WORKSHOP

USP – UNICAMP

REPRESENTATION THEORY, ALGEBRAIC GEOMETRY AND MATHEMATICAL PHYSICS

JUNE, 11th-12th 2008

Organizers:

V. Futorny (IME-USP), M. Jardim (IMECC-UNICAMP), A. Moura (IMECC-UNICAMP)

 

PROGRAM:

WEDNESDAY, JUNE 11TH at IME-USP

10:00 – 10:50 Roman Bezrukavnikov (MIT, USA)

TBA

10:50 – 11:20 Coffee break

11:20 – 12:10 Anton Malkin (Univ. Illinois Urbana-Champaign, USA)

Local geometric quantization

12:10 – 14:30 Lunch break

14:30 – 15:20 Alistair Savage (University of Ottawa, Canada)

Moduli spaces of sheaves and the boson-fermion correspondence

15:30 – 16:20 Prasad Senesi (University of Ottawa, Canada)

Finite-dimensional representations of a twisted loop algebra and sigma-equivariance

 

THURSDAY, JUNE 12TH at IMECC-UNICAMP

10:00 – 10:50 Vera Serganova (UC Berkeley, USA)

Weight representations of Lie algebras and quivers

10:50 – 11:20 Coffee break

11:20 – 12:10 Yuri Drozd (Institute of Mathematics, Kiev, Ukraine)

Koszul duality for extension algebras of standard modules

12:10 – 14:30 Lunch break

14:30 – 15:20 Dijana Jakelic (Univ. Illinois Chicago, USA)

Finite-dimensional representations of hyper loop algebras

15:30 – 16:20 Ivan Dimitrov (Queen’s University, Canada)

Cup product on homogeneous varieties and PRV components of tensor products of irreducible modules

ABSTRACTS:

Ivan Dimitrov (Queen’s University)

Cup product on homogeneous varieties and PRV components of tensor products of irreducible modules

Abstract:

 

Yuri Drozd (Institute of Mathematics, Academy of Sciences of Ukraine)
Koszul duality for extension algebras of standard modules
Abstract: We define a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the derived category of graded modules
and the derived category of graded modules over the extension algebra of standard modules. Examples of such algebras include, in particular, the
multiplicity free blocks of the BGG category. This is a joint work with V.Mazorchuk.

 

 

Dijana Jakelic (University of Illinois Chicago, USA)

Finite-dimensional representations of hyper loop algebras

Abstract: Hyper loop algebras are certain Hopf algebras associated to affine Kac-Moody algebras. We will focus on finite-dimensional representations of
hyper loop algebras over arbitrary fields. The main results concern the classification of the irreducible representations, their tensor products, the construction
of the Weyl modules, and base change. Several of the results are related to the study of irreducible representations of polynomial algebras and Galois theory.
We will also address multiplicity problems for the underlying tensor category.
 

 

Anton Malkin (University of Illinois Urbana-Champaign)
Local geometric quantization.
Abstract: I'll talk about stacks of differential characters and explain their role in geometric quantization and its higher degree versions.
 
 

Alistair Savage (University of Ottawa)

Moduli spaces of sheaves and the boson-fermion correspondence

Abstract: The boson-fermion correspondence is a fundamental result in mathematical physics that relates the state spaces of fermions (particles with

 half-integer spin) and bosons (particles with integer spin).  It is also one of the basic examples of the use of vertex operators in representation theory.

We will describe how one can give a geometric realization of the (r-colored) boson-fermion correspondence using the equivariant cohomology of the

moduli space of framed torsion-free sheaves on the projective plane.  In this framework, the vertex operators are realized as Chern classes of vector

 bundles.  We expect this work to allow one to develop geometric realizations of other representation theoretic constructions such as the homogeneous

and principle realizations of the basic representations of affine Lie algebras, thus yielding new insight into these theories.  This is joint work with

Anthony Licata.

 

 

Prasad Senesi (University of Ottawa, Canada)

Finite-dimensional representations of a twisted loop algebra and sigma-equivariance

Abstract: Let $\frak{g}$ be a finite--dimensional complex simple Lie algebra, $L(\frak{g}) = \frak{g} \otimes \mathbb{C}\left[ t^\pm \right]$ its loop

algebra and $L^\sigma(\frak{g})$ the twisted loop (sub)algebra corresponding to a Dynkin diagram automorphism $\sigma$ of $\frak{g}$.  We will

review the representation theory of finite--dimensional representations of $L(\frak{g})$ and of $L^\sigma(\frak{g})$.  In particular we will discuss results

concerning parametrization of irreducible representations and of blocks of these algebras, but we will re--state these classifications using finitely supported

functions and $\sigma$--equivariant finitely supported functions (for $L(\frak{g})$ and $L^\sigma(\frak{g})$, respectively).  This approach provides a

more unified way to classify these representations and simplifies the relationship between the representation theory of $L(g)$ and of $L^\sigma(\frak{g})$. 

 
 

Vera Serganova (UC Berkeley, USA)

Weight representations of Lie algebras and quivers

Abstract: Representation theory of an algebraic structure (a group, a ring, a quiver) essentially amounts to the following two questions: classification

of irreducible representations and classification of indecomposable ones. I concentrate on the particular example of weight representations of the Lie

algebra sl(n). By definition the weight representations decompose into direct sum of finite-dimensional eigenspaces with respect to the diagonal subalgebra

of sl(n). The study of weight representations was initiated in 80-s by Britten, Lemire, Benkart. Fernando and Futorny (independently) proved the important

parabolic induction theorem, using it in 1998 Mathieu classified irreducible weight modules. In the  first part of my talk I review this classification and

explain connection with geometry of projective space. The second part of the talk concerns some recent developments in classification of indecomposable

weight modules (joint work with D. Grantcharov). I plan to illustrate how representation theory of quivers can be used for classification of indecomposable

weight modules. The quiver which appears in our case is the well-known Gelfand-Ponomarev quiver, it has a beautiful representation theory.