Tensor categories
Tensor categories
Dr. Johannes Flake (Max-Planck-Institut für Mathematik, Bonn)
Tuesday, 3 October, 10:15-12:00, HG G 43
Friday, 6 October, 10:15-12:00, HG G 43
Abstract
1. What are tensor categories?
Roughly speaking, a tensor category is a category with two operations, a direct sum and a tensor product, similar in spirit to the addition and multiplication operation in a ring. We will explore the fundamental properties of tensor categories, and get to know some first examples, including categories of group representations.
2. Are all tensor categories representation-theoretic?
A deep and central result by Deligne establishes that tensor categories in characteristic 0 satisfying certain conditions are automatically categories of (super)group representations. We will discuss these conditions and the ideas going into Deligne's theorem.
3. Tensor categories in positive characteristics
Naive reformulations of Deligne's theorem for positive characteristics fail, and correct analogs of this crucial result were found only recently. We will discuss the case of fusion categories in positive characteristics, and some current research directions involving generalized Verlinde categories.
4. Interpolation tensor categories
Even in characteristic 0, not too much is known about tensor categories beyond the scope of Deligne's theorem. As some interesting examples, we will review interpolation tensor categories, certain universal tensor categories constructed using cobordisms, and recent examples coming from oligomorphic groups.