Graded quantum cluster algebras and an application to quantum Grassmannians

TL;DR

This paper develops a framework for compatible gradings on quantum cluster algebras, enabling the construction of new quantum cluster structures on quantum Grassmannians, extending classical results and previous quantum matrix work.

Contribution

It introduces a grading framework compatible with mutation for quantum cluster algebras and applies it to establish quantum Grassmannians as quantum cluster algebras.

Findings

Quantum Grassmannians admit quantum cluster algebra structures.

The grading framework produces new quantum cluster algebras with modified quasi-commutation.

Extension of quantum matrix cluster algebra structures to Grassmannians.

Abstract

We introduce a framework for $\mathbb{Z}$-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation. The resulting grading has the property that every (quantum) cluster variable is homogeneous. In the quantum setting, we use this grading framework to give a construction that behaves somewhat like twisting, in that it produces a new quantum cluster algebra with the same cluster combinatorics but with different quasi-commutation relations between the cluster variables. We apply these results to show that the quantum Grassmannians admit quantum cluster algebra structures, as quantizations of the cluster algebra structures on the classical Grassmannian coordinate ring found by Scott. This is done by lifting the quantum cluster algebra structure on quantum matrices due to Gei{\ss}, Leclerc and Schr\"{o}er and completes earlier work of the authors on the finite-type cases.