Representations of non-splittable extension algebras
type: Item
instance of: scholarly article
title: Representations of non-splittable extension algebras
zbMATH Open document ID: 0644.16020
DOI: 10.1016/0021-8693(88)90281-5
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let \(A\) be a basic finite dimensional algebra over a field \(k\) whose ordinary quiver contains no oriented cycles. Denote by \(\text{mod\,}A\) the category of finitely generated left \(A\)-modules, by \(\text{mod}_ pA\) the full subcategory of \(\text{mod\,}A\) formed by modules without projective direct summands and by \(D:\text{mod\,}A\rightleftharpoons\text{mod\,}A^{op}\) a self-duality. Moreover, let \(T\) be an extension over \(A\) with kernel \(DA\) and \(A\ltimes DA\) be the trivial extension of \(A\) by \(DA\). It is shown that \(\text{mod}_ pT\) is equivalent to \(\text{mod}_ pA\ltimes DA\), and the factor algebra \(T/soc T\) is isomorphic to the trivial extension \(A\ltimes(DA/\text{soc\,}DA)\). Moreover, the Auslander-Reiten quivers of \(T\) and \(A\ltimes DA\) are isomorphic. As a consequence one obtains that \(T\) is of finite representation type iff so is \(A\ltimes DA\).
reviewed by: Andrzej Skowroński
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16S20
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 4049819
zbMATH Keywords: basic finite dimensional algebras
zbMATH Keywords: categories of finitely generated left modules
zbMATH Keywords: self-dualities
zbMATH Keywords: trivial extensions
zbMATH Keywords: Auslander-Reiten quivers
zbMATH Keywords: finite representation type
author: Kunio Yamagata
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90281-5
OpenAlex ID: W2025846774
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description: scientific article; zbMATH DE number 4049819