Cell algebras

A family of associative algebras called cell algebras is defined and studied. These algebras generalize the cellular algebras of Graham and Lehrer. Standard results for cellular algebras carry over nicely to the more general cell algebras, including the characterization of their irreducible modules in terms of a bilinear form, a description of their decomposition and Cartan matrices, and a description of their hereditary ideals and possible quasi-hereditary algebra structures.As examples of cell algebras which are not cellular, the semigroup algebras R[Tr]R[Tr] and R[PTr]R[PTr] corresponding to the full transformation semigroup TrTr and the partial transformation semigroup PTrPTr are shown to be cell algebras and cell algebra bases are obtained for these (and related) algebras. The general cell algebra theory is then applied to classify the irreducible representations of these algebras (when R is any field of characteristic 0 or p). In certain cases the algebras are found to be quasi-hereditary.