A note on constructing quasi modules for quantum vertex algebras from twisted Yangians

TL;DR

This paper constructs quasi modules for quantum vertex algebras using twisted Yangians related to orthogonal and symplectic Lie algebras, providing explicit formulas for central elements and invariants.

Contribution

It introduces a subalgebra of the double Yangian and employs its structure to build quasi modules for quantum affine vertex algebras, extending the understanding of their centers and invariants.

Findings

Constructed examples of quasi modules for quantum affine vertex algebras.

Derived explicit formulas for central elements in the algebra.

Identified invariants of vacuum modules related to twisted Yangians.

Abstract

In this note, we consider the twisted Yangians $\text{Y}(\mathfrak{g}_N)$ associated with the orthogonal and symplectic Lie algebras $\mathfrak{g}_N=\mathfrak{o}_N,\mathfrak{sp}_N$. First, we introduce a certain subalgebra $\text{A}_c(\mathfrak{g}_N)$ of the double Yangian for $\mathfrak{gl}_N$ at the level $c\in\mathbb{C}$, which contains the centrally extended $\text{Y}(\mathfrak{g}_N)$ at the level $c$ as well as its vacuum module $\mathcal{M}_c(\mathfrak{g}_N)$. Next, we employ its structure to construct examples of quasi modules for the quantum affine vertex algebra $\mathcal{V}_c(\mathfrak{gl}_N)$ associated with the Yang $R$-matrix. Finally, we use the description of the center of $\mathcal{V}_c(\mathfrak{gl}_N)$ to obtain explicit formulae for families of central elements for a certain completion of $\text{A}_c(\mathfrak{g}_N)$ and invariants of $\mathcal{M}_c(\mathfrak{g}_N)$.