On the periodicity of the Auslander-Reiten translation and the Nakayama functor for the enveloping algebra of self-injective Nakayama algebras

Abstract

In this paper, we describe the structures of the left $B^e$-modules ${\tau }_{B^e}^i(B)$ and ${\mathcal{N}}_{B^e}^i(B)$ for $i⩾0$, where $B$ is a certain finite dimensional selfinjective Nakayama algebra, $B^e$ is the enveloping algebra of $B$, ${\tau }_{B^e}$ is the Auslander-Reiten translation in the category mod $(B^e)$ of finitely generated left $B^e$-modules and ${\mathcal{N}}_{B^e}:\text{mod}(B^e)\to \text{mod}(B^e)$ is the Nakayama functor. Moreover, we compute the ${\tau }_{B^e}$-period and the ${\mathcal{N}}_{B^e}$-period of $B$.