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Working Seminar in Algebra

Fall 2008

Wednesdays, 3:00–3:50 p.m., Milner 317

Seminar History

This semester we plan to discuss some recent work and open problems in tensor categories, as well as the related topics of Hopf algebras and quantum groups. (Other topics as interest and time dictate.) The main sources are:

  • Tensor categories: A selective guided tour, by Mueger. (with many good references)
  • Some details may be found in On fusion categories, by Etingof, Nikshych, and Ostrik.
  • Some background material is in the book by B. Bakalov and A. Kirillov, Jr., Lectures on Tensor Categories and Modular Functors, University Lecture Series, Volume 21, American Mathematical Society, Providence, 2001.

September 3
Eric Rowell
Introductory lecture on fusion categories

Fusion categories are abstractions of representation categories. They provide us with an algebraic link between many areas including low-dimensional topology (e.g. topological quantum field theory), representation theory (e.g. of Hopf algebras), mathematical physics (e.g. conformal field theory), condensed matter physics (quantum Hall liquids) and even quantum computing. I will discuss some of the basic structure motivated by examples and open problems. This talk should be accessible to graduate students.

September 10
Deepak Naidu
More on Fusion Categories

In this talk we will present a few more examples of fusion categories. We will also discuss examples on realization of fusion rings, i.e., given a fixed fusion ring we will classify fusion categories whose Grothendieck rings are equivalent to the prescribed fusion ring. Time permitting, we will introduce the notion of Frobenius-Perron dimension for fusion categories.

No seminar September 17

September 24
Sarah Witherspoon
Algebras in Tensor Categories I

Many important algebras are objects in tensor categories. We will define algebras in tensor categories, and discuss such algebras in some or all of the following contexts: quantum McKay correspondence, connections to module categories over tensor categories, Frobenius algebras, and Nichols algebras (also known as quantum symmetric algebras).

October 1
Sarah Witherspoon
Algebras in Tensor Categories II

We will discuss a version of McKay correspondence for quantum sl(2).

October 8
Informal discussions

October 15
Eric Rowell
Braid representations

I will discuss some recent work on the representation theory of the braid group motivated by the problem of determining the images of those representations coming from braided fusion categories.

October 22
Deepak Naidu
Dimension theory for fusion categories

October 29
Discussion about Frobenius-Perron dimension and open problems

November 5
Sarah Witherspoon
Hopf algebras in braided monoidal categories

I will introduce Hopf algebras in braided monoidal categories, featuring those in the category of Yetter-Drinfeld modules over a Hopf algebra, and the bosonization (or biproduct) construction by which they become ordinary Hopf algebras. Some examples are the quantum symmetric algebras, or Nichols algebras. These will be the subject of my talk in the algebra and combinatorics seminar this week, in which I will give an independent, more combinatorial approach.

No seminar November 12

The organizer this semester is Sarah Witherspoon.