Cluster-tilted algebras of type $D_n$

TL;DR

This paper characterizes when two cluster-tilted algebras of type D_n are isomorphic, showing they are so if and only if their tilting objects are related by specific automorphisms or translations.

Contribution

It provides a complete classification of isomorphism classes of cluster-tilted algebras of type D_n based on tilting object relations.

Findings

Isomorphism of cluster-tilted algebras corresponds to tilting objects related by $ au$ or $ au$ combined with automorphism $\sigma$.

The result applies for hereditary algebras of Dynkin type D_n with n ≥ 5.

The paper characterizes the automorphisms and translations that relate tilting objects leading to isomorphic algebras.

Abstract

Let $H$ be a hereditary algebra of Dynkin type $D_n$ over a field $k$ and $\mathscr{C}_H$ be the cluster category of $H$. Assume that $n\geq 5$ and that $T$ and $T'$ are tilting objects in $\mathscr{C}_H$. We prove that the cluster-tilted algebra $\Gamma=\mathrm{End}_{\mathscr{C}_H}(T)^{\rm op}$ is isomorphic to $\Gamma'=\mathrm{End}_{\mathscr{C}_H}(T')^{\rm op}$ if and only if $T=\tau^iT'$ or $T=\sigma\tau^jT'$ for some integers $i$ and $j$, where $\tau$ is the Auslander-Reiten translation and $\sigma$ is the automorphism of $\mathscr{C}_H$ defined in section 4.