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Abstracts for Fusion categories seminar

Alternatively have a look at the program.

Introduction

Posted in
Speaker: 
Ehud Meir, Orit Davidovich
Date: 
Wed, 08/02/2012 - 14:00 - 15:30
Parent event: 
Fusion categories seminar

The seminar will deal with fusion and modular categories. These are abelian tensor
categories which gained an increasing amount of interest in the last few years.
Fusion categories may be viewed as a categorification of the notion of an algebra.
Indeed, one fundamental example, that of a fusion category with underlying abelian
category of G-graded vector spaces (G a finite group), is a categorification of the group
algbra k[G]. Fusion and modular categories arise in areas of mathematics such as:

Introduction - Part II

Posted in
Speaker: 
Orit Davidovich
Affiliation: 
MPI
Date: 
Wed, 15/02/2012 - 14:00 - 15:30
Parent event: 
Fusion categories seminar

Dimensions in Fusion Categories

Posted in
Speaker: 
Orit Davidovich
Affiliation: 
MPI
Date: 
Wed, 22/02/2012 - 14:00 - 15:30
Parent event: 
Fusion categories seminar

In this talk we will investigate the various notions of dimensions in fusion categories,
such as the Frobenius-Perron dimension and the global dimension.  We will introduce
the notions of pivotal and spherical categories and discuss quantum dimensions.  We
will discuss Etingof, Nikshych and Ostrik's result on the non-vanishing of global
dimension, which is important for Ocneanu rigidity.

 

Module categories and Morita equivalence in fusion categories

Posted in
Speaker: 
Ehud Meir
Affiliation: 
Technion/MPI
Date: 
Wed, 29/02/2012 - 14:00 - 15:30
Parent event: 
Fusion categories seminar

We will present the notion of module category over a fusion category.
This is a categorification of the notion of a module over a ring.
We will use this in order to describe the notion of Morita equivalence
for fusion categories. This will enable us to construct new examples
of fusion categories (for example- the family of group theoretical
categories)

On Morita equivalence of fusion categories

Posted in
Speaker: 
Ehud Meir
Affiliation: 
Technion/MPI
Date: 
Wed, 07/03/2012 - 14:00 - 15:30
Parent event: 
Fusion categories seminar

We will describe the notion of Morita equivalence of fusion categories.
We will describe certain dual pairs of fusion categories, and the
class of group theoretical fusion categories.
The Drinfeld center will also be described as a dual category.
We will show that, like in the case of algebras, two Morita equivalent
categories have equivalent centers.
 

Ocneanu Rigidity

Posted in
Speaker: 
Orit Davidovich
Affiliation: 
MPI
Date: 
Wed, 14/03/2012 - 14:00 - 15:30
Parent event: 
Fusion categories seminar

Ocneanu Rigidity for fusion and modular categories is
roughly the statement that such categorical structures cannot be
infinitesimally deformed, hence rigidity.  As a consequence, only
finitely many fusion or modular categories share the same Grothendieck
ring.  We follow A. Kitaev in a proof applicable to the unitary case,
in the course of which we introduce the notion of Davydov-Yetter
cohomology of a fusion category.
 

Group theoretical fusion categories

Posted in
Speaker: 
Ehud Meir
Affiliation: 
Technion/MPI
Date: 
Wed, 21/03/2012 - 14:00 - 15:30
Parent event: 
Fusion categories seminar

In this talk we will define the family of group theoretical fusion categories.
We will study some basic properties of these categories, and describe
them as categories of bimodules.
Also, we will explain their role in the theory of finite dimensional
Hopf algebras.
 

Finite groups grading and actions on fusion categories

Posted in
Speaker: 
Ehur Meir (Technion/MPI)
Date: 
Wed, 28/03/2012 - 14:00 - 15:30
Parent event: 
Fusion categories seminar

In this talk we will present two dual notions: finite group actions and finite group gradings
on a fusion category. We will explain how to use these notion to construct new fusion
categories from old ones. We will describe the equivariantization category, and extension theory
for fusion categories. We will also describe the Tambara Yamagami fusion categories, and use
them to give examples for non-group theoretical fusion categories.
 

Introduction to quantum groups and their representation theory

Posted in
Speaker: 
G. Williamson
Affiliation: 
MPI
Date: 
Wed, 04/04/2012 - 14:00 - 15:30
Parent event: 
Fusion categories seminar

This talk will start with a gentle introduction to quantum
groups and their representations. I will recall the representation
theory of complex semi-simple Lie algebras, and explain how quantum
groups can be seen as a deformation of this theory. Then we will turn
to integral forms, and how this enables one to consider
representations at roots of unity. By considering a quotient one can
define certain braided semi-simple tensor categories, which eventually
lead to Reshetikhin-Turaev invariants. (Probably there won't be enough
time to cover everything this week.)

Introduction to quantum groups and their representation theory II

Posted in
Speaker: 
G. Williamson
Affiliation: 
MPI
Date: 
Wed, 11/04/2012 - 14:00 - 15:30
Parent event: 
Fusion categories seminar

Having explored some of the representation theory of quantum
sl_2, I will give the definition of the quantum group in general. I
will then discuss R-matrices and the braided monoidal structure. We
will then describe the integral forms of quantum groups, and how this
allows one to consider representation theory at a root of unity. If
time permits I will discuss tilting modules, and how a quotient
construction allows one to construct finite semi-simple braided tensor
categories.
 

 

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