Affinization of shifted quantum affine ${\mathfrak{g}\mathfrak{l}}_2$

We give a realization ${\mathcal{A}}_0$ of quantum toroidal algebra associated to ${\mathfrak{g}\mathfrak{l}}_2$ which can be viewed as an affinization of the Drinfeld new realization of quantum affine ${\mathfrak{g}\mathfrak{l}}_2$. We use this realization to define an affinization ${\mathcal{A}}_N$, $N\in \mathbb{Z}$, of shifted quantum affine ${\mathfrak{g}\mathfrak{l}}_2$. 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 $\mathfrak{g}\mathfrak{l}(N+2\mathrm{\mid }1)$.