Thursday, August 18 |
07:30 - 08:30 |
Breakfast (Restaurant at your assigned hotel) |
09:00 - 10:00 |
Hans Wenzl: Centralizer Algebras for Quantum Groups ↓ It has been well-known for some time that there are generalizations of
Schur-Weyl duality between vector representations of quantum groups of classical Lie types
and the braid groups. More recently, this has been extended to tensor powers of spinor
representations, where it is more convenient to replace the braid groups by certain
non-standard deformations of orthogonal groups. We review these results and discuss
applications towards identifying and classifying categories for certain fusion rings. (Conference Room San Felipe) |
10:00 - 10:30 |
Coffee Break (Conference Room San Felipe) |
10:30 - 11:30 |
Scott Morrison: Modular data for centres ↓ The classification of small index subfactors has resulted in the discovery of some rather unusual fusion categories. Those coming from the extended Haagerup subfactor seem particularly interesting --- at this point we know of no relationship to any family or standard construction. As part of the effort to understand these unusual objects, we have computed the modular data for the centres of these fusion categories. As it turns out, we need to know remarkably little about the fusion categories; the conditions on modular data are so restrictive that we can leverage information about the Galois action and the representation theory of SL(2,Z) to completely determine the modular data. Time permitting, I'll indicate the range of examples we've since tested these techniques against. (Conference Room San Felipe) |
11:30 - 12:00 |
Costel Bontea: Classification of non-semisimple pointed braided tensor categories and their Brauer-Picard groups ↓ A well-known result in the theory of tensor categories states that the category of braided equivalence classes of pointed braided fusion categories is equivalent to the category of pre-metric groups. Objects in this latter category are pairs (Γ,q), where Γ is a finite abelian group and q is a quadratic form on Γ. (Conference Room San Felipe) |
12:00 - 13:00 |
Emily Peters (Conference Room San Felipe) |
13:00 - 14:30 |
Lunch (Restaurant Hotel Hacienda Los Laureles) |
14:30 - 15:30 |
Terry Gannon: Vector-valued modular forms and modular tensor categories ↓ In my talk i'll explain how to find vector-valued modular forms
whose multiplier is the modular data of a modular tensor category, and
how that can help us reconstruct a rational vertex operator algebra from
that category. (Conference Room San Felipe) |
15:30 - 16:00 |
James Tener: On classification of vertex operator algebras by their representation categories ↓ In this talk, I will present work in progress with Zhenghan Wang on the subject of classification of vertex operator algebras whose representation theory is given by a fixed modular tensor category. Perhaps the best studied problem in this area is the classification of "holomorphic" vertex operator algebras, i.e. those VOAs whose representation theory is simply Vec, with small central charge. However in this talk we will focus on a complementary problem, that of classifying, modulo holomorphic VOAs, those VOAs whose representation theory is given by a fixed non-trivial modular tensor category. Using techniques of Terry Gannon, we will explore the landscape of this classification problem for some small modular tensor categories. (Conference Room San Felipe) |
16:00 - 16:30 |
Coffee Break (Conference Room San Felipe) |
16:30 - 17:00 |
Juan Cuadra: On Frobenius tensor categories ↓ A Hopf algebra H over a field k is called co-Frobenius if it possesses a nonzero
right (or left) integral R
: H → k. The existence of a nonzero integral amounts to
each one of the following conditions on the category C of finite-dimensional right (or
left) H-comodules:
• C has a nonzero injective object.
• C has injective hulls.
• C has a nonzero projective object.
• C has projective covers.
A tensor category satisfying one of these (equivalent) conditions is called Frobenius.
In this case, every injective object is projective and vice versa.
Radford showed in [3] that a co-Frobenius Hopf algebra whose coradical is a subalgebra
has finite coradical filtration. Andruskiewitsch and D˘asc˘alescu proved later in
[2] that a Hopf algebra with finite coradical filtration is necessarily co-Frobenius and
they conjectured that any co-Frobenius Hopf algebra has finite coradical filtration.
In this talk we will show that this conjecture admits a categorical formulation and
we will answer it in the affirmative. The idea of the proof is to provide a uniform
bound on the composition length of the indecomposable injective objects in terms
of the composition series of the injective hull of the unit object. This result is joint
with Nicol´as Andruskiewitsch and Pavel Etingof and appears in [1].
References
[1] N. Andruskiewitsch, J. Cuadra, and P. Etingof, On two finiteness conditions
for Hopf algebras with nonzero integral. Ann. Sc. Norm. Super. Pisa Cl.
Sci. (5) Vol. XIV, 401-440.
[2] N. Andruskiewitsch and S. D˘asc˘alescu, Co-Frobenius Hopf algebras and the
coradical filtration. Math. Z. 243 (2003), 145-154.
[3] D. E. Radford, Finiteness conditions for a Hopf algebra with a nonzero integral.
J. Algebra 46 (1977), 189-195. (Conference Room San Felipe) |
18:00 - 20:00 |
Dinner (Restaurant Hotel Hacienda Los Laureles) |