Semi-infinite construction for the double Yangian of type $A_1^{(1)}$

TL;DR

This paper constructs semi-infinite bases for level 1 modules of the double Yangian of type A1^(1), using the Iohara-Kohno realization, and explores their applications in quantum affine vertex algebra representation theory.

Contribution

It introduces a new semi-infinite monomial basis for modules of the double Yangian of type A1^(1), advancing the understanding of their structure and applications.

Findings

Existence of Feigin-Stoyanovsky-type bases for these modules

Bases expressed in terms of semi-infinite monomials of integrable operators

Applications to the representation theory of quantum affine vertex algebras

Abstract

We consider certain infinite dimensional modules of level 1 for the double Yangian $\text{DY}(\mathfrak{gl}_2)$ which are based on the Iohara-Kohno realization. We show that they possess topological bases of Feigin-Stoyanovsky-type, i.e. the bases expressed in terms of semi-infinite monomials of certain integrable operators which stabilize and satisfy the difference two condition. Finally, we give some applications of these bases to the representation theory of the corresponding quantum affine vertex algebra.