east) .. controls (3,1.5) .. node\[above\] [$\theta$]{} (r.north west) ; (-1.north) .. controls (3,1) .. (1.north) ; (0) edge\[&lt;-&gt;, loop above\] (0);

The algebra ${\U^{\imath}}_q(\mathfrak{sl}_{2r+2})$ is defined to be the associative algebra over $\Q(q)$ generated by $\be_{\alpha_i}$, $\bff_{\alpha_i}$, $\bk_{\alpha_i}$, $\bk^{-1}_{\alpha_i}$ ($i \in \I^{\imath}_{2r+1}$) , and $\bt$, subject to the following relations for $i$, $j \in \I^{\imath}_{2r+1}$: $$\begin{aligned}
 \bk_{\alpha_i} \bk_{\alpha_i}^{-1} &= \bk_{\alpha_i}^{-1} \bk_{\alpha_i} =1,\displaybreak[0]\\
  \bk_{\alpha_i} \bk_{\alpha_j} &= \bk_{\alpha_j} \bk_{\alpha_i}, \displaybreak[0]\\
  \bk_{\alpha_i} \be_{\alpha_j} \bk_{\alpha_i}^{-1} &= q^{(\alpha_i-\alpha_{-i}, \alpha_j)} \be_{\alpha_j}, \displaybreak[0]\\
  \bk_{\alpha_i} \bff_{\alpha_j} \bk_{\alpha_i}^{-1} &= q^{-(\alpha_i-\alpha_{-i}, \alpha_j)}
 \bff_{\alpha_j}, \displaybreak[0]\\
  \bk_{\alpha_i}\bt\bk^{-1}_{\alpha_i} &= \bt, \displaybreak[0]\\
  \be_{\alpha_i} \bff_{\alpha_j} -\bff_{\alpha_j} \be_{\alpha_i} &= \delta_{i,j} \frac{\bk_{\alpha_i}
 -\bk^{-1}_{\alpha_i}}{q-q^{-1}}, \displaybreak[0]\\
  \be_{\alpha_i}^2 \be_{\alpha_j} +\be_{\alpha_j} \be_{\alpha_i}^2 &= (q+q^{-1}) \be_{\alpha_i} \be_{\alpha_j} \be_{\alpha_i}, 
   \ \ \quad\quad &\text{if }& |i-j|=1, \displaybreak[0]\\
  \be_{\alpha_i} \be_{\alpha_j} &= \be_{\alpha_j} \be_{\alpha_i}, \ \qquad\qquad\ \ \ \ \qquad &\text{if }& |i-j|>1, \displaybreak[0]\\
  \bff_{\alpha_i}^2 \bff_{\alpha_j} +\bff_{\alpha_j} \bff_{\alpha_i}^2 &= (q+q^{-1}) \bff_{\alpha_i} \bff_{\alpha_j} \bff_{\alpha_i},
    \ \ \quad\quad &\text{if }& |i-j|=1,\displaybreak[0]\\
  \bff_{\alpha_i} \bff_{\alpha_j}  &= \bff_{\alpha_j}  \bff_{\alpha_i},  \ \qquad\qquad\ \ \ \ \qquad &\text{if }& |i-j|>1, \displaybreak[0]\\
  \be_{\alpha_i}\bt &=\bt\be_{\alpha_i}, \quad\qquad\quad &\text{if }&  i > 1, \displaybreak[0]\\
  \be_{\alpha_1}^2\bt + \bt\be_{\alpha_1}^2 &= (q+q^{-1}) \be_{\alpha_1}\bt\be_{\alpha_1},\displaybreak[0]\\
  \bt^2\be_{\alpha_1} + \be_{\alpha_1}\bt^2 &= (q + q^{-1}) \bt\be_{\alpha_1}\bt + \be_{\alpha_1},\displaybreak[0]\\
  \bff_{\alpha_i}\bt &=\bt\bff_{\alpha_i}, \quad\qquad&\text{if }&  i > 1, \displaybreak[0]\\
  \bff_{\alpha_1}^2\bt + \bt\bff_{\alpha_1}^2 &= (q+q^{-1}) \bff_{\alpha_1}\bt\bff_{\alpha_1},\displaybreak[0]\\
  \bt^2\bff_{\alpha_1} + 