Physics
There is a rich interplay between physics and various aspects of representation theory. My interest in this area is from the algebraic side, in some of the methods and examples in representation theory that have been inspired by physical problems.
Even from this restricted point of view, I will only consider two particular topics here; diagram algebras and quantum groups. There are many other connections between algebra and physics, but a proper discussion of these is beyond the scope of these notes. A more extended discussion of the physical ideas behind some of these topics (and the kinds of algebras that arise) can be found in my Benin lecture notes.
Diagram algebras
While there is no definitive definition of a diagram algebra, they are essentially algebras with a basis given by some set of diagrams, such that multiplication of basis elements corresponds to some graphical operation (typically concatenation), along with some elementary relations between diagrams.
The archetypal example of such an algebra is the Temperley-Lieb algebra. This first arose in algebraic statistical mechanics, and has been generalised in a variety of different directions. In most cases, the graphical description of such algebras is motivated by some physical model; however, by their algebraic description (in terms of generators and relations) they can be seem to be closely connected to algebras of long-standing interest, such as the group algebras of symmetric groups.
These algebras are typically quasi-hereditary, and come in families ordered by inclusion. Together with Paul Martin, Alison Parker, and Changchang Xi, I have given an axiomatisation covering many such examples in terms of towers of recollement.
With John Graham and Paul Martin I have also analysed the representation theory of a particular family of algebras, the blob algebras. This work is of interest in the study of affine Hecke algebras, as it has been shown that our results are enough to determine decomposition numbers for certain standard modules in that setting.
Although not traditionally considered in this manner, the group algebras of a symmetric group can also be realised as a diagram algebra. Generalising this realisation, one can define the Brauer algebra. Implicit in recent work with Maud De Visscher and Paul Martin on this algebra is an application of diagrammatic methods (and in particular of towers of recollement).
Quantum groups
Quantum groups first arose in the 1980s, and have since been widely studied both in algebraic and physically motivated contexts. There are a number of definitions of these in the literature, but the most widely used is in terms of certain deformations (depending on some parameter q) of the universal enveloping algebra associated to a semi-simple (or Kac-Moody) Lie algebra.
Physically, such algebras are of interest because they give rise to solutions of the Yang-Baxter equation. Algebraists on the other hand have been very interested in quantum groups due to the very close relationship between their representation theory and that of the corresponding algebraic group.
Representations of algebraic groups over the complex numbers are very well understood; over fields with positive characteristic very little is known. Quantum groups over the complex numbers with q a root of unity have an intermediate degree of complexity, and can be used to derive results for algebraic groups in positive characteristic. For example, this approach has proved the most powerful tool yet found for understanding simple modules.
Almost every aspect of the study of representations of algebraic groups can be replicated in the quantum setting. Examples include my calculations (and those done jointly with Karin Erdmann) of certain Ext-groups for quantum general linear groups
The close relationship of the symmetric group to the general linear group (via the Schur algebra) has a corresponding relation in the quantum world involving Hecke algebras. Many of the diagram algebras considered so far can be realised as quotients of these same Hecke algebras (or affine versions thereof), which provides another motivation for their study.
Anton Cox
(A.G.Cox@city.ac.uk)
Last modified: Tue 7 Feb 2006