Quantum cluster algebras and their specializations

TL;DR

This paper demonstrates that under certain conditions, quantum cluster algebras can be seen as flat deformations of classical cluster algebras, linking their algebraic structures.

Contribution

It establishes a connection between classical and quantum cluster algebras via flat deformations when specific conditions are met.

Findings

Quantum cluster algebra is a flat deformation of the classical cluster algebra.

The result applies when the cluster algebra equals its upper cluster algebra and admits a suitable grading.

Provides a new perspective on the relationship between classical and quantum cluster algebras.

Abstract

We show that in case a cluster algebra coincides with its upper cluster algebra and the cluster algebra admits a grading with finite dimensional homogeneous components, the corresponding Berenstein-Zelevinsky quantum cluster algebra can be viewed as a flat deformation of the classical cluster algebra.