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

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 ${\text{A}}_c({\mathfrak{gl}}_N)$ of the completed double Yangian ${\widetilde{\text{DY}}}_c({\mathfrak{gl}}_N)$ 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 ${\tilde{\text{A}}}_c({\mathfrak{gl}}_N)$.

24 pages

References in corpus (5)

• Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence
• On higher order Sugawara operators
• Center of the quantum affine vertex algebra in type A
• Higher Sugawara operators for the quantum affine algebras of type A
• Modules-at-infinity for quantum vertex algebras

Cited by in corpus (4)

• $h$-adic quantum vertex algebras associated with rational $R$-matrix in types $B$, $C$ and $D$
• Double Yangian and reflection algebras of the Lie superalgebra ${\mathfrak{gl}}_{m|n}$
• Quantum current algebras associated with rational $R$-matrix
• A note on constructing quasi modules for quantum vertex algebras from twisted Yangians