τ2τ2-Stable tilting complexes over weighted projective lines

Let XX be a weighted projective line and cohX the associated category of coherent sheaves. We classify the tilting complexes T   in Db(cohX) such that τ2T≅Tτ2T≅T, where τ   is the Auslander–Reiten translation in Db(cohX). 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.