Quasi modules for the quantum affine vertex algebra in type $A$

TL;DR

This paper constructs quasi modules for the quantum affine vertex algebra of type A using subalgebras related to the reflection equation, and derives formulas for central elements in the completed algebra.

Contribution

It introduces subalgebras associated with the reflection equation and constructs quasi modules for the quantum affine vertex algebra in type A.

Findings

Construction of quasi modules for the quantum affine vertex algebra.

Explicit formulas for central elements in the completed algebra.

Connection between subalgebras and the structure of the vertex algebra.

Abstract

We consider the quantum affine vertex algebra $\mathcal{V}_{c}(\mathfrak{gl}_N)$ associated with the rational $R$-matrix, as defined by Etingof and Kazhdan. We introduce certain subalgebras $\textrm{A}_c (\mathfrak{gl}_N)$ of the completed double Yangian $\widetilde{\textrm{DY}}_{c}(\mathfrak{gl}_N)$ at the level $c\in\mathbb{C}$, associated with the reflection equation, and we employ their structure to construct examples of quasi $\mathcal{V}_{c}(\mathfrak{gl}_N)$-modules. Finally, we use the quasi module map, together with the explicit description of the center of $\mathcal{V}_{c}(\mathfrak{gl}_N)$, to obtain formulae for families of central elements in the completed algebra $\widetilde{\textrm{A}}_c (\mathfrak{gl}_N)$.