Quantum vertex algebra associated to quantum toroidal ${\mathfrak{gl}}_N$

Abstract

In this paper, we associate the quantum toroidal algebra ${\mathcal{E}}_N$ of type ${\mathfrak{gl}}_N$ with quantum vertex algebra through equivariant $\ddot{I}•$-coordinated quasi modules. More precisely, for every $\ell \in \mathbb{C}$, by deforming the universal affine vertex algebra of ${\mathfrak{sl}}_{\infty }$, we construct an $\hslash$-adic quantum $\mathbb{Z}$-vertex algebra $V_{{\widehat{\mathfrak{sl}}}_{\infty },\hslash }(\ell ,0)$. Then we prove that the category of restricted ${\mathcal{E}}_N$-modules of level $\ell$ is canonically isomorphic to that of equivariant $\ddot{I}•$-coordinated quasi $V_{{\widehat{\mathfrak{sl}}}_{\infty },\hslash }(\ell ,0)$-modules.