A Note on Constructing Quasi Modules for Quantum Vertex Algebras from 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 \(\mathcal {M}_c(\mathfrak {g}_N)\).