Algebra
Algebra underpins all of my work; here I will briefly review the main techniques from algebra that are most ubiquitous in this area.
Many of the algebras that arise in my work are finite dimensional, and there is a well-developed general theory of such algebras. However, most of the algebras that I study have an additional quasi-hereditary (or cellular) structure, and these are the typical methods used in their study.
Quasi-hereditary algebras
Quasi-hereditary algebras are a class of algebras which have many features in common with Lie objects. In Lie theory one often has some naturally arising 'standard modules' (such as Weyl modules) which are relatively well understood, and which can be used to study the simple modules. Indeed, the study of simple modules is usually recast as the determination of decomposition numbers for such standard modules.
Quasi-heredity abstracts some of the key properties of standard modules to a general algebraic setting. This allows for the use of the language of Lie theory (such as indexing representations by weights), and also provides some powerful tools for analysing such algebras. For example, any quasi-hereditary algebra has finite global dimension.
Many of the algebras which I study are quasi-hereditary, and also fall naturally into towers. With Martin, Parker, and Xi I have given an axiom scheme for studying such algebras which we called towers of recollement. Recent work with De Visscher and Martin has also developed translation functors in this general setting, again inspired by Lie theory.
Cellular algebras
Cellular algebras were introduced to identify a class of algebras with representation theories similar to that of the symmetric group. As first introduced, this was an entirely combinatorial definition, although later versions are more algebraic.
Cellular algebras can be quasi-hereditary, but are in general harder to understand. Often we study classes of algebras which are quasihereditary except at certain 'degenerate' parameter choices, and at these values they are typically cellular. Further, the combinatorial conditions for cellularity, which are often difficult to check in general, are usually obvious for a diagram algebra. Thus diagram algebras provide a large class of algebras amenable to these techniques.
Anton Cox
(A.G.Cox@city.ac.uk)
Last modified: Mon 26 May 2008