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arXiv · 2504.10182

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

Danting Yang·Xueqing Chen·Ming Ding·Fan Xu
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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.

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Danting Yang, Xueqing Chen, Ming Ding, Fan Xu. 2025-04-14. Explicit cluster multiplication formulas for the quantum cluster algebra of type $A_2^{(1)}$. https://arxiv.org/abs/2504.10182

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