Spring, 2013
- Speaker Siddhartha Sahi, Rutgers University
- Title Intertwining operators for Grassmannians
- Time/place 2/22/2013, Friday, 12:00 in Hill 705
- Abstract Let G be the group GL(n,F) where F is a local
field, let Gr(k,n) be the Grassmannian of
k-dimensional subspaces of F^n, and let L be
a G-equivariant line bundle on Gr(k,n). Then
we get a (principal series) representation
\pi of G on the space of smooth sections of L.
Let \pi_1, \pi_2 be two such representations
corresponding to line bundles L_1 and L_2 on
two Grassmannians Gr(k_1,n) and Gr(k_2,n). We
give an explicit description of the space of
G-intertwiners from \pi_1 to \pi_2, and show
that it has dimension at most 1.
This is joint work with Dmitry Gourevitch.
- Speaker Jim Lepowsky, Rutgers University
- Title Braided tensor categories and extensions of vertex operator
algebras
- Time/place 3/1/2013, Friday, 12:00 in Hill 705
- Abstract The notion of commutative associative
algebra in a braided tensor
category is analogous to the classical notion of commutative
associative algebra, but based on the braided tensor category
structure. Under suitable conditions, the module category of a vertex
operator algebra has a natural vertex tensor category structure, in
the sense of Huang-Lepowsky. This is a subtle enhancement, requiring
complex variables, of braided tensor category structure. Using this
structure, we discuss theorems of Huang-Kirillov-Lepowsky relating
commutative associative algebras in the braided tensor category of
modules for a suitable vertex operator algebra V to vertex operator
algebras containing V as a subalgebra (extensions of V).
- Speaker Yi-Zhi Huang, Rutgers University
- Title Modular tensor categories in mathematics and physics
- Time/place 3/8/2013, Friday, 12:00 in Hill 705
- Abstract Modular tensor categories were first discovered in
physics by Moore and Seiberg in 1988.
Later, they were formulated and constructed mathematically and
found applications in many different branches of mathematics and
physics. This is an expository talk on modular tensor categories
and their applications in, in particular, representation theory,
knot theory, string theory and quantum computation. The talk will be
accessible to advanced undergraduates and beginning
graduate students.
- Speaker Yi-Zhi Huang, Rutgers University
- Title The Verlinde formula, rigidity and modular transformations
- Time/place 3/15/2013, Friday, 12:00 in Hill 705
- Abstract Let g be a finite-dimensional simple Lie algebra and \hat{g}
the corresponding affine Lie algebra. The pioneering work of Moore and
Seiberg on conformal field theory in 1988 led to the conjecture that
the category of \hat{g}-modules generated by the integrable highest
weight \hat{g}-modules of a fixed positive integral level k could be
given a structure of modular tensor category. For a long time (at
least twenty years), many mathematicians have believed that these
modular tensor categories must have been constructed either by using
the works of Tsuchiya-Ueno-Yamada, Beilinson-Feigin-Mazur and
Bakalov-Kirillov, or by using the works of Kazhdan-Lusztig and
Finkelberg. In particular, the famous Verlinde formula would be an
easy consequence of such a construction.
The present talk is on a recent discovery showing that this belief has
been wrong. It has been known for a while that, despite a statement in
the book of Bakalov-Kirillov, the works of Tsuchiya-Ueno-Yamada,
Beilinson-Feigin-Mazur and Bakalov-Kirillov cannot be used to prove
the rigidity of these tensor categories or to identify the S-matrices
for these tensor categories with the modular transformations
associated to \tau \mapsto -1/\tau on the space spanned by the
characters of the integrable highest weight \hat{g}-modules of level
k. Most recently, it has been discovered, and graciously acknowledged by
Finkelberg, that the works of
Kazhdan-Lusztig and Finkelberg alone also did not prove the rigidity
of these tensor categories and thus also did not identify these S-matrices; for
such a proof and for such an identification, one in fact needs the
Verlinde formula proved by Faltings, Teleman and me. Moreover, in the
cases (i) g=E_6, k=1, (ii) g=E_7, k=1 and (iii) g=E_8, k=1 or 2, even
the Verlinde formula proved by Faltings and Teleman does not help
because, as has long been known, the works of Kazhdan-Lusztig and
Finkelberg simply do not apply to these cases and were never claimed
to apply to these cases. In these cases, especially in the deep case
g=E_8, k=2, the only proof of the rigidity and the only identification
of the S-matrices mentioned above were given by me based on (i) the
general vertex-algebraic tensor category theory constructed by
Lepowsky and me and (ii) my vertex-algebraic theorems on modular
invariance for compositions of intertwining operators and on the
Verlinde conjecture, which gave, in particular, a much stronger
version of the Verlinde formula relating the fusion rules, modular
transformations, and braiding and fusing matrices. These general
vertex-algebraic theorems applied to all k and all affine Lie
algebras, as well as to the minimal models and many other families of
conformal-field-theoretic structures.
