Precellular Algebras

In this thesis we define a class of associative algebras ("precellular") by means of multiplicative properties of a basis and this class of algebras generalize cellular algebras introduced by Graham and Lehrer. We prove that precellualr algebras have cell representations and the structure of these cell representations depends on certain invariant bilinear forms. Next we discuss some questions about the homomorphisms between free modules of precellular algebras and the filtrations for projective modules of precellular algebras. In terms of the properties of certain invariant bilinear forms on the cell representations of precellular algebras, we give a complete parametrisation of irreducible modules of precellular algebras (up to equivalence), and obtain a general description of irreducible representations of precellular algebras. In addition, we give a sufficient condition to determine a smash product algebra to be a precellular algebra and under this condition we also give a construction of a precellular basis of this smash product algebra. Finally we give another kind of definition of a precellular algebra from the point of ring theory and prove that these two kinds of definitions are equivalent.