Approximations of algebras by standardly stratified algebras.
type: Item
instance of: scholarly article
title: Approximations of algebras by standardly stratified algebras.
zbMATH Open document ID: 1147.16008
author: István Ágoston
author: Vlastimil Dlab
author: Erzsébet Lukács
published in: Journal of Algebra
publication date: 2008-07-29T00:00:00Z
review text: Let \(A\) be a finite-dimensional associative algebra with a fixed (natural) linear order on the set \(e_1,\dots,e_n\) of primitive orthogonal idempotents. For every \(i=1,\dots,n\) we define the standard module \(\Delta(i)\) as the quotient of \(Ae_i\) modulo the trace of all \(Ae_j\), \(j>i\). The proper standard module \(\overline\Delta(i)\) is defined as the quotient of \(\Delta(i)\) modulo the trace of \(Ae_i\) in the radical of \(\Delta(i)\). Denote by \(\mathcal F(\Delta,A)\) and \(\mathcal F(\overline\Delta,A)\) the full subcategories of \(A\)-mod, consisting of all modules which have a filtration, whose subquotients are standard or proper standard modules, respectively. The principal question addressed in the paper under review is: Given some algebra \(A\), does there exist a quasi-hereditary algebra \(B\) such that the subcategories \(\mathcal F(\Delta,A)\) and \(\mathcal F(\Delta,B)\) are equivalent? The authors show that for any \(A\) there is a unique basic algebra \(\Sigma(A)\) such that \(\Sigma(A)\) is standardly stratified and the categories \(\mathcal F(\Delta,A)\) and \(\mathcal F(\Delta,\Sigma(A))\) are equivalent. Further, there is a unique basic algebra \(\Omega(A)\) such that \(\Omega(A)^{op}\) is standardly stratified and the categories \(\mathcal F(\overline\Delta,A)\) and \(\mathcal F(\overline\Delta,\Sigma(A))\) are equivalent. It is further shown that the algebra \((\Omega\Sigma)^{n-1}(A)\) is always properly stratified. The paper contains several interesting examples.
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 5304643
zbMATH Keywords: stratified algebras
zbMATH Keywords: quasi-hereditary algebras
zbMATH Keywords: approximations
zbMATH Keywords: standard modules
zbMATH Keywords: basic algebras
zbMATH Keywords: category equivalences
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.02.017
OpenAlex ID: W2015331097
cites work: Stratifying endomorphism algebras
cites work: Q4338466
cites work: Properly stratified algebras
cites work: Q4028607
cites work: On Standardly Stratified Algebras
cites work: Stratifying systems via relative simple modules.
cites work: Stratifying Systems via Relative Projective Modules
Wikidata QID: Q105657283
DOI: 10.1016/J.JALGEBRA.2008.02.017
description: scientific article; zbMATH DE number 5304643