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

Abstract: We introduce a family of algebras $\mathcal{A}<em>{M,N}$, $M,N\in\mathbb{Z}$, as an extension of a pair of commuting quantum toroidal $\mathfrak{gl}_1$ 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 $\mathcal{A}</em>{\pm1,N}$ is a shifted quantum toroidal $\mathfrak{gl}<em>2$ algebra introduced in [FJM2]. Conjecturally there is a coproduct homomorphism $$\mathcal{A}</em>{M,N_1+N_2}\to\mathcal{A}<em>{M,N_1}\hat\otimes\mathcal{A}</em>{M,N_2}$$ to a completed tensor product, whose restriction to the subalgebras $\mathcal{E}<em>1,\check{\mathcal{E}}_1$ coincides with the standard coproduct of the latter. We give examples of $\mathcal{A}</em>{M,N}$ modules constructed on certain direct sums of tensor products of Fock modules of $\mathcal{E}_1\otimes\check{\mathcal{E}}_1$.