Recursive formulas for the Kronecker quantum cluster algebra with principal coefficients

TL;DR

This paper develops recursive formulas for the Kronecker quantum cluster algebra with principal coefficients using quantum Chebyshev polynomials, leading to new bar-invariant positive bases.

Contribution

It introduces explicit recursive formulas for the algebra and constructs two new bar-invariant positive bases, including the atomic basis.

Findings

Explicit recursive formulas for the algebra

Construction of two bar-invariant positive bases

Identification of the atomic basis

Abstract

We use the quantum version of Chebyshev polynomials to explicitly construct the recursive formulas for the Kronecker quantum cluster algebra with principal coefficients. As a byproduct, we obtain two bar-invariant positive $\mathbb{ZP}$-bases with one being the atomic basis.