A self-injective cellular algebra is weakly symmetric
type: Item
instance of: scholarly article
title: A self-injective cellular algebra is weakly symmetric
zbMATH Open document ID: 0953.16004
author: Changchang Xi
author: Steffen Koenig
published in: Journal of Algebra
publication date: 2000-11-07T00:00:00Z
review text: This paper is concerned with a hierarchy of four classes of finite-dimensional algebras: the class of semisimple algebras, which is contained in the class of symmetric algebras, which is contained in turn in the class of weakly symmetric algebras, which is contained in the class of quasi-Frobenius (self-injective) algebras. In general, each of these containments is proper. The authors restrict the above situation to the case where the finite-dimensional algebra is a cellular algebra in the sense of Graham-Lehrer. The main result is that a self-injective cellular algebra over a field is weakly symmetric, which shows in particular that the notions of ``symmetric'' and ``weakly symmetric'' coincide in the case of a cellular algebra. However, the other two containments in the hierarchy are shown to be strict. Some consequences of the main theorem given include results about cell filtrations for the projective indecomposable modules associated to self-injective cellular algebras. There are also explicit results on the structure of certain left ideals, each of which is generated by a cellular basis element.
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16D60
zbMATH DE Number: 1475237
zbMATH Keywords: quasi-Frobenius algebras
zbMATH Keywords: finite-dimensional algebras
zbMATH Keywords: semisimple algebras
zbMATH Keywords: weakly symmetric algebras
zbMATH Keywords: cell filtrations
zbMATH Keywords: projective indecomposable modules
zbMATH Keywords: self-injective cellular algebras
zbMATH Keywords: cellular bases
reviewed by: R. M. Green
MaRDI profile type: Publication
OpenAlex ID: W1992666380
Wikidata QID: Q122938815
cites work: Q5340151
cites work: Cellular algebras
cites work: Q4217072
cites work: On the number of cells of a cellular algebra
cites work: When is a cellular algebra quasi-hereditary?
DOI: 10.1006/JABR.1999.8037
description: scientific article; zbMATH DE number 1475237