Repetitive cluster-tilted algebras

TL;DR

This paper studies the properties of cluster tilting objects and the structure of repetitive cluster-tilted algebras in the context of hereditary algebras, extending known results to a more general setting.

Contribution

It generalizes a key theorem about cluster-tilted algebras to the repetitive cluster category setting and proves the connectedness of the tilting graph in this context.

Findings

Properties of cluster tilting objects in repetitive cluster categories

Structure of repetitive cluster-tilted algebras

Connectedness of the tilting graph in this setting

Abstract

Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field $k$ and $\mathscr{C}_{F^m}$ be the repetitive cluster category of $H$ with $m\geq 1$. We investigate the properties of cluster tilting objects in $\mathscr{C}_{F^m}$ and the structure of repetitive cluster-tilted algebras. Moreover, we generalized Theorem 4.2 in \cite{bmrrt} (Buan A, Marsh R, Reiten I. Cluster-tilted algebra. Trans. Amer. Math. Soc., 359(1)(2007), 323-332.) to the situation of $\mathscr{C}_{F^m}$, and prove that the tilting graph $\mathscr{K}_{\mathscr{C}_{F^m}}$ of $\mathscr{C}_{F^m}$ is connected.