On the quasi-heredity and the semi-simplicity of cellular algebras.
type: Item
instance of: scholarly article
title: On the quasi-heredity and the semi-simplicity of cellular algebras.
zbMATH Open document ID: 1039.16006
DOI: 10.1016/S0021-8693(03)00259-X
author: Yongzhi Cao
published in: Journal of Algebra
publication date: 2003-08-20T00:00:00Z
review text: Cellular algebras were introduced by \textit{J. J. Graham} and \textit{G. I. Lehrer} [in Invent. Math. 123, No. 1, 1--34 (1996; Zbl 0853.20029)] by the existence of a basis with a certain property. It turns out that this class of algebras appears very commonly in many fields in mathematics and physics. In fact, many cellular algebras even possess a strong homological property, that is, they are quasi-hereditary. In the present paper the author uses the basis-free definition of cellular algebras given by \textit{S. König} and the reviewer [in Algebras and modules II, CMS Conf. Proc. 24, 365--386 (1998; Zbl 0926.16016)] to characterize the quasi-heredity and the semisimplicity of cellular algebras. This is a very nice paper: One of the main results in the present paper generalizes the Ext\(^1\)-homological description of quasi-heredity given by the reviewer [Math. Proc. Camb. Philos. Soc. 133, No. 1, 37--53 (2002; Zbl 1026.16001)] in a substantial way. This new description is simpler but surprising. The other main result provides a set of equivalent conditions for a cellular algebra to be semisimple. They are given in terms of the vanishing of the first cohomology groups between cell modules and co-cell modules. This can be considered as a homological characterization of semisimplicity of cellular algebras. For a numerical characterization one may see the earlier papers of \textit{J. J. Graham} and \textit{G. I. Lehrer} [loc. cit.] and the reviewer and \textit{D.-J. Xiang} [Linear Algebra Appl. 365, 369--388 (2003; Zbl 1067.16023)].
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16E30
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16E60
zbMATH DE Number: 1967931
zbMATH Keywords: cellular algebras
zbMATH Keywords: quasi-hereditary algebras
zbMATH Keywords: semisimple algebras
zbMATH Keywords: cell modules
reviewed by: Changchang Xi
MaRDI profile type: Publication
cites work: Q3804882
cites work: Quasi-hereditary algebras
cites work: Q4028607
cites work: Cellular algebras
cites work: Q4217072
cites work: When is a cellular algebra quasi-hereditary?
cites work: Q3495525
cites work: The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences
cites work: On the quasi-heredity of Birman-Wenzl algebras
cites work: Standardly stratified algebras and cellular algebras
cites work: Cellular algebras and Cartan matrices.
description: scientific article; zbMATH DE number 1967931