[Paper Review] A geometric realization of tame categories

Karin Baur, Hermund André Torkildsen|arXiv (Cornell University)|Feb 23, 2015

Algebraic structures and combinatorial models11 references3 citations

TL;DR

This paper provides a geometric realization of the module category and cluster category of type $\tilde{A}_n$ using oriented arcs in an annulus. It introduces 'long moves' between arcs to combinatorially model elements of the infinite radical, establishing an isomorphism between a geometric quiver with long moves and the truncated module categories $\mathcal{J}_m(Q_{g,h})$, thereby offering a complete geometric description of the infinite radical and extending to cluster categories via unoriented arcs.

ABSTRACT

We give a geometric realization of module categories of type $ ilde{A}_n$. We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type $ ilde{A}_n$. To get a description of the module category, we introduce long moves between arcs. These allow us to include the infinite radical in the geometric description. Finally, our results can also be used to describe the corresponding cluster categories by taking unoriented arcs instead.

Motivation & Objective

• To provide a geometric, combinatorial description of the infinite radical in the module category of type $\tilde{A}_n$.
• To extend geometric models of module categories to include elements beyond the Auslander-Reiten quiver by introducing long moves between arcs.
• To establish a correspondence between the geometric quiver with long moves and the truncated module categories $\mathcal{J}_m(Q_{g,h})$.
• To adapt the geometric model to describe subcategories of the cluster category of type $\tilde{A}_n$ using unoriented arcs.

Proposed method

• Model indecomposable modules in preprojective and preinjective components using oriented arcs in an annulus with marked boundary points.
• Define a translation quiver $\Gamma$ isomorphic to the Auslander-Reiten quiver of $\operatorname{mod} kQ_{g,h}$ via arc moves.
• Introduce 'long moves' between arcs as a new geometric tool to represent morphisms in the infinite radical $\operatorname{rad}^\infty(\operatorname{mod} \tilde{A})$.
• Construct a new quiver $\overline{\Gamma}$ by adding arrows corresponding to long moves to the original quiver $\Gamma$, preserving the structure of the AR quiver.
• Prove that the full subquiver $\overline{\Gamma}_m$ of $\overline{\Gamma}$ is isomorphic to the truncated quiver $Q_m$ from previous work, thus realizing $\mathcal{J}_m(Q_{g,h})$ geometrically.
• Extend the model to cluster categories by replacing oriented arcs with unoriented arcs and adding a new slice of vertices to model the orbit category structure.

Experimental results

Research questions

• RQ1 How can the infinite radical of the module category of type $\tilde{A}_n$ be geometrically realized using arc configurations in an annulus?
• RQ2 What combinatorial operation on arcs corresponds to morphisms in the infinite radical, and how can it be formalized?
• RQ3 Can the truncated module categories $\mathcal{J}_m(Q_{g,h})$ be fully described via a geometric quiver model incorporating long moves?
• RQ4 How does the geometric model extend to describe subcategories of the cluster category of type $\tilde{A}_n$?
• RQ5 What is the role of unoriented arcs and the added slice of vertices in modeling the cluster category’s structure?

Key findings

• The geometric model using oriented arcs in an annulus realizes the Auslander-Reiten quiver of $\operatorname{mod} kQ_{g,h}$ as a translation quiver $\Gamma$ isomorphic to the AR quiver.
• Long moves between arcs are introduced as a geometric representation of morphisms in the infinite radical $\operatorname{rad}^\infty(\operatorname{mod} \tilde{A})$, providing a combinatorial interpretation of these elements.
• The quiver $\overline{\Gamma}_m$, formed by adding long move arrows to $\Gamma$, is isomorphic to the truncated quiver $Q_m$ from [4], thus fully realizing $\mathcal{J}_m(Q_{g,h})$ geometrically.
• The construction extends to the cluster category of type $\tilde{A}_n$ by using unoriented arcs and adding a new slice of vertices $\{\underline{\eta_i}\}$, resulting in a stable translation quiver $\underline{\Gamma}_m$.
• The $k$-category of $\underline{\Gamma}_m$ is equivalent to the full subcategory $\mathcal{J}'_m(\mathcal{C}_{\tilde{A}})$ of the cluster category, establishing a geometric description of its subcategories.