A categorification of acyclic principal coefficient cluster algebras

TL;DR

We address categorifying acyclic cluster algebras with polarised principal coefficients by constructing a frozen Jacobian algebra $A_{Q,W}$ from $(\widetilde{Q},\widetilde{F},\widetilde{W})$, proving it is bimodule internally $3$-Calabi–Yau with respect to the frozen idempotent and hence yields a boundary algebra $B_Q$ with a Frobenius category $\mathcal{E}_Q=\operatorname{GP}(B_Q)$. The stable category $\underline{\mathcal{E}}_Q$ is triangle equivalent to the cluster category $\mathcal{C}_Q$, and a Fu–Keller cluster character gives a bijection between indecomposable rigid objects and cluster variables of the polarised principal coefficient cluster algebra $\widetilde{\mathscr{A}}_{Q}$, with seeds in bijection with cluster-tilting objects, and mutation commuting with endomorphism algebras. By partial stabilisation, we obtain an extriangulated categorification $\mathcal{E}^+_Q$ of the principal coefficient cluster algebra $\mathscr{A}_{Q}^{\bullet}$, equivalent to Fu–Keller's realisation via a subcategory $\mathcal{U}_Q$, and show how to recover g-vectors and dimension vectors from indices and Euler characteristics. The paper also provides explicit boundary-algebra presentations, proves mutation compatibility, and presents concrete examples (including $A_2$, $A_3$, and Grassmannian cases) illustrating the framework and its connection to the bimodule–Calabi–Yau perspective of Van den Bergh.

Abstract

In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi-Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi-Yau algebras for cluster algebras with polarised principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables), and obtain Frobenius categorifications in the acyclic case. Via partial stabilisation, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh's result that the Ginzburg dg-algebra of a quiver with potential is bimodule 3-Calabi-Yau.

Figures (3)

• Figure 1: The Auslander--Reiten quiver of $\operatorname{GP}(B_Q)$ for $Q$ of type $\mathsf{A}_2$.
• Figure 2: The Auslander--Reiten quiver of $\operatorname{GP}(B_Q)$ for $Q$ linearly oriented of type $\mathsf{A}_3$. In addition to the the usual mesh relations coming from Auslander--Reiten sequences, the length two path from $P_2$ to $P_5$ represents the zero map.
• Figure 3: The Auslander--Reiten quiver of $\operatorname{GP}(B_{Q,W})$, where $(Q,W)$ is a $3$-cycle and its usual potential, shown as the Grassmannian cluster category $\operatorname{CM}(B_{2,6})$.

Theorems & Definitions (91)

• Theorem 1: Categorification for polarised principal coefficients
• Theorem 2: Corollary of Theorem \ref{['t:bi3cy']}
• Corollary 1: Categorification for principal coefficients
• Conjecture 1
• Definition 2.1
• Remark 2.2
• Proposition 2.3
• proof
• Definition 2.4: Pressland-iCY
• Remark 2.5