Categorification and the quantum Grassmannian

TL;DR

This paper extends the categorification of Grassmannian cluster algebras to quantum cluster algebras using Cohen-Macaulay modules, establishing a link to quantum Grassmannians.

Contribution

It constructs a compatible pair from a cluster tilting object in Cohen-Macaulay modules, enabling the definition of quantum cluster algebras related to quantum Grassmannians.

Findings

Quantum cluster algebra is isomorphic to the quantum Grassmannian in certain cases.

Constructs a compatible pair (B, L) from a cluster tilting object.

Extends categorification to quantum setting for Grassmannian cluster algebras.

Abstract

In \cite{JKS} we gave an (additive) categorification of Grassmannian cluster algebras, using the category $\CM(A)$ of Cohen-Macaulay modules for a certain Gorenstein order $A$. In this paper, using a cluster tilting object in the same category $\CM(A)$, we construct a compatible pair $(B, L)$, which is the data needed to define a quantum cluster algebra. We show that when $(B, L)$ is defined from a cluster tilting object with rank 1 summands, this quantum cluster algebra is (generically) isomorphic to the corresponding quantum Grassmannian.