Stable equivalence of representation-finite trivial extension algebras (Q1079640)

If the trivial extension of a finite-dimensional algebra over an algebraically closed field is representation-finite, then its Cartan class is one of the Dynkin diagrams \(A_ n\), \(D_ n\), \(E_ 6\), \(E_ 7\), \(E_ 8\) [see \textit{C. Riedtmann}, Comment. Math. Helv. 55, 199-224 (1980; Zbl 0444.16018)]. The authors show that if two trivial extensions are also basic and connected, then they are stably equivalent if and only if they have the same Cartan class. The proof utilizes the techniques of Auslander-Reiten sequences and of tilted algebras and rests on the results of \textit{D. Hughes} and \textit{J. Waschbüsch} [Proc. Lond. Math. Soc., III. Ser. 46, 347-364 (1983; Zbl 0488.16021)] and \textit{H. Tachikawa} [J. Algebra 90, 98-118 (1984; Zbl 0542.16029)].