = \mathcal{W}_+(u)\mathcal{W}_-(v) + \frac{1}{V - U}\left(\frac{q - q^{-1}}{\rho(q + q^{-1})}(\mathcal{G}_+(u)\mathcal{G}_-(v) - \mathcal{G}_+(v)\mathcal{G}_-(u))\right. \\ \qquad \left. + \frac{1}{q + q^{-1}}(\mathcal{G}_+(u) - \mathcal{G}_-(u) + \mathcal{G}_-(v) - \mathcal{G}_+(v))\right),$$

$$(C.3) \qquad \mathcal{W}_-(v)\mathcal{G}_+(u) = \frac{q}{U - V}\left((Uq^{-1} - Vq)\mathcal{G}_+(u)\mathcal{W}_-(v) - (q - q^{-1})(\mathcal{W}_+(u)\mathcal{G}_+(v) - \mathcal{W}_+(v)\mathcal{G}_+(u))\right),$$

$$(C.4) \qquad \mathcal{W}_-(v)\mathcal{G}_-(u) = \frac{1}{q(V - U)}\left((Vq^{-1} - Uq)\mathcal{G}_-(u)\mathcal{W}_-(v) - (q - q^{-1})(\mathcal{W}_+(u)\mathcal{G}_-(v) - \mathcal{W}_+(v)\mathcal{G}_-(u))\right. \\ \qquad \left. - U\mathcal{G}_-(v)\mathcal{W}_-(u)\right) + \rho(U\mathcal{W}_-(u) - V\mathcal{W}_-(v) - \mathcal{W}_+(u) + \mathcal{W}_+(v)),$$

$$(C.5) \qquad \mathcal{G}_+(v)\mathcal{W}_+(u) = \frac{q}{U - V}\left((Uq - Vq^{-1})\mathcal{W}_+(u)\mathcal{G}_+(v) - (q - q^{-1})(\mathcal{G}_+(v)\mathcal{W}_-(u) - \mathcal{W}_-(v)\mathcal{G}_-(u))\right. \\ \qquad \left. + \rho(U\mathcal{W}_+(u) - V\mathcal{W}_+(v) - \mathcal{W}_-(u) + \mathcal{W}_-(v))\right),$$

$$(C.6) \qquad \mathcal{G}_-(v)\mathcal{W}_+(u) = \frac{1}{q(V - U)}\left((Vq - Uq^{-1})\mathcal{W}_+(u)\mathcal{G}_-(v) - (q - q^{-1})(\mathcal{G}_-(v)\mathcal{W}_-(u) - \mathcal{G}_-(u)\mathcal{W}_-(v)\right. \\ \qquad \left. + V\mathcal{W}_+(v)\mathcal{G}_-(u)\right) + \rho(U\mathcal{W}_+(u) - V\mathcal{W}_+(v) - \mathcal{W}_-(u) + \mathcal{W}_-(v)).$$

*Proof.* The first two ordering relations (C.1), (C.2), are obtained directly from (5.9) and (5.10). The third relation (C.3) follows from (5.11) and (5.13) by replacing the element which is not in the chosen order. The relations (C.4)–(C.6) are derived similarly. $\square$
## APPENDIX D. PROOF OF THEOREM 5.7

For $(j_1, j_2) = (\frac{1}{2}, \frac{1}{2})$, the reflection equation (5.30) holds for $\mathcal{K}(\frac{1}{2})(u)$ due to [BS09]. The proof is divided into three parts: We first show the case $(j_1, j_2) = (\frac{1}{2}, j + \frac{1}{2})$, assuming the equation (5.30) holds for $(j_1, j_2) = (\frac{1}{2}, j)$. Then, we prove the case $(j + \frac{1}{2}, \frac{1}{2})$, assuming (5.30) holds for $(j_1, j_2) = (j, \frac{1}{2})$. Finally, the generic case $(j_1, j_2)$ is proven considering two distinct cases $j_1 \ge j_2$ and $j_1 \le j_2$, using that the reflection equation (5.30) holds for $(j_1, j_2) = (j + \frac{1}{2}, \frac{1}{2})$ and $(\frac{1}{2}, j + \frac{1}{2})$.

