Homological Aspects Of Cellular Algebras

Cellular algebras were introduced by Graham and Lehrer as a convenient axiomatization of Ariki-Koiki Hecke algebras and related algebras. These arc a class of algebras which come equipped with a distinguished basis which is particularly well-adapted to the representation theory of the algebra. A number of well-known algebras such as the group algebras and Hecke algebras of symmetric groups, Brauer algebras, Temperley-Lieb algebras, and Birman-Wenzl algebras have been shown to be cellular.This dissertation focuses on the homological aspects of cellular algebras, including the quasi-heredity, the semi-simplicity, and the self-duality of projec-tive modules. The approach adopted here is largely homological, based upon the cell module filtration of projective modules and the dimension shifting of some exact sequences involving cell modules. The following are our main results. Some simpler homological characterizations of quasi-hereditary algebras inside the class of cellular algebras are presented in terms of cell modules. By using the cohomology groups of cell modules and simple modules, we then give some new criteria for the semi-simplicity of cellular algebras. Later, an interesting formula is provided to compute the Cartan determinants of cellular algebras. By applying this formula we obtain some descriptions of the cellular algebras with Cartan determinant two. A sufficient condition for a cellular algebra to be standardly stratified and an inductive construction of standardly stratified cellular algebras are also introduced subsequently. We next establish certain conditions which arc necessary and sufficient for a projective module over a cellular algebra to be injective. These conditions are also shown to be necessary and sufficient for an injective module over a cellular algebra to be projective. The final work is devoted to investigating a new class of algebras which we will refer to as cyclotomic blob algebras. After describing the cellular structure of such algebras, we classify all irreducible representations and determine for which parameters the algebras are quasi-hereditary. Finally, we show a branching rule for the cell modules.