  We study Serre structures on categories enriched in pivotal monoidal
categories, and apply this to study Serre structures on two types of graded
k-linear categories: categories with group actions and categories with graded
hom spaces. We check that Serre structures are preserved by taking orbit
categories and skew group categories, and describe the relationship with graded
Frobenius algebras. Using a formal version of Auslander-Reiten translations, we
show that the derived category of a d-representation finite algebra is
fractionally Calabi-Yau if and only if its preprojective algebra has a graded
Nakayama automorphism of finite order. This connects various results in the
literature and gives new examples of fractional Calabi-Yau algebras.
