Selfinjective algebras of Euclidean type with almost regular nonperiodic Auslander-Reiten components (Q2717707)

Start with a quiver of Euclidean type \(\Delta\) (without an oriented cycle). Take a tilting \(K\Delta\)-module \(T\). Then the endomorphism ring \(B=\text{ End}_{K\Delta}(T)\) is called a tilted algebra of type \(\Delta\). From this one now can construct a self-injective algebra by dividing the so-called repetitive algebra \(\widehat B\) by some admissible group of \(K\)-linear automorphisms \(G\). The resulting algebras are called self-injective algebras of tilted Euclidean type \(\Delta\).NEWLINENEWLINENEWLINEConsider now the Auslander-Reiten quiver of such a self-injective algebra of Euclidean type. A connected component of this Auslander-Reiten quiver is called almost regular if its stable part is obtained by omitting exactly one projective-injective module.NEWLINENEWLINENEWLINEThe main goal of the paper is, as the title suggests, to describe the selfinjective algebras of Euclidean tilted type whose nonperiodic components are all almost regular. Using results of the second author and \textit{K. Yamagata} [Proc. Lond. Math. Soc., III. Ser. 72, No. 3, 545-566 (1996; Zbl 0862.16001)] one obtains a complete description of all tame self-injective algebras which admit nonperiodic Auslander-Reiten components, and whose nonperiodic Auslander-Reiten components are all almost regular and generalized standard.