First of all, the following Yang-Baxter equation holds for any $j_1, j_2 \in \frac{1}{2}\mathbb{N}$:

$$ (D.1) \quad R_{12}^{(j_1, j_2)}(u_1/u_2) R_{13}^{(j_1, j_3)}(u_1/u_3) R_{23}^{(j_2, j_3)}(u_2/u_3) = R_{23}^{(j_2, j_3)}(u_2/u_3) R_{13}^{(j_1, j_3)}(u_1/u_3) R_{12}^{(j_1, j_2)}(u_1/u_2). $$

This is due to the fact that $\mathcal{R}^{(j_1, j_2)}(u)$ is proportional to $R^{(j_1, j_2)}(u)$, see (4.39), and it satisfies the Yang-Baxter equation in (4.11). In what follows we use the shorthand notation

$$ (D.2) \quad R_{k\ell} = R_{k\ell}^{(j_k, j_\ell)}(u_k/u_\ell), \quad \bar{R}_{k\ell} = R_{k\ell}^{(j_k, j_\ell)}(u_\ell/u_k), \quad \hat{R}_{k\ell} = R_{k\ell}^{(j_k, j_\ell)}(u_k u_\ell), \quad \mathcal{K}_\ell = \mathcal{K}_\ell^{(j_\ell)}(u_\ell), $$

and denote by $R_{\langle\ell\ell+1\rangle k}^{(j+\frac{1}{2}, j_k)}(u)$, $R_{k\langle\ell\ell+1\rangle}^{(j_k, j+\frac{1}{2})}(u)$, $\mathcal{K}_{\langle\ell\ell+1\rangle}^{(j+\frac{1}{2})}(u)$ the resulting R-matrices and K-operators obtained by fusing the space at position $\ell$ with the one at position $\ell+1$, for $k, \ell \in \mathbb{N}_+$.

According to the notation (D.2), the Yang-Baxter equation (D.1) reads

$$ (D.3) \quad R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}, \quad \text{or} \quad R_{12}\hat{R}_{13}\hat{R}_{23} = \hat{R}_{23}\hat{R}_{13}R_{12}, \quad \text{or} \quad \hat{R}_{12}\hat{R}_{13}\bar{R}_{23} = \bar{R}_{23}\hat{R}_{13}\hat{R}_{12}, $$

where the first two relations in (D.3) are related by $u_3 \to u_3^{-1}$, and the last two relations in (D.3) are related by $u_2 \to u_2^{-1}$. In the following, we will need relations derived from the Yang-Baxter equation:

$$ (D.4) \quad \hat{R}_{13}\hat{R}_{12}R_{23} = R_{23}\hat{R}_{12}\hat{R}_{13}, \quad R_{13}\hat{R}_{12}\hat{R}_{23} = \hat{R}_{23}\hat{R}_{12}R_{13}. $$

The first equality is obtained from the first relation in (D.3) by multiplying both sides on the left and on the right by $\bar{R}_{23}$, using the unitarity property of the R-matrix in (4.55), that is $R_{23}\bar{R}_{23} \propto \mathbb{I}$, and substituting $u_2 \to u_2^{-1}$, $u_3 \to u_3^{-1}$. The two equations in (D.4) are related by $u_3 \to u_3^{-1}$.

First, we give a relation derived from the reflection equation that will be used later.

