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 \(\mathfrak {gl}_1\) 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 \(\mathcal {A}_{\pm 1,N}\) is a shifted quantum toroidal \(\mathfrak {gl}_2\) algebra introduced in Feigin et al (Affinization of shifted quantum affine gl2. arXiv:2511.12178). Conjecturally there is a coproduct homomorphism \(\mathcal {A}_{M,N_1+N_2}\rightarrow \mathcal {A}_{M,N_1}{\hat{\otimes }}\mathcal {A}_{M,N_2}\) 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 \({\mathcal {E}}_1\otimes \check{{\mathcal {E}}}_1\) .
