Coil enlargements of algebras
type: Item
instance of: scholarly article
title: Coil enlargements of algebras
zbMATH Open document ID: 0860.16014
author: Ibrahim Assem
author: Bertha Tomé
author: Andrzej Skowroński
published in: Tsukuba Journal of Mathematics
publication date: 1997-04-20T00:00:00Z
review text: 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 \(A\) whic…
reviewed by: Hagen Meltzer
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G30
zbMATH DE Number: 895949
zbMATH Keywords: finite dimensional basic connected algebras
zbMATH Keywords: orthogonal components
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: weakly separating families of components
zbMATH Keywords: indecomposable modules
zbMATH Keywords: coil enlargements
zbMATH Keywords: admissible operations
zbMATH Keywords: stable tubes
zbMATH Keywords: corays
zbMATH Keywords: rays
zbMATH Keywords: tubes
zbMATH Keywords: category of modules
zbMATH Keywords: branch coextensions
zbMATH Keywords: tame concealed algebras
zbMATH Keywords: tameness
zbMATH Keywords: multicoil algebras
zbMATH Keywords: Tits forms
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496162881
OpenAlex ID: W1539296059
DOI: 10.21099/TKBJM/1496162881
description: scientific article; zbMATH DE number 895949