$t$-Analog of $q$-Characters, Bases of Quantum Cluster Algebras, and a Correction Technique

TL;DR

This paper introduces a new geometric framework for quantum cluster algebras using graded quiver varieties and a $t$-deformation, enabling the construction of various bases through a correction technique.

Contribution

It generalizes previous results to all acyclic quantum cluster algebras with arbitrary coefficients, introducing a correction method for basis construction.

Findings

Constructed a new family of graded quiver varieties.

Developed a $t$-deformation of Grothendieck rings.

Established bases for quantum cluster algebras using correction techniques.

Abstract

We first study a new family of graded quiver varieties together with a new $t$-deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint paper by Yoshiyuki Kimura and the author. We further generalize the result of that paper to any acyclic quantum cluster algebra with arbitrary nondegenerate coefficients. In particular, we obtain the generic basis, the dual PBW basis, and the dual canonical basis. The method consists in a correction technique, which works for general quantum cluster algebras.