Extensions of a Commuting Pair of Quantum Toroidal gl1

We introduce a family of algebras AM,N, M,N∈Z, as an extension of a pair of commuting quantum toroidal gl1 subalgebras E1,Eˇ1, wherein the parameters are tuned in a specific way according to M, N. In the case M=±1, algebra A±1,N is a shifted quantum toroidal gl2 algebra introduced in Feigin et al (Affinization of shifted quantum affine gl2. arXiv:2511.12178). Conjecturally there is a coproduct homomorphism AM,N1+N2→AM,N1⊗^AM,N2 to a completed tensor product, whose restriction to the subalgebras E1,Eˇ1 coincides with the standard Drinfeld coproduct. We give examples of AM,N modules constructed on certain direct sums of tensor products of Fock modules of E1⊗Eˇ1.