Presentations of trivial extensions of finite dimensional algebras and a theorem of Sheila Brenner
type: Item
instance of: scholarly article
title: Presentations of trivial extensions of finite dimensional algebras and a theorem of Sheila Brenner
zbMATH Open document ID: 1009.16013
author: Elsa A. Fernández
author: M. I. Platzeck
published in: Journal of Algebra
publication date: 2003-02-03T00:00:00Z
full work available at URL: http://hdl.handle.net/11336/78775
review text: Let \(Q\) be a finite connected quiver and \(k\) an algebraically closed field. Consider a finite dimensional algebra \(\Lambda\) of the form \(kQ/I\), where \(I\) is an admissible ideal in the path algebra \(kQ\) of \(Q\). If \(D(\Lambda)\) denotes the dual algebra then the trivial extension of \(\Lambda\) is the algebra \(T(\Lambda)\) equal to \(\Lambda\oplus D(\Lambda)\) as a \(k\)-vector space, equipped with the multiplication given by \((\lambda,f)(\mu,g)=(\lambda\mu,\lambda g+f\mu)\) [see \textit{Y. Iwanaga} and \textit{T. Wakamatsu} in Representation theory II, Lect. Notes Math. 832, 295-301 (1980; Zbl 0445.16024)]. The main results of the paper describe the presentation of \(T(\Lambda)\) and of the repetitive algebra \(\widehat\Lambda\) [see \textit{D. Hughes} and \textit{J. Waschbüsch}, Proc. Lond. Math. Soc., III. Ser. 46, 347-364 (1983; Zbl 0488.16021)] by quivers with relations, under the assumption that every oriented cycle in \(Q\) is zero in \(\Lambda\). A result of \textit{S. Brenner} [Arch. Math. 62, No. 3, 203-206 (1994; Zbl 0796.16016)] is applied in this situation to obtain a characterization of the number of indecomposable direct summands of the middle term of the almost split sequence starting at a simple \(T(\Lambda)\)-module.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 1750358
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: quivers
zbMATH Keywords: admissible ideals
zbMATH Keywords: trivial extensions
zbMATH Keywords: repetitive algebras
zbMATH Keywords: path algebras
zbMATH Keywords: almost split sequences
zbMATH Keywords: simple modules
reviewed by: Stanisław Kasjan
MaRDI profile type: Publication
OpenAlex ID: W2021391520
cites work: Q4318498
cites work: The derived equivalence classification of representation-finite selfinjective algebras
cites work: The almost split sequence starting with a simple module
cites work: Trivial extensions of Abelian categories. Homological algebra of trivial extensions of Abelian categories with applications to ring theory
cites work: Q3859949
cites work: Universal Galois coverings of self-injective algebras by repetitive algebras and Hochschild cohomology
cites work: Q3931576
DOI: 10.1006/JABR.2001.9056
description: scientific article; zbMATH DE number 1750358