Cluster-tilted algebras of type $D_n$

Abstract

Let $H$ be a hereditary algebra of Dynkin type $D_n$ over a field $k$ and ${\mathcal{C}}_H$ be the cluster category of $H$. Assume that $n\ge 5$ and that $T$ and $T^{\prime }$ are tilting objects in ${\mathcal{C}}_H$. We prove that the cluster-tilted algebra $\Gamma ={\mathrm{End}}_{{\mathcal{C}}_H}(T{)}^{\mathrm{op}}$ is isomorphic to ${\Gamma }^{\prime }={\mathrm{End}}_{{\mathcal{C}}_H}(T^{\prime }{)}^{\mathrm{op}}$ if and only if $T={\tau }^i{T}^{\prime }$ or $T=\sigma {\tau }^j{T}^{\prime }$ for some integers $i$ and $j$, where $\tau$ is the Auslander-Reiten translation and $\sigma$ is the automorphism of ${\mathcal{C}}_H$ defined in section 4.

21 pages, 13 figures, accepted for publication in Comm.Algebra

References in corpus (2)

• The mutation class of $D_n$ quivers
• The number of elements in the mutation class of a quiver of type $D_n$

Cited by in corpus (1)

• The number of elements in the mutation class of a quiver of type $D_n$