**Lemma D.1.** *Assuming the reflection equation holds for any $j_1, j_2, j_3$, then*

$$ (D.5) \quad \mathcal{K}_1 \hat{R}_{12} \mathcal{K}_2 \bar{R}_{12} = \bar{R}_{12} \mathcal{K}_2 \hat{R}_{12} \mathcal{K}_1. $$

*Proof.* According to the notation (D.2), the reflection equation (5.30) reads

$$ (D.6) \quad R_{12} \mathcal{K}_1 \hat{R}_{12} \mathcal{K}_2 = \mathcal{K}_2 \hat{R}_{12} \mathcal{K}_1 R_{12}. $$

The equation (D.5) is obtained by multiplying on both sides of (D.6) on the left and on the right by $\bar{R}_{12}$ and using the unitarity property (4.55). $\square$

In the following, we will need various expressions of the fused R-matrices.

**Lemma D.2.** *The fused R-matrices defined in (4.37) satisfy the following relations:*

$$ (D.7) \quad R^{(j_1, j_2)}(u) = \mathcal{F}_{\langle 12 \rangle}^{(j_1)} R_{23}^{(j_1 - \frac{1}{2}, j_2)} (u q^{-\frac{1}{2}}) R_{13}^{(\frac{1}{2}, j_2)} (u q^{j_1 - \frac{1}{2}}) \mathcal{E}_{\langle 12 \rangle}^{(j_1)}, $$

$$ (D.8) \quad = \mathcal{F}_{\langle 23 \rangle}^{(j_2)} R_{12}^{(j_1, \frac{1}{2})} (u q^{-j_2 + \frac{1}{2}}) R_{13}^{(j_1, j_2 - \frac{1}{2})} (u q^{\frac{1}{2}}) \mathcal{E}_{\langle 23 \rangle}^{(j_2)}, $$
$$ (D.9) \qquad = \mathcal{F}_{(23)}^{(j_2)} R_{13}^{(j_1, j_2 - \frac{1}{2})} (uq^{-\frac{1}{2}}) R_{12}^{(\frac{j_1}{2}, \frac{1}{2})} (uq^{\frac{j_1}{2} - \frac{1}{2}}) \mathcal{E}_{(23)}^{(j_2)}. $$

*Proof.* First, note that the r.h.s. of (D.7)-(D.8) are directly derived from the expressions for the R-matrices in (4.47) and (4.46), while the r.h.s. of (D.9) is a generalization<fnref n="21" /> of (4.18). Recall that $R^{(j_1, j_2)}(u)$ is proportional to $\mathcal{R}^{(j_1, j_2)}(u)$, see (4.39) with (4.44), and using (4.29) one finds that the coefficient of proportionality reads

$$ (D.10) \qquad f^{(j_1, j_2)}(u) = \prod_{k=0}^{2j_1-1} \prod_{\ell=0}^{2j_2-1} \pi^{\frac{1}{2}}(\mu(uq^{j_1+j_2-1-k-\ell})). $$

Thus, we have $R^{(j_1, j_2)}(u) = \mathcal{R}^{(j_1, j_2)}(u)/f^{(j_1, j_2)}(u)$. Then, inserting in the r.h.s. of the latter relation the corresponding R-matrices, the equations (D.7)-(D.9) are obtained provided the following equalities hold

$$ \begin{aligned} \frac{f^{(j_1 - \frac{1}{2}, j_2)}(uq^{-\frac{1}{2}}) f^{(\frac{1}{2}, j_2)}(uq^{j_1 - \frac{1}{2}})}{f^{(j_1, j_2)}(u)} &= \frac{f^{(j_1, \frac{1}{2})}(uq^{-j_2 + \frac{1}{2}}) f^{(j_1, j_2 - \frac{1}{2})}(uq^{\frac{1}{2}})}{f^{(j_1, j_2)}(u)} \\ &= \frac{f^{(j_1, j_2 - \frac{1}{2})}(uq^{-\frac{1}{2}}) f^{(j_1, \frac{1}{2})}(uq^{j_2 - \frac{1}{2}})}{f^{(j_1, j_2)}(u)} = 1. \end{aligned} $$

Finally, it is easily checked that they indeed hold using (D.10). □

We now show by induction that the reflection equation (5.30) holds for $(j_1, j_2) = (\frac{1}{2}, j + \frac{1}{2})$ using the R-matrix decomposition in (3.46) and Corollary 3.6 together with the second result in Lemma 4.13.

