Explicit cluster multiplication formulas for the quantum cluster algebra of type $A_2^{(1)}$

TL;DR

This work provides explicit quantum cluster multiplication formulas for the affine quiver $A_2^{(1)}$ with principal coefficients, enabling exact expressions of every quantum cluster variable as a polynomial in the initial cluster variables from one-step mutations. It constructs a bar-invariant positive $\\mathbb{ZP}$-basis by combining quantum cluster monomials with families $u_n\omega^{k}$ and $u_n z^{k}$, offering a concrete positive basis in the quantum setting. The results extend classical atomic-basis constructions to the quantum domain and supply computational tools for studying quantum cluster algebras of affine type. Together, these contributions enhance our ability to compute and understand the structure constants and positivity phenomena in quantum cluster algebras of type $A_2^{(1)}$ with principal coefficients.

Abstract

Let $Q$ be an affine quiver of type $A_2^{(1)}$. We explicitly construct the cluster multiplication formulas for the quantum cluster algebra of $Q$ with principal coefficients. As applications, we obtain: (1)\ an exact expression for every quantum cluster variable as a polynomial in terms of the quantum cluster variables in clusters which are one-step mutations from the initial cluster; (2)\ an explicit bar-invariant positive $\mathbb{ZP}$-basis.

Theorems & Definitions (29)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• proof
• Lemma 3.5