Topological Quantum Groups in Action

Supervisor: Dr Matthew Daws

This is a project at the interface of Mathematical Analysis with Algebra.  The classical area of Abstract Harmonic Analysis is interested in studying groups and algebras built from these groups, and properties of these algebras.  Typically our groups have some notion of topology (which might be the trivial discrete topology for an infinite group, or Lie groups coming from geometry, or more exotic topological groups).  As our groups are infinite, the algebras we study are topological, which is where Analysis becomes important.

A motivating example in this area is that of Pontryagin Duality, which generalises the idea of Fourier Series and Transforms.  This says that each abelian locally compact group admits a dual group, the bidual is the original group, and there is an analogue of the Fourier transform defined on function spaces of the group and its dual.  In particular, we obtain an equivalence between the L^1 algebra of a group and the Fourier algebra of its dual.  For non-abelian groups, we now longer have this theory.  However, we can still talk about the L^1 algebra and the Fourier algebra, and the formal idea that these algebras are dual to one another is highly profitable.  Various attempts have been made to turn this vague idea into a mathematical theory, culminating in the theory of Locally Compact Quantum Groups, a complicated but complete generalisation of Pontryagin Duality.  Surprisingly, this theory also encompasses many examples of “topological” quantum groups coming from motivation from Physics.

This PhD project is part of a programme of developing a theory of Abstract Harmonic Analysis for such locally compact quantum groups.  In particular, we are interested in the idea of a quantum group “action”, which is a generalisation of the notion of an action of a group on a space.  The mathematical tools involved are the theory of Operator Algebras, Banach Algebras, associated techniques from functional analysis, and some techniques from pure algebra.