Quantum current algebras associated with rational $R$-matrix

Abstract

We study quantum current algebra $\textrm{A}(\overline{R})$ associated with the rational $R$-matrix of $\mathfrak{gl}_N$ and we give explicit formulae for the elements of its center at the critical level. Due to Etingof--Kazhdan's construction, the level $c$ vacuum module $\mathcal{V}_c(\overline{R})$ for the algebra $\textrm{A}(\overline{R})$ possesses a quantum vertex algebra structure for any complex number $c$. We prove that any module for the quantum vertex algebra $\mathcal{V}_c(\overline{R})$ is naturally equipped with a structure of restricted $\textrm{A}(\overline{R})$-module of level $c$ and vice versa.