Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems

We present a shuffle realization of the GKLO-type homomorphisms for shifted quantum affine, toroidal, and quiver algebras in the spirit of Feigin and Odesskii (Funktsional. Anal. Prilozhen. 31(3):57–70, 1997), thus generalizing its rational version of Frassek and Tsymbaliuk (Commun. Math. Phys. 392:545–619, 2022) and the type A construction of Finkelberg and Tsymbaliuk (Arnold Math. J. 5(2–3):197–283, 2019). As an application, this allows us to construct large families of commuting and q-commuting difference operators, in particular, providing a convenient approach to the Q-systems where it proves a conjecture of Di Francesco and Kedem (Commun. Math. Phys. 369(3):867–928, 2019).