**Lemma D.3.** The following relation holds for $j \in \frac{1}{2}\mathbb{N}$:

$$ (D.11) \qquad R_{1(23)}^{(\frac{1}{2}, j+\frac{1}{2})}(u/v) K_1^{(\frac{1}{2})}(u) R_{1(23)}^{(\frac{1}{2}, j+\frac{1}{2})}(uv) K_{(23)}^{(j+\frac{1}{2})}(v) = \\ K_{(23)}^{(j+\frac{1}{2})}(v) R_{1(23)}^{(\frac{1}{2}, j+\frac{1}{2})}(uv) K_1^{(\frac{1}{2})}(u) R_{1(23)}^{(\frac{1}{2}, j+\frac{1}{2})}(u/v). $$

*Proof.* We proceed by induction. Recall the reflection equation (5.30) holds for $(j_1, j_2) = (\frac{1}{2}, \frac{1}{2})$ due to [BS09]. Assume that (5.30) holds for $(\frac{1}{2}, j)$ with a fixed value of $j \in \frac{1}{2}\mathbb{N}_+$. Consider the l.h.s. of (D.11) and multiply on the right by $\mathbb{I}_{2(2j+2)} = \mathcal{H}_{(23)}^{(\frac{j+1}{2})}[\mathcal{H}_{(23)}^{(\frac{j+1}{2})}]^{-1}$. Then, using the relations (D.9), (D.8) and the fused K-operator given in (5.29), we get (with the necessary steps of the calculation underlined):

$$ (D.12) \qquad R_{1(23)}^{(\frac{1}{2}, j+\frac{1}{2})}(u/v) K_1^{(\frac{1}{2})}(u) R_{1(23)}^{(\frac{1}{2}, j+\frac{1}{2})}(uv) K_{(23)}^{(j+\frac{1}{2})}(v) \mathcal{H}_{(23)}^{(j+\frac{1}{2})} [\mathcal{H}_{(23)}^{(j+\frac{1}{2})}]^{-1} \\ = \mathcal{F}_{(23)} R_{13} R_{12} \mathcal{E}_{(23)} K_1 \mathcal{F}_{(23)} \hat{R}_{12} \hat{R}_{13} \mathcal{E}_{(23)} \mathcal{F}_{(23)} K_2 \hat{R}_{23} K_3 \mathcal{E}_{(23)} \mathcal{H}_{(23)} [\mathcal{H}_{(23)}]^{-1}, $$

where we fix in the notations from (D.2)

$$ (D.13) \qquad j_1 = j_2 = \frac{1}{2}, \quad j_3 = j, \quad u_1 = u, \quad u_2 = vq^{-j}, \quad u_3 = vq^{\frac{1}{2}}. $$

<footnote n="21">²¹The relation (4.18) is given for $j_1 = \frac{1}{2}$ but it can be generalized to any $j_1$ by computing $\mathcal{R}^{(j_1, j_2)}(u) = (\pi^{j_1} \otimes \mathrm{id})(\mathbf{L}^{(j_2)}(u))$ with $\mathbf{L}^{(j_2)}(u)$ in (4.14).</footnote>
First, we remove the products $\mathcal{E}_{\langle 23 \rangle} \mathcal{F}_{\langle 23 \rangle}$. Recall that $R^{(\frac{1}{2},j)}(u)$ from (4.38) agrees with (3.41), see Lemma 4.13, then using (3.48) we find that (D.12) equals

