A self-injective cellular algebra is weakly symmetric
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.
scientific article; zbMATH DE number 1475237