Modular tensor categories from semisimple algebras

  • Date: 04/30/2007

Hendryk Pfeiffer (Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge)


University of British Columbia


Modular tensor categories are finitely semisimple Abelian categories
with some additional structure that allows us to obtain combinatorial
invariants of oriented framed tangles. Reshetikhin and Turaev invented
them in order to construct their 3-manifold invariant.

Algebraically, these categories can be constructed as follows: There
exist very special finite-dimensional and non-semisimple Hopf algebras.
Restrict the category H-Mod of such a Hopf algebra H to its full
subcategory of tilting modules. The modular tensor category then
appears as a quotient of the category of tilting modules.

I find this recipe too complicated. Question: Is every modular tensor
category equivalent to A-Mod for some finite-dimensional semismple
algebra A? I show how to fix Tannaka-Krein reconstruction in order to
give an affirmative answer to this question.

As the term 'modular' suggests, there are plenty of similarities with
representation theory in positive characteristic p. Using quantum
groups, however, one obains equivalent categories by merely working in
characteristic zero and one can even treat the case in which `p' is not

Other Information: 

Algebraic Geometry Seminar 2007