Affinization of shifted quantum affine $\mathfrak{gl}_2$

TL;DR

This paper introduces a new realization of quantum toroidal algebra for rak{gl}_2, defines its affinization for shifted quantum affine algebras, and constructs a broad class of their representations with applications to deformed W-algebras.

Contribution

It provides a novel realization of quantum toroidal algebra and constructs a family of representations for shifted quantum affine rak{gl}_2, linking to deformed W-algebras.

Findings

Realization A_0 of quantum toroidal algebra for rak{gl}_2

Definition of affinization A_N of shifted quantum affine rak{gl}_2

Construction of a large family of representations for A_N

Abstract

We give a realization $\mathcal{A}_0$ of quantum toroidal algebra associated to $\mathfrak{gl}_2$ which can be viewed as an affinization of the Drinfeld new realization of quantum affine $\mathfrak{gl}_2$. We use this realization to define an affinization $\mathcal{A}_N$, $N\in{\mathbb Z}$, of shifted quantum affine $\mathfrak{gl}_2$. We construct a large family of representations of dominantly shifted algebra $\mathcal A_N$, $N>0$. The examples of representations with even positive $N$ appear in the study of extensions of deformed $W$-algebras of type $\mathfrak{gl}(N+2|1)$.