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

TL;DR

This paper constructs a quantum vertex algebra associated with the quantum toroidal algebra of type rak_N and establishes an isomorphism between their module categories, advancing the understanding of quantum algebraic structures.

Contribution

It introduces a new quantum vertex algebra linked to quantum toroidal rak_N and proves a categorical equivalence with restricted modules of the algebra.

Findings

Construction of rak_ ext{hbar}-adic quantum rak_ ext{infinity} vertex algebra

Categorical isomorphism between restricted rak_N-modules and equivariant oordinated quasi modules

Deformation of universal affine vertex algebra of rak_ ext{infinity}

Abstract

In this paper, we associate the quantum toroidal algebra $\mathcal{E}_N$ of type $\mathfrak{gl}_N$ with quantum vertex algebra through equivariant $\phi$-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 $\hbar$-adic quantum $\Z$-vertex algebra $V_{\widehat{\mathfrak{sl}}_{\infty},\hbar}(\ell,0)$. Then we prove that the category of restricted $\mathcal{E}_N$-modules of level $\ell$ is canonically isomorphic to that of equivariant $\phi$-coordinated quasi $V_{\widehat{\mathfrak{sl}}_{\infty},\hbar}(\ell,0)$-modules.