Associating modules for the $h$-Yangian and quantum elliptic algebra in type $A$ with $h$-adic quantum vertex algebras

We consider the Etingof-Kazhdan quantum vertex algebra ${\mathcal{V}}^c(R)$ associated with the trigonometric and elliptic $R$-matrix of type $A\mathrm{.}$ We establish a connection between (restricted) modules for the $h$-Yangian ${\text{Y}}_h({\mathfrak{g}\mathfrak{l}}_N)$ and the elliptic quantum algebra ${\mathcal{A}}_{h,p}({\widehat{\mathfrak{g}\mathfrak{l}}}_2)$ of level zero, and deformed (twisted) $\phi$-coordinated ${\mathcal{V}}^c(R)$-modules. As its application, in the trigonometric case, we construct new families of central elements of ${\mathcal{V}}^c(R)$ at the critical level $c=-N,$ which we then use to derive commutative families in the $h$-Yangian ${\text{Y}}_h({\mathfrak{g}\mathfrak{l}}_N)\mathrm{.}$