$$
\begin{align*}
& \mathcal{F}_{\langle 23 \rangle} R_{13} R_{12} K_1 \mathcal{E}_{\langle 23 \rangle} \mathcal{F}_{\langle 23 \rangle} \hat{R}_{12} \hat{R}_{13} \mathcal{E}_{\langle 23 \rangle} \mathcal{F}_{\langle 23 \rangle} \underline{\mathcal{K}_2 \hat{R}_{23} \mathcal{K}_3 \bar{R}_{23}} \mathcal{E}_{\langle 23 \rangle} [\mathcal{H}_{\langle 23 \rangle}]^{-1} \\
& \stackrel{(\text{D.5})}{=} \mathcal{F}_{\langle 23 \rangle} R_{13} R_{12} K_1 \mathcal{E}_{\langle 23 \rangle} \mathcal{F}_{\langle 23 \rangle} \hat{R}_{12} \hat{R}_{13} \mathcal{E}_{\langle 23 \rangle} \mathcal{F}_{\langle 23 \rangle} \bar{R}_{23} \mathcal{K}_3 \hat{R}_{23} \mathcal{K}_2 \mathcal{E}_{\langle 23 \rangle} [\mathcal{H}_{\langle 23 \rangle}]^{-1} \\
& \stackrel{(\text{D.50})}{=} \mathcal{F}_{\langle 23 \rangle} R_{13} R_{12} K_1 \mathcal{E}_{\langle 23 \rangle} \mathcal{F}_{\langle 23 \rangle} \underline{\hat{R}_{12} \hat{R}_{13} \bar{R}_{23}} \mathcal{K}_3 \hat{R}_{23} \mathcal{K}_2 \mathcal{E}_{\langle 23 \rangle} [\mathcal{H}_{\langle 23 \rangle}]^{-1} \\
& \stackrel{(\text{D.3})}{=} \mathcal{F}_{\langle 23 \rangle} R_{13} R_{12} K_1 \mathcal{E}_{\langle 23 \rangle} \mathcal{F}_{\langle 23 \rangle} \bar{R}_{23} \hat{R}_{13} \hat{R}_{12} \mathcal{K}_3 \hat{R}_{23} \mathcal{K}_2 \mathcal{E}_{\langle 23 \rangle} [\mathcal{H}_{\langle 23 \rangle}]^{-1} \\
& \stackrel{(\text{D.50})}{=} \mathcal{F}_{\langle 23 \rangle} R_{13} R_{12} K_1 \bar{R}_{23} \hat{R}_{13} \hat{R}_{12} \mathcal{K}_3 \hat{R}_{23} \mathcal{K}_2 \mathcal{E}_{\langle 23 \rangle} [\mathcal{H}_{\langle 23 \rangle}]^{-1} \\
& \stackrel{\phantom{(\text{D.50})}}{=} \mathcal{F}_{\langle 23 \rangle} R_{13} R_{12} K_1 \hat{R}_{12} \underline{\hat{R}_{13} \mathcal{K}_2 \hat{R}_{23} \mathcal{K}_3} \mathcal{E}_{\langle 23 \rangle},
\end{align*}
$$

where we used (D.3), (D.5) and (3.48) to get the last line. Secondly, rearranging the different R-matrices and K-operators using $[R_{23}, K_1] = [R_{13}, K_2] = [\hat{R}_{13}, K_2] = [\hat{R}_{12}, K_3] = 0$ and (D.4), (D.3), (D.6), we get:

