Quantization of canonical bases and the quantum symplectic double

TL;DR

This paper introduces a q-deformation of the canonical basis for cluster varieties related to type A quivers and extends it to the symplectic double, enriching the algebraic structure with quantum features.

Contribution

It presents a natural q-deformation of the canonical basis and extends the construction to the symplectic double cluster variety, advancing the understanding of quantum cluster algebras.

Findings

Defined a q-deformation of the canonical basis for type A cluster varieties.

Extended the construction to include the symplectic double cluster variety.

Provided a framework for quantum deformations in cluster algebra theory.

Abstract

We describe a natural $q$-deformation of Fock and Goncharov's canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type $A$. We then describe an extension of this construction involving a cluster variety called the symplectic double.