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

TL;DR

This paper introduces a shuffle algebra approach to GKLO-type homomorphisms for various quantum algebras, enabling the construction of commuting difference operators and proving a conjecture related to quantum Q-systems.

Contribution

It generalizes previous rational and type A constructions to shifted quantum affine, toroidal, and quiver algebras using shuffle realizations.

Findings

Constructed large families of commuting difference operators.

Provided a new approach to quantum Q-systems.

Proved a conjecture on quantum Q-systems.

Abstract

We present a shuffle realization of the GKLO-type homomorphisms for shifted quantum affine, toroidal, and quiver algebras, thus generalizing its rational version of arXiv:2104.14518 and the type A construction of arXiv:1811.12137. 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 arXiv:1704.00154.