Toroidal Grothendieck rings and cluster algebras

TL;DR

This paper introduces toroidal cluster algebras as deformations of classical cluster algebras with quantum parameters, connecting them to Grothendieck rings of quantum affine algebra representations.

Contribution

It constructs toroidal Grothendieck rings and proves they are flat deformations, establishing a link between these rings and toroidal cluster algebra structures.

Findings

Toroidal Grothendieck rings are flat deformations of classical Grothendieck rings.

Toroidal cluster algebra structures are established for certain monoidal categories.

The work connects quantum affine algebra representations with toroidal cluster algebras.

Abstract

We study deformations of cluster algebras with several quantum parameters, called toroidal cluster algebras, which naturally appear in the study of Grothendieck rings of representations of quantum affine algebras. In this context, we construct toroidal Grothendieck rings and we establish these are flat deformations of Grothendieck rings. We prove that for a family of monoidal categories $\mathscr{C}_1$ of simply-laced quantum affine algebras categorifying finite-type cluster algebras, the toroidal Grothendieck ring has a natural structure of a toroidal cluster algebra.