[Paper Review] Relative cluster categories and Higgs categories with infinite-dimensional morphism spaces

Chris Fraser, Bernhard Keller+1 more|arXiv (Cornell University)|Jul 23, 2023

Algebraic structures and combinatorial modelsMathematics3 citations

TL;DR

This paper extends the categorification of cluster algebras with coefficients by constructing Higgs categories and relative cluster categories in the relative Jacobi-infinite setting, generalizing previous Frobenius and cluster category frameworks. It introduces a canonical cluster character that lifts cluster variables and clusters to the categorical level, enabling a categorical realization of cluster combinatorics even when morphism spaces are infinite-dimensional.

ABSTRACT

Cluster algebras *with coefficients* are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells, ... . The approach of Geiss-Leclerc-Schröer often yields Frobenius exact categories which allow to categorify such cluster algebras. In previous work, the third-named author has constructed Higgs categories and relative cluster categories in the relative Jacobi-finite setting (arXiv:2109.03707). Higgs categories generalize the Frobenius categories used by Geiss-Leclerc-Schröer. In this article, we construct the Higgs category and the relative cluster category in the relative Jacobi-infinite setting under suitable hypotheses. These cover for example the case of Jensen-King-Su's Grassmannian cluster category. As in the relative Jacobi-finite case, the Higgs category is no longer exact but still extriangulated in the sense of Nakaoka-Palu. We also construct a cluster character refining Plamondon's. In the appendix, Chris Fraser and the second-named author categorify quasi-cluster morphisms using Frobenius categories. A recent application of this result is due to Matthew Pressland, who uses it to prove a conjecture by Muller-Speyer.

Motivation & Objective

• To generalize the categorification of cluster algebras with coefficients beyond the Jacobi-finite case, where Frobenius categories fail.
• To construct Higgs categories and relative cluster categories in the relative Jacobi-infinite setting, covering important geometric examples like Grassmannian and positroid cluster categories.
• To define a canonical cluster character in this generalized setting that lifts cluster variables and clusters to the categorical level.
• To establish a categorical framework for quasi-cluster morphisms using Frobenius categories, enabling applications to cluster algebra isomorphisms.
• To provide a categorical foundation for recent conjectures, such as those by Muller–Speyer, via the derived category and K-theory techniques.

Proposed method

• Construct the relative Ginzburg algebra $\bm{\Gamma}(Q,F,W)$ from an ice quiver with potential $(Q,F,W)$, using the relative Jacobian algebra $J(Q,F,W) = H^0(\bm{\Gamma})$.
• Define the Higgs category $\mathcal{H}(Q,F,W)$ as a full subcategory of the perfect derived category $\mathrm{per}\,\bm{\Gamma}$, closed under extensions and satisfying extriangulated structure via Nakaoka–Palu theory.
• Define the relative cluster category $\mathcal{C}(Q,F,W)$ as the idempotent completion of the quotient of $\mathrm{per}\,\bm{\Gamma}$ by the thick subcategory generated by simple $H^0(\bm{\Gamma})$-modules.
• Introduce a canonical cluster character refining Plamondon’s construction, mapping isomorphism classes of objects in $\mathcal{H}(Q,F,W)$ to Laurent polynomials in the cluster algebra.
• Use K-theory isomorphisms $K_0(\mathcal{P}) \xrightarrow{\sim} K_0(\mathcal{P}')$ and triangle functors between bounded derived categories to induce ring isomorphisms between cluster algebras.
• Apply the appendix result to categorify quasi-cluster morphisms via Frobenius categories, enabling a proof of a conjecture by Muller–Speyer.

Experimental results

Research questions

• RQ1 Can Higgs categories and relative cluster categories be constructed in the relative Jacobi-infinite setting, beyond the previously known Jacobi-finite case?
• RQ2 Does a canonical cluster character exist in the relative Jacobi-infinite setting that lifts cluster variables and clusters to the categorical level?
• RQ3 Can the categorical framework support the categorification of quasi-cluster morphisms, particularly in the context of cluster algebra isomorphisms?
• RQ4 How do K-theory isomorphisms and triangle functors between derived categories relate to cluster algebra isomorphisms in this generalized setting?
• RQ5 Can this framework be applied to prove open conjectures in cluster algebra theory, such as those by Muller–Speyer?

Key findings

• The Higgs category $\mathcal{H}(Q,F,W)$ is extriangulated in the sense of Nakaoka–Palu, even though it is not exact in the Quillen sense, generalizing Frobenius categories to infinite-dimensional morphism spaces.
• The relative cluster category $\mathcal{C}(Q,F,W)$ is well-defined in the relative Jacobi-infinite setting under suitable hypotheses, covering examples such as Jensen–King–Su’s Grassmannian cluster category.
• A canonical cluster character is constructed that generalizes Plamondon’s character and lifts cluster variables and clusters to the categorical level.
• The cluster character induces a bijection between isomorphism classes of reachable rigid indecomposable objects in $\mathcal{H}(Q,F,W)$ and cluster variables in the associated cluster algebra.
• A triangle functor $F: \mathcal{D}^b(\mathcal{E}) \to \mathcal{D}^b(\mathcal{E}')$ inducing a $K_0$-isomorphism and a triangle equivalence on stable categories gives rise to a unique quasi-cluster isomorphism $f: \mathcal{A} \xrightarrow{\sim} \mathcal{A}'$.
• The appendix establishes a Frobenius categorification of quasi-cluster morphisms, which Matthew Pressland uses to prove a conjecture by Muller–Speyer, demonstrating the framework’s applicability to open problems.