- Speaker Lilit Martirosyan, University of California, Berkeley
- Title Representation theory of exceptional Lie superalgebras
F(4) and G(3)
- Time/place 3/29/2013, Friday, 12:00 in Hill 705
- Abstract In this talk, I will talk about the problem of
classifying all indecomposable representations of exceptional Lie
superalgebras F(4) and G(3) and finding (super)-character or
(super)-dimension formulae for the simple modules. I will discuss
the problem of constructing the "superanalogue" of Borel-Weil-Bott
theorem for these Lie superalgebras. I will start with basic
definitions and explain the ideas in the proof.
- Speaker Haisheng Li, Rutgers University - Camden
- Title On quasi modules at infinity for vertex algebras
- Time/place 4/19/2013, Friday, 12:00 in Hill 705
- Abstract
A notion of quasi module at infinity for quantum vertex algebras was
previously introduced, in order to associate quantum vertex algebras
to certain algebras including double Yangians. In this talk, we shall
discuss quasi modules at infinity for vertex $\Gamma$-algebras with
$\Gamma$ a group and we shall present a commutator formula for the
vertex operators for quasi modules at infinity for a vertex
$\Gamma$-algebra. We then show how to use this commutator formula to
establish an equivalence between the category of lowest weight type
modules for a certain family of Lie algebras and that of quasi modules
at infinity for certain vertex $\Gamma$-algebras. This talk is
partially based on joint work with Qiang Mu.
- Speaker Gestur Olafsson, Louisiana State University
- Title The Cos^\lambda transform and intertwining operators
- Time/place 4/26/2013, Friday, 12:00 in Hill 705
- Abstract Cos^\lambda transform has been
widely studied during the last few years because of it's connection
to convex geometry and to some classical integral transforms, like the
Funk and Radon transform on the
sphere and their generalizations to Grassmann manifolds.
For the sphere S^n, the Cos^\lambda transform is defined
on L^2(\rS^n) by
C^\lambda (f)(\omega )=\int_{S^n} |(x,\omega )|^{\lambda - (n+1)/2}\,
f(x)\, d\sigma (x)
where $d\sigma$ is the rotational invariant probability measure on
S^n and (\cdot,\cdot) stands
for the usual inner product on \mathbb{R}^{n+1}. The factor
(n+1)/2 is included so that C^\lambda agrees with a standard
intertwining operator between certain principal series
representations of SL(n+1,\mathcal{R}).
The name cosine transform was first introduced by E. Ludwag
for the case \lambda-(n+1)/2=1 the integral
kernel is a power of the cosine of the angle between x and \omega.
We will in this talk start by discussing the Cos^\lambda-transform
on the sphere and it's connection to the Funk transform. Then we discuss
the generalization to the Grassmanians of p-planes in (n+1)-dimensional
space and its connection to intertwining operators. We then use the
spectrum generating operator introduced by Branson-{\'O}lafsson-{\O}rsted
in 1996 to determine the spectrum of the Cos^\lambda-transform.
- Speaker Antun Milas, SUNY Albany
- Title Beyond C_2-cofinite vertex algebras
- Time/place 5/3/2013, Friday, 12:00 in Hill 705
- Abstract
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