A categorification of combinatorial Auslander-Reiten quivers

TL;DR

This paper categorifies combinatorial Auslander-Reiten quivers using derived categories of Ginzburg dg algebras, generalizing classical results and connecting to quantum Cartan matrices.

Contribution

It introduces a new categorification framework for combinatorial AR quivers within derived categories, extending classical results and linking to quantum algebra.

Findings

Defined subcategories $C([i])$ and $D([i])$ related to Weyl group elements

Extended mesh-additivity to arbitrary commutation classes

Connected the Euler form to the Cartan-Killing form and related to quantum Cartan matrices

Abstract

We provide a categorification of Oh and Suh's combinatorial Auslander-Reiten quivers in the simply laced case. We work within the perfectly valued derived category $\mathrm{pvd}(\Pi_Q)$ of the 2-dimensional Ginzburg dg algebra of a Dynkin quiver $Q$. For any commutation class $[i]$ of reduced words in the corresponding Weyl group, we define a subcategory $C([i])$ of $\mathrm{pvd}(\Pi_Q)$ whose objects are obtained by applying a sequence of spherical twist functors to the simple objects. We describe the Hom-order for $C([i])$ in terms of $[i]$, generalizing a result of B\'edard. Furthermore, when $[i]$ is a commutation class for the longest element, we construct a category $D([i])$ generalizing the bounded derived category of $Q$. It is realized as a certain subquotient of $\mathrm{pvd}(\Pi_Q)$. We demonstrate the existence of particular distinguished triangles in $\mathrm{pvd}(\Pi_Q)$ with corners in $D([i])$, which allows us to extend the classical mesh-additivity to arbitrary commutation classes. Additionally, we define an analog of the Euler form and prove that its symmetrization yields the corresponding Cartan-Killing form. For commutation classes $[i]$ arising from Q-data, a generalization of Dynkin quivers with a height function introduced by Fujita and Oh, we establish the existence of a partially Serre functor on $D([i])$. Lastly, we apply our results to reinterpret a formula by Fujita and Oh for the inverse of the quantum Cartan matrix.