Citation: Van Daele, Alfons. "An algebraic framework for group duality." Advances in Mathematics 140.2 (1998): 323-366.
Web: https://core.ac.uk/download/pdf/82490226.pdf
Tags: Non-finite/semisimple, Hopf-algebras
This paper introduces a lovely algebraic framework for group duality based on multiplier Hopf algebras. The idea is that the dual of a Hopf algebra is not a Hopf algebra because the dual of tensor products is not equal to the tensor product of the duals. The framework used by the author is thus based on multiplier Hopf algebras. In a multiplier Hopf algebra, the comultiplication is not required to have codomain equal to the tensor product of it with itself. It is only required to have codomain equal to the space of "multipliers" on the tensor product with itself. This definition is nice because the dual of multiplier Hopf algebras is a multiplier Hopf algebra.
As a subcase of this theory, one gets the standard duality for abelian groups. Plugging in non-abelian groups, one gets a generalized theory. There is a notion of a discrete and a compact multiplier Hopf algebra. This duality establishes a bijection between discrete and compact multiplier Hopf algebras. A Hopf algebra is discrete and compact if and only if it is finite dimensional. There is a well-defined notion of the Drinfeld double of Hopf algebras. A Hopf algebra has discrete double if and only if it is both discrete and compact.
An interesting philosophical point in topological quantum information is why the groups have to be finite. Why couldn't they be infintie discrete groups? The point is that topological quantum information is not based on a group, it's based on the quantum double of a group. We need the quantum double to be discrete. In this theory of duality, we see that the quantum double being discrete implies that the original group must have been finite. This is just like how Z is discrete, but its quantum double would be Z \times U(1) which has a non-discrete part.
I'll note that this is not the only approach to duality for Hopf algebras. There is also the approach (which I think is more standard now) to define the dual to be the space of linear functionals whose kernel contains an ideal of finite codimension.