Cluster-tilted algebras of type $D_n$

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 $Γ=\mathrm{End}_{\mathscr{C}_H}(T)^{\rm op}$ is isomorphic to $Γ'=\mathrm{End}_{\mathscr{C}_H}(T')^{\rm op}$ if and only if $T=τ^iT'$ or $T=στ^jT'$ for some integers $i$ and $j$, where $τ$ is the Auslander-Reiten translation and $σ$ is the automorphism of $\mathscr{C}_H$ defined in section 4.

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