Repetitive cluster categories of type \(D_n\) (Q2107651)

Given an hereditary finite dimensional algebra \(\mathcal{H}\) over an algebraically closed field, the cluster category \(\mathcal{C}_{\mathcal{H}}\) is obtained from the derived category \(\mathcal{D}^b(\mathcal{H})\) by identifying the shift functor \([1]\) with the Auslander-Reiten translation \(\tau\). Let \(p\) be a positive integer and \(\mathcal{H}\) a hereditary abelian category with tilting objects. The repetitive cluster category \(\mathcal{C}_{F^p}(\mathcal{H})\) is defined as the orbit category of the bounded derived category \(\mathcal{D}^b(\mathcal{H})\) under the action of the cyclic group generated by the auto-equivalence \(F^p=(\tau^{-1}[1])^p\), where \(\tau\) is the Auslander--Reiten translation and \([1]\) is the shift functor. Repetitive cluster categories are triangulated and fractionally Calabi-Yau of dimension \(\frac{2p}{p}\). V. Gubitosi shows that the repetitive cluster category of type \(D_n\), defined as the orbit category \(\mathcal{D}^b(\mathrm{mod}\mathsf{k}\: D_n)/(\tau^{-1}[1])^p\), is equivalent to a category defined on a subset of tagged edges in a regular punctured polygon.