τ²-stable tilting complexes over weighted projective lines

Let X be a weighted projective line and coh X the associated category of coherent sheaves. We classify the tilting complexes T in D^b(coh X) such that τ²T ≅ T , where τ is the Auslander–Reiten translation in D^b(coh X). As an application of this result, we classify the 2-representation-finite algebras which are derived-equivalent to a canonical algebra. This complements Iyama-Oppermann’s classification of the iterated tilted 2-representation-finite algebras. By passing to 3-preprojective algebras, we obtain a classification of the selfinjective cluster-tilted algebras of canonical-type. This complements Ringel’s classification of the selfinjective cluster-tilted algebras.