Monday, January 13 at 2:00pm to 3:00pm
Kaprielian Hall (KAP), 245
3620 South Vermont Avenue, Los Angeles, CA 90089
Kenichi Shimizu, Shibaura Institute of Technology
A modular tensor category is an interesting class of tensor categories studied from the viewpoint of, e,g., representation theory, low-dimensional topology, quantum physics and quantum computing. Although a modular tensor category is defined to be a semisimple
tensor category with some extra structures, we are interested in a "non-semisimple" generalization of a modular tensor category. We expect such a "non-semisimple" modular tensor category gives a new perspective for the above-mentioned areas of mathematics and mathematical physics.
In this talk, I will review recent developments on the study of "non-semisimple" modular tensor categories and their applications. We shall remark that a non-semisimple analogue of a modular tensor category has been introduced by Lyubashenko in mid 90's. I will,
especially, talk about the fact that a ribbon finite tensor category C is modular (in the sense of Lyubashenko) if and only if the Mueger center of C is trivial. This fact allows us to give new examples of non-semisimple modular tensor categories. For the proof of this fact, we need some categorical techniques originating from the theory of Hopf algebras: the character theory and the integral theory.