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

Abstract

We consider the Etingof-Kazhdan quantum vertex algebra $\mathcal{V}^c(R)$ associated with the trigonometric and elliptic $R$-matrix of type $A.$ We establish a connection between (restricted) modules for the $h$-Yangian $\textrm{Y}_h(\mathfrak{gl}_N)$ and the elliptic quantum algebra $\mathcal{A}_{h,p}(\widehat{\mathfrak{gl}}_2)$ 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 $\textrm{Y}_h(\mathfrak{gl}_N).$