On the quasi-heredity and the semi-simplicity of cellular algebras.
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)].
scientific article; zbMATH DE number 1967931