m,j}(w) E^{\pm,i}(z_2) \nonumber\\
& &~~~~~+ E^{\pm,j}(w) E^{\pm,i}(z_1) E^{\pm,i}(z_2) +
(\mbox{replacement:}~z_1 \leftrightarrow z_2) = 0
~\mbox{for}~a_{ij} = -1, \label{8}\end{aligned}$$

where $a_{ij}$ are elements of the Cartan matrix of the type $A_{N-1}$ and

$$\begin{aligned}
\delta(x)= \sum_{n\in {\Bbb Z}} x^n.\end{aligned}$$

In this paper we only consider $q$-affine algebra and Yangian double as associative algebras and do not care about the Hopf algebra aspect.

The Yangian double $DY_\hbar(sl_N)$
-----------------------------------

As an associative algebra, the Yangian double $DY_\hbar(sl_N)$ is generated by the Drinfeld generators $\{h_{il},~e^{\pm}_{il} |
i=1,~2,~...,~N-1;~l \in {\Bbb Z}_{ \geq 0} \}$ and the center $c$. In terms of the formal power series (Drinfeld currents)

$$\begin{aligned}
& & H^{+}_i(u) = 1 + \hbar \sum_{l \geq 0} h_{il} u^{-l-1},~
H^{-}_i(u) = 1 - \hbar \sum_{l < 0} h_{il} u^{-l-1}, \\
& & E^{\pm}(u) = \sum_{l \in {\Bbb Z}} e^{\pm}_{il} u^{-l-1}\end{aligned}$$

we can write the generating relations for $DY_\hbar(sl_N)$ as follows [@iohara],

$$\begin{aligned}
& & [ H_i^\pm(u),~H_j^\pm(v) ] =0, \label{y1}\\
& & (u_\mp-v_\pm + B_{ij} \hbar) (u_\pm-v_\mp - B_{ij} \hbar)
H_i^+(u) H_j^-(v) \nonumber\\
& &~~~~= (u_\mp-v_\pm - B_{ij} \hbar) (u_\pm-v_\mp + B_{ij} \hbar)
H_j^-(v) H_i^+(u), \label{y2} \\
& & (u_\pm-v \mp B_{ij} \hbar) H_i^+(u) E^{\pm}_j(v)
= (u_\pm-v \pm B_{ij} \hbar) E^{\pm}_j(v) H_i^+(u), \label{y3} \\
& & (u_\mp-v \mp B_{ij} \hbar) H_i^-(u) E^{\pm}_j(v)
= (u_\mp-v \pm B_{ij} \hbar) E^{\pm}_j(v) H_i^-(u), \label{y4} \\
& & (u-v \mp B_{ij} \hbar) E^{\pm}_i(u) E^{\pm}_j(v)
=  (u-v \pm B_{ij} \hbar) E^{\pm}_j(v) E^{\pm}_i(u), \label{y5} \\
& & [ E^+_i(u),~E^-_j(v) ] =
\frac{1}{\hbar} \delta_{ij} \left(
\delta( u_- - v_+) H^+_i(v_+) - \delta( u_+ - v_-) H^-_i(v_-) \right),
\label{y6} \\
& & E^{\pm}_i(u_1) E^{\pm}_i(u_2) E^{\pm}_j(v)
-2 E^{\pm}_i(u_1) E^{\pm}_j(v) E^{\pm}_i(u_2) \nonumber\\
& &~~~~~+ E^{\pm}_j(v) E^{\pm}_i(u_1) E^{\pm}_i(u_2) +
(\mbox{replacement:}~u_1