On an orbit algebra induced by the Auslander-Reiten translation for the enveloping algebra of a self-injective Nakayama algebra

Abstract

Let $A$ be a basic self-injective Nakayama algebra over an algebraically closed field. In this paper, we investigate the ring structure of the orbit algebra $\mathbb{A}\left({\tau }_{A^e};A\right)={\oplus }_{i\ge 0}{\underset{\_}{\text{Hom}}}_{A^e}({\tau }_{A^e}^i(A),A)$, where $A^e$ is the enveloping algebra of $A$ and ${\tau }_{A^e}$ is the Auslander-Reiten translation for $A^e$.