Tame algebras with sincere directing modules
This paper gives a significant contribution to the study of tame algebras \(\Lambda\), in case they have a sincere directing module [see \textit{C. M. Ringel}, ``Tame algebras and integral quadratic forms'' (Lect. Notes Math. 1099, 1984; Zbl 0546.16013)]. In \S 1, a good description is given of the Auslander-Reiten quiver of \(\Lambda\). This can also be obtained from the results of \textit{A. Skowroński} and \textit{S. Smalø} [J. Algebra 147, 137-146 (1992; Zbl 0746.16008)]. In \S2, it is shown that the Euler form of \(\Lambda\) controls \(\text{mod} \Lambda\) (and, hence, it determines the dimension vectors of all the indecomposable modules). The main theorem is the following (see \S3) Theorem: Let \(\Lambda\) be a tame algebra with a sincere directing module. Then, \(\Lambda\) is domestic with at most two 1-parameters. It is also shown that, in case \(\Lambda\) is domestic in two 1-parameters, \(\Lambda\) is a glueing of two tilted algebras that are domestic in one 1- parameter. The author intends to use these and other results to give, in near future, a classification of all tame algebras admitting a sincere directing module.
scientific article; zbMATH DE number 554268