Let $U_q(\widehat{\mathfrak g})$ be the quantum affine algebra over a simple Lie algebra $\mathfrak g$. I am trying to understand and compare the so called simple $\ell$-roots $A_{i,a}$ seen in both Hernandez and Jimbo's paper arXiv:1104.1891 and Young's paper arXiv:1206.6657.
Let $I$ be an index set for the Dynkin diagram of $\mathfrak g$ and let $B$ be a symmetric matrix so that $B= DC$, where $C$ is the Cartan matrix of $\mathfrak g$ and $D = \operatorname{diag}(d_i)$ is a diagonal matrix of integers. For $q$ not a root of unity, we set $q_i = q^{d_i}$ (cf. page 4 arXiv:1104.1891 for a refresher on these conventions).
On page 14 of Hernandez and Jimbo's paper arxiv:1104.1891, the $\ell$-root $A_{i,a}$ is defined as follows:
For $i\in I, a \in \mathbb C^\times$, one can define an $I$-tuple of rational functions $Y_{i,a}$ by $(Y_{i,a})_i(z) = q_i\frac{1-q_i^{-1}az}{1-q_iaz} $ and $(Y_{i,a})_j(z) = 1$ for $j\neq i$. Then the simple $\ell$-root $A_{i,a}$ is defined as the monomial
$$A_{i,a} = Y_{i,aq_i}Y_{i,aq_i^{-1}}\prod_{j: C_{ji} = -1} Y^{-1}_{j,a}\prod_{j: C_{ji} = -2} Y^{-1}_{j,aq}Y^{-1}_{j,aq^{-1}} \prod_{j: C_{ji} = -3} Y^{-1}_{j,aq^2}Y^{-1}_{j,a}Y^{-1}_{j,aq^{-2}}. \tag{1}\label{485701_1}$$Rmk This is the same defition of the original papers on the subject of $q$-characters arXiv:9911112, arXiv:9810055, in which simple roots are the columns of the Cartan matrix $C_{ij} = 2(\alpha_i, \alpha_j)/(\alpha_i,\alpha_i)$.
Now, on page 8 of Young's paper arXiv:1206.6657, the $A_{i,a}$ are defined as the $I$-tuple of rational functions:$$(A_{i,a})_j(z) = q^{B_{ij}}\frac{1-q^{-B_{ij}}}{1-q^{B_{ij}}},\tag{2}\label{485701_2}$$where $B_{ij}$ are the entries of $B=DC$, so in fact $q^{B_{ij}} = q_i^{C_{ij}}$.
Right after this definition on page 8 of Young's paper, it is warned that this version of $A_{j,a}$ just defined should coincide with $A_{j,aq_j}$ defined in Hernandez and Jimbo's paper. However, I don't see that being the case here. For a short example, consider $\mathfrak g = B_2$ with cartan matrix $C = \begin{bmatrix}2 & -1 \\-2 & 2 \end{bmatrix}$. Then, according to Hernandez and Jimbo's definition \eqref{485701_1}, we should have $A_{1,aq_1} = Y_{1,a{q_1}^2}Y_{1,a}Y_{2,aq_1q}^{-1}Y_{2,aq_1q^{-1}}^{-1}$, whence$$(A_{1,aq_1})_2(z) = q_2^{-2}\frac{(1-aq_1qq_2z)(1-aq_1q^{-1}q_2z)}{(1-aq_1qq_2^{-1}z)(1-aq_1q^{-1}q_2^{-1}z)}, $$using the fact that $(Y^{-1}_{2,a})_2(z) = q_2^{-1}(1-aq_2z)/(1-aq_2^{-1}z)$. However, this is not equal to the corresponding rational function given by Young's definition$$q^{B_{12}} \frac{1-q^{-B_{12}}az}{1-q^{B_{12}}az} = q_1^{-1}\frac{1-aq_1z}{1-aq_1^{-1}z} \ \ \ \ \ \ \ \ (\text{note: } q^{B_{12}} = q_1^{C_{12}} = q_1^{-1}). $$
I am not sure what is happening here. It seems that Young's definition should instead be changed to$(A_{i,a})_j(z) = q^{B_{ji}}(1-aq^{-B_{ji}})/(1-aq^{B_{ji}})$ (actually, in the introduction he defines it like this). In this way, for $\mathfrak g = B_2$, we find that
$$q^{B_{12}} \frac{1-q^{-B_{12}}az}{1-q^{B_{12}}az} = q_2^{-2}\frac{1-aq_2^2z}{1-q_2^{-2}az}.$$Then, I find that this equals to $$(Y^{-1}_{2,aq_2^{-1}}Y^{-1}_{2,aq_2})_2(z) = q_2^{-2} \frac{(1-aq_2^{-1}q_2z)(1-aq_2q_2z)}{(1-aq_2^{-1}q_2^{-1}z)(1-aq_2q_2^{-1}z)} = q_2^{-2}\frac{1-aq_2^2z}{1-aq_2^{-2}z}.$$Although close, this still doesn't satisfy the formula given on \eqref{485701_1}.
Could someone try to explain what is the relation between these two definitions? Any help would be very much appreciated.