d-Representation-finite self-injective algebras

In this paper, we initiate the study of higher-dimensional Auslander-Reiten theory of self-injective algebras. We give a systematic construction of (weakly) $d$-representation-finite self-injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the Serre functor. The condition holds, in particular, for all fractionally Calabi-Yau algebras of global dimension at most $d$. This generalizes Riedtmann's classical construction of representation-finite self-injective algebras. Our method is based on an adaptation of Gabriel's covering theory for $k$-linear categories to the setting of higher-dimensional Auslander-Reiten theory. Applications include $n$-fold trivial extensions and (classical and higher) preprojective algebras, which are shown to be $d$-representation-finite in many cases. We also get a complete classification of all $d$-representation-finite self-injective Nakayama algebras for arbitrary $d$.