Quantum cluster algebras and fusion products

TL;DR

This paper explores the connection between quantum cluster algebras, quantum $Q$-systems, and fusion products, providing new proofs and reformulations of key identities and coefficients in the context of quantum affine algebras.

Contribution

It establishes a relationship between quantum $Q$-systems and graded tensor products, proving the graded $M=N$ identities and expressing fusion coefficients as matrix elements in quantum $Q$-system representations.

Findings

Proved the graded $M=N$ identities.

Expressed fusion coefficients as matrix elements.

Reformulated quantum $Q$-system relations in the context of fusion products.

Abstract

$Q$-systems are recursion relations satisfied by the characters of the restrictions of special finite-dimensional modules of quantum affine algebras. They can also be viewed as mutations in certain cluster algebras, which have a natural quantum deformation. In this paper, we explain the relation in the simply-laced case between the resulting quantum $Q$-systems and the graded tensor product of Feigin and Loktev. We prove the graded version of the $M=N$ identities, and write expressions for these as non-commuting evaluated multi-residues of suitable products of solutions of the quantum $Q$-system. This leads to a simple reformulation of Feigin and Loktev's fusion coefficients as matrix elements in a representation of the quantum $Q$-system algebra.