Generic Variables in Acyclic Cluster Algebras and Bases in Affine Cluster Algebras

Abstract

Let $Q$ be a finite quiver without oriented cycles and $\mathcal{A}(Q)$ be the coefficient-free cluster algebra with initial seed $(Q,\text{u})$. Using the Caldero-Chapoton map, we introduce and investigate a family of generic variables in $\mathbb{Z}[{\text{u}}^{\pm 1}]$ containing the cluster monomials of $\mathcal{A}(Q)$. The aim of these generic variables is to give an explicit new method for constructing $\mathbb{Z}$-bases in the cluster algebra $\mathcal{A}(Q)$. If $Q$ is an affine quiver with minimal imaginary root $\delta$, we investigate differences between cluster characters associated to indecomposable representations of dimension vector $\delta$. We define the notion of \emph{difference property} which gives an explicit description of these differences. We prove in particular that this property holds for quivers of affine type $\tilde{A}$. When $Q$ satisfies the difference property, we prove that generic variables span the cluster algebra $\mathcal{A}(Q)$. If $\mathcal{A}(Q)$ satisfies some gradability condition, we prove that generic variables are linearly independent over $\mathbb{Z}$ in $\mathcal{A}(Q)$. In particular, this implies that generic variables form a $\mathbb{Z}$-basis in a cluster algebra associated to an affine quiver of type $\tilde{A}$.

63 pages. v2: Title changed since the first part of this article can now be found as an independent article under the initial title

References in corpus (5)

• On triangulated orbit categories
• Acyclic Calabi-Yau categories
• Cluster algebra structures and semicanonical bases for unipotent groups
• Denominators of cluster variables
• Cluster multiplication in regular components via generalized Chebyshev polynomials

Cited by in corpus (12)

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• Quiver Grassmannians associated with string modules
• Bases for cluster algebras from orbifolds
• Atomic bases in cluster algebras of types A and $\tilde{A}$
• Dual canonical bases and quantum cluster algebras
• Cluster multiplication in regular components via generalized Chebyshev polynomials
• Quasi-cluster algebras from non-orientable surfaces
• Notes on the cluster multiplication formulas for 2-Calabi-Yau categories
• Positivity for Regular Cluster Characters in Acyclic Cluster Algebras
• The multiplication theorem and bases in finite and affine quantum cluster algebras
• Cluster algebras and their bases
• Transverse Quiver Grassmannians and Bases in Affine Cluster Algebras