$$
\begin{align*}
&= \mathcal{F}_{\langle 23 \rangle} R_{13} \underline{R_{12} K_1 \hat{R}_{12} K_2 \hat{R}_{13} \hat{R}_{23} K_3 \mathcal{E}_{\langle 23 \rangle}} \stackrel{(\text{D.6})}{=} \mathcal{F}_{\langle 23 \rangle} \underline{R_{13} K_2 \hat{R}_{12} K_1 \underline{R_{12} \hat{R}_{13} \hat{R}_{23} K_3 \mathcal{E}_{\langle 23 \rangle}}} \\
&\stackrel{(\text{D.3})}{=} \mathcal{F}_{\langle 23 \rangle} \underline{K_2 \hat{R}_{13} \hat{R}_{12} K_1 \hat{R}_{23} \hat{R}_{13} \underline{R_{12} K_3 \mathcal{E}_{\langle 23 \rangle}}} \stackrel{(\text{D.4})}{=} \mathcal{F}_{\langle 23 \rangle} \underline{K_2 \hat{R}_{23} \hat{R}_{12} \underline{R_{13} K_1 \hat{R}_{13} K_3 R_{12} \mathcal{E}_{\langle 23 \rangle}}} \\
&\stackrel{(\text{D.6})}{=} \mathcal{F}_{\langle 23 \rangle} \underline{K_2 \hat{R}_{23} \hat{R}_{12} K_3 \hat{R}_{13} K_1 R_{13} R_{12} \mathcal{E}_{\langle 23 \rangle}} = \mathcal{F}_{\langle 23 \rangle} \underline{K_2 \hat{R}_{23} K_3 \hat{R}_{12} \hat{R}_{13} K_1 R_{13} R_{12} \mathcal{E}_{\langle 23 \rangle}} \\
&= \mathcal{F}_{\langle 23 \rangle} \underline{K_2 \hat{R}_{23} K_3 \mathcal{E}_{\langle 23 \rangle} \mathcal{F}_{\langle 23 \rangle} \hat{R}_{12} \hat{R}_{13} \mathcal{E}_{\langle 23 \rangle} K_1 \mathcal{F}_{\langle 23 \rangle} R_{13} R_{12} \mathcal{E}_{\langle 23 \rangle}},
\end{align*}
$$

where we put back the products $\mathcal{E}_{\langle 23 \rangle} \mathcal{F}_{\langle 23 \rangle}$ by inserting $\mathbb{I}_{2(2j+2)} = \mathcal{H}_{\langle 23 \rangle}^{(j+\frac{1}{2})} [\mathcal{H}_{\langle 23 \rangle}^{(j+\frac{1}{2})}]^{-1}$ and using (D.4), (D.5), (3.48)-(3.50). Finally, identifying the fused R-matrices and the fused K-operator, the equation (D.11) follows. $\square$

Then, we show by induction that the reflection equation (5.30) holds for $(j_1, j_2) = (j + \frac{1}{2}, \frac{1}{2})$.

**Lemma D.4.** The following relation holds for $j \in \frac{1}{2}\mathbb{N}$

$$
\begin{align*}
(D.14) \qquad & R_{\langle 12 \rangle 3}^{(\frac{j+1}{2}, \frac{1}{2})} (u/v) K_{\langle 12 \rangle}^{(\frac{j+1}{2})} (u) R_{\langle 12 \rangle 3}^{(\frac{j+1}{2}, \frac{1}{2})} (uv) K_3^{(\frac{1}{2})} (v) = \\
& \qquad K_3^{(\frac{1}{2})} (v) R_{\langle 12 \rangle 3}^{(\frac{j+1}{2}, \frac{1}{2})} (uv) K_{\langle 12 \rangle}^{(\frac{j+1}{2})} (u) R_{\langle 12 \rangle 3}^{(\frac{j+1}{2}, \frac{1}{2})} (u/v) .
\end{align*}
$$

