Tame triangular matrix algebras over self-injective algebras (Q1100262)

Let A be a basic connected finite dimensional algebra over an algebraically closed field and \(T_ 2(A)\) be the algebra of \(2\times 2\) upper triangular matrices over A. It is known [the reviewer, Bull. Pol. Acad. Sci., Math. 34, 517-523 (1986; Zbl 0612.16016)] that, if \(T_ 2(A)\) is tame, then A is representation-finite. Here, one proves that, for A selfinjective, \(T_ 2(A)\) is representation-infinite and tame if and only if A is of Dynkin class \(A_ 3\). The proof applies \textit{C. M. Ringel}'s theory of tubular extensions [Tame algebras and integral quadratic forms (Lect. Notes Math. 1099, 1984; Zbl 0546.16013)] and Galois covering techniques [\textit{P. Dowbor}, \textit{H. Lenzing} and the reviewer, Lect. Notes Math. 1177, 91-93 (1986; Zbl 0567.16022)].