Fundamental domains of cluster categories inside module categories

TL;DR

This paper describes the fundamental domain of cluster categories within module categories for hereditary algebras, providing a new way to understand cluster-tilting objects and cluster-tilted algebras.

Contribution

It offers a novel description of the fundamental domain and cluster-tilting objects in terms of modules over a tilted algebra, linking cluster categories to module categories.

Findings

Explicit description of the fundamental domain in the derived category.

Characterization of cluster-tilting objects via modules over tilted algebras.

Method to determine the quiver of any cluster-tilted algebra.

Abstract

Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field, and let $\mathcal{C}_{H}$ be the corresponding cluster category. We give a description of the (standard) fundamental domain of $\mathcal{C}_{H} $ in the bounded derived category $\mathcal{D}^{b}(H)$, and of the cluster-tilting objects, in terms of the category $\mod\Gamma $\ of finitely generated modules over a suitable tilted algebra $% \Gamma .$ Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra.