Graded quantum cluster algebras of infinite rank as colimits

TL;DR

This paper develops a framework for infinite-rank graded quantum cluster algebras, showing they can be constructed as colimits of finite-rank cases, and applies this to build an infinite quantum Grassmannian with a cluster structure.

Contribution

It introduces a graded quantum version of cluster algebras and demonstrates that infinite-rank cases are colimits of finite-rank algebras, extending previous constructions.

Findings

Infinite-rank graded quantum cluster algebras can be expressed as colimits of finite-rank algebras.

Constructed a graded quantum infinite Grassmannian with a cluster algebra structure for each k.

Extended earlier work on quantum Grassmannians to the infinite case.

Abstract

We provide a graded and quantum version of the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler and show that every graded quantum cluster algebra of infinite rank can be written as a colimit of graded quantum cluster algebras of finite rank. As an application, for each k we construct a graded quantum infinite Grassmannian admitting a cluster algebra structure, extending an earlier construction of the authors for k=2.