*Proof.* We proceed by induction similarly to the proof of Lemma D.3. Recall the reflection equation (5.30) holds for $(j_1, j_2) = (\frac{1}{2}, \frac{1}{2})$ due to [BS09]. Assume that (5.30) holds for $(j, \frac{1}{2})$ with a fixed value of $j \in \frac{1}{2}\mathbb{N}_+$. Consider the l.h.s. of (D.14) and multiply on the right by $\mathbb{I}_{2(2j+2)} = \mathcal{H}_{\langle 12 \rangle}^{(j+\frac{1}{2})} [\mathcal{H}_{\langle 12 \rangle}^{(j+\frac{1}{2})}]^{-1}$. Then, using the fused R-matrix (4.37) and the fused K-operator given in (5.29), we get (with the necessary steps of the calculation underlined):

$$
\begin{align}
& R_{\langle 12 \rangle 3}^{(j+\frac{1}{2}, \frac{1}{2})} (u/v) K_{\langle 12 \rangle}^{(j+\frac{1}{2})} (u) R_{\langle 12 \rangle 3}^{(j+\frac{1}{2}, \frac{1}{2})} (uv) K_3^{(\frac{1}{2})} (v) \mathcal{H}_{\langle 12 \rangle}^{(j+\frac{1}{2})} [\mathcal{H}_{\langle 12 \rangle}^{(j+\frac{1}{2})}]^{-1} \nonumber \\
& \quad = \mathcal{F}_{\langle 12 \rangle} R_{13} R_{23} \mathcal{E}_{\langle 12 \rangle} \mathcal{F}_{\langle 12 \rangle} K_1 \hat{R}_{12} K_2 \mathcal{E}_{\langle 12 \rangle} \mathcal{F}_{\langle 12 \rangle} \hat{R}_{13} \hat{R}_{23} \mathcal{E}_{\langle 12 \rangle} \mathcal{H}_{\langle 12 \rangle} K_3 [\mathcal{H}_{\langle 12 \rangle}]^{-1}, \tag{D.15}
\end{align}
$$
where we fix in the notations from (D.2)

$$ (D.16) \qquad j_1 = j_3 = \frac{1}{2}, \quad j_2 = j, \quad u_1 = uq^{-j}, \quad u_2 = uq^{\frac{1}{2}}, \quad u_3 = v. $$

Firstly, similarly to Lemma D.3, we remove the products $\mathcal{E}_{\langle 12 \rangle} \mathcal{F}_{\langle 12 \rangle}$ using (3.48)-(3.50), (D.3) and (D.6). Then, we find that (D.15) equals

$$ (D.17) \qquad \mathcal{F}_{\langle 12 \rangle} R_{13} R_{23} K_1 \hat{R}_{12} \underline{K_2 \hat{R}_{13}} \hat{R}_{23} K_3 \mathcal{E}_{\langle 12 \rangle}. $$

Secondly, we rearrange the R-matrices and K-operators in (D.17) as follows:

$$ \begin{align*} \mathcal{F}_{\langle 12 \rangle} R_{13} K_1 \underline{R_{23} \hat{R}_{12} \hat{R}_{13}} K_2 \hat{R}_{23} K_3 \mathcal{E}_{\langle 12 \rangle} &\stackrel{(D.4)}{=} \mathcal{F}_{\langle 12 \rangle} R_{13} K_1 \hat{R}_{13} \hat{R}_{12} \underline{R_{23} K_2 \hat{R}_{23} K_3} \mathcal{E}_{\langle 12 \rangle} \\ &\stackrel{(D.6)}{=} \mathcal{F}_{\langle 12 \rangle} R_{13} K_1 \hat{R}_{13} \hat{R}_{12} \underline{K_3 \hat{R}_{23} K_2 R_{23}} \mathcal{E}_{\langle 12 \rangle} \stackrel{(D.6)}{=} \underline{\mathcal{F}_{\langle 12 \rangle} K_3 \hat{R}_{13} K_1 R_{13} \hat{R}_{12} \hat{R}_{23} K_2 R_{23}} \mathcal{E}_{\langle 12 \rangle} \\ &\stackrel{(D.4)}{=} K_3 \mathcal{F}_{\langle 12 \rangle} \hat{R}_{13} \underline{K_1 \hat{R}_{23} R_{12} R_{13} K_2 R_{23}} \mathcal{E}_{\langle 12 \rangle} = K_3 \mathcal{F}_{\langle 12 \rangle} \hat{R}_{13} \hat{R}_{23} K_1 \hat{R}_{12} K_2 R_{13} R_{23} \mathcal{E}_{\langle 12 \rangle} \\ &= K_3 \mathcal{F}_{\langle 12 \rangle} \hat{R}_{13} \hat{R}_{23} \mathcal{E}_{\langle 12 \rangle} K_1 \hat{R}_{12} K_2 \mathcal{E}_{\langle 12 \rangle} \mathcal{F}_{\langle 12 \rangle} R_{13} R_{23} \mathcal{E}_{\langle 12 \rangle}, \end{align*} $$

