Extensions of a commuting pair of quantum toroidal ${\mathfrak{g}\mathfrak{l}}_1$

We introduce a family of algebras ${\mathcal{A}}_{M,N}$, $M,N\in \mathbb{Z}$, as an extension of a pair of commuting quantum toroidal ${\mathfrak{g}\mathfrak{l}}_1$ subalgebras ${\mathcal{E}}_1,{\check{\mathcal{E}}}_1$, wherein the parameters are tuned in a specific way according to $M,N$. In the case $M=\pm 1$, algebra ${\mathcal{A}}_{\pm 1,N}$ is a shifted quantum toroidal ${\mathfrak{g}\mathfrak{l}}_2$ algebra introduced in [FJM2]. Conjecturally there is a coproduct homomorphism ${\mathcal{A}}_{M,N_1+N_2}\to {\mathcal{A}}_{M,N_1}\widehat{\otimes }{\mathcal{A}}_{M,N_2}$ to a completed tensor product, whose restriction to the subalgebras ${\mathcal{E}}_1,{\check{\mathcal{E}}}_1$ coincides with the standard Drinfeld coproduct. We give examples of ${\mathcal{A}}_{M,N}$ modules constructed on certain direct sums of tensor products of Fock modules of ${\mathcal{E}}_1\otimes {\check{\mathcal{E}}}_1$.