Tilted string algebras
In the paper under review the authors give combinatorial criteria for string algebras to be tilted or quasitilted. Recall that a string algebra is an algebra \(A=kQ/I\) where \(k\) is a field and the ideal \(I\) is generated by paths and the quiver \(Q\) has the property that each vertex is the start-point of at most two arrows and the end-point of at most two arrows and, moreover, for each arrow \(\alpha\) there is at most one arrow \(\beta\) such that \(\alpha\beta\not\in I\) and at most one arrow \(\gamma\) such that \(\gamma\alpha\not\in I\) [\textit{M. C. R. Butler} and \textit{C. M. Ringel}, Commun. Algebra 15, 145-179 (1987; Zbl 0612.16013)]. It is shown that a string algebra \(A=kQ/I\) is quasitilted if and only if there is no sectional pair of zero-relations in \((Q,I)\). Moreover, \(A\) is tilted if and only if the following two conditions are satisfied: (1) there is no sectional pair of zero-relations in \((Q,I)\). (2) If \(\Theta\) is a simple cycle of \(Q\) containing no zero-relation, then the arrows attached to \(\Theta\) either all enter \(\Theta\) or all leave \(\Theta\).
scientific article; zbMATH DE number 1538663
scientific article