Tame biserial algebras
A basic connected finite dimensional algebra \(A\) over an algebraically closed field \(K\) is called special biserial if \(A\) is isomorphic to a bounden quiver algebra \(KQ/I\) where \((Q,I)\) satisfies the following two conditions: (1) the number of arrows starting, respectively ending, at a fixed vertex of \(Q\) is bounded by 2; (2) for any arrow \(\alpha\) of \(Q\) there is at most one arrow \(\gamma\) and at most one arrow \(\sigma\) such that \(\alpha\gamma\) and \(\sigma\alpha\) are not in \(I\). It was shown by \textit{A. Skowroński} and \textit{J. Waschbüsch} [J. Reine Angew. Math. 345, 172--181 (1983; Zbl 0511.16021)] that the class of representation-finite special biserial algebras coincides with the class of representation-finite algebras \(A\) with \(\beta(A)\leq 2\), where \(\beta (A)\) records the largest possible number of non-projective-injective indecomposable direct summands in the middle terms of Auslander-Reiten sequences over \(A\). Here, using combinatorial Gelfand-Ponomarev methods [\textit{I. M. Gel'fand} and \textit{V. A. Ponomarev}, Usp. Mat. Nauk 23, No. 2(140), 3--60 (1968; Zbl 0236.22012)], one proves that any special biserial algebra \(A\) is tame and \(\beta(A)\leq 2\). An alternative proof of this result using new covering techniques, has been recently obtained by the reviewer and \textit{P. Dowber} in the paper ``Galois coverings of representation-infinite algebras'' [Comment. Math. Helv. 62, 311--337 (1987; Zbl 0628.16019)].
basic connected finite dimensional algebras
representation-finite special biserial algebras
non-projective-injective indecomposable direct summands
scientific article; zbMATH DE number 3904786
scientific article