Fundamental domains of cluster categories inside module categories

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.