Coil enlargements of algebras
The authors generalize the notion of a separating tubular family introduced by \textit{C. M. Ringel} [Tame algebras and integral quadratic forms (Lect. Notes Math. 1099, 1984; Zbl 0546.16013)] in the following way. Let \(A\) be a finite dimensional, basic and connected \(k\)-algebra over an algebraically closed field \(k\). A family of standard, pairwise orthogonal components \({\mathcal T}=({\mathcal T}_i)_{i\in I}\) of the Auslander-Reiten quiver of \(A\) is called a weakly separating family if the indecomposable modules not in \(\mathcal T\) split into two classes \(\mathcal P\) and \(\mathcal Q\) such that there is no non-zero morphism from \(\mathcal Q\) to \(\mathcal P\), from \(\mathcal Q\) to \(\mathcal T\), or from \(\mathcal T\) to \(\mathcal P\), while any non-zero morphism from \(\mathcal P\) to \(\mathcal Q\) factors through the additive subcategory generated by \(\mathcal T\). Given an algebra \(A\) having a weakly separating family, an algebra \(B\) is called a coil enlargement of \(A\) using modules from \(\mathcal T\) if \(B\) is obtained from \(A\) by an iteration of admissible operations (see the recent papers of the first two authors [Math. Scand. 71, No. 1, 31-61 (1992; Zbl 0796.16014) and CMS Conf. Proc. 14, 29-68 (1993; Zbl 0827.16010)]) performed either on a stable tube of \(\mathcal T\) or on a coil obtained from a stable tube of \(\mathcal T\) by means of the operations done so far. There are also introduced numerical invariants \(c^-_B\) and \(c^+_B\) which measure respectively the number of corays and rays inserted in the tubes of \(\mathcal T\) by this sequence of admissible operations. The authors prove the following results for the category of modules over a coil enlargement algebra \(B\). (a) \(B\) has a weakly separating family \({\mathcal T}'\) of coils obtained from the stable family of tubes \(\mathcal T\) by the corresponding sequence of admissible operations, (b) there is a unique maximal branch coextension \(B^-\) of \(A\) which is a full subcategory of \(B\), and \(c^-_B\) is the coextension type of \(B^-\), (c) there is a unique maximal branch extension \(B^+\) of \(A\) which is a full subcategory of \(B\), and \(c^+_B\) is the extension type of \(B^+\), (d) each indecomposable \(B\)-module is in \({\mathcal P}'\), \({\mathcal T}'\) or \({\mathcal Q}'\), where \({\mathcal P}'\) consists of indecomposable \(B^-\)-modules, and \({\mathcal Q}'\) consists of indecomposable \(B^+\)-modules. If, in particular, \(A\) is a tame concealed algebra and \(\mathcal T\) is its separating tubular family, the following criteria for tameness are obtained: \(B\) is tame if and only if \(B^-\) and \(B^+\) are tame if and only if \(B\) is a multicoil algebra, or if and only if the Tits form of \(B\) is weakly non-negative. A generalization of the algebras characterized here was given by the third author [in Fundam. Math. 146, No. 3, 251-266 (1995)].
finite dimensional basic connected algebras
weakly separating families of components
scientific article; zbMATH DE number 895949