where we put back the products $\mathcal{E}_{\langle 12 \rangle} \mathcal{F}_{\langle 12 \rangle}$ inserting $\mathbb{I}_{2(2j+2)} = \mathcal{H}_{\langle 12 \rangle}^{(j+\frac{1}{2})}[\mathcal{H}_{\langle 12 \rangle}^{(j+\frac{1}{2})}]^{-1}$ and using (3.48)-(3.50), (D.4) and (D.5). Finally, identifying the fused R-matrices and the fused K-operator, the equation (D.14) follows. $\square$

**Proof of Theorem 5.7.** We are now ready to show that the reflection equation (5.30) is satisfied by the fused K-operator (5.29) for all $j_1, j_2 \in \frac{1}{2}\mathbb{N}_+$. We consider separately (5.30) for two distinct cases $j_1 \ge j_2$ and $j_1 \le j_2$:

$$ (i) \quad (j_1, j_2) = (j + \frac{1}{2}, k + \frac{1}{2}) \text{ with } 0 \le k \le j. $$

$$ (ii) \quad (j_1, j_2) = (l + \frac{1}{2}, j + \frac{1}{2}) \text{ with } 0 \le l \le j. $$

The first case is shown by induction on $k$. For $k=0$, the equation (5.30) holds for $(j_1, j_2) = (j + \frac{1}{2}, \frac{1}{2})$ due to Lemma D.4. Now, assume (5.30) holds for $(j_1, j_2) = (j + \frac{1}{2}, k)$ with a fixed value of $k \le j$. The case (i) is shown similarly to the proof of Lemma D.3. Indeed, fix in (D.12) $j_1 = j + \frac{1}{2}$, $j_2 = \frac{1}{2}$, $j_3 = k$ and $u_1 = u$, $u_2 = vq^{-k}$, $u_3 = vq^{\frac{1}{2}}$ (instead of (D.13)). Then, the rest of the proof is the same as for Lemma D.3, using now (D.11), (D.14), (4.39) and our assumption.

The second case is also shown by induction on $\ell$. For $\ell=0$, the equation (5.30) holds for $(j_1, j_2) = (\frac{1}{2}, j + \frac{1}{2})$ due to Lemma D.3. Assuming that (5.30) holds for $(j_1, j_2) = (\ell, j + \frac{1}{2})$ with a fixed value of $\ell \le j$, then the proof of the case (ii) is similar to Lemma D.4. Fix in (D.15) $j_1 = \frac{1}{2}$, $j_2 = \ell$, $j_3 = j + \frac{1}{2}$ and $u_1 = uq^{-\ell}$, $u_2 = uq^{\frac{1}{2}}$, $u_3 = v$ (instead of (D.16)). Then, the rest of the proof is the same as for Lemma D.4, using now (D.11), (D.14), (4.39) and our assumption. This concludes the proof of Theorem 5.7. $\square$

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INSTITUT DENIS-POISSON CNRS/UMR 7013 - UNIVERSITÉ DE TOURS - UNIVERSITÉ D'ORLÉANS PARC DE GRAM-MONT, 37200 TOURS, FRANCE

Email address: pascal.baseilhac@idpoisson.fr, guillaume.lemarthe@idpoisson.fr

Email address: azat.gainutdinov@lmpt.univ-tours.fr